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Element partition trees for h-refined meshes to optimize direct solver performance. Part I: Dynamic programming

Aboueisha H., Calo V. M., Jopek K., Moshkov M., Paszyńska A., Paszyński M., Skotniczny M.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 2

351--365

EN

We consider a class of two- and three-dimensional h-refined meshes generated by an adaptive finite element method. We introduce an element partition tree, which controls the execution of the multi-frontal solver algorithm over these refined grids. We propose and study algorithms with polynomial computational cost for the optimization of these element partition trees. The trees provide an ordering for the elimination of unknowns. The algorithms automatically optimize the element partition trees using extensions of dynamic programming. The construction of the trees by the dynamic programming approach is expensive. These generated trees cannot be used in practice, but rather utilized as a learning tool to propose fast heuristic algorithms. In this first part of our paper we focus on the dynamic programming approach, and draw a sketch of the heuristic algorithm. The second part will be devoted to a more detailed analysis of the heuristic algorithm extended for the case of hp-adaptive grids.

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Hypergrammar-based parallel multi-frontal solver for grids with point singularities

Gurgul P., Paszyński M., Paszyńska A.

Computer Science

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2015

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Vol. 16 (1)

75--102

EN

This paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. Such graph grammar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution. In particular,the partial order of execution of graph grammar productions can be found, and the sets of independent graph grammar productions can be localized. They can be scheduled set by set into a shared memory parallel machine. The graph-grammar-based solver has been implemented with NVIDIA CUDA for GPU. Graph grammar productions are accompanied by numerical results for a 2D case. We show that our graph-grammar-based solver with a GPU accelerator is, by order of magnitude, faster than the state-of-the-art MUMPS solver.

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Adaptive analysis of inelastic problems with Bodner-Partom constitutive model

Cecot W.

Computer Assisted Mechanics and Engineering Sciences

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2006

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Vol. 13, No. 4

513-521

EN

The Bodner-Partom elastic-visco-plastic constitutive eąuations [4] were used for numerical analysis of inelastic problems. This rate-dependent model makes it possible to describe elastic, plastic and viscous processes in metals, including temperaturę and continuum damage effects. The adaptive finite element method [9] was applied to approximate solution of the governing eąuations with two a posteriori error es-timates that control accuracy of time and space discretization of displacements and internal variables. The paper addresses a further development of the methodology proposed by the author in previous works [7, 8] and used in [6]. We present here certain additional theoretical background and propose a novel strategy of adaptation as well as verify the method of solution transfer.