Error Analysis for Quadtree-Type Mesh-Coarsening Algorithms Adapted to Pixelized Heterogeneous Microstructures

TL;DR

This paper introduces adaptive quadtree mesh-coarsening algorithms for pixelized microstructures, analyzing their error and efficiency trade-offs to optimize finite element simulations in computational materials science.

Contribution

It proposes two novel quadtree-based mesh-coarsening methods and a modified stress recovery scheme for improved error estimation in microstructure analysis.

Findings

Error analysis quantifies the trade-off between accuracy and efficiency.

The modified stress recovery scheme improves error estimation near interfaces.

Adaptive mesh-coarsening enhances computational efficiency without significant accuracy loss.

Abstract

Pixel- and voxel-based representations of microstructures obtained from tomographic imaging methods is an established standard in computational materials science. The corresponding highly resolved, uniform discretitization in numerical analysis is adequate to accurately describe the geometry of interfaces and defects in microstructures and, therefore, to capture the physical processes in these regions of interest. For the defect-free interior of phases and grains however, the high resolution is in view of only weakly varying field properties not necessary such that mesh-coarsening in these regions can improve efficiency without severe losses of accuracy in simulations. The present work proposes two different variants of adaptive, quadtree-based mesh-coarsening algorithms applied to pixelized images that serves the purpose of a preprocessor for consecutive finite element analyses, here, in the context of numerical homogenization. Error analysis is carried out on the microscale by error estimation which itself is assessed by true error computation. A modified stress recovery scheme for a superconvergent error estimator is proposed which overcomes the deficits of the standard recovery scheme for nodal stress computation in cases of interfaces with stiffness jump. By virtue of error analysis the improved efficiency by the reduction of unknowns is put into relation to the increase of the discretization error. This quantitative analysis sets a rational basis for decisions on favorable meshes having the best trade-off between accuracy and efficiency as will be underpinned by various examples.