Discontinuous Galerkin method for numerical simulation of dynamic processes in solids

V. A. Miryaha ; A. V. Sannikov ; I. B. Petrov

Matematičeskoe modelirovanie, Tome 27 (2015) no. 3, pp. 96-108

Résumé

The aim of this paper is applying of Discontinuous Galerkin Method to deformation and destruction problems of elastoplastic bodies as well as to hydroelastic problems. Two-sided crack model for destruction, Prandtl–Reuss elastoplastic model, dynamic contact of bodies, wave propagation through coupled elastic-acoustic media for marine seismology, comparison of reflected waves in fluid-filled and infinite thin crack models and underwater objects detection are described. Method was implemented for HPC infrastructure.

Détail

Keywords: Discontinuous Galerkin Method, mechanics of deformable solids, HPC, wave problems in coupled elastic-acoustic media, marine seismology, fluid-filled crack, dynamic contact of bodies
Mots-clés : destruction.

Bibliographie

[1] J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54, Springer, 2008 | DOI

[2] M. Käser, M. Dumbser, “An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes I: The two-dimensional isotropic case with external source terms”, Geophys. J. Int., 166 (2006), 855–877 | DOI

[3] J. Virieux, “SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method”, Geophys., 49:11, November (1984), 1933–1957 | DOI

[4] L. C. Wilcox, G. Stadler, C. Burstedde, O. Ghattas, “A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media”, Journal of Computational Physics, 229 (2010), 9373–9396 | DOI

[5] M. Käser, M. Dumbser, “A Highly Accurate Discontinuous Galerkin Method for Complex Interfaces Between Solids and Moving Fluids”, Geophysics, 73:3 (2008) | DOI

[6] C. E. Castro, M. Käser, G. B. Brietzke, “Seismic waves in heterogeneous material: subcell resolution”, Geophys. J. Int., 182 (2010), 250–264

[7] R. L. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002

[8] E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, 2-nd ed., Springer, 1999

[9] J. A. Trangenstein, Numerical solution of hyperbolic partial differential equations, Cambridge University Press, 2008

[10] E. Sonnendrücker, Numerical methods for hyperbolic systems, Lecture notes (Sommersemester, 2013)

[11] D. A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer, 2012

[12] A. G. Kulikovskii, N. V. Pogorelov, A. Y. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Chapman and Hall/CRC, Boca Raton, 2001

[13] C. Pelties, A. A. Gabriel, J. P. Ampuero, “Verification of an ADER-DG method for complex dynamic rupture problems”, Geosci. Model Dev. Discuss., 6 (2013), 5981–6034 | DOI

[14] M. Hochbruck, T. Pažur, A. Schulz, E. Thawinan, C. Wieners, Efficient time integration for discontinuous Galerkin approximations of linear wave equations, , 2013 http://na.math.kit.edu/download/papers/TI-DG-Wave.pdf

[15] C. Pelties, M. Käser, “Dynamic Rupture Modelling on Unstructured Meshes Using a Discontinuous Galerkin Method”, Computational Methods in Structural Dynamics and Earthquake Engineering, 2011, 3201–3209

[16] M. Käser, C. Pelties, C. E. Castro, H. Djikpesse, M. Prange, “Wave Field Modeling in Exploration Seismology Using the Discontinuous Galerkin Finite Element Method on HPC-infrastructure”, The Leading Edge, 29 (2010), 76–85 | DOI

[17] L. Noels, R. Radovitzky, “An explicit discontinuous Galerkin method for non-linear solid dynamics: Formulation, parallel implementationand scalability properties”, International journal for numerical methods in engineering, 74 (2008), 1393–1420 | DOI

[18] V. A. Miryaha, A. V. Sannikov, “O programmnoi realizatsii parallelnogo algoritma razryvnogo metoda Galerkina dlia chislennogo modelirovaniia volnovykh protsessov v geterogennykh tverdykh deformiruemykh sredakh”, Trudy 56-i nauchnoi konferentsii MFTI, v. 2, 2013, 135

[19] J. R. Shewchuk, “Delaunay Refinement Algorithms for Triangular Mesh Generation”, Computational Geometry: Theory and Applications, 22:1–3, May (2002), 21–74 | DOI

[20] G. Karypis, V. Kumar, “A Fast and Highly Quality Multilevel Scheme for Partitioning Irregular Graphs”, SIAM Journal on Scientific Computing, 20:1 (1999), 359–392 | DOI

[21] S. Tirupathi, J. S. Hesthaven, Y. Liang, M. Parmentier, “Multilevel and Local Timestepping Discontinuous Galerkin Methods for Magma Dynamics”, Geophys. J. Int., 2013, 1–12

[22] A. Demirel, J. Niegemann, K. Busch, M. Hochbruck, “Efficient Multiple Time-Stepping Algorithms of Higher Order”, Journal of Computational Physics, 2014 (to appear)

[23] I. B. Petrov, A. V. Favorskaya, A. V. Sannikov, I. E. Kvasov, “Setochno-kharakteristicheskii metod s ispolzovaniem interpolyatsii vysokikh poryadkov na tetraedralnykh ierarkhicheskikh setkakh s kratnym shagom po vremeni”, Matem. modelirovanie, 25:2 (2013), 42–52

[24] V. I. Golubev, I. B. Petrov, N. I. Khokhlov, “Chislennoe modelirovanie seismicheskoi aktivnosti setochno-kharakteristicheskim metodom”, Zh. vychisl. matem. i matem. fiz., 53:10 (2013), 1709–1720 | DOI

[25] M. V. Muratov, I. B. Petrov, “Raschet volnovykh otklikov ot sistem subvertikalnykh makrotreschin s ispolzovaniem setochno-kharakteristicheskogo metoda”, Matem. modelirovanie, 25:3 (2013), 89–104

[26] I. E. Kvasov, I. B. Petrov, “Chislennoe modelirovanie volnovykh protsessov v geologicheskikh sredakh v zadachakh seismorazvedki s pomoschyu vysokoproizvoditelnykh EVM”, Zhurnal vychislitelnoi matematiki i matem. fiziki, 52:2 (2012), 330–341

[27] M. Frehner, “Krauklis wave initiation in fluid-filled fractures by seismic body waves”, Geophysics, 79:1 (2014), 27–35

[28] P. C. Etter, Underwater Acoustic Modelling and Simulation, 3rd edition, Spon Press, London, 2003

[29] Acad. Press, New York–London, 1964

[30] I. B. Petrov, F. B. Chelnokov, “Chislennoe issledovanie volnovykh protsessov i protsessov razrusheniya v mnogosloinykh pregradakh”, Zh. vychisl. matem. i matem. fiz., 43:10 (2003), 1562–1579

[31] V. I. Golubev, D. P. Grigorievikh, I. B. Petrov, N. I. Khokhlov, “Otsenka seismostoikosti kupolnykh konstruktsii na osnove rezultatov polnovolnovogo modelirovaniia”, International Journal for Computational Civil and Structural Engineering, 10:1 (2014), 65–71

[32] M. A. Schweitzer, Meshfree and Generalized Finite Element Methods, Habilitationsschrift an der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn, 2008

[33] A. V. Gerasimov et al., Teoreticheskie i eksperimentalnye issledovaniia vysokoskorostnogo vzaimodeistviia tel, Izd-vo Tomskogo universiteta, Tomsk, 2007, 177–187

[34] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Issledovanie vliianiia limitera na poriadok tochnosti resheniia razryvnym metodom Galerkina”, Preprinty IPM im. M. V. Keldysha RAN, 2012, 034, 31 pp.

[35] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Impact of different limiting functions on the order of solution obtained by RKDG”, Math. Models and Comp. Simulations, 5:4 (2013), 346–349 | DOI

[36] M. E. Ladonkina, O. A. Nekliudova, V. F. Tishkin, “Limiter povyshennogo poriadka tochnosti dlia razryvnogo metoda Galerkina na treugolnykh setkakh”, Preprinty IPM im. M. V. Keldysha RAN, 2013, 053, 26 pp.