Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

TL;DR

This paper introduces an adaptive isogeometric analysis framework for 2D trimmed domains using hierarchical splines, enabling efficient and accurate solutions for complex engineering problems with singularities.

Contribution

It develops an error-driven adaptive method with a posteriori error estimation and hierarchical spline refinement for trimmed geometries, validated on various engineering problems.

Findings

Achieves optimal convergence rates for smooth and singular solutions.

Significantly improves accuracy per degree of freedom over uniform refinement.

Demonstrates applicability to industrial-like shell analysis in commercial software.

Abstract

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.