TSPEED is a library for the high order approximation of the elastodynamics equation on triangular unstructured meshes.
The compilation of the library requires CMake. To install, run
git clone https://github.com/carlomr/tspeed.git
cd tspeed
mkdir build
cmake ..
make
make install
make doc
This will by default install the library in /usr/local/lib and the header files in /usr/local/include/tspeed. To choose a different installation directory, use
cmake .. -DCMAKE_INSTALL_PREFIX=/path/to/dir
Two test are generated as Examples/Lamb and Examples/Wedge. For the source generating them, see Examples/src/Wedge.cpp and Examples/src/Lamb.cpp.
The analysis of elastic and acoustic wave propagation phenomena has been widely studied by mathematicians and scientists since the XIX century. Elastic waves in solids have also been historically studied, though analytic solutions are available only for simple domains and settings. The approximation of the solution to the elastodynamics equation is therefore of critical importance for the analysis of the propagation of seismic waves in complex scenarios.
In recent years, seismological, geophysical and technological advances have allowed for a greater insight into physical seismological events and this has contributed to the growth of the demand for accurate and flexible numerical methods. Those, indeed, permit the comparison between the empirical observations and accurate numerical wave fields in complex domains.
In this work we consider the development of a C++ library for the application of the discontinuous spectral element method on meshes made of simplicial elements to the approximation of the elastodynamic equation. The library was designed with extensibility and ease of use as primary goals. It is made of three layers: the geometrical layer, whose purposes are common to any finite or spectral element code; a functional space layer, where the specific setting of discontinuous methods on triangular elements are implemented and a layer which deals with the approximation of the elastodynamics equation.
Spectral element methods were introduced in the computational fluid
dynamics field (A. T. Patera 1984; Maday and Patera 1989) and they
combine the high order accuracy of spectral methods with the flexibility
and computational feasibility of finite elements methods. They are
strictly related with the h-p version of the finite element method and
they have been extensively used for computational geodynamics in the
last two decades (Komatitsch and Vilotte 1998; Komatitsch and Tromp
2002). Spectral element methods have been introduced and are currently
built on mashes made of tensor product elements (i.e. deformed squares
and cubes), since this is the context in which the extension from one
spatial dimension to
Discontinuous methods were introduced for hyperbolic equations, have been extended to to the elliptic case and developed independently in both environments. The advantages of discontinuous methods lies in the fact that they allow for the accurate approximation of sharp gradients in the solution, that they can be used to develop h-p adaptive strategy and that the computational cost can be distributed without much overhead.
Dubiner, Moshe. 1991. “Spectral methods on triangles and other domains.” Journal of Scientific Computing 6 (4): 345–390.
Hesthaven, Jan S., and Tim Warburton. 2008. Nodal discontinuous Galerkin methods. Texts in Applied Mathematics. Vol. 54. New York: Springer. http://dx.doi.org/10.1007/978-0-387-72067-8.
Karniadakis, George Em, and Spencer J. Sherwin. 2005. Spectral/$hp$ element methods for computational fluid dynamics. Numerical Mathematics and Scientific Computation. Second.. New York: Oxford University Press. http://dx.doi.org/10.1093/acprof:oso/9780198528692.001.0001.
Komatitsch, Dimitri, and Jeroen Tromp. 2002. “Spectral-element simulations of global seismic wave propagation—I. Validation.” Geophysical Journal International 149 (2): 390–412. doi:10.1046/j.1365-246X.2002.01653.x. http://dx.doi.org/10.1046/j.1365-246X.2002.01653.x.
Komatitsch, Dimitri, and Jean-Pierre Vilotte. 1998. “The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures.” Bulletin of the Seismological Society of America 88 (2): 368–392. http://www.bssaonline.org/content/88/2/368.abstract.
Koornwinder, Tom. 1975. “Two-variable analogues of the classical orthogonal polynomials.” In Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), 435–495. New York: Academic Press.
Maday, Y., and A. T. Patera. 1989. “Spectral element methods for the incompressible Navier-Stokes equations.” In State-of-the-art surveys on computational mechanics (A90-47176 21-64), 71–143. American Society of Mechanical Engineers.
Mercerat, Enrique D., Jean–Pierre Vilotte, and Francisco J. Sánchez-Sesma. 2006. “Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids.” Geophysical Journal International 166 (2): 679–698. doi:10.1111/j.1365-246X.2006.03006.x. [<Go to ISI>://WOS:000239004900015](://WOS:000239004900015 "://WOS:000239004900015").
Patera, Anthony T. 1984. “A spectral element method for fluid dynamics: Laminar flow in a channel expansion .” *Journal of Computational Physics *54 (3): 468–488. doi:http://dx.doi.org/10.1016/0021-9991(84)90128-1. http://www.sciencedirect.com/science/article/pii/0021999184901281.
Pelties, Christian, Josep de la Puente, Jean-Paul Ampuero, Gilbert B. Brietzke, and Martin Käser. 2012. “Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes.” Journal of Geophysical Research: Solid Earth 117 (B2). doi:10.1029/2011JB008857. http://dx.doi.org/10.1029/2011JB008857.