Parallel Iterative Substructuring in Structural Mechanics

Rheinbach, Oliver

doi:10.1007/s11831-009-9035-4

Cited by 23 publications

(23 citation statements)

References 84 publications

(197 reference statements)

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“…3.1(a) is the one typically followed by most of the existing code implementations of (ML)BDDC and related DD algorithms [8,26,29]. It comes naturally into mind as it reflects the multilevel structure of the preconditioner.…”

Section: Multilevel Balancing Domain Decompositionmentioning

confidence: 99%

Multilevel Balancing Domain Decomposition at Extreme Scales

Badia¹,

Martı́n²,

Principe³

2016

SIAM J. Sci. Comput.

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Abstract. In this paper we present a fully-distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely inter-level overlapping). Essentially, the main communicator is split into as many non-overlapping subsets of MPI tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful re-scheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping to be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrary number of levels can be built while re-using significant and recurrent parts of the codes. This approach leads to excellent weak scalability results as soon as level-1 tasks can mask coarser-levels duties. We provide a model to indicate how to choose the number of levels and coarsening ratios between consecutive levels and determine qualitatively the scalability limits for a given choice. We have carried out a comprehensive weak scalability analysis of the proposed implementation for the 3D Laplacian and linear elasticity problems. Excellent weak scalability results have been obtained up to 458,752 IBM BG/Q cores and 1.8 million MPI tasks, being the first time that exact domain decomposition preconditioners (only based on sparse direct solvers) reach these scales.1. Introduction. The simulation of scientific and engineering problems governed by partial differential equations (PDEs) involves the solution of sparse linear systems. The time spent in an implicit simulation at the linear solver relative to the overall execution time grows with the size of the problem and the number of cores [22]. In order to satisfy the ever increasing demand of reality and complexity in the simulations, scientific computing must advance in the development of numerical algorithms and implementations that will efficiently exploit the largest amounts of computational resources, and a massively parallel linear solver is a key component in this process.The growth in computational power passes now through increasing the number of cores in a chip, instead of making cores faster. The next generation of supercomputers, able to reach 1 exaflop/s, is expected to reach billions of cores. Thus, the future of scientific computing will be strongly related to the ability to efficiently exploit these extreme core counts [1].Only numerical algorithms with all their components scalable will efficiently run on extreme scale supercomputers. On extreme core counts, it will be a must to reduce communication and synchronization among cores, and overlap communication ...

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Section: Multilevel Balancing Domain Decompositionmentioning

confidence: 99%

Multilevel Balancing Domain Decomposition at Extreme Scales

Badia¹,

Martı́n²,

Principe³

2016

SIAM J. Sci. Comput.

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“…Among the extensions are inexact FETI-DP methods which were introduced in [28]. Their parallel scalability has been demonstrated in [17,33] for up to 65 000 processors. Recently, new scalable nonlinear versions of the FETI-DP have been introduced in [34].…”

Section: The Feti-dp Methodsmentioning

confidence: 99%

“…For an introduction to domain decomposition methods, see, e.g., [35,36]. The parallel FETI-DP implementation used in this paper is based on [19,33] and uses PETSc [13,14] and UMFPACK [37]. There is proven robustness of FETI-DP for standard finite element discretizations of second order self adjoint elliptic partial differential equations, including (almost incompressible) linear elasticity, when the discontinuities occur only inside of each subdomain; see Gippert, Klawonn, and Rheinbach [38].…”

Section: The Feti-dp Methodsmentioning

confidence: 99%

The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results

Heinlein

Hetmaniuk

Klawonn

et al. 2015

Journal of Computational and Applied Mathematics

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a b s t r a c tA special finite element method based on approximate component mode synthesis (ACMS) was introduced in Hetmaniuk and Lehoucq (2010). ACMS was developed for second order elliptic partial differential equations with rough or highly varying coefficients. Here, a parallel implementation of ACMS is presented and parallel scalability issues are discussed for representative examples. Additionally, a parallel domain decomposition preconditioner (FETI-DP) is applied to solve the ACMS finite element system. Weak parallel scalability results for ACMS are presented for up to 1024 cores. Our numerical results also suggest a quadratic-logarithmic condition number bound for the preconditioned FETI-DP method applied to ACMS discretizations.

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“…The initial guess may also be the solution of a modified nonlinear elasticity problem such as the solution of the same nonlinear model but with modified parameters, e.g., a reduced penalty parameter κ, or modified boundary conditions, e.g., a reduced pressure on the surface. The latter is equivalent to an incremental load stepping scheme with a parameter τ ∈ (0, 1], τ → 1, so that (25) Klawonn and Rheinbach [24] used a load stepping scheme of this kind, for more information on load stepping and global Newton methods, see [48,47]. The standard finite element method (FEM) now yields a linear system of equations which is equivalent to the discretized variational formulation (23).…”

Section: Linearization and Discretizationmentioning

confidence: 99%

Classical and all‐floating FETI methods for the simulation of arterial tissues

Augustin

Holzapfel

Steinbach

2014

Numerical Meth Engineering

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High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is -as most other biological tissues -characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.

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Parallel Iterative Substructuring in Structural Mechanics

Cited by 23 publications

References 84 publications

Multilevel Balancing Domain Decomposition at Extreme Scales

Multilevel Balancing Domain Decomposition at Extreme Scales

The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results

Classical and all‐floating FETI methods for the simulation of arterial tissues

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