TrackError Custom Task

A custom task to archive simulations results compared to theoretical predictions

Introduction

This tasks can compare particle values to a theoretical model and archive some metric for accuracy of the simulation. It is useful for tracking convergence of MPM simulations.

Simulation Error Metrics

Imagine a generic distance from particle value to a theoretical expectation for that value equal to [math]\displaystyle{ |x_p| }[/math]. For example, the distance could be difference between particle stress in the x directions and expected stress:

      [math]\displaystyle{ |x_p| = |\sigma_{p,xx} - \sigma_{xx}({\rm theory})| }[/math]

If the particle value is a vector, the distance would be a vector distance. For example, velocity distance would be:

      [math]\displaystyle{ |x_p| = \sqrt{\bigl(\vec V_p-\vec V({\rm theory})\bigr)\cdot\bigl(\vec V_p-\vec V({\rm theory})\bigr)} }[/math]

The P-norm for a give distance is defined as:

      [math]\displaystyle{ ||x||_P = \left( \sum_{p=1}^{N_p} |x_p|^P\right)^{1/P} }[/math]

where [math]\displaystyle{ N_p }[/math] is the number of particles. The root-mean-squared distance is defined as:

      [math]\displaystyle{ {\rm RMS} = \sqrt{{1\over N_p} \sum_{p=1}^{N_p} |x_p|^2} }[/math]

This task can archive mean P-norm values and RMS distances for any entered function of [math]\displaystyle{ |x_p|^P }[/math]. These mean values can be for each time step or cumulatively averaged over all time steps.