3D volumetric isotopological meshing for finite element and isogeometric based reduced order modeling

Maquart, Tristan; Wenfeng, Y.; Elguedj, Thomas; Gravouil, Anthony; Rochette, Michel

doi:10.1016/j.cma.2019.112809

Cited by 24 publications

(9 citation statements)

References 25 publications

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“…To show the viability of the quadrature-free approach to handle complex 3D geometries, we consider the two CAD models shown in Figs. 15 and 16). In order to build the finite element operators, the presented quadrature-free approach is applied.…”

Section: Poisson's Problem On Complex 3d Trimmed-geometriesmentioning

confidence: 95%

“…To achieve this goal, two different strategies can be undertaken: The first one consists in generating a fully boundary-conforming and matching geometric model such that standard analysis procedures can be directly applied. Generating those spline meshes is however a quite challenging task for geometries with complex topologies, especially when only tensor-product splines are considered [12][13][14][15]. For those cases, unstructured spline meshes [16][17][18][19][20] constitute an appealing alternative.…”

Section: Introductionmentioning

confidence: 99%

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Quadrature-free immersed isogeometric analysis

Antolín

Hirschler

2022

Engineering with Computers

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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. It relies on a newly developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the divergence theorem, those integrals over B-Reps are transformed into the first surface and then line integrals with polynomials integrands. Eventually, these line integrals are evaluated analytically with machine precision accuracy. 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. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

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Section: Poisson's Problem On Complex 3d Trimmed-geometriesmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Quadrature-free immersed isogeometric analysis

Antolín

Hirschler

2022

Engineering with Computers

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“…This simple spline model can be enriched in many ways: e.g., geometries can be described as a collections of spline patches (multipatch geometries), or via NURBS in the case of conic sections. We refer the interested readers to the following non-exhaustive list of works which deal with key points regarding the modeling with splines: theoretical description and practical use of B-Spline and NURBS [22,25,32,59], the issue of trimming procedures and boundary representation [1,14,21,24,52], and the generation of analysis-suitable geometric models for complex structures [4,23,51,54].…”

Section: Spline Modelingmentioning

confidence: 99%

Fast and multiscale formation of isogeometric matrices of microstructured geometric models

2021

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The matrix formation associated to high-order discretizations is known to be numerically demanding. Based on the existing procedure of interpolation and lookup, we design a multiscale assembly procedure to reduce the exorbitant assembly time in the context of isogeometric linear elasticity of complex microstructured geometries modeled via spline compositions. The developed isogeometric approach involves a polynomial approximation occurring at the macro-scale and the use of lookup tables with pre-computed integrals incorporating the micro-scale information. We provide theoretical insights and numerical examples to investigate the performance of the procedure. The strategy turns out to be of great interest not only to form finite element operators but also to compute other quantities in a fast manner as for instance sensitivity analyses commonly used in design optimization.

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“…To achieve this goal, two different strategies can be undertaken: The first one consists in generating a fully conformal multi-patch geometric model such that standard analysis procedures can be directly employed. Generating these conformal meshes is however a quite challenging task in the case of geometries with complex topologies [12][13][14], especially when only tensor-product splines are considered [15][16][17][18]. On the contrary, the second approach aims to directly use standard CAD models which may contain non-conforming and trimmed surfaces and present geometric defects, as water leaks or surface overlaps, and to recall to high-end analysis procedures [19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Quadrature-free Immersed Isogeometric Analysis

Antolín¹,

Hirschler²

2021

Preprint

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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. It relies on a new developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. 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. Eventually these line integrals are evaluated analytically with machine precision accuracy. 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. Finally, the methodology is illustrated for 3D CAD models with an industrial level of complexity.

show abstract

3D volumetric isotopological meshing for finite element and isogeometric based reduced order modeling

Cited by 24 publications

References 25 publications

Quadrature-free immersed isogeometric analysis

Quadrature-free immersed isogeometric analysis

Fast and multiscale formation of isogeometric matrices of microstructured geometric models

Quadrature-free Immersed Isogeometric Analysis

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