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1

Evaluation of partial factorization for reduction of finite element matrices

Jarzębski P., Wiśniewski K.

Engineering Transactions

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2017

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Vol. 65, iss. 1

163--170

EN

In this paper, we present the concept of Partial Factorization [1] and discuss its possible applications to the Finite Element method. We consider: (1) reduction of the element tangent matrix, which is particularly important for mixed/enhanced elements and (2) reduction of the sub-domain matrices of the Domain Decomposition (DD) equation solvers run either sequentially on a single machine or in parallel on a cluster of computers. We demonstrate that Partial Factorization can be beneficial for these applications.

2

On the convergence of domain decomposition algorithm for the body with thin inclusion

Styahar A., Savula Y.

Acta Mechanica et Automatica

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2015

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Vol. 9, no. 1

27--32

EN

We consider a coupled 3D model that involves computation of the stress-strain state for the body with thin inclusion. For the description of the stress-strain state of the main part, the linear elasticity theory is used. The inclusion is modelled using Timoshenko theory for shells. Therefore, the dimension of the problem inside the inclusion is decreased by one. For the numerical solution of this problem we propose an iterative domain decomposition algorithm (Dirichlet-Neumann scheme). This approach allows us to decouple problems in both parts and preserve the structure of the corresponding matrices. We investigate the convergence of the aforementioned algorithm and prove that the problem is well-posed.

3

Parallel solution of electrostatic and static magnetic field problems by domain decomposition metod

Marcsa D., Kuczmann M.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 2b

49--52

EN

The paper presents a parallel approach for the efficient solution of a one-dimensional and a two-dimensional problem by parallel finite element method. These problems are case studies. The non-overlapping domain decomposition method has been used to cut the problem into subregions or also called sub-domains, and it reduces the large mass matrix into smaller parts. The independent sub-domains, and the assembling of these equation systems can be handled by the independent processors of a supercomputer, i.e. in a parallel way. The execution time and speedup of parallel finite element method have been compared to the serial one.

PL

W artykule opisano metodę efektywnego rozwiązywania problemów jedno- i dwuwymiarowych, poprzez równoległe analizy metodą elementów skończonych. Analizowany obiekt jest dzielony na podregiony co zmniejsza rozmiary jego macierzy i dzieli ją na mniejsze (poddziedziny). Te z kolei mogą być obliczane przez niezależne procesory superkomputera.

4

Parallel Implementation of Ray Tracing Procedure in Anisotropic Medium

Pieta A., Dwornik M.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2012

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Vol. 16, No 1-2

135-143

EN

This article describes a parallel implementation of a ray tracing algorithm in a heterogeneous anisotropic geological medium. The shortest path method, which was used for calculations, can give ray path and travel time of seismic wave propagation even for a random and discontinuous velocity field. The high precision required in such calculations was obtained by employing a dense computational grid. This led to a significant increase in the computational effort of the algorithm. The procedure was parallelized using domain decomposition. The results show that the parallel performance of the ray tracing procedure strongly depends on the assumed geological method and differs between media with and without anisotropy of seismic wave propagation.

5

Comparison of domain decomposition methods for elliptic partial differential problems with unstructured meshes

Marcsa D., Kuczmann M.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 12b

1-4

EN

The paper presents the parallelisation of sequential (single-processor) finite element simulations with the use of domain decomposition methods. These domain decomposition methods are the Schur complement method and the Finite Element Tearing and Interconnecting (FETI) method. The execution time and speedup of these parallel finite element methods have been compared to each other and to the sequential one.

PL

W artykule zaprezentowano paralelizację sekwencyjnych (jednoprocesorowych) symulacji metodą elementów skończonych z wykorzystaniem metod dekompozycji obszaru. Metody te to metoda uzupełnienia Schura oraz metoda FETI (Finite Element Tearing and Interconnecting). Czas obliczeń i przyspieszenie paralelizowanych metod elementów skończonych zostały porównae między sobą oraz z procesem sekwencyjnym. (Porównanie metod dekompozycji obszaru dla rozwiązywania równań różniczkowych cząstkowych eliptycznych z niestrukturalną siatką)

6

3D Multi-Domain MFS Analysis of Sound Pressure Level Reduction Between Connected Enclosures

Godinho L., Branco F. G., Mendes P. A. P. A.

Archives of Acoustics

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2011

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Vol. 36, No. 3

575-601

EN

In this paper, the authors study the 3D propagation of sound waves between two closed spaces. The separation element between the two rooms is considered to include either a small opening or a homogeneous lightweight panel, coupling the two spaces. A numerical study of this configuration is performed, trying to understand the influence of the position and geometry of this opening in the sound pressure level reduction curve at low and midfrequencies. Additionally, the coupling effect between the two acoustic spaces is analyzed, in order to better understand its importance when determining the sound pressure level reduction. Different boundary conditions are ascribed to the walls of these rooms, simulating both the completely reflecting and partially absorbing surfaces. The numerical modelling was performed using a multi-domain formulation of the Method of Fundamental Solutions (MFS). The system is composed of two coupled rooms, limited by rigid or by absorbing walls, and separated by a thin wall (tending to null thickness) with a small opening. An experimental validation of the proposed model is presented, comparing its results with those found experimentally for a reduced-scale model. It is important to note that, for such a configuration, a traditional single-domain approach using methods like the MFS or the BEM would lead to undetermined equation systems, and thus the proposed model makes use of a domain decomposition technique.

7

Single and Double Lagrange multipliers approaches applied to Scalar potential formulation in magnetostatic FEM

Henneron T., Aubertin M., Boiteau O., Piriou F., Guerin P., Mipo J.-C.

Przegląd Elektrotechniczny

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2010

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Nr 5

119-122

EN

In this paper, we propose to introduce the single and double Lagrange multipliers approaches in the case of the finite element method (FEM). These approaches allow non-conforming meshes to be linked together. The methods introduced are developed in the case of a magnetostatic problem solved by the scalar potential formulation. An application is studied and the results obtained by both approaches are compared.

PL

W artykule przedstawiono zastosowanie pojedynczych i podwójnych mnożników Lagrangea stosowanych w metodzie elementów skończonych. To podejście pozwala na połączenie niezgodnych siatek. Metodę rozwinięto dla problemu magnetostatycznego rozwiązywanego z użyciem potencjału skalarnego. Porównano wyniki otrzymane z zastosowaniem proponowanej metody i metod klasycznych.

8

Comparison between substructuring and overlapping domain decomposition methods for diffuse optical tomography problems

Grzywacz T., Sikora J.

Przegląd Elektrotechniczny

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2010

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R. 86, nr 1

164-165

EN

The forward problem in DOT can be modelled in an frequency domain as a diffusion equation with Robin boundary conditions. In case of multilayered geometries the forward problem can be treated as a set of coupled equations. In this paper we present the solution for diffuse light propagation in a four-layer spherical model using Boundary Element Method. Additionally, we compare overlapping with non-overlapping domain decomposition methods applied to this problem to improve its efficiency.

PL

Zagadnienie proste w dyfuzyjnej tomografii optycznej może być modelowane w dziedzinie częstotliwości równaniem dyfuzji z warunkami brzegowymi Robina. W artykule zostanie zaprezentowany wynik rozwiązania równania dyfuzji jako modelu propagacji światła w czterowarstwowym obiekcie sferycznym przy użyciu metody elementów brzegowych. Dodatkowo zostanie porównana jedna z metod dekompozycji obszarowej z nakładaniem z metodą dekompozycji bez nakładania, które zostały użyte do przyspiesznia obliczeń rozważanego zagadnienia prostego.

9

The domain decomposition approach with contact problem

Podesva J.

Modelling and Optimization of Physical Systems

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2010

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z. 9

65--70

CS

Modelování valivého ložiska přináší dva významné problémy. U kontaktního problému předpokládame, že dva povrchy se budou dotýkat. Dotyk přenáší tlakovou sílu, ne však tahovou. Není dopředu jasné ani jak velka bude kontaktní plocha, ani jaké bude rozložení kontaktního tlaku. Druhým problémem může být velikost úlohy. Dnešní hardware sice umožňuje rychlé řešení velkých úloh. Přesto při poćtu stupňů volnosti v řádu 105 až 106 a při použití algoritmů, vyžadujících opakování výpočtu, může toto představovat problém. Účinnou cestou řešení může být dekompozice modelu na tzv. „super-prvky“. Příspěvek přináší popis modelování s použitím super-prvků a doplnění modelu o kontaktní prvky.

EN

The large structure with contact problem appears while modeling the roller bearing. Nowadays the hardware allows solving the large systems of equations. Nevertheless if the number of DOF goes over 105 or 106 and the algorithm necessitates the iterative approach this could be the problem. The effective way of solution can be domain decomposition. The contact problem means that two surfaces touch one another. The contact transmits the pressure force but not tensile force. The paper brings the description of modeling with super-elements and solving the contact problem.

10

Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids

Bresch D., Koko J.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 4

419-429

EN

We present a numerical simulation of two coupled Navier-Stokes flows, using operator-splitting and optimization-based nonoverlapping domain decomposition methods. The model problem consists of two Navier-Stokes fluids coupled, through a common interface, by a nonlinear transmission condition. Numerical experiments are carried out with two coupled fluids; one with an initial linear profile and the other in rest. As expected, the transmission condition generates a recirculation within the fluid in rest.

11

Some remarks on the concept of stream tubes for numerical simulations of complex fluid flows - applications

Clermont J., Normandin M., Radu D.

Control and Cybernetics

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2000

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Vol. 29, no 2

535-553

EN

This paper presents new theoretical elements for numerical simulation of two- and three-dimensional flows, based on the concept of streamlines and domain decomposition. The so-called "stream-tube method", considered previously particularly for flows inolving open streamlines, is extended to general streamline comfigurations. It is shown how local transformation functions may be defined in order to simulate flows of complex fluids, notably those requiring evaluation of particle time history. The specific features (for example : mass conservation, simplicity in handling time-dependent constitutive equations) of the stream-tube methods previously investigated numerically are still preserved in the new formulation. An example of calculations is given in the case of the two-dimensional flow of a Newtonian fluid between two eccentric cylinders where results are found to be consistent with literature data.

12

Domain decomposition in exact controllability of second order hyperbolic systems on 1-d networks

Lagnese J.

Control and Cybernetics

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1999

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Vol. 28, no 3

531-556

EN

This paper is concerned with domain decomposition in exact controllability of a class of linear second order hyperbolic systems on one-dimensional graphs in [R^3] that in particular serve as descriptive models of the dynamics of various multi-link structures consisting of one-dimensional elements, such as networks of Timoshenko beams in [R^3]. We first consider a standard unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the global problem from a given target state. A convergent domain decomposition for the optimality system associated with this problem was recently given by G. Leugering. This decomposition depends on the penalty parameter. On each edge of the graph and at each iteration level the local problem is itself the optimality system associated with an unconstrained optimal control problem in which the cost functional penalizes the deviation of the final state of the particular edge from the target state for that edge. The main purpose of this paper is to show that at each iteration level and on each edge the local optimality system converges as the penalty parameter approaches its limit and that the limit system is a domain decomposition for the problem of norm minimum exact control to the target state.