Abstract
The Finite Cell Method (FCM) together with Isogeometric analysis (IGA) has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid–structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme. We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed isogeometric analysis. Additionally, we present a detailed numerical investigation of the proposed optimal integration algorithm and its application to immersed isogeometric analysis using two- and three-dimensional linear elasticity problems.
Highlights
Immersed finite element methods – such as, e.g., the finite cell method (FCM) [1], CutFEM [2] and immersogeometric analysis [3,4,5] – have been demonstrated to be suitable for computational problems for which the performance of mesh-fitting finite element methods is impeded by complications in the meshing procedure
In recent years, immersed finite element methods have been successfully combined with isogeometric analysis (IGA) [6], a spline-based higher-order finite element framework targeting the integration of finite element analysis (FEA) and computer aided design (CAD)
We have developed an algorithm to construct quadrature rules for cut-elements in which the integration points are distributed over the element in such a way that the integration error is minimized
Summary
Immersed finite element methods – such as, e.g., the finite cell method (FCM) [1], CutFEM [2] and immersogeometric analysis [3,4,5] – have been demonstrated to be suitable for computational problems for which the performance of mesh-fitting finite element methods is impeded by complications in the meshing procedure. This is the case for octree integration, the accuracy of which can be controlled by the bisectioning depth and the integration orders used on the sub-cells Optimization of these parameters to reduce the computational expense of octree integration without compromising its robustness with respect to configurations was proposed in Ref. The idea of reduced integration in finite element methods has been studied extensively over the last decades, with applications in the analysis of plates and shells [46] and mixed finite element methods [47, 48] being prominent examples In this manuscript we propose an error-estimation-based adaptive algorithm to obtain optimal octree integration rules for cut-cells.
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