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Quadrature-free immersed isogeometric analysis

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arXiv 2021年

作者： Antolin, Pablo Hirschler, Thibaut Institute of Mathematics Chair of Numerical Modelling Simulation École Polytechnique Fédérale de Lausanne Switzerland

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. Copyright © 2021, The Authors. All rights reserved.

关键词： Computer aided design

Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow

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arXiv 2021年

作者： Burman, Erik Puppi, Riccardo Chair of Computational Mathematics University College London London United Kingdom Chair of Modelling and Numerical Simulation École Polytechnique Fédérale de Lausanne Lausanne Switzerland

We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as O(h−1), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as O(h−k−1), k being the order of the Raviart-Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual L2-norm. However, we are still able to recover the optimal a priori L2-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart-Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited. © 2021, CC0.

关键词： Boundary conditions

Tree-Cotree-Based Tearing and Interconnecting for 3D Magnetostatics: A Dual-Primal Approach

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arXiv 2024年

作者： Mally, Mario Kapidani, Bernard Merkel, Melina Schöps, Sebastian Vázquez, Rafael Computational Electromagnetics Group Technische Universität Darmstadt Darmstadt64289 Germany Centre for Computational Engineering Technische Universität Darmstadt Darmstadt64289 Germany Chair of Numerical Modelling and Simulation École Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland Department of Applied Mathematics Universidade de Santiago de Compostela Santiago de Compostela15782 Spain Santiago de Compostela15782 Spain

The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree-cotree decomposition. The classical tree-cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable. Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual-primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence. Copyright © 2024, The Authors. All rights reserved.

关键词： Magnetostatics

Fast and multiscale formation of isogeometric matrices of microstructured geometric models

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arXiv 2021年

作者： Hirschler, T. Antolin, P. Buffa, A. Institute of Mathematics Chair of Numerical Modelling and Simulation École Polytechnique Fédérale de Lausanne Switzerland Istituto di Matematica Applicata e Tecnologie Informatiche "Enrico Ma-genes" Consiglio Nazionale delle Ricerche Pavia Italy

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. Copyright © 2021, The Authors. All rights reserved.

关键词： Table lookup

A projected super-penalty method for the C1-coupling of multi-patch isogeometric Kirchhoff plates

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arXiv 2020年

作者： Coradello, Luca Loli, Gabriele Buffa, Annalisa Chair of Numerical Modelling and Simulation École Polytechnique Fédérale de Lausanne Lausanne Switzerland Pavia Italy Dipartimento di Matematica ‘F. Casorati’ Università di Pavia Pavia Italy

This work focuses on the development of a super-penalty strategy based on the L2-projection of suitable coupling terms to achieve C1-continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from locking also on very coarse meshes. We demonstrate the applicability of the proposed coupling algorithm to Kirchhoff plates by studying several benchmark examples discretized by nonconforming meshes. In all cases, we recover the optimal rates of convergence achievable by B-splines where we achieve a substantial gain in accuracy per degree-of-freedom compared to other choices of the penalty parameters. Copyright © 2020, The Authors. All rights reserved.

关键词： Lagrange multipliers

Overlapping multi-patch isogeometric method with minimal stabilization

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arXiv 2019年

作者： Antolin, Pablo Buffa, Annalisa Puppi, Riccardo Wei, Xiaodong Chair of Modelling and Numerical Simulation Institute of Mathematics École Polytechnique Fédérale de Lausanne Lausanne Switzerland Istituto di Matematica Applicata e Tecnologie Informatiche"Enrico Magenes" del CNR Pavia Italy

We present a novel method for isogeometric analysis (IGA) to directly work on geometries constructed by Boolean operations including difference (i.e., trimming), union and intersection. Particularly, this work focuses on the union operation, which involves multiple independent, generally non-conforming and trimmed spline patches. Given a series of patches, we overlay one on top of another in a certain order. While the invisible part of each patch is trimmed away, the visible parts of all the patches constitute the entire computational domain. We employ the Nitsche's method to weakly couple independent patches through visible interfaces. Moreover, we propose a minimal stabilization method to address the instability issue that arises on the interfaces shared by small trimmed elements. We show in theory that our proposed method recovers stability and guarantees well-posedness of the problem as well as optimal error estimates. In the end, we numerically verify the theory by solving the Poisson's equation on various geometries that are obtained by the union operation. Copyright © 2019, The Authors. All rights reserved.

关键词： Stabilization

A minimal stabilization procedure for isogeometric methods on trimmed geometries

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arXiv 2019年

作者： Buffa, A. Puppi, R. Vázquez, R. Istituto di Matematica Applicata e Tecnologie Informatiche" Enrico Magenes" del CNR Pavia Italy Chair of Numerical Modelling and Simulation Institute of Mathematics École Polytechnique Fédérale de Lausanne Lausanne Switzerland

Trimming is a common operation in CAD, and, in its simplest formulation, consists in removing superfluous parts from a geometric entity described via splines (a spline patch). After trimming the geometric description of the patch remains unchanged, but the underlying mesh is unfitted with the physical object. We discuss the main problems arising when solving elliptic PDEs on a trimmed domain. First we prove that, even when Dirichlet boundary conditions are weakly enforced using Nitsche’s method, the resulting method suffers lack of stability. Then, we develop novel stabilization techniques based on a modification of the variational formulation, which allow us to recover well-posedness and guarantee accuracy. Optimal a priori error estimates are proven, and numerical examples confirming the theoretical results are *** Codes 65N12, 65N15, 65N30, 65N85 Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

Multipatch approximation of the de rham sequence and its traces in isogeometric analysis

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arXiv 2018年

作者： Buffa, Annalisa Dölz, Jürgen Kurz, Stefan Schöps, Sebastian Vázquez, Rafael Wolf, Felix TU Darmstadt Institute TEMF Graduate School CE Darmstadt Germany École polytechnique fédérale de Lausanne Chair of Numerical Modelling and Simulation Lausanne Switzerland

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straight forward generalisation to finite element *** Codes 65D07, 65N38 Copyright © 2018, The Authors. All rights reserved.

关键词： Interpolation

BPX preconditioners for isogeometric analysis using (truncated) hierarchical B-splines

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arXiv 2019年

作者： Bracco, Cesare Cho, Durkbin Giannelli, Carlotta Vázquez, Rafael Department of Mathematics and Computer Science University of Florence Viale Morgagni 67/A Florence50134 Italy Department of Mathematics Dongguk University Pil-dong 3-ga Jung-gu Seoul04620 Korea Republic of Chair of Modelling and Numerical Simulation Institute of Mathematics École Polytechnique Fédérale de Lausanne Station 8 Lausanne1015 Switzerland Istituto di Matematica Applicata Tecnologie Informatiche"Enrico Magenes" del CNR Pavia Italy

We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes, the condition number of the preconditioned system is bounded independently of the number of levels. A selection of numerical examples validates the theoretical results and the performance of the preconditioner. Copyright © 2019, The Authors. All rights reserved.

关键词： Interpolation