A Certified Goal-Oriented A Posteriori Defeaturing Error Estimator for Elliptic PDEs

TL;DR

This paper develops mathematically certified, goal-oriented a posteriori error estimators for geometry defeaturing in elliptic PDEs, enabling precise control of errors in specific quantities of interest during model simplification.

Contribution

It introduces new goal-oriented error estimators for defeaturing in elliptic PDEs, applicable to multiple boundary conditions and combined with the DWR method for accurate QoI error estimation.

Findings

Effective error bounds for geometry simplifications in PDEs.

Reliable estimates for quantities of interest demonstrated numerically.

Extension of estimators to multiple boundary conditions and features.

Abstract

Defeaturing, the process of simplifying computational geometries, is a critical step in industrial simulation pipelines for reducing computational cost. Rigorous a posteriori estimators exist for the global energy-norm error introduced by geometry simplifications. However, practitioners are usually more concerned with the accuracy of specific quantities of interest (QoIs) in the solution. This paper bridges that gap by developing mathematically certified, goal-oriented a posteriori defeaturing error estimators for Poisson's equation, linear elasticity, and Stokes flow. First, we derive new reliable energy-norm estimators for features subject to Dirichlet boundary conditions in linear elasticity and Stokes flow, based on existing results for Poisson's equation. Second, we formulate general energy-norm estimators for multiple negative features, subject to either Dirichlet or Neumann boundary conditions for the first time. Finally, we combine these estimators with the dual-weighted residual (DWR) method to obtain reliable estimates for linear QoIs and demonstrate their effectiveness across a range of numerical experiments.