Martin Lanser

NONLINEAR FETI-DP AND BDDC METHODS ∗

New nonlinear FETI-DP (dual-primal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton--Krylov--FETI-DP or BDDC approach. Moreover, the coarse space of the nonlinear FETI-DP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETI-DP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the p-Laplace operator are presented.

TOWARD EXTREMELY SCALABLE NONLINEAR DOMAIN DECOMPOSITION METHODS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods toward extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an algebraic multigrid (AMG) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyp...

On an Adaptive Coarse Space and on Nonlinear Domain Decomposition

Two different aspects of FETI-DP domain decomposition methods are considered. In the first part, the adaptive construction of coarse spaces from local eigenvalue problems for the solution of heterogeneous, e.g., multiscale, problems is considered. This strategy to construct a coarse space is implemented using a deflation approach. In the second part of the proceedings article, new domain decomposition approaches for nonlinear problems are introduced and numerical results are presented.

Scalability of Classical Algebraic Multigrid for Elasticity to Half a Million Parallel Tasks

The parallel performance of several classical Algebraic Multigrid (AMG) methods applied to linear elasticity problems is investigated. These methods include standard AMG approaches for systems of partial differential equations such as the unknown and hybrid approaches, as well as the more recent global matrix (GM) and local neighborhood (LN) approaches, which incorporate rigid body modes (RBMs) into the AMG interpolation operator. Numerical experiments are presented for both two- and three-dimensional elasticity problems on up to 131,072 cores (and 262,144 MPI processes) on the Vulcan supercomputer (LLNL, USA) and up to 262,144 cores (and 524,288 MPI processes) on the JUQUEEN supercomputer (JSC, Julich, Germany). It is demonstrated that incorporating all RBMs into the interpolation leads generally to faster convergence and improved scalability.

A Nonlinear FETI-DP Method with an Inexact Coarse Problem

A new nonlinear version of the well-known FETI-DP method (Finite Element Tearing and Interconnecting Dual-Primal) is introduced. In this method, the nonlinear problem is decomposed before linearization. Nonlinear approaches to domain decomposition can be viewed as a strategy to localize computational work for the efficient use with future extreme-scale supercomputers. As opposed to known nonlinear FETI-DP algorithms, in the new method the coarse solver can be replaced by a preconditioner, i.e., the coarse solve can be inexact. It is expected that the new method can show a superior parallel scalability if the number of subdomains is large. If the coarse solver is exact and the method is applied to linear problems then the method is equivalent to the standard FETI-DP method. Numerical results for up to 32,768 cores are presented using cycles of an algebraic multigrid for the coarse problem of the new method.

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

A variant of a nonlinear FETI-DP domain decomposition method is considered. It is combined with a parallel algebraic multigrid method (BoomerAMG) in a way which completely removes sparse direct solvers from the algorithm. Scalability to 524,288 MPI ranks is shown for linear elasticity and nonlinear hyperelasticity using more than half of the JUQUEEN supercomputer (JSC, Julich; TOP500 rank: 11th).

Hybrid MPI/OpenMP Parallelization in FETI-DP Methods

We present an approach to hybrid MPI/OpenMP parallelization in FETI-DP methods using OpenMP with PETSc+MPI in the finite element assembly and using the shared memory parallel direct solver Pardiso in the FETI-DP solution phase. Our approach thus uses OpenMP parallelization on subdomains and MPI in between subdomains. We investigate the efficiency of this approach for a benchmark problem from two dimensional nonlinear hyperelasticity. We observe good scalability for up to four threads for each MPI rank on a state-of-the-art Ivy Bridge architecture and incremental improvements for up to ten OpenMP threads for each MPI rank.

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

Parallel Newton--Krylov FETI-DP (Finite Element Tearing and Interconnecting---Dual-Primal) domain decomposition methods are fast and robust solvers, e.g., for nonlinear implicit problems in structural mechanics. In these methods, the nonlinear problem is first linearized and then decomposed into loosely coupled (linear) problems, which can be solved in parallel. By changing the order of the operations, new parallel communication can be constructed, where the loosely coupled local problems are nonlinear. We discuss different nonlinear FETI-DP methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems. Moreover, a new unified framework is introduced which casts all nonlinear FETI-DP domain decomposition approaches discussed in the literature into a single algorithm. Furthermore, the equivalence of nonlinear FETI-DP methods to specific nonlinearly right-preconditioned Newton--Krylov methods is shown. For the methods using nested Newton iteration...

New Nonlinear FETI-DP Methods Based on a Partial Nonlinear Elimination of Variables

We introduce two new nonlinear FETI-DP (Finite Element Tearing and Interconnecting—Dual-Primal) methods based on a partial nonlinear elimination of variables and provide a comparison to Newton-Krylov-FETI-DP, Nonlinear-FETI-DP-1, and Nonlinear-FETI-DP-2, which have already been described earlier. In contrast to classical Newton-Krylov-FETI-DP methods, where a geometrical decomposition after a Newton linearization is performed, in nonlinear FETI-DP methods, the discretized nonlinear problem is decomposed before linearization. The approaches helps to localize work and reduce communication and thus are better suited for modern computer architectures.

One-Way and Fully-Coupled FE2 Methods for Heterogeneous Elasticity and Plasticity Problems: Parallel Scalability and an Application to Thermo-Elastoplasticity of Dual-Phase Steels

In this paper, aspects of the two-scale simulation of dual-phase steels are considered. First, we present two-scale simulations applying a top-down one-way coupling to a full thermo-elastoplastic model in order to study the emerging temperature field. We find that, for our purposes, the consideration of thermo-mechanics at the microscale is not necessary. Second, we present highly parallel fully-coupled two-scale FE2 simulations, now neglecting temperature, using up to 458, 752 cores of the JUQUEEN supercomputer at Forschungszentrum Julich. The strong and weak parallel scalability results obtained for heterogeneous nonlinear hyperelasticity exemplify the massively parallel potential of the FE2 multiscale method.