Oliver Rheinbach

TU Bergakademie Freiberg

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Axel Klawonn

University of Cologne

Alexander Heinlein

Delft University of Technology

Martin Lanser

University of Cologne

Jörg Schröder

University of Duisburg-Essen

Friederike Röver

TU Bergakademie Freiberg

Daniel Balzani

Ruhr University Bochum

Martin J. Kühn

University of Bonn

Dominik Brands

University of Duisburg-Essen

Bjöern Kiefer

TU Bergakademie Freiberg

Jascha Knepper

University of Cologne

Cooperative Institutions

University of Duisburg-Essen

TU Bergakademie Freiberg

University of Cologne

TU Dresden

Friedrich-Alexander-Universität Erlangen-Nürnberg

Delft University of Technology

Technical University of Darmstadt

Ruhr University Bochum

Johann Radon Institute for Computational and Applied Mathematics

Essen University Hospital

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A Comparison Of Direct Solvers In FROSch Applied To Chemo-Mechanics

arXiv (Cornell University) (2023)

Alexander Heinlein Bjöern Kiefer Stefan Prüger Oliver Rheinbach Friederike Röver

Sparse direct linear solvers are at the computational core of domain decomposition preconditioners and therefore have a strong impact on their performance. In this paper, we consider the Fast and Robust Overlapping Schwarz (FROSch) solver framework of the Trilinos software library, which contains a parallel implementations of the GDSW domain decomposition preconditioner. We compare three different sparse direct solvers used to solve the subdomain problems in FROSch. The preconditioner is applied to different model problems; linear elasticity and more complex fully-coupled deformation diffusion-boundary value problems from chemo-mechanics. We employ FROSch in fully algebraic mode, and therefore, we do not expect numerical scalability. Strong scalability is studied from 64 to 4096 cores, where good scaling results are obtained up to 1728 cores. The increasing size of the coarse problem increases the solution time for all sparse direct solvers.

Solver

10.48550/arxiv.2310.12659

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Citations (0)

A Deflation Based Coarse Space in Dual-Primal FETI Methods for Almost Incompressible Elasticity

Lecture notes in computational science and engineering (2014)

Sabrina Gippert Axel Klawonn Oliver Rheinbach

FETI

Deflation

Elasticity

Dual space

10.1007/978-3-319-10705-9_56

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Citations (3)

Exact and Inexact FETI-DP Methods for Spectral Elements in Two Dimensions

Lecture notes in computational science and engineering (2008)

Axel Klawonn Oliver Rheinbach Luca F. Pavarino

FETI

Quadrature (astronomy)

Degree (music)

Tearing

10.1007/978-3-540-75199-1_32

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Citations (2)

Nonlinear BDDC Methods with approximate solvers

ETNA - Electronic Transactions on Numerical Analysis (2018)

Axel Klawonn Martin Lanser Oliver Rheinbach

New nonlinear BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods using inexact solvers for the subdomains and the coarse problem are proposed. In nonlinear domain decomposition methods, the nonlinear problem is decomposed before linearization to improve concurrency and robustness. For linear problems, the new methods are equivalent to known inexact BDDC methods. The new approaches are therefore discussed in the context of other known inexact BDDC methods for linear problems. Relations are pointed out, and the advantages of the approaches chosen here are highlighted. For the new approaches, using an algebraic multigrid method as a building block, parallel scalability is shown for more than half a million (524288) MPI ranks on the JUQUEEN IBM BG/Q supercomputer (JSC JÃ¼lich, Germany) and on up to 193600 cores of the Theta Xeon Phi supercomputer (ALCF, Argonne National Laboratory, USA), which is based on the recent Intel Knights Landing (KNL) many-core architecture. One of our nonlinear inexact BDDC domain decomposition methods is also applied to three-dimensional plasticity problems. Comparisons to standard Newton-Krylov-BDDC methods are provided.

Multigrid method

10.1553/etna_vol49s244

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Citations (9)

On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies

Archive of Applied Mechanics (2009)

Daniel Balzani Dominik Brands Axel Klawonn Oliver Rheinbach Jörg Schröder

FETI

Hyperelastic material

10.1007/s00419-009-0379-x

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Citations (22)

Effective hyperelastic material parameters from microstructures constructed using the planar Boolean model

Computational Mechanics (2022)

Matthias Brändel Dominik Brands Simon Maike Oliver Rheinbach Jörg Schröder

Abstract We construct two-dimensional, two-phase random heterogeneous microstructures by stochastic simulation using the planar Boolean model, which is a random collection of overlapping grains. The structures obtained are discretized using finite elements. A heterogeneous Neo-Hooke law is assumed for the phases of the microstructure, and tension tests are simulated for ensembles of microstructure samples. We determine effective material parameters, i.e., the effective Lamé moduli $$\lambda ^*$$ λ ∗ and $$\mu ^*$$ μ ∗ , on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters $$\lambda ^*$$ λ ∗ and $$\mu ^*$$ μ ∗ are considered as functions of the microscale material parameters and the geometric parameters of the Boolean model including the grain shape. We also consider the size of the Representative Volume Element (RVE) given a precision and an ensemble size. We use structured and unstructured meshes and also provide a comparison with the FE $$^2$$ 2 method.

10.1007/s00466-022-02142-5

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Citations (3)

Comparison of a Minimization and a Saddle Point Formulation of a Coupled Mechanics‐Diffusion Problem Implemented in deal.II

PAMM (2025)

Akshay Balachandran Jeeja Bjöern Kiefer Stefan Prüger Oliver Rheinbach Friederike Röver

Abstract For a problem in chemo‐mechanics, stated as a rate‐type variational principle, a minimization and a saddle point formulation are considered. Both formulations couple mechanical balance equations to mass diffusion and are intended to accurately model the diffusion‐induced swelling of hydrogels. The variational formulation is discretized by finite elements and implemented using the deal.II software library. The linearized systems in the incremental‐iterative solution procedure are solved using the unsymmetric multifrontal package (UMFPACK) sparse direct solver. The two formulations, employing three different, conforming finite element types, are considered for the case of mechanically‐induced diffusion and compared with respect to the evolution of the reaction force, mesh convergence, and the solution time. This initial step, comparing the performance of sparse direct solvers, sets the stage for future assessments involving domain decomposition, where sparse direct solvers are a computational kernel.

Saddle point

Saddle

Minification

10.1002/pamm.202400188

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A Highly Scalable Implementation of Inexact Nonlinear FETI-DP Without Sparse Direct Solvers

Lecture notes in computational science and engineering (2016)

Axel Klawonn Martin Lanser Oliver Rheinbach

FETI

Multigrid method

10.1007/978-3-319-39929-4_25

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Citations (10)

FETI‐DP Methods for P‐Elasticity

PAMM (2008)

Axel Klawonn Patrizio Neff Oliver Rheinbach Stefanie Vanisy

Abstract A FETI–DP method is introduced for the problem of linear P–elasticity which arises from linear elasticity by the introduction of a matrix P and which is motivated by micromorphic models. Numerical results as well as a condition number estimate are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

FETI

Elasticity

Linear elasticity

10.1002/pamm.200810859

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Citations (0)

Newton-Krylov-FETI-DP with Adaptive Coarse Spaces

Lecture notes in computational science and engineering (2017)

Axel Klawonn Martin Lanser Balthasar Niehoff Patrick Radtke Oliver Rheinbach

FETI

Krylov subspace

10.1007/978-3-319-52389-7_19

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Citations (2)