Improved conditioning of isogeometric analysis matrices for trimmed geometries

Marussig, Benjamin; Hiemstra, René R.; Hughes, Thomas J.R.

doi:10.1016/j.cma.2018.01.052

Cited by 37 publications

(28 citation statements)

References 35 publications

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“…where h u refers to the average knot span size of the initial level 0 of the basis and c e is a user-defined constant. According to [11], a good choice for this constant is c e = p /2. Local refinement along trimming curves is performed by adding hierarchical www.gamm-proceedings.com levels until the related extrapolation length d ( ) e complies with condition (11).…”

Section: Enhanced Control By Local Refinementmentioning

confidence: 99%

“…According to [11], a good choice for this constant is c e = p /2. Local refinement along trimming curves is performed by adding hierarchical www.gamm-proceedings.com levels until the related extrapolation length d ( ) e complies with condition (11). For a detailed discussion the interested reader is referred to [11].…”

Section: Enhanced Control By Local Refinementmentioning

confidence: 99%

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Isogeometric Analysis with Trimmed CAD Models

Marussig

2018

Proc Appl Math and Mech

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Section: Enhanced Control By Local Refinementmentioning

confidence: 99%

Section: Enhanced Control By Local Refinementmentioning

confidence: 99%

Isogeometric Analysis with Trimmed CAD Models

Marussig

2018

Proc Appl Math and Mech

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“…A similar framework was first studied in [34,5,35] for hierarchical B-splines defined on a rectangular planar grid, where no surface parametrization is used and local refinement is performed using a priori knowledge of the features of the problem. More recently, in [42,22], the use of truncated hierarchical B-splines in the context of the Laplace problem and linear elasticity was studied, where the emphasis is put on the issues of stability of the basis and bad conditioning of the system matrix caused by trimming and no real error indicator is employed. Lastly in [43], a fourth-order PDE, namely the Biharmonic problem, formulated on implicit domains was analyzed, where again no error-driven adaptive simulation is performed but refinement is steered a priori towards geometric features of interest.…”

Section: Introductionmentioning

confidence: 99%

Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

Coradello

Antolín

Vázquez

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second-and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.

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“…Without dedicated treatments, the severe ill-conditioning of linear systems derived from immersed finite element methods generally forestalls convergence of iterative solution procedures. Multiple resolutions for these conditioning problems have been proposed, the most prominent of which are the ghost penalty, e.g., [4,5,37], constraining, extending, or aggregation of basis functions, e.g., [13,[38][39][40][41][42][43][44][45][46], and preconditioning, which is discussed in detail below.…”

Section: Introductionmentioning

confidence: 99%

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

et al. 2019

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Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems in parallel, at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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Improved conditioning of isogeometric analysis matrices for trimmed geometries

Cited by 37 publications

References 35 publications

Isogeometric Analysis with Trimmed CAD Models

Isogeometric Analysis with Trimmed CAD Models

Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

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