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We quantify the contribution of the integration error using the theoretical basis provided by Strang’s first lemma. In this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme. 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 particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. 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. The results in terms of accuracy and numerical cost make us believe that the presented method is a viable strategy to build a robust framework for shape optimization. The numerical performances of the analytical sensitivities are compared with approximate sensitivities. In order to highlight the versatility of this sensitivity analysis method, we perform eight benchmark optimization examples with different types of objective functions (compliance, displacement field, stress field, and natural frequencies), different types of isogeometric element (2D and 3D standard solids, and a Kirchhoff–Love shell), and different types of structural analysis (static and vibration). We present new theoretical developments, algorithms, and quantitative results regarding the analytical calculation of discrete adjoint-based sensitivities. We explain how these two models interact during the optimization, and especially during the sensitivity analysis. The principal feature is the use of two refinement levels of the same geometry: a coarse level where the shape updates are imposed and a fine level where the analysis is performed. Based on all the researches related to isogeometric shape optimization, we present a global overview of the process which has emerged. This contribution aims at compiling the key ingredients within this promising framework, with a particular attention to sensitivity analysis. Isogeometric shape optimization has been now studied for over a decade. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity. The performance of the proposed method is demonstrated by means of numerical experiments in the context of 2D and 3D elliptic problems, retrieving optimal error convergence order in all cases. Eventually these line integrals are evaluated analytically with machine precision accuracy. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into first surface and then line integrals with polynomials integrands.
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First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. It relies on a new developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes.