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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

Overview of adaptive and low-rank approximation algorithms for modeling influence of electromagnetic waves generated by cellphone antenna on human head

Głut Barbara, Paszyński Maciej

Computer Science

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2021

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T. 22 (4)

433--461

EN

This paper presents an overview of formulations and algorithms that are dedicated to modeling the influence of electromagnetic waves on the human head. We start from h adaptive approximation of a three-dimensional MRI scan of the human head. Next, we solve the time-harmonic Maxwell equations with a 1.8 GHz cellphone antenna. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation modeling the heat generated by EM waves inside the head. We propose an adaptive algorithm mixed with time-stepping iterations where we simultaneously refine the computational mesh, solve the Maxwell and Pennes equations, and iterate the time steps. We employ the sparse Gaussian elimination algorithm with the low-rank compres-sion of the off-diagonal matrix blocks for the factorization of the matrices. We conclude with the statement that 15 minutes of talking with a 1.8 GHz antenna with one watt of power results in increased brain tissue temperatures (up to 38.4◦C).

4

Hypergraph grammar based multi-thread multi-frontal direct solver with Galois scheduler

Jopek Konrad, Paszyński Maciej, Paszyńska Anna, Hassan Muhammad Amber, Pingali Keshav

Computer Science

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2019

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Vol. 20 (1)

27--55

EN

In this paper, we analyze two-dimensional grids with point and edge singularities in order to develop an eficient parallel hypergraph grammar-based multi- frontal direct solver algorithm. We express these grids by a hypergraph. For these meshes, we define a sequence of hypergraph grammar productions expressing the construction of frontal matrices, eliminating fully assembled nodes, merging the resulting Schur complements, and repeating the process of elimination and merging until a single frontal matrix remains. The dependency relationship between hypergraph grammar productions is analyzed, and a dependency graph is plotted (which is equivalent to the elimination tree of a multi- frontal solver algorithm). We utilize a classical multi-frontal solver algorithm; the hypergraph grammar productions allow us to construct an eficient elimination tree based on the graph representation of the computational mesh (not the global matrix itself). The hypergraph grammar productions are assigned to nodes on a dependency graph, and they are implemented as tasks in the GALOIS parallel environment and scheduled according to the developed dependency graph over the shared memory parallel machine. We show that our hypergraph grammar-based solver outperforms the parallel MUMPS solver.

5

Framework for topographic mesh generation and its application to the pollution simulations in Kraków area

Podsiadło Krzysztof, Serra Albert Oliver, Paszyński Maciej

Computer Methods in Materials Science

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2019

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Vol. 19, No. 1

21--28

EN

Air pollution is receiving a lot of interest nowadays. It is visible especially in the Kraków area, as this is one of the most polluted cities in Europe. People living there are more and more aware of the problem, what causes raising of various movements and NGOs that are trying to improve air quality. Unfortunately, this is not as simple as people usually think: air pollution grows because of multiple factors including traffic, climate, heating of buildings in winter, city’s architecture, etc. In this paper, we simulate and predict pollution with high resolution, as air quality can vary significantly over a distance of even a few hundreds of meters. Air quality simulation is a multidisplinary endeavor, comprising physical models (meteorological and chemistry) as well as numerical methods (geometry discretization, time and space discretization, etc.). It has been found that there is no proper method for automatic terrain mesh generation, so an algorithm for doing it is presented in this document as its significant part.

PL

Problem zanieczyszczenia powietrza staje się coraz bardziej zauważalny w czasach dzisiejszych. Dotyczy to szczególnie okolic Krakowa, będącego jednym z najbardziej zanieczyszczonych miast w Europie. Ludzie mieszkający w rejonie Krakowa stali się bardziej świadomi problemu zanieczyszczeń, co zmotywowało do powstawania różnych organizacji publicznych oraz zainicjowało różne działania władz na szczeblu lokalnym oraz globalnym w celu polepszenia jakości powietrza. Niestety problem powstawania i propagacji zanieczyszczeń jest bardzo skomplikowany, i łączy w sobie wiele czynników takich jak transport miejski, lokalny klimat, problem ogrzewania budynków w okresie zimowym, struktura architektury miejskiej, itp. W artykule tym zajmujemy się symulacjami i predykcją zanieczyszczeń z wysoką rozdzielczością, z uwagi na fakt, iż jakość powietrza może się istotnie zmieniać z jednego miejsca do drugiego na odległości kilkuset metrów. Symulacje zanieczyszczeń to zadanie wielodyscyplinarne, łączące ze sobą modele fizyczne (meteorologiczne i chemiczne), oraz modele numeryczne (dyskretyzacja geometrii, dyskretyzacja czasowoprzestrzenna). W szczególności zajęliśmy się skonstruowaniem nowego algorytmu automatycznej generacji topograficznych siatek obliczeniowych i jego wykorzystaniem w zagadnieniach symulacji zanieczyszczeń.

6

Framework for fast simulations of material science phenomena with Cahn-Hilliard equations

Gurgul Grzegorz, Paszyński Maciej, Szeliga Danuta, Puzyrev Vladimir

Computer Methods in Materials Science

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2019

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Vol. 19, No. 1

12--20

EN

This paper presents the framework for executing Cahn-Hilliard simulations through a web interface which is based on a popular continuous integration tool called Jenkins. This setup allows launching computations from any machine, in the client mode, and without the need to sustain a connection to the computational environment. It also isolates the researcher from the complexity of the underlying infrastructure and reduces the number of steps necessary to perform the simulations. Moreover, the results of the computations are automatically post-processed and stored upon job completion for future retrieval in the form of raw data, a sequence of bitmaps, as well as a video sequence illustrating changes in the material structure over time. The Cahn-Hilliard equations are parameterized with mobility and chemical potential function, allowing for several numerical applications. The discretization is performed with Isogeometric finite element method, and it is parameterized with the number of time steps, the time step size, the mesh size, and the order of the B-spline basis functions using for the approximation of the solution. The interface is linked with the alternating direction semi-implicit solver, resulting in a linear computational cost of the simulation.

PL

W niniejszej pracy przedstawiamy framework służący do przeprowadzania symulacji opartych o wzory Cahna-Hilliarda poprzez wygodny interfejs webowy. Wykorzystujemy do tego popularne narzędzie służące do ciągłej integracji o nazwie Jenkins. Tego typu konfiguracja pozwala na uruchamianie obliczeń z dowolnej maszyny w trybie klienckim bez konieczności utrzymywania połączenia do środowiska obliczeniowego. Dzięki temu naukowiec wykonujący obliczenia jest odizolowany od skomplikowanej infrastruktury obliczeniowej, a uruchomienie symulacji wymaga mniejszej liczby czynności. Ponadto, wyniki symulacji są automatycznie przetwarzane i prezentowane w formie tabularycznej, sekwencji bitmap oraz filmu, który odzwierciedla zmiany zachodzące w strukturze badanego materiału w czasie. Równania Cahna-Hilliarda są parametryzowane poprzez funkcje mobilności i potencjału chemicznego, co pozwala na przeprowadzanie symulacji wybranych zjawisk dla wielu materiałów. Dyskretyzacja jest wykonywana z wykorzystaniem Izogeometrycznej Metody Elementów Skończonych i jest uzależniona od liczby i rozmiaru kroków czasowych, wielkości siatki oraz rzędu krzywych B-sklejanych, użytych do aproksymacji rozwiązania. Interfejs, o którym mowa, konfiguruje solwer zmienno-kierunkowy z dyskretyzacją czasową schematem wprost, co skutkuje liniowym kosztem obliczeniowym symulacji.

7

Using the Hypergraph Grammar for generation of quasi optimal element partition trees in two dimensions

Ryzner Jakub, Paszyński Maciej, Paszyńska Anna

Computer Methods in Materials Science

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2018

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Vol. 18, No. 1

29--40

EN

The paper presents the graph grammar model of Finite Element Method allowing for speeding up performed numerical simulations. In the presented approach, the finite element mesh operations are performed together with operations generating so-called element partition tree. The element partition tree sets the ordering of matrix operations performed by solver in order to solve the computational problem. The quality of element partition tree influences the computational time of the solver. Our method allows for generation good quality element partition trees for h-adaptive Finite Element Method. The paper is concluded with numerical results confirming the quality of generated element partition trees.