Abstract
Methods based on Discontinuous Finite Element approximation (DG FEM) are basically well-adapted to specifics of wave propagation problems in complex media, due to their numerical accuracy and flexibility. However, they still lack of computational efficiency, by reason of the high number of degrees of freedom required for simulations.
The Trefftz-DG solution methodology investigated in this work is based on a formulation which is set only at the boundaries of the mesh. It is a consequence of the choice of test functions that are local solutions of the problem. It owns the important feature of involving a space-time approximation which requires using elements defined in the space-time domain.
Herein, we address the Trefftz-DG solution of the Elastodynamic System. We establish its well-posedness which is based on mesh-dependent norms. It is worth noting that we employ basis functions which are space-time polynomial. Some numerical experiments illustrate the proper functioning of the method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
E. Trefftz, Ein Gegenstuck zum Ritzschen Verfahren, in Proceedings of the 2nd International Congress of Applied Mechanics, Zurich (1926), pp. 131–137
J. Jirousek, Basis for development of large finite elements locally satisfying all field equations. Comput. Methods Appl. Mech. Eng. 14(1), 65–92 (1978)
O.C. Zienkiewic, Trefftz type approximation and the generalized finite element method - history and development. Comput. Assist. Mech. Eng. Sci. 4, 305–316 (1997)
C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192(11), 1389–1419 (2003)
R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Methods Eng. 66(5), 796–815 (2006)
G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225(2), 1961–1984 (2007)
Z. Badics, Trefftz-discontinuous Galerkin and finite element multi-solver technique for modeling time-harmonic EM problems with high-conductivity regions. IEEE Trans. Magn. 50(2), 401–404 (2014)
R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49(1), 264–284 (2011)
A. Moiola, R. Hiptmair, I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62(5), 809–837 (2011)
R. Hiptmair, A. Moiola, I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82(281), 247–268 (2013)
S. Petersen, C. Farhat, R. Tezaur, A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain. Int. J. Numer. Methods Eng. 78(3), 275–295 (2009)
F. Kretzschmar, S.M. Schnepp, I. Tsukerman, T. Weiland, Discontinuous Galerkin methods with Trefftz approximations. J. Comput. Appl. Math. 270, 211–222 (2014)
H. Egger, F. Kretzschmar, S.M. Schnepp, T. Weiland, A space-time discontinuous Galerkin Trefftz method for time dependent Maxwell’s equations. SIAM J. Sci. Comput. 37(5), B689–B711 (2015)
F. Kretzschmar, A. Moiola, I. Perugia, S. Schnepp, A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems. IMA J. Numer. Anal. 36(4), 1599–1635 (2015)
D. Wang, R. Tezaur, C. Farhat, A hybrid discontinuous in space and time Galerkin method for wave propagation problems. Int. J. Numer. Methods Eng. 99(4), 263–289 (2014)
L. Banjai, E. Georgoulis, O. Lijoka, A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation. SIAM J. Numer. Anal. 55(1), 63–86 (2017)
A. Moiola, I. Perugia, A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math. 138(2), 389–435 (2018)
P. Le Tallec, Modélisation et calcul des milieux continus (Editions Ecole Polytechnique, Palaiseau, 2009)
J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer Science & Business Media, Berlin, 2007)
H. Barucq, H. Calandra, J. Diaz, E. Shishenina, Space-time Trefftz - discontinuous Galerkin approximation for elasto-acoustics. Inria Bordeaux Sud-Ouest, LMAP CNRS, Total EP, RR-9104 (2017)
A. Maciag, J. Wauer, Solution of the two-dimensional wave equation by using wave polynomials. J. Eng. Math. 51(4), 339–350 (2005)
Acknowledgements
This work is supported by the Inria—Total S.A. strategic action “Depth Imaging Partnership” (http://dip.inria.fr).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Barucq, H., Calandra, H., Diaz, J., Shishenina, E. (2019). Trefftz-Discontinuous Galerkin Approach for Solving Elastodynamic Problem. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-96415-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96414-0
Online ISBN: 978-3-319-96415-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)