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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

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.

3

Applications of alternating direction solver for simulations of time-dependent problems

Łoś M., Paszyński M.

Computer Science

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2017

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Vol. 18 (2)

117--128

EN

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p.

4

Open source JAVA implementation of the parallel multi-thread alternating direction isogeometric L2 projections solver for material science simulations

Gurgul G., Woźniak M., Łoś M., Szeliga D., Paszyński M.

Computer Methods in Materials Science

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2017

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Vol. 17, No. 1

1--11

EN

This paper describes multi-thread parallel open source JAVA implementation of an alternating directions isogeometric L2 projections solver. The solver enables for fast numerical simulations of time dependent problems. To apply our solver, the time-dependent problem must be discretized using isogeometric finite element method with B-spline basis functions in spatial domain. The problem is solved using explicit method with respect to time. The application of the explicit method with B-spline based spatial discretization results in a sequence of isogeometric L2 projections that can be solved using our fast solver. The computational cost of solution of either 2D or 3D problem is linear O(N) in every time step. This cost is lower than the cost of traditional multi-frontal solvers, delivering O(N1.5) computational cost for 2D problems and O(N2) computational cost for 3D problems. This cost is also lower from any iterative solver, delivering O(Nk) computational cost, where k is the number of iterations, which depends on the particular iterative solver algorithm. Our algorithm is used for numerical solution of 3D elasticity problem.

PL

W artykule tym opisujemy otwarte oprogramowanie zawierające implementację w języku JAVA równoległego solwera wielowątkowej metody izogeometrycznych L2 projekcji. Solwer ten umożliwia przeprowadzanie szybkich symulacji problemów zależnych od czasu. Aby dało się zastosować proponowany solwer, problem niestacjonarny musi zostać zdyskretyzowany za pomocą izogeometrycznej metody elementów skończonych zawierającej funkcje bazowe B-spline w dziedzinie przestrzennej. Problem niestacjonarny rozwiązywany jest za pomocą metody explicite w dziedzinie czasowej. Zastosowanie metody explicite względem czasu redukuje problem obliczeniowy do sekwencji izogeometrycznych L2 projekcji które mogą zostać rozwiązane za pomocą naszego szybkiego solwera. Koszt obliczeniowy szybkiego solwera projekcji izogeometrycznych dla dwu- oraz trójwymiarowych problemów niestacjonarnych jest liniowy O(N) w każdym kroku czasowym. Koszt ten jest niższy niż koszt klasycznego solwera wielo-frontalnego O(N1.5) dla problemów 2D oraz O(N2) dla problemów 3D. Koszt ten jest również niższy niż koszt solwera iteracyjnego O(Nk) gdzie k oznacza liczbę iteracji, która zależy od rodzaju algorytmu solwera iteracyjnego. Solwer nasz zastosowany jest do rozwiązania trójwymiarowego problemu 3D liniowej sprężystości.

5

One-dimensional fully automatic h-adaptive isogeometric finite element method package

Lipski P., Paszyński M

Computer Science

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2016

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Vol. 17 (4)

439--459

EN

his paper deals with an adaptive finite element method originally developed by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with B-spline basis functions. We restrict ourselves to h-adaptivity, since the polynomial order of approximation must be fixed in the isogeometric case. The classical variant of the adaptive FEM algorithm, as delivered by the group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and fine grids (the latter utilized as a reference solution). The problem is solved independently over a coarse mesh and a fine mesh. The fine-mesh solution is then utilized as a reference to estimate the relative error of the coarse-mesh solution and to decide which elements to refine. Prof. Demkowicz uses hierarchical basis functions, which (though locally providing Cp−1 continuity) ensure only C0 on the interfaces between elements. The CUDA C library described in this paper switches the basis to B-spline functions and proposes a one-dimensional isogeometric version of the h-adaptive FEM algorithm to achieve global Cp−1 continuity of the solution.

6

On the computational cost and complexity of stochastic inverse solvers

Faliszewski P., Smołka M., Schaefer R., Paszyński M.

Computer Science

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2016

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Vol. 17 (2)

225--264

EN

The goal of this paper is to provide a starting point for investigations into a mainly underdeveloped area of research regarding the computational cost analysis of complex stochastic strategies for solving parametric inverse problems. This area has two main components: solving global optimization problems and solving forward problems (to evaluate the misfit function that we try to minimize). For the first component, we pay particular attention to genetic algorithms with heuristics and to multi-deme algorithms that can be modeled as ergodic Markov chains. We recall a simple method for evaluating the first hitting time for the single-deme algorithm and we extend it to the case of HGS, a multi-deme hierarchic strategy. We focus on the case in which at least the demes in the leaves are well tuned. Finally, we also express the problems of finding local and global optima in terms of a classic complexity theory. We formulate the natural result that finding a local optimum of a function is an NP-complete task, and we argue that finding a global optimum is a much harder, DP-complete, task. Furthermore, we argue that finding all global optima is, possibly, even harder (#P-hard) task. Regarding the second component of solving parametric inverse problems (i.e., regarding the forward problem solvers), we discuss the computational cost of hp-adaptive Finite Element solvers and their rates of convergence with respect to the increasing number of degrees of freedom. The presented results provide a useful taxonomy of problems and methods of studying the computational cost and complexity of various strategies for solving inverse parametric problems. Yet, we stress that our goal was not to deliver detailed evaluations for particular algorithms applied to particular inverse problems, but rather to try to identify possible ways of obtaining such results.

7

Application of projection-based interpolation algorithm for non-stationary problem

Woźniak M., Paszyński M.

Computer Science

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2016

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Vol. 17 (3)

297--319

EN

In this paper, we present a solver for non-stationary problems using L 2 projec- tion and h -adaptations. The solver utilizes the Euler time integration scheme for time evolution mixed with projection-based interpolation techniques for solving the L 2 projection problem at every time step. The solver is tested on the model problem of a heat transfer in an L-shape domain. We show that our solver delivers linear computational cost at every time step.

8

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.

9

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.

10

An agent-oriented hierarchic strategy for solving inverse problems

Smołka M., Schaefer R., Paszyński M., Pardo D., Álvarez-Aramberri J.

International Journal of Applied Mathematics and Computer Science

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2015

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Vol. 25, no. 3

483--498

EN

The paper discusses the complex, agent-oriented hierarchic memetic strategy (HMS) dedicated to solving inverse parametric problems. The strategy goes beyond the idea of two-phase global optimization algorithms. The global search performed by a tree of dependent demes is dynamically alternated with local, steepest descent searches. The strategy offers exceptionally low computational costs, mainly because the direct solver accuracy (performed by the hp-adaptive finite element method) is dynamically adjusted for each inverse search step. The computational cost is further decreased by the strategy employed for solution inter-processing and fitness deterioration. The HMS efficiency is compared with the results of a standard evolutionary technique, as well as with the multi-start strategy on benchmarks that exhibit typical inverse problems’ difficulties. Finally, an HMS application to a real-life engineering problem leading to the identification of oil deposits by inverting magnetotelluric measurements is presented. The HMS applicability to the inversion of magnetotelluric data is also mathematically verified.

11

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.

12

Graph Transformation Systems for Modeling Three Dimensional Finite Element Method. Part 2

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

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.

13

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.

14

Multi-frontal parallel direct solver for one dimensional isogeometric collocation method

Lipski P., Paszyński M

Computer Methods in Materials Science

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2015

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Vol. 15, No.1

192--197

EN

In this paper wc present a new multi-frontal solver for the isogeometric collocation method (ISO-C) on GPU. The ISO-C method constitutes an alternative for the isogeometric finite element method (ISO-FEM). The key advantage of ISO-C over ISO-FEM is that it does not include the computationally intensive operation of integrating the variational formulation. The ISO-C method requires using only a single collocation point per one basis function, whereas in ISO-FEM, Gaussian quadrature is applied on many points at each finite element. The presented multi-frontal solver for collocation method results in logarithmic execution time assuming that large enough number of GPU processors is available. In this article, the method is employed for an exemplary ID nanolithography problem of Step-and-Flash Imprint Lithography (SFIL). The algorithm, however, may be applied to a wide class of 2D and 3D problems.

PL

W artykule przedstawiamy nowy solwer wielofrontalny dla izogeometrycznej metody kolokacji (ISO-C) na GPU. Metoda ISO-C stanowi alternatywę dla izogeometrycznej metody elementów skończonych (ISO-FEM). Główną zaletą metody ISO-C jest redukcja znacznego kosztu obliczeniowego całkowania sformułowania wariacyjnego występującego w metodzie ISO-FEM. Metoda ISO-C wymaga bowiem użycia tylko jednego punktu kolokacji dla jednej funkcji bazowej, podczas gdy metoda ISO-FEM wiąże się z zastosowaniem kwadratury Gaussa w wielu punktach na każdym elemencie skończonym. Prezentowany solwer wielofrontalny dla metody kolokacji uzyskuje logarytmiczną złożoność obliczeniową przy założeniu odpowiednio dużej liczby procesorów graficznych GPU. Niniejsza publikacja przedstawia proste wykorzystanie metody dla jednowymiarowego przykładowego problemu nanolitografii Step-and-Flash Imprint Lithography (SFIL). Zaprezentowany algorytm znajduje jednak ogólnie zastosowanie dla szerokiej klasy problemów w dwóch i trzech wymiarach.

15

Dealing with periodic boundary conditions for 1D, 2D and 3D isogeometric finite element method

Łoś M., Paszyński M, Dalcin L., Calo V.

Computer Methods in Materials Science

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2015

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Vol. 15, No.1

213--218

EN

In this paper we analyze the problem of implementing periodic boundary conditions in the isogeomotric finite element method (ISO-FEM). The ISO-FEM method uses the B-spline-based basis functions, which facilitates usage of the same basis functions for approximation of the geometry as well as for the numerical solution of the modeled physical phenomena. The usage of the B-spline based basis functions results in CA(p-l) global continuity of the solution. The drawback is a difficulty in implementing the periodic boundary conditions, and special dedicated methods are necessary. In this paper we present two algorithms implementing the periodic boundary conditions. The first one is an iterative algorithm that utilizes widely available block-diagonal LAPACK solver. The second one is a modification of the multi-frontal solver algorithm itself, and it requires a dedicated solver with its source code modified accordingly. The presented methods can be applied in one, two or three-dimensional isogeometric finite element method.

PL

W artykule analizujemy sposób implementacji periodycznych warunków brzegowych w izogeometrycznej metodzie elementów skończonych (ISO-FEM). Metoda ISO-FEM cechuje się użyciem B-spline'ów jako funkcji bazowych, co pozwala na zastosowanie takiej samej bazy wielomianów do odwzorowania geometrii jak również do rozwiązania modelowanego zagadnienia fizycznego. Baza zbudowana z B-spline'ów stopnia p posiada globalną ciągłość C۸(p-l). Z tego też powodu sposób wymuszania periodycznych warunków brzegowych nie jest oczywisty, i konieczne jest zastosowanie specjalnych technik. W artykule tym prezentujemy dwa algorytmy wymuszające periodyczne warunki brzegowe. Pierwszy algorytm iteracyjny umożliwia wykorzystanie powszechnie dostępnych solwerów (takich jak LAPACK) dla macierzy blokowo-diagonalnych, drugi algorytm polega na modyfikacji kodu solwera wielofrontalnego, i z tego względu wymaga dedykowanej implementacji algorytmu solwera. Przedstawione sposoby implementacji periodycznych warunków brzegowych można zastosować w jedno, dwu i trójwymiarowej isogeometrycznej metodzie elementów skończonych.

16

A hybrid algorithm for solving inverse problems in elasticity

Barabasz B., Gajda-Zagórska E., Migórski S., Paszyński M., Schaefer R., Smołka M.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 4

865--886

EN

The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.

17

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.

18

Two-dimensional hp-adaptive algorithm for continuous approximations of material data using space projection

Gurgul P., Sieniek M., Paszyński M., Madej Ł., Collier N.

Computer Science

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2013

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Vol. 14 (1)

97--112

EN

In this paper we utilize the concept of the L 2 and H 1 projections used to adaptively generate a continuous approximation of an input material data in the finie element (FE) base.This approximation, along with a corresponding FE mesh, can be used as material data for FE solvers. We begin with abrief theoretical background, followed by description of the hp -adaptive algorithm adopted here to improve gradually quality of the projections. We investigate also a few distinct sample problems, apply the aforementioned algorithms and conclude with numerical results evaluation.

19

HP-HGS strategy for inverse AC/DC resistivity logging measurement simulations

Gajda-Zagórska E., Paszyński M, Schaefer R., Pardo D.

Computer Science

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2013

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Vol. 14 (4)

629--644

EN

In this paper, we present resistivity-logging-measurement simulation with the use of two types of borehole logging devices: those which operate with zero frequency (direct current, DC) and those with higher frequencies (alternate current, AC). We perform simulations of 3D resistivity measurements in deviated wells, with a sharp angle between the borehole and formation layers. We introduce a hierarchical adaptive genetic strategy hp−HGS interfaced with an adaptive finite element method. We apply a strategy for the solution of the inverse problem, where we identify the resistivities of the formation layers based on a given measurement. We test the strategy on both direct and alternate current cases.

20

Grammar based multi-frontal solver for isogeometric analysis in 1d

Kuźnik K., Paszyński M, Calo V.

Computer Science

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2013

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Vol. 14 (4)

589--613

EN

In this paper, we present a multi-frontal direct solver for one-dimensional iso-geometric finite element method. The solver implementation is based on the graph grammar (GG) model. The GG model allows us to express the entire solver algorithm, including generation of frontal matrices, merging, and eliminations as a set of basic undividable tasks called graph grammar productions. Having the solver algorithm expressed as GG productions, we can find the partial order of execution and create a dependency graph, allowing for scheduling of tasks into shared memory parallel machine. We focus on the implementation of the solver with NVIDIA CUDA on the graphic processing unit (GPU). The solver has been tested for linear, quadratic, cubic, and higher-order B-splines, resulting in logarithmic scalability.