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

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.

An a posteriori error estimator for isogeometric analysis on trimmed geometries

“…Closely following the framework of adaptive finite elements [46] for elliptic partial differential equations, and [23] in the context of hierarchical trimmed geometries, we recall the four main building blocks of adaptivity, composing one iteration of the iterative process:…”

“…Since for all K ∈ Q h such that K ∩ Ω = ∅, E K (u h ) = 0, then the elements outside of the trimmed domain will never be marked. But to guarantee that a basis function B ∈ H Ω is deactivated when all the elements in supp (B) ∩ Ω are marked for refinement, we also need to add the so-called ghost elements [23] to M h . That is, when a trimmed element K is marked for refinement, we add to M h all the elements in…”

“…More precisely, we are interested in the study of an adaptive framework using hierarchical Bsplines (HB-splines) [19,20] and their variant called truncated hierarchical B-splines (THB-splines) [21,22], in the context of trimmed geometries. In [23], using [24,Section 4.5], it is shown that in the presence of trimming, (T)HB-splines yield a linearly independent basis suitable for the analysis.…”

“…The use of THB-splines in the context of Poisson problem and linear elasticity is studied in [25,26], where an emphasis is put on stability issues and bad conditioning of the system matrix caused by trimming, but no study of error indicators is given. To provide an approximation of the error, an implicit error estimator is introduced in [23] in the context of (T)HB-spline isogeometric analysis on trimmed surfaces, extending their previous work [27] on error estimation for linear fourth-order elliptic partial differential equations on non-trimmed geometries. The estimator relies on the solution of an additional residual-like system, but its reliability is not demonstrated.…”

Coupling of non-conforming trimmed isogeometric Kirchhoff-Love shells via a projected super-penalty approach

“…In the following section, we summarize the basic mathematical foundation of isogeometric methods defined on trimmed domains, following closely the notation used in [3,17]. For a detailed review of trimming and the current state-of-the-art in IGA we refer to [41] and references therein.…”