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1

Applications of a hyper-graph grammar system in adaptive finite-element computations

Gurgul P., Jopek K., Pingali K., Paszyńska A.

International Journal of Applied Mathematics and Computer Science

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2018

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Vol. 28, no. 3

569--582

EN

This paper describes application of a hyper-graph grammar system for modeling a three-dimensional adaptive finite element method. The hyper-graph grammar approach allows obtaining a linear computational cost of adaptive mesh transformations and computations performed over refined meshes. The computations are done by a hyper-graph grammar driven algorithm applicable to three-dimensional problems. For the case of typical refinements performed towards a point or an edge, the algorithm yields linear computational cost with respect to the mesh nodes for its sequential execution and logarithmic cost for its parallel execution. Such hyper-graph grammar productions are the mathematical formalism used to describe the computational algorithm implementing the finite element method. Each production indicates the smallest atomic task that can be executed concurrently. The mesh transformations and computations by using the hyper-graph grammar-based approach have been tested in the GALOIS environment. We conclude the paper with some numerical results performed on a shared-memory Linux cluster node, for the case of three-dimensional computational meshes refined towards a point, an edge and a face.

2

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.

3

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.

4

Using the system of graph grammar for generation of quasi optimal element partition trees in two dimensions

Paszyńska A., Świderska I., Woźniak M., Jopek K., Paszyński M., Grabska E., Lenhart A., Nguyen D., Pingali K.

Computer Methods in Materials Science

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2016

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Vol. 16, No. 3

143--155

EN

The paper presents a graph grammar based approach for h-adaptive finite element method and multi-frontal solver algorithm. The multi-frontal solver is used for solving systems of linear equations created by finite element method. The multi-frontal solver is controlled by so-called ordering. The quality of ordering influences hardly the solver effectiveness. In our approach, the finite element mesh is represented by means of a hypergraph and corresponding element partition tree. The finite element operations like mesh generation or h-adaptation are modelled by graph grammar production. Additionally graph grammar productions have corresponding productions for the construction of the element partition tree. The element partition trees are transformed into the ordering that controlls execution of the solver algorithm. We show that the ordering resulting from our element partititon tree results in better performance of the parallel solver than the state of the art nested-dissection ordering available through MUMPS interface on the class of grids refined towards singularities.

PL

W artykule tym prezentujemy gramatykę grafową do modelowania algorytmów h adaptacyjnej metody elementów skończonych oraz solwera wielofrontalncgo. Solwer wielofronatlny używany jest do rozwiązania układu równań liniowych stworzonych przez metodę elementów skończonych. Solwer ten kontrolowany jest przez tak zwany porządek eliminacji. Jakość porządku eliminacji wpływa na efektywność solwera wielofrontalnego. W naszym podejściu siatka metody elementów skończonych reprezentowana jest przez hipergraf oraz związane z nim drzewo podziałów siatki. Operacje na elementach skończonych takie jak generacja siatki oraz h adaptacja modelowane są przez produkcję gramatyki grafowej. Dodatkowo, gramatyka grafowa posiada powiązane produkcje do generacji drzewa podziałów siatki. Drzewo podziałów siatki z kolei transformowane jest w porządek eliminacji, który kontroluje wykonanie algorytmu solwera. Pokazujemy że porządek eliminacji uzyskany na podstawie naszego drzewa podziałów siatki daje lepszą wydajność algorytmu solwera równoległego w porównaniu z klasycznym porządkiem nested- disseetions dostępnym w solwerze MUMPS, dla klas siatek adaptowalnych do lokalnych osobliwości.