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1
Content available 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
|
2018
|
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
Content available 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
|
2018
|
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
Content available remote Supporting Product Modelling with “GraphTool”
Palacz w., Paszyńska A., Ślusarczyk G., Strug B., Grabska E.
Machine Dynamics Research
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2017
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Vol. 41, No. 4
135--147
EN
This paper presents the GraphTool system which supports all steps required to define graph grammars and to control their application. It provides graphical editors for graphs, graph transformation rules, and control diagrams. The considered graph grammars are based on different types of graphs (composite graphs, hierarchical graphs, hypergraphs and hierarchical hypergraphs) which can be labelled and attributed. In this tool, the standard approach to graphs and graph grammars is extended to graph grammar systems and graphs with layers. Adding layers allows the user to model graph structures composed of disjoint substructures, while adopting grammar systems allows for defining groups of grammars working together in a single derivation process. Graph structures obtained as the result of the graph derivations are used as knowledge representation in different application fields. This paper shows the versatility of GraphTool by presenting examples of its use in four different areas: computational grids, computer game states, Finite Element Method computations and architectural designs.
4
Content available 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
|
2017
|
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.
5
Content available remote 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
|
2016
|
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.
6
Content available remote 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.
7
Content available remote Graph Transformation Systems for Modeling Three Dimensional Finite Element Method. Part I
Ryszka I., Paszyńska A., Grabska E., Sieniek M., Paszyński M.
Fundamenta Informaticae
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2015
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Vol. 140, nr 2
129--172
EN
In this paper we present several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. In our approach a computational mesh is represented by a composite graph and mesh operations are expressed by the graph transformation rules. Each graph transformation system is responsible for different kind of operations. In particular, there is a graph transformation system expressing generation of an initial mesh, generating element matrices and elimination trees for interfacing with direct solver algorithm, a graph transformation system deciding which elements have to be further refined, as well as a graph transformation system responsible for execution of mesh refinements. These graph transformation systems are tested using a graph transformation tool (called GRAGRA), which provides a graphical environment for defining graphs, graph transformation rules and graph transformation systems. In this paper we illustrate the concepts by using an exemplary derivation for a three dimensional projection problem, based on a set of graph transformation rules.
8
Content available remote Graph Transformation Systems for Modeling Three Dimensional Finite Element Method. Part 2
Ryszka I., Paszyńska A., Grabska E., Sieniek M., Paszyński M.
Fundamenta Informaticae
|
2015
|
Vol. 140, nr 2
173--203
EN
In this paper we introduce formal definitions for several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. We introduce a composite graph representation of the computational mesh and graph transformation rules expressing the mesh operations. In particular, there are graph transformation rules expressing the generation of the initial mesh consisting with tetrahedral finite elements, graph transformation rules expressing the construction of an elimination tree for interfacing with multi-frontal direct solver algorithm, graph transformation rules selecting sub-graph representing finite elements for further refinements, graph transformation rules responsible for execution of mesh refinements. We also discuss several benefits of using graph transformation system instead of classical FEM approach, including the benefits from the viewpoint of multi-frontal direct solvers.
9
Content available Hypergrammar-based parallel multi-frontal solver for grids with point singularities
Gurgul P., Paszyński M., Paszyńska A.
Computer Science
|
2015
|
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.
10
Content available 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
|
2013
|
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.
11
Content available remote Graph grammar for three dimensional multi physics adaptive finite element method simulations
Paszyńska A., Paszyński M
Computer Methods in Materials Science
|
2013
|
Vol. 13, No. 1
9--15
EN
In this paper we present an application of graph grammar for modelling a three dimensional hp adaptive finite element method for multi-physics simulations. The graph grammar model involves the process of generation of the mesh with tetrahedral finite elements, the process of mesh adaptation by means of h refinement as well as interfacing with the multi-frontal solver algorithm by translating the finite elements into a sequence of frontal matrices, being an input for the multi-frontal solver algorithm. The paper is concluded with numerical results concerning an advanced multi-physics problems, the propagation of an acoustic waves over the model of the human head.
PL
W artykule tym przedstawiamy gramatykę grafową do modelowania adaptacyjnej metody elementów skończonych służącej do wykonywania trójwymiarowych symulacji problemów wielo-fizycznych. W szczególności model gramatyki grafowej obejmuje proces generacji siatki obliczeniowej zbudowanej z elementów czworościennych, proces zagęszczania siatki obliczeniowej za pomocą mechanizmu h adaptacji, a także interfejs do solwera wielo-frontalnego uzyskany poprzez generowanie ciągu macierzy frontalnych dla poszczególnych elementów skończonych. Artykuł podsumowują przykładowe wyniki symulacji dotyczące zagadnienia propagacji fal akustycznych na modelu głowy ludzkiej.
12
Content available remote A Graph GrammarModel of the hp Adaptive Three Dimensional Finite Element Method. Part I
Paszyńska A., Grabska E., Paszyński M.
Fundamenta Informaticae
|
2012
|
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).
13
Content available remote A Graph GrammarModel of the hp Adaptive Three Dimensional Finite Element Method. Part II
Paszyńska A., Grabska E., Paszyński M.
Fundamenta Informaticae
|
2012
|
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.
14
Content available remote Parallel genetic algorithm finding optimal initial mesh for three dimensional self-adaptive hp finite element metho
Paszyńska A., Paszyński M.
Computer Methods in Materials Science
|
2011
|
Vol. 11, No. 1
34-40
EN
In this paper we investigate the parallel version of the hierarchical chromosome based genetic algorithm (HCBGA) for finding the optimal initial mesh for self-adaptive hp-Finite Element Method (hp-FEM). The HCBGA algorithm solves the global optimization problem of r refinements in order to provide optimal starting mesh for the hp-FEM that will fit material data and singularities, and will result in the exponential convergence of the hp-FEM. The parallel algorithm is tested on the damaged Step-and-Flash Imprint Lithography problem, modeled by linear elasticity with thermal expansion ':, coefficient.
PL
W artykule tym przedstawiono wersję równoległą algorytmu genetycznego bazującego na hierarchicznym chromosomie (AGBHC) służącego do znajdowania optymalnych siatek począt-kowych dla hp adaptacyjnej metody elementów skończonych (hp-MES). Zaproponowany algorytm AGBHC rozwiązuje problem r adaptacji. Jest to problem globalnej optymalizacji polegający na znalezieniu optymalnej siatki początkowej dla algorytmu automa-tycznej hp adaptacji. Poszukiwana siatka początkowa powinna pasować do przyjętych stałych materiałowych oraz do osobliwości rozwiązania. W efekcie algorytm automatycznej hp adaptacji uruchomiony na takiej optymalnej siatce początkowej powinien dostarczyć eksponencjalną zbieżność dokładności rozwiązania względem rozmiaru siatki obliczeniowej. Algorytm równoległy testowany jest na problemie nanolitografii poprzez wyciskanie i naświetlanie, modelowanym z pomocą liniowej sprężystości ze współczynnikiem rozszerzalności cieplnej.
15
Content available remote Cellular Automata and Self-Adaptive hp Finite Element Method Multi-Component Diffusion Models
Paszyńska A., Paszyński M.
Computer Methods in Materials Science
|
2009
|
Vol. 9, No. 2
221-227
PL
W artykule zaprezentowany został przegląd opracowanych modeli dyfuzji wieloskładnikowej. Problem dyfuzji wieloskładnikowej opisany został za pomocą modeli stochastycznych, bazujących na automatach komórkowych, jak również modeli ciągłych, bazujących na równaniach różniczkowych cząstkowych. Rozważane modele stochastyczne opierają się na automatach komórkowych z regułami wymiany lub rotacji cząstek wielofazowych. Podczas symulacji, cząstki różnych faz poddawane są rotacji zgodnie lub przeciwnie do ruchu wskazówek zegara, zgodnie z przyjętym prawdopodobieństwem. Współczynnik dyfuzji symulowanego układu cząstek definiowany jest poprzez modyfikację prawdopodobieństw wykonania reguł międzycząsteczkowych. Opracowane modele zostały zweryfikowane poprzez wykonanie szeregu symulacji numerycznych, oraz porównane do uprzednio opracowanych modeli ciągłych, bazujących na adaptacyjnej metodzie elementów skończonych.
EN
The paper presents a survey of developed multi-component diffusion models. Stochastic and continuous PDE models for multi component diffusion problems are studied by performing a sequence of numerical simulations. The stochastic models are based on Cellular Automata with interchange and block-rotation model. The multi-component particles are either intechanged or rotated clockwise or counter-clockwise with prescribed probability. The resulting diffusion coefficients are obtained by statistical tools, and can be modified by adjusting the probabilities of interchange or rotations. The continuous PDE models are based on classical variational formulations solved by the self-adaptive hp Finite Element Method. Numerical results are compared and discussed.
16
An application of hierarchical chromosome based genetic algorithm to the optimization pf platform shape
Paszyńska A., Paszyński M.
Prace Naukowe Politechniki Warszawskiej. Elektronika
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2007
|
z. 160
225-232
EN
In this paper we present hierarchical chromosome based genetic algorithm and its application to solving of exemplary design problem-optimization of platform with minimal overall volume and the stress value not greater then the maximum admissible value, when the platform is under the external load. The algorithm uses hierarchical chromosome representation and hierarchical genetic operators, instead of the binary representation and traditional genetic operators. It allows for coding of phenotypes with different number of components as well as performing reproduction with such individuals. In order to solve the problem, the application with hierarchical chromosome based genetic algorithm was interfaced with the parallel fully automatic hp adaptive 3D Finite Element Method package (parph3D). The parph3D application was used to compute the stress values for individuals generated by genetic algorithm.
17
Content available Hierarchical genetic computation in optimal design
Kołodziej J., Schaefer R., Paszyńska A.
Journal of Theoretical and Applied Mechanics
|
2004
|
Vol. 42 nr 3
519-538
EN
The paper presents two examples of genetic hierarchic global optimization methods. They are two different goals of introducing hierarchy into the computational model: to perform the multi-scale search with the adapted accuracy and to better express the structure geometry in the optimal shape design. Results of the formal analysis and simple computational examples are also attached.
PL
Praca przedstawia dwa przykłady hierarchicznych, genetycznych metod optymalizacji. Sklasyfikowano dwa główne powody wprowadzenia hierarchii do modelu obliczeniowego: dla uzyskania wieloskalowego przeszukania z adaptowaną dokładnością oraz dla lepszego odwzorowania kształtu konstrukcji w zadaniach optymalnego projektowania kształtu. Zamieszczono rezultaty formalnej analizy proponowanych strategii oraz proste przykłady obliczeniowe.
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