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1

Heuristic algorithm to predict the location of C0 separators for efficient isogeometric analysis simulations with direct solvers

Paszyńska A., Jopek K., Woźniak M., Paszyński M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 6

907--917

EN

We focus on two and three-dimensional isogeometric finite element method computations with tensor product Ck B-spline basis functions. We consider the computational cost of the multi-frontal direct solver algorithm executed over such tensor product grids. We present an algorithm for estimation of the number of floating-point operations per mesh node resulting from the execution of the multi-frontal solver algorithm with the ordering obtained from the element partition trees. Next, we propose an algorithm that introduces C0 separators between patches of elements of a given size based on the stimated number of flops per node. We show that the computational cost of the multi-frontal solver algorithm executed over the computational grids with C0 separators introduced is around one or two orders of magnitude lower, while the approximability of the functional space is improved. We show O(NlogN) computational complexity of the heuristic algorithm proposing the introduction of the C0 separators between the patches of elements, reducing the computational cost of the multi-frontal solver algorithm.

2

Bisection weighted by element size ordering algorithm for multi-frontal solver executed over 3D h refined grids

Skotniczny M., Paszyński M, Paszyńska A.

Computer Methods in Materials Science

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2016

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Vol. 16, No. 1

54--61

EN

In this paper we present an algorithm for generation of ordering over 3D grids h refined towards singularities. The ordering controls the execution of multi-frontal direct solver algorithm on systems of linear equations generated by 3D h adaptive finite element method. The proposed ordering algorithm outperforms other state-of-the-art orderings available through MUMPS interface, namely nested-dissections, AMD and PORD. Our algorithm uses additional knowledge about the structure of the computational mesh, not available to alternative ordering algorithms.

PL

W artykule prezentujemy algorytm generacji porządku eliminacji kierujący wykonaniem solwera wielo-frontalncgo dla trójwymiarowych siatek li adaptowanych do osobliwości punktowych, krawędziowych i ścianowych. Wygenerowany porządek generuje permutacje macierzy układu równań liniowych uzyskanych podczas obliczeń trójwymiarową metodą elementów skończonych. Proponowany algorytm dostarcza porządku eliminacji który pozwala wykonywać faktoryzację z mniejszą liczbą operacji zmienno-przecinkowych niż klasyczne algorytmy generacji porządku dostępne za pośrednictwem solwera MUMPS, takie jak nested-disseetions, AMD oraz PORD. Nasz algorytm wykorzystuje dodatkową wiedzę o strukturze siatki obliczeniowej, nie dostępną dla alternatywnych algorytmów generacji porządku.

3

Minimizing the memory usage with parallel out-of-core multi-frontal direct solver

Paszyński M.

Computer Assisted Methods in Engineering and Science

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2013

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Vol. 20, no. 1

15--41

EN

This paper presents the out-of-core solver for three-dimensional multiphysics problems. In particular, our study focuses on the three-dimensional simulations of the linear elasticity coupled with acoustics. The out-of-core solver is designed with three principles in mind. First, to store the dense matrices associated with the nodes of the elimination tree with blocks related to nodes of the mesh, where many degrees of freedom may be located in the case of multiphysics computations with high order polynomials. The second principle is to minimize the memory usage. This is obtained by dumping out all local systems from the entire elimination tree to the disk during the elimination stage. The local systems are reutilized later during the backward substitution stage. The third principle is that the communication in the parallel version of the out-of-core solver occurs through the parallel file system. The memory usage of the solver is compared against the state-of-the-art MUMPS solver.