CutIGA with basis function removal

Elfverson, Daniel; Larson, Mats G.; Larsson, Karl

doi:10.1186/s40323-018-0099-2

Cited by 44 publications

(56 citation statements)

References 11 publications

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“…Possible alternatives to the diagonal scaling preconditioning are the ghost penalty techniques [14,15]; the use of additive Schwarz preconditioners as proposed in [19,32] for finite cell and isogeometric discretizations, and extended in [20] to case of immersed IGA combined with multigrid methods; or the stable removal of basis functions, as proposed in [24] in the context of CutIGA methods.…”

Section: Linear System Preconditioningmentioning

confidence: 99%

Isogeometric Analysis on V-reps: First results

Antolín

Buffa

Martinelli

2019

Computer Methods in Applied Mechanics and Engineering

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Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parameterization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justifies our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffusion and linear elasticity.

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Section: Linear System Preconditioningmentioning

confidence: 99%

Isogeometric Analysis on V-reps: First results

Antolín

Buffa

Martinelli

2019

Computer Methods in Applied Mechanics and Engineering

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“…We denote by V x any of the FE spaces V std , or V agg , when it is not necessary to distinguish between them. We approximate problem (8) in V x with the following variational equation:…”

Section: Model Problemmentioning

confidence: 99%

Distributed-memory parallelization of the aggregated unfitted finite element method

Verdugo

Martı́n

Badia

2019

Computer Methods in Applied Mechanics and Engineering

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The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.

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“…To obtain a positive definite stiffness matrix we apply basis function removal. This approach was analyzed in [10] and builds on the the idea of systematically removing basis functions that have sufficiently small intersection with the domain. This is done in such a way that optimal order accuracy in a specified norm is retained.…”

Section: Basis Function Removalmentioning

confidence: 99%

“…The first alternative rests on a complete theoretical basis; the second is common in practice but optimal order a priori bounds can not be established in general since the penalty parameter may become very large, see the discussion in [7]; and the third alternative was considered in [10] where a least squares term was added in the vicinity of the Dirichlet part of the boundary to provide the additional stability necessary to establish a priori error bounds. We refer to the overview article [5], and the recent conference proceedings [3] for an overview of current research on cut element methods.…”

Section: Introductionmentioning

confidence: 99%

“…We utilize the added smoothness of the splines in our derivations and therefore only consider finite element spaces with at least C 1 regularity. The bulk part of the new least squares stabilization was used in [10] in combination with the non symmetric Nitsche formulation. Here we show that adding some additional stabilization also on the boundary leads to a natural extension of these results to the symmetric Nitsche formulation.…”

Section: Introductionmentioning

confidence: 99%

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A new least squares stabilized Nitsche method for cut isogeometric analysis

Elfverson

Larson

Larsson

2019

Computer Methods in Applied Mechanics and Engineering

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We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider C 1 splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in H 2 (Ω) and optimal order a priori error estimates in the energy and L 2 norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.

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CutIGA with basis function removal

Cited by 44 publications

References 11 publications

Isogeometric Analysis on V-reps: First results

Isogeometric Analysis on V-reps: First results

Distributed-memory parallelization of the aggregated unfitted finite element method

A new least squares stabilized Nitsche method for cut isogeometric analysis

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