# Modeling Uncertainty

## Contents |

# Uncertainty

## Objective

This page will provide information on how to model uncertainty using the MEAM parameter calibration (MPC) tool and Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). On this page, the central finite difference approximation is used as an example to help users to understand how to model the uncertainty of the response of your system with respect to certain variables.

In this example, the "response" of our system will be the dislocation velocity determined from LAMMPS. Additionally, "variables" can be inputs that contribute to the response of your system. In this example, these variables will be will be the MEAM parameters that are used to input into LAMMPS.

## Theory

The uncertainty of the response of your system can be approximated using a one-factor-at-atime perturbation methodology. This method uses the central difference approximation to estimate the sensitivity of your response with respect to input variables. This sensitivity can be expressed as:

where f() is the model function, Xi is the model input parameter, X0,i is the base value of a parameter, +/-i is the perturbation size around the base parameter, and DeltaXi is the difference between the perturbed input parameters. The perturbation size typically assumes a +/-1% perturbed factor. The uncertainty based on the sensitivity of an input can be determined from the following equation:

where Uf is the total uncertainity propagated through the model, df/dx is the model sensitivity in the equation prescribed above, N is the total number of parameters, and Uxi is the input parameter uncertainty. This parameter uncertainty term will have to depend on previous studies with respect to it's variance on the response your system. Conservatively, it can be assumed that a 5% parameter uncertainty can be used.

## Example

In this example, we will vary a single MEAM parameter and look at the influence with respect to the dislocation velocity of a single material, called "Material 1". In this study, we will vary the parameter b2 by approximately +/-1% and assume a parameter uncertainty of approximately 5%. In Figure 1, the final result to calculate the uncertainty is shown.

### Step 1

Calibrate your MEAM potential with respect to Density Functional Theory. This requires the use of elastic constants from experiments or literature to calibrate your material to DFT. The MEAM parameters we will be using is shown below for our "Material".

### Step 2

We will need to run LAMMPS to determine the dislocation velocity for an edge dislocation. This was done at several applied shear stress levels (10 to 1200 MPa) for the following test cases:

- a nominal MEAM parameter test case
- a 1% increase in b2 test case
- a 1% decrease b2 test case

In Figure 2, a flow chart to run LAMMPS is shown. First, the volume of atoms needs to be generated for an edge dislocation. Secondly, the dislocation velocity specifying the applied stress and temperature needs to be written. Lastly, the MEAM input parameters needs to be written with respect to the test cases prescribed above. Please refer to the LAMMPS page on how this is performed. Once LAMMPS is used to calculate the displacement of the atoms, a dislocation velocity can be calculated using a single defect velocity script.

### Step 3

In table 2, the uncertainty with respect to the MEAM parameters is determined. Note here that since we are only evaluating the uncertainty with respect to one variable, there is no summation. Essentially, N=1. If we were varying more than one parameter, a summation would need to be performed to determine the total accumulated uncertainty. Lastly, the uncertainty is added or subtracted to determine the uncertainty bands with respect to the mean response.

## References

J.M. Hughes, M.F. Horstemeyer, R. Carino, N. Sukhija, W.B. Lawrimore, S. Kim, and M.I. Baskes. "Hierarchical Bridging Between Ab Initio and Atomistic Level Computations: Sensitivity and Uncertainty Analysis for the Modified Embedded-Atom Method (MEAM) Potential (Part B)." JOM, Vol. 67, No. 1, 2015. DOI: 10.1007/s11837-014-1205-7