An a posteriori error estimator for isogeometric analysis on trimmed geometries

TL;DR

This paper introduces a reliable a posteriori error estimator for isogeometric analysis on trimmed geometries, enabling adaptive mesh refinement without reconstructing the global parametrization, validated through numerical experiments.

Contribution

It develops a residual a posteriori error estimator for hierarchical B-spline based isogeometric analysis on trimmed geometries, independent of mesh trimming complexity.

Findings

Estimator is reliable regardless of hierarchical levels

Effectivity index is independent of trimmed mesh size

Numerical validation confirms theoretical properties

Abstract

Trimming consists of cutting away parts of a geometric domain, without reconstructing a global parametrization (meshing). It is a widely used operation in computer aided design, which generates meshes that are unfitted with the described physical object. This paper develops an adaptive mesh refinement strategy on trimmed geometries in the context of hierarchical B-spline based isogeometric analysis. A residual a posteriori estimator of the energy norm of the numerical approximation error is derived, in the context of the Poisson equation. The estimator is proven to be reliable, independently of the number of hierarchical levels and of the way the trimmed boundaries cut the underlying mesh. Numerical experiments are performed to validate the presented theory, and to show that the estimator's effectivity index is independent of the size of the active part of the trimmed mesh elements.