https://icme.hpc.msstate.edu/mediawiki/api.php?action=feedcontributions&user=Kyle&feedformat=atomEVOCD - User contributions [en]2019-09-22T06:18:29ZUser contributionsMediaWiki 1.19.1https://icme.hpc.msstate.edu/mediawiki/index.php/High_Rate_Testing_and_Simulation_of_Ram%27s_HornHigh Rate Testing and Simulation of Ram's Horn2016-04-20T16:14:56Z<p>Kyle: Replaced content with "
Category:Biomaterials
Category:Animal Tissue Research"</p>
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<div><br />
<br />
[[Category:Biomaterials]]<br />
[[Category:Animal Tissue Research]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/High_Rate_Testing_and_Simulation_of_Ram%27s_HornHigh Rate Testing and Simulation of Ram's Horn2016-04-11T22:37:38Z<p>Kyle: </p>
<hr />
<div>This research investigates the effects of moisture, anisotropy, stress state, and strain rate on the mechanical properties of the bighorn sheep (Ovis Canadensis) horn keratin under high strain rate loading. The horns consist of fibrous keratin tubules extending along the length of the horn and are contained within an amorphous keratin matrix. Samples were tested in the rehydrated and ambient dry conditions along the longitudinal and radial directions under tension and compression. Increased moisture content was found to increase ductility and decrease strength. The horn keratin demonstrates a clear strain rate dependence in both tension and compression, and also showed increased energy absorption in the hydrated condition at high strain rates when compared to quasi-static data, with increases of 114% in tension and 192% in compression. Compressive failure occurred by lamellar buckling in the longitudinal orientation followed by shear delamination. Tensile failure in the longitudinal orientation occurred by lamellar delamination combined with tubule pullout and fracture. The structure-property relationships quantified here for bighorn sheep horn keratin can be used to help validate finite element simulations of ram’s impacting each other as well as being useful for other analysis regarding horn keratin on other animals.<br />
<br />
[[Category:Biomaterials]]<br />
[[Category:Animal Tissue Research]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/High_Rate_Testing_and_Simulation_of_Ram%27s_HornHigh Rate Testing and Simulation of Ram's Horn2016-04-11T22:36:24Z<p>Kyle: </p>
<hr />
<div>This research investigates the effects of moisture, anisotropy, stress state, and strain rate on the mechanical properties of the bighorn sheep (Ovis Canadensis) horn keratin under high strain rate loading. The horns consist of fibrous keratin tubules extending along the length of the horn and are contained within an amorphous keratin matrix. Samples were tested in the rehydrated and ambient dry conditions along the longitudinal and radial directions under tension and compression. Increased moisture content was found to increase ductility and decrease strength. The horn keratin demonstrates a clear strain rate dependence in both tension and compression, and also showed increased energy absorption in the hydrated condition at high strain rates when compared to quasi-static data, with increases of 114% in tension and 192% in compression. Compressive failure occurred by lamellar buckling in the longitudinal orientation followed by shear delamination. Tensile failure in the longitudinal orientation occurred by lamellar delamination combined with tubule pullout and fracture. The structure-property relationships quantified here for bighorn sheep horn keratin can be used to help validate finite element simulations of ram’s impacting each other as well as being useful for other analysis regarding horn keratin on other animals.</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/High_Rate_Testing_and_Simulation_of_Ram%27s_HornHigh Rate Testing and Simulation of Ram's Horn2016-04-11T22:36:06Z<p>Kyle: Created page with "This paper investigates the effects of moisture, anisotropy, stress state, and strain rate on the mechanical properties of the bighorn sheep (Ovis Canadensis) horn keratin und..."</p>
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<div>This paper investigates the effects of moisture, anisotropy, stress state, and strain rate on the mechanical properties of the bighorn sheep (Ovis Canadensis) horn keratin under high strain rate loading. The horns consist of fibrous keratin tubules extending along the length of the horn and are contained within an amorphous keratin matrix. Samples were tested in the rehydrated and ambient dry conditions along the longitudinal and radial directions under tension and compression. Increased moisture content was found to increase ductility and decrease strength. The horn keratin demonstrates a clear strain rate dependence in both tension and compression, and also showed increased energy absorption in the hydrated condition at high strain rates when compared to quasi-static data, with increases of 114% in tension and 192% in compression. Compressive failure occurred by lamellar buckling in the longitudinal orientation followed by shear delamination. Tensile failure in the longitudinal orientation occurred by lamellar delamination combined with tubule pullout and fracture. The structure-property relationships quantified here for bighorn sheep horn keratin can be used to help validate finite element simulations of ram’s impacting each other as well as being useful for other analysis regarding horn keratin on other animals.</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Bighorn_Ram%27s_HornBighorn Ram's Horn2016-04-11T22:34:58Z<p>Kyle: </p>
<hr />
<div>* [[Modeling Ram's Horn]]<br />
*[[Experiments-Structure-Mechanical Property Relations]]<br />
*[[High Rate Testing and Simulation of Ram's Horn]]<br />
*[[Geometric effects on stress wave propagation]]<br />
<br />
<br />
[[Category:Biomaterials]]<br />
[[Category:Animal Tissue Research]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:Structural_ScaleCategory:Structural Scale2016-03-14T20:47:12Z<p>Kyle: </p>
<hr />
<div><br />
[[Image:PC_2_difference.gif|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[Image:HelmetHeadImpactPicICME.png|thumb|right|250px| Football helmet impact simulation]]<br />
<br />
<br />
= Overview =<br />
The key to structural scale applications is employing the "best" numerical method for the application. Typically, for solid mechanics, finite element methods are employed and used mostly for the engineering applications described in this CyberInfrastructure. <br />
<br />
<br />
= Tutorials =<br />
<br />
Below are several codes that are used for simulating events at the Structural Scale. <br />
<br />
[[Code:_ABAQUS_FEM| Abaqus]] - Standard (Implicit)[http://www.3ds.com/products-services/simulia/products/abaqus/abaqusstandard/ ] , Dynamic (Explicit)[http://www.3ds.com/products-services/simulia/products/abaqus/abaqusexplicit/ ] <br><br />
[[Code:_CALCULIX| CALCULIX]][http://www.calculix.de] <br><br />
[https://www.comsol.com/ COMSOL] <br><br />
EPIC <ref name="johnson_1990"> Gordon R. Johnson, Robert A. Stryk, Tim J. Holmquist, and Omar A. Souka. Recent epic code developments for high velocity impact: 3d element arrangements and 2d fragment distributions. International Journal of Impact Engineering, 10(14):281-294, 1990. </ref> <br><br />
[http://www.lstc.com/products/ls-dyna LS-DYNA] <br><br />
[[Code:_MOOSE| MOOSE]][http://mooseframework.org] <br><br />
[http://www.mscsoftware.com/products NASTRAN] <br><br />
[https://www.esi-group.com/software-solutions/virtual-performance/virtual-performance-solution PAM-CRASH] <br><br />
[https://www.esi-group.com/software-solutions/virtual-manufacturing/sheet-metal-forming PAM-STAMP] <br><br />
[https://www.esi-group.com/software-solutions/virtual-manufacturing/casting-simulation-suite ProCAST] <br><br />
[http://www.solidworks.com SOLIDWORKS] <br><br />
[http://www.tc-liv.eu/index.php?option=com_content&view=article&id=55&Itemid=59&lang=en SysWELD] <br><br />
[[Code:_TAHOE| TAHOE]][http://tahoe.sourceforge.net] <br><br />
<br />
= Structural Scale Research =<br />
<br />
== Biomaterials ==<br />
* Rams Horn<br />
**[[Experiments-Structure-Mechanical Property Relations]]<br />
**[[Geometric effects on stress wave propagation|Geometric effects on stress wave propagation]]<br />
* Porcine Brain<br />
**[[Coupled Dynamic Experiments/Modeling]]<br />
* Woodpecker Beak<br />
**[[Structural scale of the beak]]<br />
<br />
== Ceramics ==<br />
<br />
== Metals ==<br />
The structural scale information essentially requires the constitutive model that is received from the [[Macroscale|macroscale]]. Although common practice finite element analysis does not include heterogeneities from microstructures, defects, and inclusions within the mesh related to the constitutive model, the [[Code: DMG|MSU plasticity-damage 1.0 model]] allows the incorporation of such materials science information. The quantities that can be included in this version of the constitutive model are the grain size, particle size and volume fraction of particles, pore size and volume fraction or pores (porosity level), nearest neighbor distances of pores and particles. Hence, each element in the finite element mesh would have a different value for each of the quantities and hence the strength and ductility of the material in those domains. Several examples that show that by not using the heterogenous distributions of microstructures, defects, and inclusions include the redesign of a Cadillac control arm <ref name="one">[http://dx.doi.org/10.1023/B:JCAD.0000024171.13480.24 Horstemeyer, M.F., Wang, P., “Cradle-to-Grave simulation-Based Design Incorporating Multiscale Microstructure-Property Modeling: Reinvigorating Design with Science,” ''J. Computer-Aided Materials Design'', Vol. 10, pp. 13-34, 2003.]</ref>, the Corvette engine cradle <ref>M.F. Horstemeyer, D. Oglesby, J. Fan, P.M. Gullett, H. El Kadiri, Y. Xue, C. Burton, K. Gall, B. Jelinek, M.K. Jones, S. G. Kim, E.B. Marin, D.L. McDowell, A. Oppedal, N. Yang, “From Atoms to Autos: Designing a Mg Alloy Corvette Cradle by Employing Hierarchical Multiscale Microstructure-Property Models for Monotonic and Cyclic Loads,” MSU.CAVS.CMD.2007-R0001, 2007</ref>, and a powder metal steel engine bearing cap <ref>Hammi, Y, Horstemeyer, MF, Stone, T., Sanderow, H., Chernenkoff, R., Weber, G., "Powder-Metal Performance Modeling of Automotive Components AMD-410, 2009</ref>.<br />
<br />
Some examples of using different finite element simulations with associated input decks using our [[Code: DMG|MSU plasticity-damage 1.0]] can be garnered from the following locations:<br />
<br />
# Cadillac control arm ([[Code:_ABAQUS|ABAQUS-Implicit]])<ref name="one"></ref> <br />
# Corvette cradle ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Dodge Neon crash ([[Code:_LS-DYNA|LS-Dyna]])<br />
# Forming of aluminum plate ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Crush of aluminum tube ([[Code:_ABAQUS|ABAQUS-Explicit]])<br />
# [[Multi-CRUSH|Axial Crushing of Multi-Cell Multi-Corner Tubes]] ([[Code:_LS-DYNA|LS-Dyna]])<br />
<br />
====[[Process_Modeling#Hydroforming|Hydroforming]]====<br />
<br />
<br />
<br />
<br />
== Polymers ==<br />
<br />
=== ISV Polymer Modeling ===<br />
<br />
[http://dx.doi.org/10.1007/s00707-010-0349-y A general inelastic internal state variable model for amorphous glassy polymers]<br />
<br />
[http://dx.doi.org/10.1016/j.ijplas.2012.10.005 An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation]<br />
<br />
=== Application ===<br />
<br />
[http://dx.doi.org/10.1016/j.engfailanal.2012.07.020 Characterization and failure analysis of a polymeric clamp hanger component]<br />
<br />
<br />
== Geomaterials ==<br />
*Earth Mantle<br />
**[[Mantle Convection Study with Lherzolite Material Modeling]]<br />
<br />
== References ==<br />
<references/><br />
<br />
[[Category:Overview]]<br />
[[Category:Multiscale Simulations]]<br />
[[Category:Metals]]<br />
[[Category:Polymers]]<br />
[[Category:Geomaterials]]<br />
[[Category:Biomaterials]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T20:17:18Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary.<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] for metals, [[CodeRepository:TP|TP]] for polymers, and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T20:12:05Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary.<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] for metals, [[CodeRepository:TP|TP]] for polymers, and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T20:11:24Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary.<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] for metals, [[TP|TP]] for polymers, and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T19:50:42Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
=Optimization Methods=<br />
<br />
==Zeroth-Order Methods==<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Bracketing Method===<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
==First-Order Methods==<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
===Steepest Descent (Cauchy) Method===<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Conjugate Gradient Method===<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
==Second-Order Methods==<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
==Population-Based Methods==<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
===Monte-Carlo Method===<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
===Genetic Algorithm===<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary.<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:MetamodelingCategory:Metamodeling2016-03-14T19:17:38Z<p>Kyle: </p>
<hr />
<div>==Overview==<br />
<br />
Metamodels are approximate mathematical models, also called surrogate models, used to predict a response given input values. They are typically used when an analytical function is not available or when obtaining a function value is computationally expensive. Metamodels mimic the behavior of the more complex analytical function while minimizing computational time. Metamodeling methods are often used in design optimization to provide function evaluations of the objective function(s) and response(s). Because of the robust nature of metamodels, they can be applied to problems at any length scale.<br />
<br />
==Metamodel Terminology==<br />
<br />
'''''Design of Experiments (DOE)''''' – selected points and their function evaluations within the design space used to build the metamodel. DOEs are typically constructed using methods such as latin hypercube sampling (lhs) to provide design points which are spread throughout the design space. The sample points provide a way to "train" the metamodel to capture simulation model behavior throughout the design space.<br />
<br />
'''''Design Point or Training Point''''' – design variable and response from the design of experiments used to construct or train the metamodel.<br />
<br />
'''''Validation Point, Test Point, or Sampling Point''''' – points within the design space where a prediction is desired, also used to check the accuracy of the metamodel.<br />
<br />
==Building a Metamodel==<br />
<br />
'''1)''' Create Design of Experiment points to use as Design points with upper and lower bounds on the variables (see links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]). The number of points depends on the problem but generally should be at least ''3xN'' where ''N'' is the number of variables. More points can produce a more accurate model but this is not always true. More points can also increase computational time, as the response from the analytical function needs to be calculated for each point. <br />
<br />
'''2)''' Obtain responses of interest at each design point and test point. This is typically done by running simulations, performing experiments, or using an analytical function.<br />
<br />
'''3)''' Build the metamodel using one of numerous techniques and check the error using an error metric.<br />
<br />
==Metamodel Techniques==<br />
===Polynomial Response Surface (PRS)===<br />
As the name implies, the PRS method uses a polynomial function of the design variables to approximate the response of the analytical model. This is a MATLAB function script file for building an PRS metamodel.[[Media:get_ypred_prs.m|MATLAB PRS function]]<br />
===Gaussian Process (GP)===<br />
This file contains a MATLAB function script for building a GP metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_gp.m|MATLAB GP function]]<br />
===Radial Basis Function (RBF)===<br />
Radial basis functions are real-valued functions whose value depends only on the distance from the origin or center. A typical metamodel will use a sum of weighted radial basis functions, each with a different center and weight to approximate the analytical model. This is a MATLAB function script file for building an RBF metamodel.[[Media:get_ypred_rbf.m|MATLAB RBF function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic RBF equation based on a previously built RBF metamodel.[[Media:write_rbf.m| Write RBF equation]]<br />
===Kriging (KR)===<br />
This file contains a MATLAB function script for building a KR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_krig.m|MATLAB KR function]]<br />
===Support Vector Regression (SVR)===<br />
This file contains a MATLAB function script for building a SVR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_svr.m|MATLAB SVR function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic SVR equation based on a previously built SVR metamodel.[[Media:write_svr.m| Write SVR equation]]<br />
===Optimized Ensemble (EN)===<br />
An Ensemble of metamodels using optimized weight factors. See the link below for a description of the method. The following file contains a MATLAB function file implementing this method. [[Media:Ensemble_valid.m|MATLAB EN function]]<br />
http://www.springerlink.com/content/u406m252480277x0/<br />
<br />
=Structural Scale=<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Numerical simulations of multiple vehicle crashes and multidisciplinary crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], K.N. Solanki, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.tandfonline.com/doi/abs/10.1533/ijcr.2005.0335<br />
<br />
'''[[Product Design Optimization with Microstructure-property Modeling and Associated Uncertainties|Product Design Optimization with Microstructure-property Modeling and Associated Uncertainties]]'''<br />
<br />
Authors: K.N. Solanki, E. Acar, [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer], and G. Steele<br />
<br />
<br />
= Macroscale=<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
= Electronic Scale=</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:MetamodelingCategory:Metamodeling2016-03-14T19:10:19Z<p>Kyle: </p>
<hr />
<div>==Overview==<br />
<br />
Metamodels are approximate mathematical models, also called surrogate models, used to predict a response given input values. They are typically used when an analytical function is not available or when obtaining a function value is computationally expensive. Metamodels mimic the behavior of the more complex analytical function while minimizing computational time. Metamodeling methods are often used in design optimization to provide function evaluations of the objective function(s) and response(s). Because of the robust nature of metamodels, they can be applied to problems at any length scale.<br />
<br />
==Metamodel Terminology==<br />
<br />
'''''Design of Experiments (DOE)''''' – selected points and their function evaluations within the design space used to build the metamodel. DOEs are typically constructed using methods such as latin hypercube sampling (lhs) to provide design points which are spread throughout the design space. The sample points provide a way to "train" the metamodel to capture simulation model behavior throughout the design space.<br />
<br />
'''''Design Point or Training Point''''' – design variable and response from the design of experiments used to construct or train the metamodel.<br />
<br />
'''''Validation Point, Test Point, or Sampling Point''''' – points within the design space where a prediction is desired, also used to check the accuracy of the metamodel.<br />
<br />
==Building a Metamodel==<br />
<br />
'''1)''' Create Design of Experiment points to use as Design points with upper and lower bounds on the variables (see links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]). The number of points depends on the problem but generally should be at least ''3xN'' where ''N'' is the number of variables. More points can produce a more accurate model but this is not always true. More points can also increase computational time, as the response from the analytical function needs to be calculated for each point. <br />
<br />
'''2)''' Obtain responses of interest at each design point and test point. This is typically done by running simulations, performing experiments, or using an analytical function.<br />
<br />
'''3)''' Build the metamodel using one of numerous techniques and check the error using an error metric.<br />
<br />
==Metamodel Techniques==<br />
===Polynomial Response Surface (PRS)===<br />
As the name implies, the PRS method uses a polynomial function of the design variables to approximate the response of the analytical model. This is a MATLAB function script file for building an PRS metamodel.[[Media:get_ypred_prs.m|MATLAB PRS function]]<br />
===Gaussian Process (GP)===<br />
This file contains a MATLAB function script for building a GP metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_gp.m|MATLAB GP function]]<br />
===Radial Basis Function (RBF)===<br />
Radial basis functions are real-valued functions whose value depends only on the distance from the origin or center. A typical metamodel will use a sum of weighted radial basis functions, each with a different center and weight to approximate the analytical model. This is a MATLAB function script file for building an RBF metamodel.[[Media:get_ypred_rbf.m|MATLAB RBF function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic RBF equation based on a previously built RBF metamodel.[[Media:write_rbf.m| Write RBF equation]]<br />
===Kriging (KR)===<br />
This file contains a MATLAB function script for building a KR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_krig.m|MATLAB KR function]]<br />
===Support Vector Regression (SVR)===<br />
This file contains a MATLAB function script for building a SVR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_svr.m|MATLAB SVR function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic SVR equation based on a previously built SVR metamodel.[[Media:write_svr.m| Write SVR equation]]<br />
===Optimized Ensemble (EN)===<br />
An Ensemble of metamodels using optimized weight factors. See the link below for a description of the method. The following file contains a MATLAB function file implementing this method. [[Media:Ensemble_valid.m|MATLAB EN function]]<br />
http://www.springerlink.com/content/u406m252480277x0/<br />
<br />
=Structural Scale=<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''[[Product Design Optimization with Microstructure-property Modeling and Associated Uncertainties|Product Design Optimization with Microstructure-property Modeling and Associated Uncertainties]]'''<br />
<br />
Authors: K.N. Solanki, E. Acar, [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer], and G. Steele<br />
<br />
<br />
= Macroscale=<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
= Electronic Scale=</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:MetamodelingCategory:Metamodeling2016-03-14T17:59:03Z<p>Kyle: </p>
<hr />
<div>Metamodels are approximate mathematical models, also called surrogate models, used to predict a response given input values. They are typically used when an analytical function is not available or when obtaining a function value is computationally expensive. Metamodels mimic the behavior of the more complex analytical function while minimizing computational time. Metamodeling methods are often used in design optimization to provide function evaluations of the objective function(s) and response(s).<br />
<br />
==Metamodel Terminology==<br />
<br />
'''''Design of Experiments (DOE)''''' – selected points and their function evaluations within the design space used to build the metamodel. DOEs are typically constructed using methods such as latin hypercube sampling (lhs) to provide design points which are spread throughout the design space. The sample points provide a way to "train" the metamodel to capture simulation model behavior throughout the design space.<br />
<br />
'''''Design Point or Training Point''''' – design variable and response from the design of experiments used to construct or train the metamodel.<br />
<br />
'''''Validation Point, Test Point, or Sampling Point''''' – points within the design space where a prediction is desired, also used to check the accuracy of the metamodel.<br />
<br />
==Building a Metamodel==<br />
<br />
'''1)''' Create Design of Experiment points to use as Design points with upper and lower bounds on the variables. The number of points depends on the problem but generally should be at least ''3xN'' where ''N'' is the number of variables. More points can produce a more accurate model but this is not always true. More points can also increase computational time, as the response from the analytical function needs to be calculated for each point. <br />
<br />
'''2)''' Obtain responses of interest at each design point and test point. This is typically done by running simulations, performing experiments, or using an analytical function.<br />
<br />
'''3)''' Build the metamodel using one of numerous techniques and check the error using an error metric.<br />
<br />
==Metamodel Techniques==<br />
===Polynomial Response Surface (PRS)===<br />
As the name implies, the PRS method uses a polynomial function of the design variables to approximate the response of the analytical model. This is a MATLAB function script file for building an PRS metamodel.[[Media:get_ypred_prs.m|MATLAB PRS function]]<br />
===Gaussian Process (GP)===<br />
This file contains a MATLAB function script for building a GP metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_gp.m|MATLAB GP function]]<br />
===Radial Basis Function (RBF)===<br />
Radial basis functions are real-valued functions whose value depends only on the distance from the origin or center. A typical metamodel will use a sum of weighted radial basis functions, each with a different center and weight to approximate the analytical model. This is a MATLAB function script file for building an RBF metamodel.[[Media:get_ypred_rbf.m|MATLAB RBF function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic RBF equation based on a previously built RBF metamodel.[[Media:write_rbf.m| Write RBF equation]]<br />
===Kriging (KR)===<br />
This file contains a MATLAB function script for building a KR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_krig.m|MATLAB KR function]]<br />
===Support Vector Regression (SVR)===<br />
This file contains a MATLAB function script for building a SVR metamodel. This function requires an outside toolbox and a link for this toolbox can be found in the comments section of the file. [[Media:get_ypred_svr.m|MATLAB SVR function]]<br />
<br />
This file contains a MATLAB script to create a text file containing an analytic SVR equation based on a previously built SVR metamodel.[[Media:write_svr.m| Write SVR equation]]<br />
===Optimized Ensemble (EN)===<br />
An Ensemble of metamodels using optimized weight factors. See the link below for a description of the method. The following file contains a MATLAB function file implementing this method. [[Media:Ensemble_valid.m|MATLAB EN function]]<br />
http://www.springerlink.com/content/u406m252480277x0/</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T17:57:21Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary.<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T17:56:26Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links: [[DOE with MATLAB 1|1]], [[DOE with MATLAB 2|2]], [[DOE with MATLAB 3|3]], and [[DOE with MATLAB 2|4]]. Once the DoE points and their objective function values are found, the data is used to "train" the metamodel. After training, the metamodel can be directly used in place of the full scale simulations to calculate objective functions in a much faster manner. Metamodels can also be used in Monte Carlo simulations to quantify [[Uncertainty|uncertainty]] when many calculations are necessary<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T17:44:26Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
One of the most prevalent uses of optimization occurs in the design of structures. Common applications in this area consist of the reduction of weight subject to strength requirements, maximizing energy absorption in crashworthiness scenarios, and topology optimization. In a simple case, analytical solutions for objective functions are solved while altering design variables subject to constraints. This optimization can be performed in a software such as MATLAB. In a more complex example, objective functions can be solved by a finite element software such as Abaqus, LS-DYNA, etc. Because the optimization algorithm sometimes requires hundreds or even thousands of iterations, optimization can become unfeasible when directly coupled to a computationally expensive finite element simulation. A suitable alternative is the use of [[Metamodeling|metamodels]]. Metamodels offer a fast analytical approximation of a complex, expensive objective function. Metamodels approximate responses of these functions over a predefined design space. In order to create a metamodel, objective function values must calculated using a full scale simulation at "training points" sampled throughout the design space. These training points can be found using a Design of Experiments (DoE). Some tutorials for DoE can be found at the following links:<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]].<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T17:28:02Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data. See [[MPC|MEAM Potential Calibration]]<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T17:17:13Z<p>Kyle: /* Structural Scale Optimization */</p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
'''A comparative study of metamodeling methods for multi objective crashworthiness optimization'''<br />
<br />
Authors: [http://mees.uncc.edu/howie-fang Howie Fang], [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]), Z. Liu, and [http://www.cavs.msstate.edu/directory/information.php?d=63 Mark Horstemeyer]<br />
<br />
http://www.sciencedirect.com/science/article/pii/S0045794905001355<br />
<br />
'''Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])'''<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T15:51:35Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the [[Code: DMG|DMGfit]] and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T15:41:41Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
*Metamodeling: A metamodel (or surrogate model) provides a quick way to approximate a function response when an analytical solution is not available, or is computationally expensive.<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
=Tutorials=<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the DMGfit and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T15:34:28Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of one or more objective functions by altering a set of design variables, and can be subject to constraints. Design optimization can be used at the different length scale models and materials similar to every ICME notion. The factors involved in the optimization process are further explained below:<br />
<br />
*Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
*Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
*Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto frontier.<br />
<br />
*Pareto Frontier: It is relatively simple to determine an optimal solution for single objective methods (solution with the lowest objective function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important physics into a few parameters.<br />
<br />
*Models: The designer must also choose models to relate the constraints and the objectives to the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
*Reliability: the probability of a component to perform its required functions under stated conditions for a specified period of time.<br />
<br />
==Metamodeling==<br />
<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Optimization Methods==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the DMGfit and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:OptimizationCategory:Optimization2016-03-14T14:37:49Z<p>Kyle: </p>
<hr />
<div>=Overview=<br />
<br />
Design Optimization deals with finding the maximum and minimum of functions subject to some constraints. Design optimization can be thought of as used at the different length scale models and materials similar to every ICME notion.<br />
<br />
==Optimization guidelines==<br />
<br />
Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weighs the various objectives and sums them to form a single objective. Other<br />
methods allow multi-objective optimization, such as the calculation of a Pareto frontier<br />
<br />
Models: The designer must also choose models to relate the constraints and the objectives to<br />
the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
Reliability: the probability of a component to perform its required functions under stated<br />
conditions for a specified period of time.<br />
<br />
==Optimization Methods==<br />
<br />
Gradient-based methods:<br />
<br />
Adjoint equation,<br />
Newton’s method,<br />
Steepest descent,<br />
Conjugate gradient, and<br />
Sequential quadratic programming<br />
<br />
Gradient-free methods:<br />
<br />
Hooke-Jeeves pattern search and<br />
Nelder-Mead method<br />
<br />
Population-based methods:<br />
<br />
Genetic algorithm,<br />
Memetic algorithm,<br />
Particle swarm optimization,<br />
Ant colony, and<br />
Harmony search<br />
<br />
Other methods:<br />
<br />
Random search,<br />
Grid search,<br />
Simulated annealing,<br />
Direct search, and<br />
IOSO(Indirect Optimization based on Self-Organization)<br />
<br />
==Convergence of Pareto Frontier== <br />
<br />
It is relatively simple to determine an optimal solution for single objective methods (solution<br />
with the lowest error function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important<br />
physics into a few parameters.<br />
<br />
==Metamodeling==<br />
<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Unconstrained Optimization==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Golden Section Search====<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
[[Category:SRCLID]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the DMGfit and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Geometric_effects_on_stress_wave_propagationGeometric effects on stress wave propagation2015-06-01T18:53:01Z<p>Kyle: </p>
<hr />
<div><br />
<br />
{{template:Research_Paper<br />
<br />
|abstract=The present study, through finite element simulations, shows the geometric effects of a bio-inspired solid on pressure and impulse mitigation for an elastic, plastic, and viscoelastic material. Because of the bio-inspired geometries, stress wave mitigation became apparent in a non-intuitive manner such that potential real-world applications in human protective gear designs are realizable. In nature, there are several toroidal designs that are employed for mitigating stress waves; examples include the hyoid bone on the back of a woodpecker’s jaw that extends around the skull to its nose and a ram’s horn. This study evaluates four different geometries with the same length and same initial cross-sectional diameter at the impact location in three dimensional finite element analyses. The geometries in increasing complexity were the following: 1. a round cylinder; 2. a round cylinder that was tapered to a point; 3. a round cylinder that was spiraled in a two dimensional plane; and 4. a round cylinder that was tapered and spiraled in a two dimensional plane. The results show that the tapered spiral geometry mitigated the greatest amount of pressure and impulse (approximately 98% mitigation) when compared to the cylinder regardless of material type (elastic, plastic, and viscoelastic) and regardless of input pressure signature. The specimen taper effectively mitigated the stress wave as a result of uniaxial deformational processes and an induced shear that arose from its geometry. Due to the decreasing cross-sectional area arising from the taper, the local uniaxial and shear stresses increased along the specimen length. The spiral induced even greater shear stresses that help mitigate the stress wave and also induced transverse displacements at the tip such that minimal wave reflections occurred. This phenomenon arose although only longitudinal waves were introduced as the initial Boundary Condition (BC). In nature, when shearing occurs within or between materials (friction), dissipation usually results helping the mitigation of the stress wave and is illustrated in this study with the taper and spiral geometries. The combined taper and spiral optimized stress wave mitigation in terms of the pressure and impulse, thus providing insight into the ram’s horn design and woodpecker hyoid designs found in nature.<br />
<br />
|authors=K.L. Johnson, M.W. Trim, M.F. Horstemeyer, N. Lee, L.N. Williams, J. Liao, H. Rhee, R. Prabhu<br />
<br />
|animation=<br />
<br />
|images=<br />
{{paper_figure|image=Stress propagation.jpg|image caption=Figure 1. Schematic representation of the four finite element meshes illustrating the four different geometric configurations with the same length (and the same bar diameter where the pressure was applied) used in the analysis.}}<br />
<br />
|methodology=Fig. 1 depicts the four geometries that were studied along with the load and prescribed boundary conditions. The total length and cross-sectional diameters at the starting end were maintained among the four geometries. The ratio of the large and small-end diameters was also consistent for the tapered geometries.<br />
The finite element program ABAQUS/Explicit v6.11 [14], a stress wave dynamics code, was used as the numerical model in this study. To demonstrate material independence, material properties for three different materials were used. The materials are metal (AM30 magnesium denoting a common plastic material), polymer (Polycarbonate denoting a common viscoelastic material), and ceramic (Silicon Carbide (SiC) denoting a common elastic material) were investigated.<br />
Post-processing of data was performed using ABAQUS/CAE v6.11 [14]. Pressure and von Mises contour plots were generated when the wave front was at distance 1/3L, 2/3L, and L. Pressure and displacement response histories were generated at a distance of 0.1 m away from the free end to avoid edge effects. The distance corresponded to a rotation of 180 degrees from the free end on the spiraled geometries. The pressure histories were created by averaging the respective output of each node lying on the cross-section at the specified offset from the free end. <br />
<br />
|material model= [[Code: ABAQUS FEM|ABAQUS/Explicit v6.11]]: Abaqus/Explicit is a finite element analysis software application that employs explicit integration scheme to solve highly nonlinear transient dynamic and quasi-static analyses.<br />
<br />
|input deck=None<br />
<br />
|results=<br />
{{paper_figure|image=Pressure contour plots in AM30.jpg |image caption=Figure 2. Pressure contour plots in AM30}}<br />
Fig. 2 show the pressure contour plots for the cylinder, tapered cylinder, spiral, and tapered spiral using AM30 material properties when the wave front is at distance 1/3L, 2/3L, and L. Fig. 2 shows indeed that the tapered cylinder geometry induced greater hydrostatic and shear stresses as the dynamic wave propagated towards the smaller end of the bar.<br />
<br />
The following conclusions can be made regarding this study:<br />
*A tapered geometry will lower the pressure and impulse due to the convergent boundary and a continually decreasing cross-sectional area such that greater uniaxial stresses and subsequent axial deformation arises. Furthermore, the tapered geometry introduces small shear stresses that also help lower the impulse by removing energy from the initial longitudinal wave.<br />
*A spiral geometry will lower the impulse due to the introduction of shear stresses along the length of the spiral. These shear stresses introduce transverse displacements that help reduce the pressure and impulse.<br />
*When both the tapered and spiral geometry are included in a design, their synergistic effects multiplicatively reduce the pressure and impulse.<br />
*Although different materials admit different wave speeds, this impulse reduction was repeated in the three material types (elastic, plastic, and viscoelastic) and loading conditions. <br><br />
<br />
[[Image:Geometrypressure.gif|thumb|900px| Figure 3. Movie showing pressure wave behavior in different geometries.]]<br />
<br />
|acknowledgement=<br />
<br />
|references=K. L. Johnson, M. W. Trim, M. F. Horstemeyer, N. Lee, L. N. Williams, J. Liao, H. Rhee, and R. Prabhu, “Geometric Effects on Stress Wave Propagation,” J. Biomech. Eng., vol. 136, no. 2, pp. 021023–021023, Feb. 2014.<br />
}}<br />
<br />
[[Category:Biomaterials]]<br />
[[Category:ABAQUS]]<br />
[[Category:Animal Tissue Research]]<br />
[[Category:Research Paper]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Geometric_effects_on_stress_wave_propagationGeometric effects on stress wave propagation2015-06-01T18:49:30Z<p>Kyle: </p>
<hr />
<div><br />
<br />
{{template:Research_Paper<br />
<br />
|abstract=The present study, through finite element simulations, shows the geometric effects of a bio-inspired solid on pressure and impulse mitigation for an elastic, plastic, and viscoelastic material. Because of the bio-inspired geometries, stress wave mitigation became apparent in a non-intuitive manner such that potential real-world applications in human protective gear designs are realizable. In nature, there are several toroidal designs that are employed for mitigating stress waves; examples include the hyoid bone on the back of a woodpecker’s jaw that extends around the skull to its nose and a ram’s horn. This study evaluates four different geometries with the same length and same initial cross-sectional diameter at the impact location in three dimensional finite element analyses. The geometries in increasing complexity were the following: 1. a round cylinder; 2. a round cylinder that was tapered to a point; 3. a round cylinder that was spiraled in a two dimensional plane; and 4. a round cylinder that was tapered and spiraled in a two dimensional plane. The results show that the tapered spiral geometry mitigated the greatest amount of pressure and impulse (approximately 98% mitigation) when compared to the cylinder regardless of material type (elastic, plastic, and viscoelastic) and regardless of input pressure signature. The specimen taper effectively mitigated the stress wave as a result of uniaxial deformational processes and an induced shear that arose from its geometry. Due to the decreasing cross-sectional area arising from the taper, the local uniaxial and shear stresses increased along the specimen length. The spiral induced even greater shear stresses that help mitigate the stress wave and also induced transverse displacements at the tip such that minimal wave reflections occurred. This phenomenon arose although only longitudinal waves were introduced as the initial Boundary Condition (BC). In nature, when shearing occurs within or between materials (friction), dissipation usually results helping the mitigation of the stress wave and is illustrated in this study with the taper and spiral geometries. The combined taper and spiral optimized stress wave mitigation in terms of the pressure and impulse, thus providing insight into the ram’s horn design and woodpecker hyoid designs found in nature.<br />
<br />
|authors=K.L. Johnson, M.W. Trim, M.F. Horstemeyer, N. Lee, L.N. Williams, J. Liao, H. Rhee, R. Prabhu<br />
<br />
|animation=<br />
<br />
|images=<br />
{{paper_figure|image=Stress propagation.jpg |image caption=Figure 1. Schematic representation of the four finite element meshes illustrating the four different geometric configurations with the same length (and the same bar diameter where the pressure was applied) used in the analysis.}}<br />
<br />
|methodology=Fig. 1 depicts the four geometries that were studied along with the load and prescribed boundary conditions. The total length and cross-sectional diameters at the starting end were maintained among the four geometries. The ratio of the large and small-end diameters was also consistent for the tapered geometries.<br />
The finite element program ABAQUS/Explicit v6.11 [14], a stress wave dynamics code, was used as the numerical model in this study. To demonstrate material independence, material properties for three different materials were used. The materials are metal (AM30 magnesium denoting a common plastic material), polymer (Polycarbonate denoting a common viscoelastic material), and ceramic (Silicon Carbide (SiC) denoting a common elastic material) were investigated.<br />
Post-processing of data was performed using ABAQUS/CAE v6.11 [14]. Pressure and von Mises contour plots were generated when the wave front was at distance 1/3L, 2/3L, and L. Pressure and displacement response histories were generated at a distance of 0.1 m away from the free end to avoid edge effects. The distance corresponded to a rotation of 180 degrees from the free end on the spiraled geometries. The pressure histories were created by averaging the respective output of each node lying on the cross-section at the specified offset from the free end. <br />
<br />
|material model= [[Code: ABAQUS FEM|ABAQUS/Explicit v6.11]]: Abaqus/Explicit is a finite element analysis software application that employs explicit integration scheme to solve highly nonlinear transient dynamic and quasi-static analyses.<br />
<br />
|input deck=None<br />
<br />
|results=<br />
{{paper_figure|image=Pressure contour plots in AM30.jpg |image caption=Figure 2. Pressure contour plots in AM30}}<br />
Fig. 2 show the pressure contour plots for the cylinder, tapered cylinder, spiral, and tapered spiral using AM30 material properties when the wave front is at distance 1/3L, 2/3L, and L. Fig. 2 shows indeed that the tapered cylinder geometry induced greater hydrostatic and shear stresses as the dynamic wave propagated towards the smaller end of the bar.<br />
<br />
The following conclusions can be made regarding this study:<br />
*A tapered geometry will lower the pressure and impulse due to the convergent boundary and a continually decreasing cross-sectional area such that greater uniaxial stresses and subsequent axial deformation arises. Furthermore, the tapered geometry introduces small shear stresses that also help lower the impulse by removing energy from the initial longitudinal wave.<br />
*A spiral geometry will lower the impulse due to the introduction of shear stresses along the length of the spiral. These shear stresses introduce transverse displacements that help reduce the pressure and impulse.<br />
*When both the tapered and spiral geometry are included in a design, their synergistic effects multiplicatively reduce the pressure and impulse.<br />
*Although different materials admit different wave speeds, this impulse reduction was repeated in the three material types (elastic, plastic, and viscoelastic) and loading conditions. <br><br />
<br />
[[Image:Geometrypressure.gif|thumb|900px| Figure 3. Movie showing pressure wave behavior in different geometries.]]<br />
<br />
|acknowledgement=<br />
<br />
|references=K. L. Johnson, M. W. Trim, M. F. Horstemeyer, N. Lee, L. N. Williams, J. Liao, H. Rhee, and R. Prabhu, “Geometric Effects on Stress Wave Propagation,” J. Biomech. Eng., vol. 136, no. 2, pp. 021023–021023, Feb. 2014.<br />
}}<br />
<br />
[[Category:Biomaterials]]<br />
[[Category:ABAQUS]]<br />
[[Category:Animal Tissue Research]]<br />
[[Category:Research Paper]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Geometric_effects_on_stress_wave_propagationGeometric effects on stress wave propagation2015-06-01T18:45:59Z<p>Kyle: </p>
<hr />
<div><br />
<br />
{{template:Research_Paper<br />
<br />
|abstract=The present study, through finite element simulations, shows the geometric effects of a bio-inspired solid on pressure and impulse mitigation for an elastic, plastic, and viscoelastic material. Because of the bio-inspired geometries, stress wave mitigation became apparent in a non-intuitive manner such that potential real-world applications in human protective gear designs are realizable. In nature, there are several toroidal designs that are employed for mitigating stress waves; examples include the hyoid bone on the back of a woodpecker’s jaw that extends around the skull to its nose and a ram’s horn. This study evaluates four different geometries with the same length and same initial cross-sectional diameter at the impact location in three dimensional finite element analyses. The geometries in increasing complexity were the following: 1. a round cylinder; 2. a round cylinder that was tapered to a point; 3. a round cylinder that was spiraled in a two dimensional plane; and 4. a round cylinder that was tapered and spiraled in a two dimensional plane. The results show that the tapered spiral geometry mitigated the greatest amount of pressure and impulse (approximately 98% mitigation) when compared to the cylinder regardless of material type (elastic, plastic, and viscoelastic) and regardless of input pressure signature. The specimen taper effectively mitigated the stress wave as a result of uniaxial deformational processes and an induced shear that arose from its geometry. Due to the decreasing cross-sectional area arising from the taper, the local uniaxial and shear stresses increased along the specimen length. The spiral induced even greater shear stresses that help mitigate the stress wave and also induced transverse displacements at the tip such that minimal wave reflections occurred. This phenomenon arose although only longitudinal waves were introduced as the initial Boundary Condition (BC). In nature, when shearing occurs within or between materials (friction), dissipation usually results helping the mitigation of the stress wave and is illustrated in this study with the taper and spiral geometries. The combined taper and spiral optimized stress wave mitigation in terms of the pressure and impulse, thus providing insight into the ram’s horn design and woodpecker hyoid designs found in nature.<br />
<br />
|authors=K.L. Johnson, M.W. Trim, M.F. Horstemeyer, N. Lee, L.N. Williams, J. Liao, H. Rhee, R. Prabhu<br />
<br />
|animation=<br />
<br />
|images=<br />
{{paper_figure|image=Stress propagation.jpg |image caption=Figure 1. Schematic representation of the four finite element meshes illustrating the four different geometric configurations with the same length (and the same bar diameter where the pressure was applied) used in the analysis.}}<br />
<br />
|methodology=Fig. 1 depicts the four geometries that were studied along with the load and prescribed boundary conditions. The total length and cross-sectional diameters at the starting end were maintained among the four geometries. The ratio of the large and small-end diameters was also consistent for the tapered geometries.<br />
The finite element program ABAQUS/Explicit v6.11 [14], a stress wave dynamics code, was used as the numerical model in this study. To demonstrate material independence, material properties for three different materials were used. The materials are metal (AM30 magnesium denoting a common plastic material), polymer (Polycarbonate denoting a common viscoelastic material), and ceramic (Silicon Carbide (SiC) denoting a common elastic material) were investigated.<br />
Post-processing of data was performed using ABAQUS/CAE v6.11 [14]. Pressure and von Mises contour plots were generated when the wave front was at distance 1/3L, 2/3L, and L. Pressure and displacement response histories were generated at a distance of 0.1 m away from the free end to avoid edge effects. The distance corresponded to a rotation of 180 degrees from the free end on the spiraled geometries. The pressure histories were created by averaging the respective output of each node lying on the cross-section at the specified offset from the free end. <br />
<br />
|material model= [[Code: ABAQUS FEM|ABAQUS/Explicit v6.11]]: Abaqus/Explicit is a finite element analysis software application that employs explicit integration scheme to solve highly nonlinear transient dynamic and quasi-static analyses.<br />
<br />
|input deck=None<br />
<br />
|results=<br />
{{paper_figure|image=Pressure contour plots in AM30.jpg |image caption=Figure 2. Pressure contour plots in AM30}}<br />
Fig. 2 show the pressure contour plots for the cylinder, tapered cylinder, spiral, and tapered spiral using AM30 material properties when the wave front is at distance 1/3L, 2/3L, and L. Fig. 2 shows indeed that the tapered cylinder geometry induced greater hydrostatic and shear stresses as the dynamic wave propagated towards the smaller end of the bar.<br />
<br />
The following conclusions can be made regarding this study:<br />
*A tapered geometry will lower the pressure and impulse due to the convergent boundary and a continually decreasing cross-sectional area such that greater uniaxial stresses and subsequent axial deformation arises. Furthermore, the tapered geometry introduces small shear stresses that also help lower the impulse by removing energy from the initial longitudinal wave.<br />
*A spiral geometry will lower the impulse due to the introduction of shear stresses along the length of the spiral. These shear stresses introduce transverse displacements that help reduce the pressure and impulse.<br />
*When both the tapered and spiral geometry are included in a design, their synergistic effects multiplicatively reduce the pressure and impulse.<br />
*Although different materials admit different wave speeds, this impulse reduction was repeated in the three material types (elastic, plastic, and viscoelastic) and loading conditions. <br><br />
<br />
[[Image:Geometrypressure.gif|thumb|900px| Figure 3. Movie showing pressure wave behavior in different geometries.]]<br />
<br />
<br />
<br />
|references=K. L. Johnson, M. W. Trim, M. F. Horstemeyer, N. Lee, L. N. Williams, J. Liao, H. Rhee, and R. Prabhu, “Geometric Effects on Stress Wave Propagation,” J. Biomech. Eng., vol. 136, no. 2, pp. 021023–021023, Feb. 2014.<br />
}}<br />
<br />
[[Category:Biomaterials]]<br />
[[Category:ABAQUS]]<br />
[[Category:Animal Tissue Research]]<br />
[[Category:Journal Article]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Geometrypressure.gifFile:Geometrypressure.gif2015-06-01T18:33:39Z<p>Kyle: Kyle uploaded a new version of &quot;File:Geometrypressure.gif&quot;</p>
<hr />
<div>Pressure wave simulation in different geometries</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:Geometrypressure.gifFile:Geometrypressure.gif2015-06-01T16:54:11Z<p>Kyle: Pressure wave simulation in different geometries</p>
<hr />
<div>Pressure wave simulation in different geometries</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/CastingCasting2013-12-17T15:37:04Z<p>Kyle: </p>
<hr />
<div>Casting is a process where a liquid material is poured into a mold based on the negative of a desired geometry after which the liquid material solidifies. This solidified part is typically referred to as a casting. Casting is the most common process for creating complex parts.<br />
<br />
[[File:Bullet Casting.jpg|300px|thumb|right|Molten metal being poured into a mold to form a bullet.]]<br />
<br />
==Metal Casting==<br />
<br />
Metal casting involves pouring liquid metal into a mold, then allowing the metal to solidify. Metal casting molds can be divided into two distinct categories: expendable and non expendable.<br />
<br />
===Expendable Mold Casting===<br />
<br />
Expendable mold casting is a generalized term for a mold that is of single use or temporary use. Examples of such molds include sand, clay, plastic, wax, and plaster. These castings are used for creating a relatively small number of parts and are very cheap to make. <br />
<br />
===Non-Expendable Mold Casting===<br />
<br />
Non-expendable mold castings are the opposite of expendable castings where after each casting the mold does not need to be reformed/rebuilt. Four methods stand out when discussing non-expendable mold casting: die, permanent, continuous, and centrifugal. These methods are highly revered for their reproducibility and lifetime of the mold.</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-09T15:58:24Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpactv3.gif| A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:Rawlings-Combined_32Percent.gif|700px|NOCSAE drop test of Riddell 360 Football helmet with and without facemask attached]]<br />
<br />
<br />
[[Image:Riddell-360-Combined_32Percent.gif|700px|NOCSAE drop test of Rawlings Quantum Plus Football helmet with and without facemask attached]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:HelmetHeadImpactv3.gifFile:HelmetHeadImpactv3.gif2013-12-09T15:57:56Z<p>Kyle: </p>
<hr />
<div></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-09T15:51:20Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|700px| A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:Rawlings-Combined_32Percent.gif|700px|NOCSAE drop test of Riddell 360 Football helmet with and without facemask attached]]<br />
<br />
<br />
[[Image:Riddell-360-Combined_32Percent.gif|700px|NOCSAE drop test of Rawlings Quantum Plus Football helmet with and without facemask attached]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-09T15:50:34Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:Rawlings-Combined_32Percent.gif|700px|NOCSAE drop test of Riddell 360 Football helmet with and without facemask attached]]<br />
<br />
<br />
[[Image:Riddell-360-Combined_32Percent.gif|700px|NOCSAE drop test of Rawlings Quantum Plus Football helmet with and without facemask attached]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME8990_Student_Contributions(Fall_2013)ICME8990 Student Contributions(Fall 2013)2013-12-09T15:48:49Z<p>Kyle: /* Student 5 */</p>
<hr />
<div>==Student 1==<br />
<br />
[[student2013_1| For further info please email @]]<br />
<br />
[[Image:PC_2_difference.gif|center|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[http://www.google.com google]]<br />
<br />
<br />
<pre>Demo[mailto:name@cavs.msstate.edu]<br />
</pre><br />
<br />
==Student 2==<br />
===Justin Carrillo===<br />
[[VASP in batch mode| Correction]]<br /><br />
[[ICME 2013 HW1| Picture - Graph]]<br /><br />
[[Electronic Scale| Electronic Scale - Overview]]<br /><br />
[[Ceramics Home| Ceramics]]<br /><br />
[[Ceramics Home| Picture - Multiscale Modeling]]<br /><br />
<br />
==Student 3==<br />
===Justin Hughes===<br />
[[Uncertainty]] - Uncertainty Analysis overview, Statistical Methods Added <br /><br />
[[Heat Treatment Processes]] - Overview of various processes<br /><br />
[[Material Models]] - Under MSF, added link to MSF Uncertainty<br /><br />
[[MSF Uncertainty]] <br /><br />
<br />
==Student 5==<br />
<br />
===Kyle Johnson===<br />
<br />
[[SRCLID:Simulation-Based Design Optimization|Macroscale section update and Genetic Algorithm section added]]<br />
<br />
[[Metamodeling|Metamodeling section corrections and update]]<br />
<br />
[[Biomaterials Home|Biomaterials and Bio-inspired section update]]<br />
<br />
[[Structural Scale|Football Helmet Simulation Image]]<br />
<br />
[[Animations List|Football Helmet Simulation Animation]]<br />
<br />
==Student 6==<br />
<br />
[[Overview of birds' beak|Overview of birds' beaks create]]<br />
<br />
[[Biomaterials Home|Bio-inspired Design section update]]<br />
<br />
[[Structural Scale|Structural scale of the woodpecker beak create]]<br />
<br />
[[File:Structural scale woodpeckerBeak.png |center|thumb|300px|image update]]<br />
<br />
==Student 9==<br />
[[Animations List|Dislocation Dynamics Simulation Animation of Frank Read Sources in Iron]]<br />
<br />
[[MDDP|Multiscale Dislocation Dynamics Plasticity]]<br />
<br />
[[MDDP|Image of Dislocation Structure Evolution from MDDP Simulations using Tecplot]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME8990_Student_Contributions(Fall_2013)ICME8990 Student Contributions(Fall 2013)2013-12-09T15:48:06Z<p>Kyle: /* Student 5 */</p>
<hr />
<div>==Student 1==<br />
<br />
[[student2013_1| For further info please email @]]<br />
<br />
[[Image:PC_2_difference.gif|center|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[http://www.google.com google]]<br />
<br />
<br />
<pre>Demo[mailto:name@cavs.msstate.edu]<br />
</pre><br />
<br />
==Student 2==<br />
===Justin Carrillo===<br />
[[VASP in batch mode| Correction]]<br /><br />
[[ICME 2013 HW1| Picture - Graph]]<br /><br />
[[Electronic Scale| Electronic Scale - Overview]]<br /><br />
[[Ceramics Home| Ceramics]]<br /><br />
[[Ceramics Home| Picture - Multiscale Modeling]]<br /><br />
<br />
==Student 3==<br />
===Justin Hughes===<br />
[[Uncertainty]] - Uncertainty Analysis overview, Statistical Methods Added <br /><br />
[[Heat Treatment Processes]] - Overview of various processes<br /><br />
[[Material Models]] - Under MSF, added link to MSF Uncertainty<br /><br />
[[MSF Uncertainty]] <br /><br />
<br />
==Student 5==<br />
[[SRCLID:Simulation-Based Design Optimization|Macroscale section update and Genetic Algorithm section added]]<br />
<br />
[[Metamodeling|Metamodeling section corrections and update]]<br />
<br />
[[Biomaterials Home|Biomaterials and Bio-inspired section update]]<br />
<br />
[[Structural Scale|Football Helmet Simulation Image]]<br />
<br />
[[Animations List|Football Helmet Simulation Animation]]<br />
<br />
==Student 6==<br />
<br />
[[Overview of birds' beak|Overview of birds' beaks create]]<br />
<br />
[[Biomaterials Home|Bio-inspired Design section update]]<br />
<br />
[[Structural Scale|Structural scale of the woodpecker beak create]]<br />
<br />
[[File:Structural scale woodpeckerBeak.png |center|thumb|300px|image update]]<br />
<br />
==Student 9==<br />
[[Animations List|Dislocation Dynamics Simulation Animation of Frank Read Sources in Iron]]<br />
<br />
[[MDDP|Multiscale Dislocation Dynamics Plasticity]]<br />
<br />
[[MDDP|Image of Dislocation Structure Evolution from MDDP Simulations using Tecplot]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/SRCLID:Simulation-Based_Design_OptimizationSRCLID:Simulation-Based Design Optimization2013-12-09T15:46:59Z<p>Kyle: /* Genetic Algorithm */</p>
<hr />
<div>Design optimization can be thought of as used at the different length scale models and materials similar to every ICME notion.<br />
<br />
=General Design Optimization=<br />
<br />
Optimization discipline deals with finding the maximum and minimum of functions subject to some constraints. <br />
<br />
==Optimization guidelines==<br />
<br />
Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other<br />
methods allow multi-objective optimization, such as the calculation of a Pareto frontier<br />
<br />
Models: The designer must also choose models to relate the constraints and the objectives to<br />
the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
Reliability: the probability of a component to perform its required functions under stated<br />
conditions for a specified period of time.<br />
<br />
==Optimization Methods==<br />
<br />
Gradient-based methods:<br />
<br />
Adjoint equation,<br />
Newton’s method,<br />
Steepest descent,<br />
Conjugate gradient, and<br />
Sequential quadratic programming<br />
<br />
Gradient-free methods:<br />
<br />
Hooke-Jeeves pattern search and<br />
Nelder-Mead method<br />
<br />
Population-based methods:<br />
<br />
Genetic algorithm,<br />
Memetic algorithm,<br />
Particle swarm optimization,<br />
Ant colony, and<br />
Harmony search<br />
<br />
Other methods:<br />
<br />
Random search,<br />
Grid search,<br />
Simulated annealing,<br />
Direct search, and<br />
IOSO(Indirect Optimization based on Self-Organization)<br />
<br />
==Convergence of Pareto Frontier== <br />
<br />
It is relatively simple to determine an optimal solution for single objective methods (solution<br />
with the lowest error function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important<br />
physics into a few parameters.<br />
<br />
==Metamodeling==<br />
<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Unconstrained Optimization==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Golden Section Search====<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure. The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage<ref>Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.</ref>. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge<ref>Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.</ref><br />
<br />
<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the DMGfit and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/SRCLID:Simulation-Based_Design_OptimizationSRCLID:Simulation-Based Design Optimization2013-12-09T15:43:08Z<p>Kyle: /* Genetic Algorithm */</p>
<hr />
<div>Design optimization can be thought of as used at the different length scale models and materials similar to every ICME notion.<br />
<br />
=General Design Optimization=<br />
<br />
Optimization discipline deals with finding the maximum and minimum of functions subject to some constraints. <br />
<br />
==Optimization guidelines==<br />
<br />
Design variables: A design variable is a specification that is controllable by the designer (eg., thickness, material, etc.) and are often bounded by maximum and minimum values. Sometimes these bounds can be treated as constraints.<br />
<br />
Constraints: A constraint is a condition that must be satisfied for the design to be feasible. Examples include physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.<br />
<br />
Objectives: An objective is a numerical value or function that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other<br />
methods allow multi-objective optimization, such as the calculation of a Pareto frontier<br />
<br />
Models: The designer must also choose models to relate the constraints and the objectives to<br />
the design variables. They may include finite element analysis, reduced order metamodels, etc.<br />
<br />
Reliability: the probability of a component to perform its required functions under stated<br />
conditions for a specified period of time.<br />
<br />
==Optimization Methods==<br />
<br />
Gradient-based methods:<br />
<br />
Adjoint equation,<br />
Newton’s method,<br />
Steepest descent,<br />
Conjugate gradient, and<br />
Sequential quadratic programming<br />
<br />
Gradient-free methods:<br />
<br />
Hooke-Jeeves pattern search and<br />
Nelder-Mead method<br />
<br />
Population-based methods:<br />
<br />
Genetic algorithm,<br />
Memetic algorithm,<br />
Particle swarm optimization,<br />
Ant colony, and<br />
Harmony search<br />
<br />
Other methods:<br />
<br />
Random search,<br />
Grid search,<br />
Simulated annealing,<br />
Direct search, and<br />
IOSO(Indirect Optimization based on Self-Organization)<br />
<br />
==Convergence of Pareto Frontier== <br />
<br />
It is relatively simple to determine an optimal solution for single objective methods (solution<br />
with the lowest error function). However, for multiple objectives, we must evaluate solutions on a “Pareto frontier.” A solution lies on the Pareto frontier when any further changes to the parameters result in one or more objectives improving with the other objective(s) suffering as a result. Once a set of solutions have converged to the Pareto frontier, further testing is required in order to determine which candidate force field is optimal for the problems of interest. Be aware that searches with a limited number of parameters might “cram” a lot of important<br />
physics into a few parameters.<br />
<br />
==Metamodeling==<br />
<br />
see [[Metamodeling|Metamodeling]] and [http://en.wikipedia.org/wiki/Metamodeling Metamodeling-Wikipedia]<br />
<br />
==Unconstrained Optimization==<br />
<br />
===Zeroth-Order Methods===<br />
<br />
These methods are referred to as “zeroth-order methods” because they require only evaluation of the function, ''f''('''X'''), in each iterative step. Some examples of zeroth-order methods are the Bracketing Method and the Golden Section Search Method. Some population based methods could also be categorized as zeroth-order methods <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Bracketing Method====<br />
<br />
The Bracketing method is a zeroth-order method which used progressively smaller intervals to converge to an optimal solution. The interval is set up such that the x value corresponding to the optimal value of f lies within the interval. The interval is then divided into any number of sub-intervals of any given length. At each dividing point the value of f is calculated. The optimum sub-interval is then chosen as the next interval. This process iterates until convergence criteria is met <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Golden Section Search====<br />
<br />
===First-Order Methods===<br />
<br />
In addition to evaluation of f(X), first-order methods require the calculation of the gradient vector ∇f(X) in each iterative step. Some examples of first-order methods are the Steepest Descent or Cauchy Method and the Conjugate Gradient Method.<br />
<br />
====Steepest Descent (Cauchy) Method====<br />
<br />
The Steepest Descent method uses a search direction of some magnitude in the negative direction of the gradient. The negative of the gradient gives the direction of maximum decrease, hence steepest descent. The magnitude of the constant for the search direction can be determined through zeroth-order methods or from direct calculation. The direct calculation is done by setting the derivative equal to zero and solving for the constant. This method is guaranteed to converge to a local minimum, but convergence may be slow as previous iterations are not considered in determining the search direction of subsequent iterations. The rate of convergence can be estimated using the condition number of the Hessian matrix. If the condition number of the Hessian is large convergence will be slow <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
====Conjugate Gradient Method====<br />
<br />
The Conjugate Gradient Method is similar to the Steepest Descent Method except that it takes into consideration previous iterations when choosing search directions. The conjugate direction is determining by adding the steepest descent direction of the previous iteration, scaled by some value, to the steepest descent direction of the current iteration. The constant used to scale the search direction of the previous iteration can be determined using either the Fletcher-Reeves formula or the Polak-Ribiere formula <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Second-Order Methods===<br />
<br />
Second-order methods take advantage of the Hessian matrix, the second derivative, of the the function to improve search direction and rate of convergence. Some examples of second-order methods are Newton's Method, Davidon-Fletcher-Powell (DFP) method, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <ref name = 'Opt'>Rais-Rohani,Masoud “Handout #3: Mathematical Programming Methods for Unconstrained Optimization,” Design Optimization Class, Mississippi State University, Spring 2012.</ref>.<br />
<br />
===Population-Based Methods===<br />
<br />
Population based methods generate a population of points throughout the design space. Some methods then specify a range of the best points and generate a new population, continuing until convergence is reached (Monte-Carlo Method). Others generate a population and then "evolve" the points. The weakest of the new population is eliminated and the remainder evolved again until convergence is reached (Genetic Algorithm).<br />
<br />
====Monte-Carlo Method====<br />
<br />
see [http://en.wikipedia.org/wiki/Monte-Carlo_method Monte-Carlo]<br />
<br />
====Genetic Algorithm====<br />
<br />
Genetic algorithms are based on the principles of natural selection and natural genetics, meaning reproduction, crossover, and mutation are involved in the search procedure.The design variables are represented as strings of binary numbers which mirror chromosomes in genetics. These strings allow for the different binary numbers, or bits, to be adjusted during the reproduction, mutation, and crossover stage1. A population of points is used, and the number of initial points is typically two to four times the number of design variables. These points are evaluated to provide a fitness value, and above average points are selected and added to a new population of points. Points in this new population undergo the second stage in the algorithm known as crossover. In this stage information from two "parent" points, or strings, is combined to produce a new "child" point. The mutation operator is optional. It selects points based on a user-defined probability and alters a bit in the points binary string, thereby maintaining diversity in the population. The process is iterated until convergence is reached. GAs differ from other optimization techniques in that they work with a coding of the parameter set and not the parameters themselves, search a population of points instead of a single point, and use objective function knowledge instead of derivatives or other auxiliary knowledge2<br />
<br />
Rao, S.S., “Genetic Algorithms,” Engineering Optimization: Theory and Practice, John Wiley and Sons, Inc., 2009, pp. 694-702.<br />
Goldberg, D.E., Genetic Algorithms in search, optimization, and machine learning, Addison Wesley Longman, 1989.<br />
[[SRCLID|back to the SRCLID home]]<br />
<br />
[[Category:Overview]]<br />
<br />
=Structural Scale Optimization=<br />
<br />
===Analytical Model for Axial Crushing of Multi-cell Multi-corner Tubes ([[Multi-CRUSH]])===<br />
Contributers: [http://www.cavs.msstate.edu/information.php?eid=317 Ali Najafi] and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
[[Media:Mohammad_Rouhi_Thesis_final.pdf| Mohammad Rouhi's MSc Thesis]]<br />
<br />
'''Topology Optimization of Continuum Structures Using Element Exchange Method'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]) and [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu])<br />
<br />
http://pdf.aiaa.org/preview/CDReadyMSDM08_1875/PV2008_1707.pdf<br />
<br />
'''Element Exchange Method for Topology Optimization'''<br />
<br />
Authors: [http://www.cavs.msstate.edu/information.php?eid=316 Mohammad Rouhi] ([mailto:rouhi@cavs.msstate.edu rouhi@cavs.msstate.edu]), [http://www.ae.msstate.edu/pages/rohani.php Masoud Rais-Rohani] ([mailto:masoud@ae.msstate.edu masoud@ae.msstate.edu]) and [http://www.cavs.msstate.edu/information.php?eid=144 Thomas Neil Williams] ([mailto:tnw7@cavs.msstate.edu tnw7@cavs.msstate.edu])<br />
<br />
http://springerlink.com/index/m30m6x1x62k252lr.pdf<br />
<br />
= Macroscale=<br />
<br />
Optimization algorithms can be used for model calibration. For example, the DMGfit and MSFfit routines employ optimization algorithms to automatically fit the plasticity-damage model and the fatigue model, respectively. The constants of interest are selected and a Monte Carlo optimization routine is performed to generate candidate constants. A single element simulation then produces the model stress-strain curve. The curve is compared to the input data for fit comparison, and this process is repeated until a satisfactory fit is achieved or a maximum number of iterations is reached. The resulting optimized constants are then output.<br />
<br />
= Mesoscale=<br />
<br />
= Microscale=<br />
<br />
= Nanoscale=<br />
<br />
The Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials can be optimized based upon on Electronics Scale calculation results and experimental data.<br />
<br />
= Electronic Scale=<br />
<br />
= Multilevel Design Optimization =<br />
This is an emerging topics at CAVS. The [[Multilevel Design Optimization|pages describing the progress]] are currently available only to the members of the research team.<br />
<br />
====References====<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Structural_ScaleStructural Scale2013-12-06T22:56:23Z<p>Kyle: </p>
<hr />
<div>__NOTOC__<br />
<br />
{{Menu_Models}}<br />
<br />
<br />
[[Image:PC_2_difference.gif|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[Image:HelmetHeadImpactPicICME.png|thumb|right|250px| Football helmet impact simulation]]<br />
<br />
<br />
== Introduction ==<br />
The key to structural scale applications is employing the "best" numerical method for the application. Typically, for solid mechanics, finite element methods are employed and used mostly for the engineering applications described in this CyberInfrastructure. <br />
<br />
<br />
== Material Models ==<br />
Finite element codes can be divided into two categories: implicit quasi-static codes and explicit dynamic (hydrodynamic) codes. Some examples of implicit quasi-static codes include commercial codes such as [[Code:_ABAQUS_FEM|ABAQUS (Simulia)]], [[Code:_NASTRAN|MSC Nastran]], and [[ESI-Pamstamp]]. Some open network implicit codes include [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]]. Some examples of explicit dynamics codes include [[Dyna]], [[Code:_LS-DYNA|LS-Dyna]], [[Pronto]], and [[Code:_ABAQUS|ABAQUS-Explicit]]. [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]] also provide some explicit dynamics solvers as well.<br />
<br />
== Metals ==<br />
The structural scale information essentially requires the constitutive model that is received from the macroscale. Although common practice finite element analysis does not include heterogeneities from microstructures, defects, and inclusions within the mesh related to the constitutive model, the MSU plasticity-damage 1.0 model allows the incorporation of such materials science information. The quantities that can be included in this version of the constitutive model are the grain size, particle size and volume fraction of particles, pore size and volume fraction or pores (porosity level), nearest neighbor distances of pores and particles. Hence, each element in the finite element mesh would have a different value for each of the quantities and hence the strength and ductility of the material in those domains. Several examples that show that by not using the heterogenous distributions of microstructures, defects, and inclusions include the redesign of a Cadillac control arm <ref name="one">[http://dx.doi.org/10.1023/B:JCAD.0000024171.13480.24 Horstemeyer, M.F., Wang, P., “Cradle-to-Grave simulation-Based Design Incorporating Multiscale Microstructure-Property Modeling: Reinvigorating Design with Science,” ''J. Computer-Aided Materials Design'', Vol. 10, pp. 13-34, 2003.]</ref>, the Corvette engine cradle <ref>M.F. Horstemeyer, D. Oglesby, J. Fan, P.M. Gullett, H. El Kadiri, Y. Xue, C. Burton, K. Gall, B. Jelinek, M.K. Jones, S. G. Kim, E.B. Marin, D.L. McDowell, A. Oppedal, N. Yang, “From Atoms to Autos: Designing a Mg Alloy Corvette Cradle by Employing Hierarchical Multiscale Microstructure-Property Models for Monotonic and Cyclic Loads,” MSU.CAVS.CMD.2007-R0001, 2007</ref>, and a powder metal steel engine bearing cap <ref>Hammi, Y, Horstemeyer, MF, Stone, T., Sanderow, H., Chernenkoff, R., Weber, G., "Powder-Metal Performance Modeling of Automotive Components AMD-410, 2009</ref>.<br />
<br />
Some examples of using different finite element simulations with associated input decks using our MSU plasticity-damage 1.0 can be garnered from the following locations:<br />
<br />
# Cadillac control arm ([[Code:_ABAQUS|ABAQUS-Implicit]])<ref name="one"></ref> <br />
# Corvette cradle ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Dodge Neon crash ([[Code:_LS-DYNA|LS-Dyna]])<br />
# Forming of aluminum plate ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Crush of aluminum tube ([[Code:_ABAQUS|ABAQUS-Explicit]])<br />
# [[Multi-CRUSH|Axial Crushing of Multi-Cell Multi-Corner Tubes]] ([[Code:_LS-DYNA|LS-Dyna]])<br />
<br />
====[[Process_Modeling#Hydroforming|Hydroforming]]====<br />
<br />
<br />
== Ceramics==<br />
<br />
<br />
== Polymers ==<br />
<br />
=== ISV Polymer Modeling ===<br />
<br />
[http://dx.doi.org/10.1007/s00707-010-0349-y A general inelastic internal state variable model for amorphous glassy polymers]<br />
<br />
[http://dx.doi.org/10.1016/j.ijplas.2012.10.005 An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation]<br />
<br />
=== Application ===<br />
<br />
[http://dx.doi.org/10.1016/j.engfailanal.2012.07.020 Characterization and failure analysis of a polymeric clamp hanger component]<br />
<br />
== Biomaterials==<br />
* Rams Horn<br />
**[[Experiments-Structure-Mechanical Property Relations]]<br />
* Porcine Brain<br />
**[[Coupled Dynamic Experiments/Modeling]]<br />
<br />
== Geomaterials ==<br />
*Earth Mantle<br />
**[[Mantle Convection Study with Lherzolite Material Modeling]]<br />
<br />
== References ==<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Structural_ScaleStructural Scale2013-12-06T22:55:52Z<p>Kyle: </p>
<hr />
<div>__NOTOC__<br />
<br />
{{Menu_Models}}<br />
<br />
<br />
[[Image:PC_2_difference.gif|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[Image:HelmetHeadImpactPicICME.png|thumb|right|200px| Football helmet impact simulation]]<br />
<br />
<br />
== Introduction ==<br />
The key to structural scale applications is employing the "best" numerical method for the application. Typically, for solid mechanics, finite element methods are employed and used mostly for the engineering applications described in this CyberInfrastructure. <br />
<br />
<br />
== Material Models ==<br />
Finite element codes can be divided into two categories: implicit quasi-static codes and explicit dynamic (hydrodynamic) codes. Some examples of implicit quasi-static codes include commercial codes such as [[Code:_ABAQUS_FEM|ABAQUS (Simulia)]], [[Code:_NASTRAN|MSC Nastran]], and [[ESI-Pamstamp]]. Some open network implicit codes include [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]]. Some examples of explicit dynamics codes include [[Dyna]], [[Code:_LS-DYNA|LS-Dyna]], [[Pronto]], and [[Code:_ABAQUS|ABAQUS-Explicit]]. [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]] also provide some explicit dynamics solvers as well.<br />
<br />
== Metals ==<br />
The structural scale information essentially requires the constitutive model that is received from the macroscale. Although common practice finite element analysis does not include heterogeneities from microstructures, defects, and inclusions within the mesh related to the constitutive model, the MSU plasticity-damage 1.0 model allows the incorporation of such materials science information. The quantities that can be included in this version of the constitutive model are the grain size, particle size and volume fraction of particles, pore size and volume fraction or pores (porosity level), nearest neighbor distances of pores and particles. Hence, each element in the finite element mesh would have a different value for each of the quantities and hence the strength and ductility of the material in those domains. Several examples that show that by not using the heterogenous distributions of microstructures, defects, and inclusions include the redesign of a Cadillac control arm <ref name="one">[http://dx.doi.org/10.1023/B:JCAD.0000024171.13480.24 Horstemeyer, M.F., Wang, P., “Cradle-to-Grave simulation-Based Design Incorporating Multiscale Microstructure-Property Modeling: Reinvigorating Design with Science,” ''J. Computer-Aided Materials Design'', Vol. 10, pp. 13-34, 2003.]</ref>, the Corvette engine cradle <ref>M.F. Horstemeyer, D. Oglesby, J. Fan, P.M. Gullett, H. El Kadiri, Y. Xue, C. Burton, K. Gall, B. Jelinek, M.K. Jones, S. G. Kim, E.B. Marin, D.L. McDowell, A. Oppedal, N. Yang, “From Atoms to Autos: Designing a Mg Alloy Corvette Cradle by Employing Hierarchical Multiscale Microstructure-Property Models for Monotonic and Cyclic Loads,” MSU.CAVS.CMD.2007-R0001, 2007</ref>, and a powder metal steel engine bearing cap <ref>Hammi, Y, Horstemeyer, MF, Stone, T., Sanderow, H., Chernenkoff, R., Weber, G., "Powder-Metal Performance Modeling of Automotive Components AMD-410, 2009</ref>.<br />
<br />
Some examples of using different finite element simulations with associated input decks using our MSU plasticity-damage 1.0 can be garnered from the following locations:<br />
<br />
# Cadillac control arm ([[Code:_ABAQUS|ABAQUS-Implicit]])<ref name="one"></ref> <br />
# Corvette cradle ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Dodge Neon crash ([[Code:_LS-DYNA|LS-Dyna]])<br />
# Forming of aluminum plate ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Crush of aluminum tube ([[Code:_ABAQUS|ABAQUS-Explicit]])<br />
# [[Multi-CRUSH|Axial Crushing of Multi-Cell Multi-Corner Tubes]] ([[Code:_LS-DYNA|LS-Dyna]])<br />
<br />
====[[Process_Modeling#Hydroforming|Hydroforming]]====<br />
<br />
<br />
== Ceramics==<br />
<br />
<br />
== Polymers ==<br />
<br />
=== ISV Polymer Modeling ===<br />
<br />
[http://dx.doi.org/10.1007/s00707-010-0349-y A general inelastic internal state variable model for amorphous glassy polymers]<br />
<br />
[http://dx.doi.org/10.1016/j.ijplas.2012.10.005 An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation]<br />
<br />
=== Application ===<br />
<br />
[http://dx.doi.org/10.1016/j.engfailanal.2012.07.020 Characterization and failure analysis of a polymeric clamp hanger component]<br />
<br />
== Biomaterials==<br />
* Rams Horn<br />
**[[Experiments-Structure-Mechanical Property Relations]]<br />
* Porcine Brain<br />
**[[Coupled Dynamic Experiments/Modeling]]<br />
<br />
== Geomaterials ==<br />
*Earth Mantle<br />
**[[Mantle Convection Study with Lherzolite Material Modeling]]<br />
<br />
== References ==<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:27:31Z<p>Kyle: </p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|600px|A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:27:11Z<p>Kyle: </p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|600px]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:22:18Z<p>Kyle: </p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|600px| A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:22:05Z<p>Kyle: </p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME8990_Student_Contributions(Fall_2013)ICME8990 Student Contributions(Fall 2013)2013-12-06T22:21:49Z<p>Kyle: /* Student 5 */</p>
<hr />
<div>==Student 1==<br />
<br />
[[student2013_1| For further info please email @]]<br />
<br />
[[Image:PC_2_difference.gif|center|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[http://www.google.com google]]<br />
<br />
<br />
<pre>Demo[mailto:name@cavs.msstate.edu]<br />
</pre><br />
<br />
==Student 2==<br />
===Justin Carrillo===<br />
[[VASP in batch mode| Correction]]<br /><br />
[[ICME 2013 HW1| Picture - Graph]]<br /><br />
[[Electronic Scale| Electronic Scale - Overview]]<br /><br />
[[Ceramics Home| Ceramics]]<br /><br />
[[Ceramics Home| Picture - Multiscale Modeling]]<br /><br />
<br />
==Student 3==<br />
<br />
==Student 5==<br />
[[SRCLID:Simulation-Based Design Optimization|Macroscale section update]]<br />
<br />
[[Metamodeling|Metamodeling section corrections and update]]<br />
<br />
[[Biomaterials Home|Biomaterials and Bio-inspired section update]]<br />
<br />
[[Structural Scale|Football Helmet Simulation Image]]<br />
<br />
[[Animations List|Football Helmet Simulation Animation]]<br />
<br />
==Student 6==<br />
<br />
[[Overview of birds' beak|Overview of birds' beaks update]]<br />
<br />
[[Biomaterials Home|Bio-inspired Design section update]]<br />
<br />
[[File:Beak properties.jpg|center|thumb|300px| image(table)update]]<br />
<br />
==Student 9==<br />
[[Animations List|Dislocation Dynamics Simulation Animation of Frank Read Sources in Iron]]<br />
<br />
[[MDDP|Multiscale Dislocation Dynamics Plasticity]]<br />
<br />
[[MDDP|Image of Dislocation Structure Evolution from MDDP Simulations using Tecplot]]</div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Structural_ScaleStructural Scale2013-12-06T22:19:54Z<p>Kyle: </p>
<hr />
<div>__NOTOC__<br />
<br />
{{Menu_Models}}<br />
<br />
<br />
[[Image:PC_2_difference.gif|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[Image:HelmetHeadImpactPicICME.png|thumb|right|300px| Football helmet impact simulation]]<br />
<br />
<br />
== Introduction ==<br />
The key to structural scale applications is employing the "best" numerical method for the application. Typically, for solid mechanics, finite element methods are employed and used mostly for the engineering applications described in this CyberInfrastructure. <br />
<br />
<br />
== Material Models ==<br />
Finite element codes can be divided into two categories: implicit quasi-static codes and explicit dynamic (hydrodynamic) codes. Some examples of implicit quasi-static codes include commercial codes such as [[Code:_ABAQUS_FEM|ABAQUS (Simulia)]], [[Code:_NASTRAN|MSC Nastran]], and [[ESI-Pamstamp]]. Some open network implicit codes include [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]]. Some examples of explicit dynamics codes include [[Dyna]], [[Code:_LS-DYNA|LS-Dyna]], [[Pronto]], and [[Code:_ABAQUS|ABAQUS-Explicit]]. [[Code:_TAHOE|TAHOE]] and [[Code:_CALCULIX|CALCULIX]] also provide some explicit dynamics solvers as well.<br />
<br />
== Metals ==<br />
The structural scale information essentially requires the constitutive model that is received from the macroscale. Although common practice finite element analysis does not include heterogeneities from microstructures, defects, and inclusions within the mesh related to the constitutive model, the MSU plasticity-damage 1.0 model allows the incorporation of such materials science information. The quantities that can be included in this version of the constitutive model are the grain size, particle size and volume fraction of particles, pore size and volume fraction or pores (porosity level), nearest neighbor distances of pores and particles. Hence, each element in the finite element mesh would have a different value for each of the quantities and hence the strength and ductility of the material in those domains. Several examples that show that by not using the heterogenous distributions of microstructures, defects, and inclusions include the redesign of a Cadillac control arm <ref name="one">[http://dx.doi.org/10.1023/B:JCAD.0000024171.13480.24 Horstemeyer, M.F., Wang, P., “Cradle-to-Grave simulation-Based Design Incorporating Multiscale Microstructure-Property Modeling: Reinvigorating Design with Science,” ''J. Computer-Aided Materials Design'', Vol. 10, pp. 13-34, 2003.]</ref>, the Corvette engine cradle <ref>M.F. Horstemeyer, D. Oglesby, J. Fan, P.M. Gullett, H. El Kadiri, Y. Xue, C. Burton, K. Gall, B. Jelinek, M.K. Jones, S. G. Kim, E.B. Marin, D.L. McDowell, A. Oppedal, N. Yang, “From Atoms to Autos: Designing a Mg Alloy Corvette Cradle by Employing Hierarchical Multiscale Microstructure-Property Models for Monotonic and Cyclic Loads,” MSU.CAVS.CMD.2007-R0001, 2007</ref>, and a powder metal steel engine bearing cap <ref>Hammi, Y, Horstemeyer, MF, Stone, T., Sanderow, H., Chernenkoff, R., Weber, G., "Powder-Metal Performance Modeling of Automotive Components AMD-410, 2009</ref>.<br />
<br />
Some examples of using different finite element simulations with associated input decks using our MSU plasticity-damage 1.0 can be garnered from the following locations:<br />
<br />
# Cadillac control arm ([[Code:_ABAQUS|ABAQUS-Implicit]])<ref name="one"></ref> <br />
# Corvette cradle ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Dodge Neon crash ([[Code:_LS-DYNA|LS-Dyna]])<br />
# Forming of aluminum plate ([[Code:_ABAQUS|ABAQUS-Implicit]])<br />
# Crush of aluminum tube ([[Code:_ABAQUS|ABAQUS-Explicit]])<br />
# [[Multi-CRUSH|Axial Crushing of Multi-Cell Multi-Corner Tubes]] ([[Code:_LS-DYNA|LS-Dyna]])<br />
<br />
====[[Process_Modeling#Hydroforming|Hydroforming]]====<br />
<br />
<br />
== Ceramics==<br />
<br />
<br />
== Polymers ==<br />
<br />
=== ISV Polymer Modeling ===<br />
<br />
[http://dx.doi.org/10.1007/s00707-010-0349-y A general inelastic internal state variable model for amorphous glassy polymers]<br />
<br />
[http://dx.doi.org/10.1016/j.ijplas.2012.10.005 An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation]<br />
<br />
=== Application ===<br />
<br />
[http://dx.doi.org/10.1016/j.engfailanal.2012.07.020 Characterization and failure analysis of a polymeric clamp hanger component]<br />
<br />
== Biomaterials==<br />
* Rams Horn<br />
**[[Experiments-Structure-Mechanical Property Relations]]<br />
* Porcine Brain<br />
**[[Coupled Dynamic Experiments/Modeling]]<br />
<br />
== Geomaterials ==<br />
*Earth Mantle<br />
**[[Mantle Convection Study with Lherzolite Material Modeling]]<br />
<br />
== References ==<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:HelmetHeadImpactPicICME.pngFile:HelmetHeadImpactPicICME.png2013-12-06T22:18:01Z<p>Kyle: </p>
<hr />
<div></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:13:08Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|600px| A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:HelmetHeadImpact2.gifFile:HelmetHeadImpact2.gif2013-12-06T22:08:18Z<p>Kyle: Kyle uploaded a new version of &quot;File:HelmetHeadImpact2.gif&quot;</p>
<hr />
<div></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:06:32Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|400px | A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/File:HelmetHeadImpact2.gifFile:HelmetHeadImpact2.gif2013-12-06T22:03:41Z<p>Kyle: </p>
<hr />
<div></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/Animations_ListAnimations List2013-12-06T22:03:05Z<p>Kyle: /* Structural Scale Animations */</p>
<hr />
<div>== Structural Scale Animations ==<br />
[[Image:110Rendummycrash.gif | A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion.]]<br />
<ref name="Horst_2009">M.F. Horstemeyer, X.C. Ren,H. Fang, E. Acar,and P.T. Wang, "A comparative study of design optimisation methodologies for side-impact crashworthiness,using injury-based versus energy-based criterion," International Journal of Crashworthiness,Vol. 14, No. 2, April 2009, 125–138. [[Media:110.Ren.dummy.crash.pdf|--link]]</ref>.<br />
<br />
[[Image:HelmetHeadImpact2.gif|400px |A study on the effect of impacts to the head in a football helmet]]<br />
<br />
[[Image:120turtlepaper2009.gif|300px | A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites ]]<br />
<br />
<ref name="HRhee2010">H. Rhee, M.F. Horstemeyer,Y. Hwang,H. Lim,H. El Kadiri, W. Trim "A study on the structure and mechanical behavior of the Terrapene carolina carapace:A pathway to design bio-inspired synthetic composites," Materials Science and Engineering,29 (2009) 2333–2339[[Media:120.turtle.paper.2009.pdf|--link]]</ref><br />
<br />
<br />
[[Image:PC_2_difference.gif|600px| Movie capturing high strain rate deformation of polycarbonate. Shown as a difference image between successive frames, so movement triggers an intensity other than gray.]]<br />
<br />
[[Image:PC_2_lowres.gif|300px| Movie capturing high strain rate deformation of polycarbonate.]]<br />
<br />
[[Image:animation_ICME5.gif|600px| Movie capturing tube forming process from sheet steel.]]<br />
<br />
== Macroscale Animations ==<br />
<br />
== Mesoscale Animations ==<br />
<br />
== Microscale Animations ==<br />
[[Image: DDD_simulation_of_FR_Source_in_Iron.gif|left]]<br />
<br />
<ref> Raabe, Dierk. [http://www.dierk-raabe.com/movies-and-animations/discrete-dislocation-dynamics-ddd/ Discrete Dislocation Dynamics Simulations (DDD)] </ref><br />
<br />
== Nanoscale Animations ==<br />
<br />
[[Image:Al_SC_100_movie2.gif|300px| [[Uniaxial_Tension | Tensile Loading of an Aluminum Single Crystal]]. Movie showing deformation of single crystal aluminum loaded in the <100> direction at a strain rate of 10<sup>10</sup> s<sup>-1</sup> and a temperature of 300 K.]]<br />
<br />
[[Image:PE_deformation.gif|300px|[[MD_PE_deformation | Polymer Atomistic Research]]. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 10<sup>10</sup> s<sup>-1</sup> and the temperature is 100 K<ref name="Hos2010">Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F.,"Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.</ref><ref name="Tsc_2010TMS">Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.</ref>.]]<br />
<br />
== Electronic Scale Animations ==<br />
<br />
== References ==<br />
<br />
<references/></div>Kylehttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME8990_Student_Contributions(Fall_2013)ICME8990 Student Contributions(Fall 2013)2013-12-06T21:36:27Z<p>Kyle: /* Student 5 */</p>
<hr />
<div>==Student 1==<br />
<br />
[[student2013_1| For further info please email @]]<br />
<br />
[[Image:PC_2_difference.gif|center|thumb|600px| Movie capturing high strain rate deformation of polycarbonate (experiment). Shown as a difference image between successive frames, so movement triggers an intensity other than gray. Experiments are used to validate macroscale models.]]<br />
<br />
[[http://www.google.com google]]<br />
<br />
<br />
<pre>Demo[mailto:name@cavs.msstate.edu]<br />
</pre><br />
<br />
==Student 2==<br />
===Justin Carrillo===<br />
[[VASP in batch mode| Correction]]<br /><br />
[[ICME 2013 HW1| Picture - Graph]]<br /><br />
[[Electronic Scale| Electronic Scale - Overview]]<br /><br />
[[Ceramics Home| Ceramics]]<br /><br />
[[Ceramics Home| Picture - Multiscale Modeling]]<br /><br />
<br />
==Student 3==<br />
<br />
==Student 5==<br />
[[SRCLID:Simulation-Based Design Optimization|Macroscale section update]]<br />
<br />
[[Metamodeling|Metamodeling section corrections and update]]<br />
<br />
[[Biomaterials Home|Biomaterials and Bio-inspired section update]]<br />
<br />
==Student 6==<br />
<br />
[[Overview of birds' beak|Overview of birds' beaks update]]<br />
<br />
[[Biomaterials Home|Bio-inspired Design section update]]<br />
<br />
[[File:Beak properties.jpg|center|thumb|300px| image(table)update]]<br />
<br />
==Student 9==<br />
[[Animations List|Dislocation Dynamics Simulation Animation of Frank Read Sources in Iron]]<br />
<br />
[[MDDP|Multiscale Dislocation Dynamics Plasticity]]<br />
<br />
[[MDDP|Image of Dislocation Structure Evolution from MDDP Simulations using Tecplot]]</div>Kyle