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1

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Łoś Marcin, Woźniak Maciej, Paszyński Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2024

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Vol. 72, nr 3

art. no. e149179

EN

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originally employed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variational splitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structure of the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally used for fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regular model parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with non-regular material data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element method simulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

2

ADI-based, conditionally stable schemes for seismic P-wave and elastic wave propagation problems

Łoś Marcin, Behnoudfar Pouria, Dobija Mateusz, Paszyński Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 5

art. no. e141985

EN

The modeling of P-waves has essential applications in seismology. This is because the detection of the P-waves is the first warning sign of the incoming earthquake. Thus, P-wave detection is an important part of an earthquake monitoring system. In this paper, we introduce a linear computational cost simulator for three-dimensional simulations of P-waves. We also generalize our formulations and derivation for elastic wave propagation problems. We use the alternating direction method with isogeometric finite elements to simulate seismic P-wave and elastic propagation problems. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along the particular coordinate axis in the sub-steps. We show that the resulting problem matrix can be represented as a multiplication of three multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. The resulting system of linear equations can be factorized in linear O (N) computational cost in every time step of the semi-implicit method. We use our method to simulate P-wave and elastic wave propagation problems. We derive the condition for the stability for seismic waves; namely, we show that the method is stable when τ < C min{ hx,hy,hz}, where C is a constant that depends on the PDE problem and also on the degree of splines used for the spatial approximation. We conclude our presentation with numerical results for seismic P-wave and elastic wave propagation problems.

3

Three-dimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver

Łoś Marcin, Woźniak Maciej, Muga Ignacio, Paszynski Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 4

art. no. e137125

EN

In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pathogens in the air. We can represent the concentration of pathogens in the air by “contamination” propagating from the source, by the mechanisms of advection (representing air movement) and diffusion (representing the spontaneous propagation of pathogen particles in the air). The three-dimensional time-dependent advection-diffusion equation is difficult to simulate due to the high computational cost and instabilities of the numerical methods. In this paper, we present alternating directions implicit isogeometric analysis simulations of the three-dimensional advection-diffusion equations. We introduce three intermediate time steps, where in the differential operator, we separate the derivatives concerning particular spatial directions. We provide a mathematical analysis of the numerical stability of the method. We show well-posedness of each time step formulation, under the assumption of a particular time step size. We utilize the tensor products of one-dimensional B-spline basis functions over the three-dimensional cube shape domain for the spatial discretization. The alternating direction solver is implemented in C++ and parallelized using the GALOIS framework for multi-core processors. We run the simulations within 120 minutes on a laptop equipped with i7 6700 Q processor 2.6 GHz (8 cores with HT) and 16 GB of RAM.

4

Linear computational cost implicit solver for parabolic problems

Gurgul Grzegorz, Łoś Marcin, Paszynski Maciej, Calo Victor

Computer Science

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2020

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T. 21 (3)

335–352

EN

In this paper, we use the alternating direction method for isogeometric finite elements to simulate transient problems. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit time-marching method to fully discretize the problem. We introduce intermediate time-steps and separate our differential operator into a summation of the blocks that act along a particular coordinate axis in the intermediate time-steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of these algebraic transformations, we get a system of linear equations that can be factorized in a linear O(N) computational cost at every time-step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

5

Handling insensitivity in multi-physics inverse problems using a complex evolutionary strategy

Sawicki Jakub, Smołka Maciej, Łoś Marcin, Schaefer Robert

Computer Methods in Materials Science

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2019

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Vol. 19, No. 1

2--11

EN

In this paper we present a complex strategy for the solution of ill posed, in-verse problems formulated as multiobjective global optimization ones. The strategy is capable of identifying the shape of objective insensitivity regions around connected components of Pareto set. The goal is reached in two phases. In the first, global one, the connected components of the Pareto set are localized and separated in course of the multi-deme, hierarchic memetic strategy HMS. In the second, local phase, the random sample uniformly spread over each Pareto component and its close neighborhood is obtained in the specially profiled evolutionary process using multiwinner selection. Finally, each local sample forms a base for the local approximation of a dominance function. Insensitivity region surrounding each connected component of the Pareto set is estimated by a sufficiently low level set of this approximation. Capabilities of the whole procedure was verified using specially-designed two-criterion benchmarks.

PL

Artykuł prezentuje złożoną strategię rozwiązywania źle postawionych problemów odwrotnych sformułowanych jako wielokryterialne zadania optymalizacji globalnej. Opisana strategia umożliwia identyfikację obszarów niewrażliwości funkcji celu wokół spójnych składowych zbioru Pareto. Cel jest osiągany w dwu etapach. W pierwszym z nich — globalnym — składowe spójne zbioru Pareto są lokalizowane i separowane przy pomocy wielopopulacyjnej hierarchicznej strategii memetycznej HMS. W etapie drugim — lokalnym — przy użyciu specjalnie sprofilowanego procesu ewolucyjnego wykorzystującego operator selekcji wyborczej z wieloma zwycięzcami produkowana jest losowa próbka rozłożona jednostajnie na każdej składowej i jej bliskim otoczeniu. Finalnie każda lokalna próbka jest użyta jako baza do zbudowania lokalnej aproksymacji funkcji dominacji. Zbiory poziomicowe tej aproksymacji dla odpowiednio niskich poziomów stanowią przybliżenie zbiorów niewrażliwości wokół składowych spójnych. Możliwości strategii zostały zweryfikowane przy użyciu specjalnie zaprojektowanych dwukryterialnych funkcji testowych.