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

• Autorzy: Łoś Marcin , Woźniak Maciej , Paszyński Maciej

Identyfikatory

DOI

10.24425/bpasts.2024.149179

• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

partial differential equations; finite element method; Maxwell equations; variational splitting; MRI scan data; magnetic resonance imaging

PL

równania różniczkowe cząstkowe; metoda elementów skończonych; równania Maxwella; podział wariacyjny; rezonans magnetyczny; dane rezonanasu magentycznego

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2024
• Tom: Vol. 72, nr 3
• Strony: art. no. e149179
• Opis fizyczny: Bibliogr. 16 poz., rys., tab.

Twórcy

autor

Łoś Marcin

• AGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland

• https://orcid.org/0000-0002-8426-6345

autor

Woźniak Maciej

• AGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland

• https://orcid.org/0000-0002-5576-5671

autor

Paszyński Maciej

• AGH University of Krakow, al. Mickiewicza 30, 30-059 Krakow, Poland

• https://orcid.org/0000-0001-7766-6052

Bibliografia

• [1] M. Hochbruck, T. Jahnke, and R. Schnaubelt, “Convergence of an ADI splitting for Maxwell’s equations,” Numer. Math., vol. 129, pp. 535–561, 2015.
• [2] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, Isogeometric Analysis: Towards Unification of Computer Aided Design and Finite Element Analysis John Wiley and Sons, 2009.
• [3] D.W. Peaceman and H.H. Rachford Jr., “The numerical solution of parabolic and elliptic differential equations,” J. Soc. Ind. Appl. Math., vol. 3, pp. 28–41, 1955.
• [4] J. Douglas and H. Rachford, "On the numerical solution of heat conduction problems in two and three space variables," Trans. Am. Math. Soc., vol. 82, pp. 421–439, 1956.
• [5] G.I. Marchuk, “Splitting and alternating direction methods,” Handbook of numerical analysis. Elsevier, 1990, vol. 1, pp. 197–462.
• [6] P.N. Vabishchevich, Additive operator-difference schemes: Splitting schemes. Walter de Gruyter, 2013.
• [7] P. Behnoudfar, V.M. Calo, Q. Deng, and P.D. Minev, “A variationally separable splitting for the generalized-𝛼 method for parabolic equations,” Int. J. Numer. Methods Eng., vol. 121, no 5, pp. 828–841, 2020.
• [8] G. Liping, “Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium,” Acta Math. Sci., vol. 32, no. 6, pp. 2341–2368, 2012.
• [9] J. Keating and P. Minev, "A fast algorithm for direct simulation of particulate flows using conforming grids," J. Comput. Phys., vol. 255, pp. 486–501, 2013.
• [10] L. Gao and V. Calo, “Fast isogeometric solvers for explicit dynamics,” Comput. Meth. Appl. Mech. Eng., vol. 274, pp. 19–41, 2014.
• [11] L. Gao and V. Calo, “Preconditioners based on the alternating-direction-implicit algorithm for the 2d steady-state diffusion equation with orthotropic heterogeneous coefficients,” J. Comput. Appl. Math., vol. 273, pp. 274–295, 2015.
• [12] M. Łoś, M. Paszyński, A. Kłusek, and W. Dzwinel, “Application of fast isogeometric l2 projection solver for tumor growth simulations,” Comput. Meth. Appl. Mech. Eng., vol. 316, pp. 1257–1269, 2017.
• [13] M. Łoś, M. Woźniak, M. Paszyński, L. Dalcin, and V. Calo, “Dynamics with matrices possessing Kronecker product structure,” Procedia Comput. Sci., vol. 51, pp. 286–295, 2015.
• [14] M. Łoś et al., “Fast parallel IGA-ADS solver for time-dependent Maxwell’s equations,” Comput. Math. Appl., vol. 151, pp. 36–49, 2023.
• [15] M. Łoś, M. Woźniak, M. Paszyński, A. Lenharth, and K. Pingali, “IGA-ADS: Isogeometric Analysis FEM using ADS solver,” Comput. Phys. Commun., vol. 217, pp. 99–116, 2017.
• [16] K. Kyunogjoo, “Finite element modeling of the radiation and induced heat transfer in the human body,” Ph.D. dissertation, The University of Texas at Austin, 2013.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).