Nonlinear FETI-DP and BDDC Methods: A Unified Framework and Parallel Results

Klawonn, Axel; Lanser, Martin; Rheinbach, Oliver; Uran, Matthias

doi:10.1137/16m1102495

Cited by 26 publications

(61 citation statements)

References 31 publications

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“…We also present results using Nonlinear-FETI-DP-1 on the Theta supercomputer later on. This is the most conservative choice among the recent Nonlinear-FETI-DP and Nonlinear-BDDC methods [35,33,40]. In our numerical experiments, if not denoted otherwise, the macroscopic problem is discretized using piecewise linear triangular elements (P1) in 2D and piecewise trilinear or quadratic brick elements (Q1 or Q2) in 3D.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For larger RVEs, efficient parallel nonlinear finite element solvers, which should also be robust for heterogeneous problems, have to be incorporated, e.g., a Newton-Krylov method with an appropriate preconditioner such as (algebraic) multigrid or domain decomposition. Recent nonlinear domain decomposition approaches can also be used as, e.g., ASPIN [13,43,31,14,30,28,27,26] or Nonlinear-FE-TI-DP [35,33,40] and the related Nonlinear FETI-1 or Neumann-Neumann methods [52,11]. In our implementation of the FE 2 method, the FE2TI package, we focus on Newton-Krylov-FETI-DP and Nonlinear-FETI-DP approaches.…”

Section: Computational Homogenizationmentioning

confidence: 99%

“…We now define nonlinear FETI-DP methods as iterative methods for the solution of the nonlinear system (see [33,40])…”

Section: Parallel Linear and Nonlinear Domain Decomposition Of Feti-dmentioning

confidence: 99%

“…The (generally) nonlinear system (12) can be solved using different strategies; see [40]. These methods are all equivalent to the classical FETI-DP method if K is a linear operator.…”

Section: Parallel Linear and Nonlinear Domain Decomposition Of Feti-dmentioning

confidence: 99%

“…However, we apply our own parallel implementation [35,41] which is earlier and also includes our most recent versions of nonlinear domain decomposition methods [40].…”

mentioning

confidence: 99%

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Computational homogenization with million-way parallelism using domain decomposition methods

et al. 2019

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Parallel computational homogenization using the well-known FE 2 approach is described and combined with fast domain decomposition and algebraic multigrid solvers. It is the purpose of this paper to show that and how the FE 2 method can take advantage of the largest supercomputers available and those of the upcoming exascale era for virtual material testing of micro-heterogeneous materials such as advanced steel. The FE 2 method is a computational micro-macro homogenization approach which incorporates micromechanical finite element simulations into macroscopic finite element simulations. In this approach, at each Gauß integration point of the macroscopic finite element problem a microscopic finite element problem, defined on a representative volume element (RVE), is attached. Note that the FE 2 method is not embarassingly parallel since the RVE problems are coupled through the macroscopic problem. Numerical results are presented considering different grids on both, the macroscopic and micro-This work was supported in part by Deutsche Forschungsgemeinschaft (DFG) through the Priority Programme 1648 "Software for Exascale Computing" (SPPEXA) under grants KL 2094/4-1, KL 2094/4-2, RH 122/2-1, and RH 122/3-2.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Computational Homogenizationmentioning

confidence: 99%

“…We now define nonlinear FETI-DP methods as iterative methods for the solution of the nonlinear system (see [33,40])…”

Section: Parallel Linear and Nonlinear Domain Decomposition Of Feti-dmentioning

confidence: 99%

“…The (generally) nonlinear system (12) can be solved using different strategies; see [40]. These methods are all equivalent to the classical FETI-DP method if K is a linear operator.…”

Section: Parallel Linear and Nonlinear Domain Decomposition Of Feti-dmentioning

confidence: 99%

“…However, we apply our own parallel implementation [35,41] which is earlier and also includes our most recent versions of nonlinear domain decomposition methods [40].…”

mentioning

confidence: 99%

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Computational homogenization with million-way parallelism using domain decomposition methods

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A dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities utilizing linear local problems

Lee

Park

2021

Numerical Meth Engineering

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We propose a novel dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities. The proposed method is based on a particular Fenchel–Rockafellar dual formulation of the target problem, which yields linear local problems despite the nonlinearity of the target problem. Since local problems are linear, each iteration of the proposed method can be done very efficiently compared with usual nonlinear domain decomposition methods. We prove that the proposed method is linearly convergent with the rate 1−r−1/2 while the convergence rate of relevant existing methods is 1−r−1, where r is proportional to the condition number of the dual operator. The spectrum of the dual operator is analyzed for the cases of two representative variational inequalities in structural mechanics: the obstacle problem and the contact problem. Numerical experiments are conducted in order to support our theoretical results.

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A novel method for solving nonlinear stochastic mechanics problems using FETI‐DP

Ajith

Ghosh

2022

Numerical Meth Engineering

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Solution of large-scale nonlinear stochastic mechanics problems such as plasticity is generally very expensive. In this work, a domain decomposition based scalable method is proposed for solving such problems. The mechanics problem and random fields are discretized using finite element (FE) bases and Karhunen-Loève expansion, respectively. The FE mesh is partitioned that offers (i) both spatial and stochastic dimensionality reduction and (ii) an inherent framework for parallelization. Then a stochastic collocation based surrogate model is built for each subdomain wherein at each collocation point a deterministic nonlinear problem is solved. The deterministic nonlinear problem is solved using Newton-Raphson method. The linear system of equations involving the Jacobian is solved using the dual-primal version of the FE tearing and interconnecting (FETI-DP) method, due to its demonstrated scalability for deterministic problems. Stochastic collocation and FETI-DP are inherently and independently parallelizable. Finally, at the post-processing stage, a statistical sampling from the surrogate model is performed by preserving the structure of the input random field. The proposed method is numerically tested for p-Laplace and plain-strain plasticity problems, and found to be computationally efficient and accurate. In parallel implementation, the method showed good scalability.

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Nonlinear FETI-DP and BDDC Methods: A Unified Framework and Parallel Results

Cited by 26 publications

References 31 publications

Computational homogenization with million-way parallelism using domain decomposition methods

Computational homogenization with million-way parallelism using domain decomposition methods

A dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities utilizing linear local problems

A novel method for solving nonlinear stochastic mechanics problems using FETI‐DP

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