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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.

2

Using a graph grammar system in the finite element method

Strug B., Paszyńska A., Paszyński M., Grabska E.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 4

839--853

EN

The paper presents a system of Composite Graph Grammars (CGGs)modelling adaptive two dimensional hp Finite Element Method (hp-FEM) algorithms with rectangular finite elements. A computational mesh is represented by a composite graph. The operations performed over the mesh are defined by the graph grammar rules. The CGG system contains different graph grammars defining different kinds of rules of mesh transformations. These grammars allow one to generate the initial mesh, assign values to element nodes and perform h- and p-adaptations. The CGG system is illustrated with an example from the domain of geophysics.

3

A Graph GrammarModel of the hp Adaptive Three Dimensional Finite Element Method. Part I

Paszyńska A., Grabska E., Paszyński M.

Fundamenta Informaticae

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2012

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Vol. 114, nr 2

149-182

EN

The first part of our paper presents a composite programmable graph grammar model for the self-adaptive two dimensional hp Finite Element Method algorithms (2D hp-FEM) with mixed triangular and rectangular finite elements. The two dimensional model is a starting point for the three dimensional model of self-adaptive hp-FEM presented in the second part of this paper. A computational mesh is represented by a composite graph. The operations performed over the mesh are expressed by the graph grammar rules. The three dimensional model is based on the extension of the two dimensional model with rectangular finite elements. In the second part of this paper, we conclude the presentation with numerical examples concerning the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL).

4

A Graph GrammarModel of the hp Adaptive Three Dimensional Finite Element Method. Part II

Paszyńska A., Grabska E., Paszyński M.

Fundamenta Informaticae

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2012

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Vol. 114, nr 2

183-201

EN

This paper presents a composite programmable graph grammar model of the three dimensional self-adaptive hp Finite Element Method (hp-FEM) algorithms. The computational mesh composed of hexahedral finite elements is represented by a composite graph. The operations performed over the mesh are expressed by composite graph grammar productions. The three dimensional model is based on the extension of the two dimensional model for rectangular finite elements. This paper is concluded with numerical examples, presenting the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL), the modern patterning process.

5

On the Parallelization of Self-Adaptive hp-Finite Element Methods Part I. Composite Programmable Graph Grammar Model

Paszyński M.

Fundamenta Informaticae

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2009

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Vol. 93, nr 4

411-434

EN

The paper presents a general methodology for an efficient parallelization of the fully automatic hp-adaptive Finite Element Method (hp-FEM). The self-adaptive hp-FEM algorithm expressed in terms of the graph grammar productions is analyzed by utilizing the Partitioning Communication Agglomeration Mapping (PCAM) model. The computational tasks are defined over a graph model of the computational mesh. It is done for all parts of the algorithm: the generation of an initial mesh, direct solver (including the integration and elimination of degrees of freedom), mesh transformations (including the h and p refinements), as well as the selection of the optimal refinements. The computation and communication complexities of the resulting parallel algorithms are analyzed. The paper is concluded with the sequence of massive parallel computations. >From the performed tests it implies that the code scales well up to 200 processors

6

On the Parallelization of Self-Adaptive hp-Finite Element Methods Part II. Partitioning Communication Agglomeration Mapping (PCAM) Analysis

Paszyński M.

Fundamenta Informaticae

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2009

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Vol. 93, nr 4

435-457

EN

The paper presents a general methodology for an efficient parallelization of the fully automatic hp-adaptive Finite Element Method (hp-FEM). The self-adaptive hp-FEM algorithm expressed in terms of the graph grammar productions is analyzed by utilizing the Partitioning Communication Agglomeration Mapping (PCAM) model. The computational tasks are defined over a graph model of the computational mesh. It is done for all parts of the algorithm: the generation of an initial mesh, direct solver (including the integration and elimination of degrees of freedom), mesh transformations (including the h and p refinements), as well as the selection of the optimal refinements. The computation and communication complexities of the resulting parallel algorithms are analyzed. The paper is concluded with the sequence of massive parallel computations. >From the performed tests it implies that the code scales well up to 200 processors.