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

• Autorzy: Łoś Marcin , Behnoudfar Pouria , Dobija Mateusz , Paszyński Maciej

Identyfikatory

DOI

10.24425/bpasts.2022.141985

• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

conditional stability; P-wave propagation problems; elastic wave propagation problems; linear computational cost; time-dependent simulations

PL

stabilność warunkowa; problemy z propagacją fali P; problemy z propagacją fali sprężystej; koszt obliczeniowy liniowy; symulacje zależne od czasu

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2022
• Tom: Vol. 70, nr 5
• Strony: art. no. e141985
• Opis fizyczny: Bibliogr. 41 poz., rys.

Twórcy

autor

Łoś Marcin

• AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, al. Mickiewicza 30, 30-059 Krakow, Poland

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

autor

Behnoudfar Pouria

• Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, Perth, Western Australia

• https://orcid.org/0000-0003-4301-3728

autor

Dobija Mateusz

• Jagiellonian University, Faculty of Astronomy, Physics and Applied Computer Science, Kraków, Poland

autor

Paszyński Maciej

• AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, al. Mickiewicza 30, 30-059 Krakow, Poland

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

Bibliografia

• [1] W. Lowrie, The Fundamentals of Geophysics. Cambridge University Press, 1997.
• [2] C.M.R. Fowler, The solid Earth: an introduction to global geophysics. Cambridge University Press, 2005.
• [3] D. Komatitsch, J. Ritsemaand, and J. Tromp, “The Spectral-Element Method, Beowulf Computing, and Global Seismology,” Science, vol. 298, no. 5599, pp. 1737–1742, 2002.
• [4] S.-J. Lee, C. How-Wei, Q. Liu, D. Komatitsch, B.-S. Huang, and J. Tromp, “Three-Dimensional Simulations of Seismic-Wave Propagation in the Taipei Basin with Realistic Topography Based upon the Spectral-Element Method,” Bull. Seismol. Soc. Am., vol. 98, no. 1, pp. 253–264, 2002.
• [5] M. Gade, and S.T.G.Raghukanth, “Seismic ground motion in micropolar elastic half-space,” Appl. Math. Modell., vol. 39, no. 23–24, pp. 7244–7265, 2015.
• [6] J.A. Cottrell, T.J.R. Hughes, and Y. Bazilevs, Isogeometric Analysis: Toward Unification of CAD and FEA. JohnWiley and Sons, 2009.
• [7] M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “High-performance computing of wind turbine aerodynamics using isogeometric analysis,” Comput. Fluids, vol. 49, no. 1, pp. 93–100, 2011.
• [8] K. Chang, T.J.R. Hughes, and V.M. Calo, “Isogeometric variational multiscale large-eddy simulation of fully-developed turbulent flow over a wavy wall,” Comput. Fluids, vol. 68, pp. 94–104, 2022.
• [9] L. Dedè,T.J.R. Hughes, S. Lipton, and V.M. Calo, “Structural topology optimization with isogeometric analysis in a phase field approach,” USNCTAM2010, 16th US National Congree of Theoretical and Applied Mechanics, 2010.
• [10] L. Dedè, M. J. Borden, and T.J.R. Hughes, “Isogeometric analysis for topology optimization with a phase field model,” ICES REPORT 11-29, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, 2011.
• [11] H. Gómez, V.M. Calo, Y. Bazilevs, and T.J.R. Hughes, “Isogeometric analysis of the Cahn-Hilliard phase-field model,” Comput. Meth. Appl. Mech. Eng., vol. 197, pp. 4333–4352, 2008.
• [12] H. Gómez, T.J.R. Hughes, X. Nogueira, and V.M. Calo, “Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations,” Comput. Meth. Appl. Mech. Eng., vol. 199, pp. 1828–1840, 2010.
• [13] R. Duddu, L. Lavier, T.J.R. Hughes, and V.M. Calo, “A finite strain Eulerian formulation for compressible and nearly incompressible hyper-elasticity using high-order NURBS elements,” Int. J. Numer. Meth. Eng., vol. 89, no. 6, pp. 762–785, 2012.
• [14] S. Hossain, S.F.A. Hossainy, Y. Bazilevs, V.M. Calo, and T.J.R. Hughes, “Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls,” Comput. Mech., vol. 49, pp. 213–242, 2011.
• [15] Y. Bazilevs, V.M. Calo, Y. Zhang, and T.J.R. Hughes: “Isogeometric fluid-structure interaction analysis with applications to arterial blood flow,” Comput. Mech., vol. 38, pp. 310–322, 2006.
• [16] Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, and G. Scovazzi, “Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows,” Comput. Meth. Appl. Mech. Eng., vol. 197, pp. 173–201, 2007.
• [17] V.M. Calo, N. Brasher, Y. Bazilevs, and T.J.R. Hughes, “Multiphysics Model for Blood Flow and Drug Transport with Application to Patient-Specific Coronary Artery Flow,” Comput. Mech., vol. 43, no. 1, pp. 161–177, 2008.
• [18] 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.
• [19] M. Łoś, A. Kłusek, M. Amber Hassam, K. Pingali, W. Dzwinel, and M. Paszyński, “Parallel fast isogeometric L2 projection solver with GALOIS system for 3D tumor growth simulations,” Comput. Meth. Appl. Mech. Eng., vol. 343, pp. 1–22, 2019.
• [20] M. Łoś, M. Woźniak, I. Muga, and M. Paszyński, “Threedimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 69, no. 4, pp. 1–8, 2021.
• [21] Y. Bazilevs, L. Beirao da Veiga, J.A. Cottrell, T.J.R. Hughes, and G. Sangalli, “Isogeometric Analysis: Approximation, stability and error estimates for h-refined meshes,” Math. Models Meth. Appl. Sci., vol. 16, no. 7, pp. 1031–1090, 2006.
• [22] 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.
• [23] J.L. Guermond and P. Minev, “A new class of fractional step techniques for the incompressible Navier-Stokes equations using direction splitting,” C. R. Math., vol. 348, no. 9–10, pp. 581–585, 2010.
• [24] J.L. Guermond, P. Minev, and J. Shen, “An overview of projection methods for incompressible flows,” Comput. Meth. Appl. Mech. Eng., vol. 195, pp. 6011—6054, 2006.
• [25] L. Gao and V.M. Calo, “Fast isogeometric solvers for explicit dynamics,” Comput. Meth. Appl. Mech. Eng., vol. 274, no. 1, pp. 19–41, 2014.
• [26] L. Gao and V.M. 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, no. 1, pp. 274–295, 2015.
• [27] L. Gao, Kronecker products on preconditioning. PhD. Thesis, King Abdullah University of Science and Technology, 2013.
• [28] G. Gurgul and M. Paszyński, “Object-oriented implementation of the Alternating Directions Implicit Solver for Isogeometric Analysis,” Adv. Eng. Software, vol. 128, pp. 187–220, 2019.
• [29] 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. https://github.com/marcinlos/iga-ads.
• [30] M. Łoś and M. Paszyński, “Applications of Alternating Direction Solver for simulations of time-dependent problems,” Comput. Sci., vol. 18, no. 2, pp. 117–128, 2017.
• [31] R.I. Fernandes and G. Fairweather, “An Alternating Direction Galerkin Method for a Class of Second-Order Hyperbolic Equations in Two Space Variables,” SIAM J. Numer. Anal., vol. 28, no.5, pp. 1265–1281, 1991.
• [32] H. Lim, S. Kim, and J. Douglas Jr., “Numerical methods for viscous and nonviscous wave equations,” Appl. Numer. Math., vol. 57, no. 2, pp. 194–212, 2007.
• [33] G. Gurgul, M. Łoś, and M. Paszyński, “Linear computational cost implicit solver for parabolic problems,” Comput. Sci., vol. 21, no. 3, pp. 335–352, 2020.
• [34] M. Paszyński, “Convergence of iterative solvers for non-linear Step-and-Flash Imprint Lithography Simulations,” Comput. Sci., vol. 12, pp. 63–83, 2011.
• [35] M. Paszyński, T. Jurczyk, and D. Pardo, “Multi-frontal solver for simulations of linear elasticity coupled with acoustics,” Comput. Sci., vol. 12, pp. 85–102, 2011.
• [36] M. Łoś, P. Behnoudfar, M. Paszyński, and V. Calo, “Fast isogeometric solvers for hyperbolic wave propagation problems,” Comput. Math. Appl., vol. 80, no. 1, pp. 109–120, 2020.
• [37] N.M. Newmark, “A method of computation for structural dynamics,” J. Eng. Mech., vol. 85 (EM3), pp. 67–94, 1959.
• [38] A. Graham, Kronecker products, matrix calculus with applications, Dover, 2018.
• [39] M. Paszyński, Fast solvers for mesh-based computations. Taylor & Francis, CRC Press, 2016.
• [40] T. Kapitaniak, M. Sofer, B. Błachowski, W. Sochacki, and S. Garus, “Vibrations, mechanical waves, and propagation of heat in physical systems,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 70, p. e140149, 2022.
• [41] S. Garus, W. Sochacki, M. Kubanek, and M. Nabiałek, “Minimizing the number of layers of the quasi one-dimensional phononic structures,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 70, p. e139394, 2022.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).