A parallel block LU decomposition method for distributed finite element matrices

Maurer, Daniel; Wieners, Christian

doi:10.1016/j.parco.2011.05.007

Cited by 22 publications

(14 citation statements)

References 15 publications

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“…We start with an initial coarse mesh with = × space-time cells which is refined 3 times in space and time to 524 288 cells. The coarse problem is solved by using a parallel direct solver [25]. Furthermore we use loworder polynomial degrees (p, q) = ( , ) as initial distribution on Q.…”

Section: Numerical Tests For Space-time Adaptivitymentioning

confidence: 99%

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Dörfler¹,

Findeisen²,

Wieners³

2016

Computational Methods in Applied Mathematics

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Abstract:We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov-Galerkin scheme in time. We show wellposedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goaloriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the efficiency of the overall adaptive solution process.

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Section: Numerical Tests For Space-time Adaptivitymentioning

confidence: 99%

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Dörfler¹,

Findeisen²,

Wieners³

2016

Computational Methods in Applied Mathematics

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“…Adaptive p-refinement The new discretization is realized in the parallel finite element software M++ [7], and the linear problems are solved with the parallel direct solver [5]. In order to resolve the corner singularity we use a sequence of graded meshes, and on every mesh we study the p-convergence by increasing adaptively the face degrees of freedom: On every level, we start with p F = q F = 1 and p K = 2.…”

Section: A Numerical Examplementioning

confidence: 99%

A hybrid weakly nonconforming discretization for linear elasticity

Krämer

Wieners

Wohlmuth

et al. 2016

Proc Appl Math and Mech

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We introduce a hybridizable discretization for problems of linear elasticity, based on a flexible weak coupling of the displacement and the normal stress across the inner element boundaries. The dual variables of the Lagrange multiplier are introduced as hybrid trace variables. A reduction of the equation system to these variables allows for an efficient parallel implementation. The performance of an adaptive choice of the hybrid degrees of freedom is demonstrated by a numerical example with a corner singularity. Hybridizable weakly nonconforming discretizationThe approximation quality of standard conforming finite elements depends sensitively on the regularity of the problem and on the material parameters. Adaptivity can resolve singularities, and nonconforming discretizations, e.g. the element proposed by Falk [3] or discontinuous (Petrov-)Galerkin methods [4,9], can reduce looking phenomena. Here, we introduce a new method for the approximation combining adaptivity and nonconformity extending the discontinuous Galerkin approaches by suitable weak interface conditions so that penalty terms can be avoided completely.We consider isotropic linear elasticity in a bounded Lipschitz domain Ω ⊂ R 2 with Dirichlet and Neumann boundary conditions on Γ D and Γ N , respectively, where Γ D has a positive Lebesgue measure,Let Ω h = K∈K K be a decomposition into convex open subdomains K ⊂ Ω, let F K be the set of faces F ⊂ ∂K, and define F h = F K . For an inner face F ⊂ ∂K let K F be the neighboring element with F = ∂K ∩ ∂K F . On each face, we define jump and average terms, taking into acount the boundary data u D , t N . The displacement jump [v] is defined as v K − v K F on inner faces, v − u D on Dirichlet faces and 0 on Neumann faces. The averaged normal stress {σ(v)n} is defined as σ(v K )n K + σ(v K F )n K F on inner faces, 0 on Dirichlet faces and σ(v K )n K − t N on Neumann faces. Depending on local degrees of freedom p K , p F , q F , r F and s F , we define the local ansatz space X h (K) = P p K (K) 2 , the Lagrange multiplier spaces W p (F ) = P p F (F ) × P q F (F ) and W d (F ) = P r F (F ) × P s F (F ) and the weakly conforming finite element space, and the bilinear and linear formsThe bilinear form is coercive, if the constraints fix the local rigid body modes; coercivity independent of the mesh size is provided, e.g., for p F , q F ≥ 1 [1]. Note that we allow for r F = s F = −1 (with P −1 = {0}).For the implementation we introduce face Lagrange parameters which allows to eliminate the inner degrees of freedom (see [8] for spectral bounds of the corresponding Schur complement). This results into a positive definite linear system of dimension A numerical exampleProblem setting The new discretization, using an adaptive strategy described in the following, is evaluated for a test problem in the L-shaped domain Ω = (−1, 1) 2 \ [0, 1] × [−1, 0] with a strong stress singularity at the re-entrant corner [6], where u r cos(ϕ − π/6) r sin ϕ − π/6) = cos(ϕ)u 1 (r, ϕ) − sin(ϕ)u 2 (r, ϕ) sin(ϕ)u 1 (r, ϕ) + cos(ϕ)u 2 ...

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“…This is because the mesh partition is only used for distributing work without in any way altering the actual sequential algorithm. This would not be the case if one would consider more complex solvers, like primal or dual Schur complement solvers [64], or more complex preconditioners, like linelet [74] or block LU [65]. Since the explicit framework is relatively straightforward to implement, in the following we only focus our attention on the implicit framework.…”

Section: Mesh Convergencementioning

confidence: 99%

Alya: Computational Solid Mechanics for Supercomputers

Casoni

Jérusalem

Samaniego

et al. 2014

Arch Computat Methods Eng

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While solid mechanics codes are now conventional tools both in industry and research, the increasingly more exigent requirements of both sectors are fuelling the need for more computational power and more advanced algorithms. For obvious reasons, commercial codes are lagging behind academic codes often dedicated either to the implementation of one new technique, or the upscaling of current conventional codes to tackle massively large scale computational problems. Only in a few cases, both approaches have been followed simultaneously. In this article, a solid mechanics simulation strategy for parallel supercomputers based on a hybrid approach is presented. Hybrid parallelization exploits the thread-level parallelism of multicore architectures, com- bining MPI tasks with OpenMP threads. This paper describes the proposed strategy, programmed in Alya, a parallel multiphysics code. Hybrid parallelization is specially well suited for the current trend of supercomputers, namely large clusters of multicores. The strategy is assessed through transient non-linear solid mechanics problems, both for explicit and implicit schemes, running on thousands of cores. In order to demonstrate the flexibility of the proposed strategy under advance algorithmic evolution of computational mechanics, a non-local parallel overset meshes method (Chimera-like) is implemented and the conservation of the scalability is demonstrated.

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A parallel block LU decomposition method for distributed finite element matrices

Cited by 22 publications

References 15 publications

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

A hybrid weakly nonconforming discretization for linear elasticity

Alya: Computational Solid Mechanics for Supercomputers

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