A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted Finite Element method

Kothari, Hardik; Krause, Rolf

doi:10.1016/j.cma.2022.114630

Cited by 10 publications

(4 citation statements)

References 49 publications

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“…For elastic and hyperelastic contact problems, there exist solution techniques for small strain and obstacle problems [44,45], and more recently, also for large deformation contact problems, cf., e.g., [21,41,62,91,92]. However, to the best of our knowledge, the robust solution of problems combining contact scenarios and finite strains [21,41,44,45,57,63,66,67,75,79,88,92] are limited, at most, to isotropic models.…”

Section: Discussionmentioning

confidence: 99%

An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver

Knodel

Nägel

et al. 2022

Theor appl mech (Belgr)

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Numerical simulations of the dynamics of soft biological tissues are highly non-trivial because tissues generally exhibit complex biological response to external and internal actions, including large deformations and remodeling. Combining the advantages of globally implicit approach (GIA) solvers with the general applicability of the semi-implicit General Plasticity Algorithm (GPA), introduced by some of us some years ago, we present a new, efficient plasticity algorithm, which we call Bio Mechanics Basis Plasticity Algorithm (BMBPA). This is fully implicit, based on a nested Newton solver, and naturally suited for massively parallel computations. The Bilby-Kr?ner-Lee (BKL) multiplicative decomposition of the deformation gradient tensor is employed to introduce the unknowns of our model. We distinguish between global and local unknowns, associated with local and global equations, which are connected by means of a resolution function. The BMBPA asks for very few conditions to be applied and thus can be easily employed to solve several types of biological and biomechanical problems. We demonstrate the efficacy of BMBPA by performing two numerical experiments of a monophasic model of fiber-reinforced tissues. In one case, we consider the shear-compression test of a cubic specimen of tissue, while, in the other case, we focus on the unconfined compression test of a cylinder. The BMBPA is capable of solving the deformation and the remodeling of anisotropic biological tissues by employing a computation time of hours, while the GPA, applied to the same problems as the BMBPA, needs a substantially longer amount of time. All computations were performed in parallel and, within all tests, the performance of the BMBPA displayed substantially higher than the one of the GPA. The results of our simulations permit to study the overall mechanical behavior of the considered tissue and enable further investigations in the field of tissue biomechanics.

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Section: Discussionmentioning

confidence: 99%

An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver

Knodel

Nägel

et al. 2022

Theor appl mech (Belgr)

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“…Dedicated multigrid routines for Nitsche-based ghost-penalty stabilized methods are presented in [177][178][179]. It is notable that [179] additionally presents multigrid routines for Lagrange multiplier based methods, see also [180,181]. Furthermore, in [182] and [183] a similar approach as in [170] (discussed above) is followed.…”

Section: Preconditioning and Literature Overviewmentioning

confidence: 99%

Stability and Conditioning of Immersed Finite Element Methods: Analysis and Remedies

Prenter

Verhoosel

Brummelen

et al. 2023

Arch Computat Methods Eng

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This review paper discusses the developments in immersed or unfitted finite element methods over the past decade. The main focus is the analysis and the treatment of the adverse effects of small cut elements. We distinguish between adverse effects regarding the stability and adverse effects regarding the conditioning of the system, and we present an overview of the developed remedies. In particular, we provide a detailed explanation of Schwarz preconditioning, element aggregation, and the ghost penalty formulation. Furthermore, we outline the methodologies developed for quadrature and weak enforcement of Dirichlet conditions, and we discuss open questions and future research directions.

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“…Dedicated multigrid routines for Nitsche-based ghost-penalty stabilized methods are presented in [168][169][170]. It is notable that [170] additionally presents multigrid routines for Lagrange multiplier based methods, see also [171,172]. Furthermore, in [173] and [174] a similar approach as in [161] (discussed above) is followed.…”

Section: Preconditioning and Literature Overviewmentioning

confidence: 99%

Stability and conditioning of immersed finite element methods: analysis and remedies

Prenter

Verhoosel

Brummelen

et al. 2022

Preprint

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A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted Finite Element method

Cited by 10 publications

References 49 publications

An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver

An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver

Stability and Conditioning of Immersed Finite Element Methods: Analysis and Remedies

Stability and conditioning of immersed finite element methods: analysis and remedies

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