Higherorder triangular spectral element method with optimized cubature points for seismic wavefield modeling
Abstract
The masslumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonalmassmatrix spectral element method (SEM) in nontensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a pnormbased optimization algorithm to obtain higherorder cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Feketebased triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubaturebased TSEM (OTSEM). A complementary study demonstrates themore »
 Authors:
 State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, 100029 (China)
 (China)
 Physics of the Earth, Sciences B, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza (Spain)
 Publication Date:
 OSTI Identifier:
 22622290
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 336; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ACCURACY; ALGORITHMS; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; CONVERGENCE; DISPERSION RELATIONS; DISPERSIONS; EFFICIENCY; INTERPOLATION; MASS; OPTIMIZATION; QUADRATURES; SCANNING ELECTRON MICROSCOPY; TOPOGRAPHY
Citation Formats
Liu, Youshan, Email: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, Email: jwteng@mail.iggcas.ac.cn, Xu, Tao, Email: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, and Badal, José, Email: badal@unizar.es. Higherorder triangular spectral element method with optimized cubature points for seismic wavefield modeling. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.01.069.
Liu, Youshan, Email: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, Email: jwteng@mail.iggcas.ac.cn, Xu, Tao, Email: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, & Badal, José, Email: badal@unizar.es. Higherorder triangular spectral element method with optimized cubature points for seismic wavefield modeling. United States. doi:10.1016/J.JCP.2017.01.069.
Liu, Youshan, Email: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, Email: jwteng@mail.iggcas.ac.cn, Xu, Tao, Email: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, and Badal, José, Email: badal@unizar.es. Mon .
"Higherorder triangular spectral element method with optimized cubature points for seismic wavefield modeling". United States.
doi:10.1016/J.JCP.2017.01.069.
@article{osti_22622290,
title = {Higherorder triangular spectral element method with optimized cubature points for seismic wavefield modeling},
author = {Liu, Youshan, Email: ysliu@mail.iggcas.ac.cn and Teng, Jiwen, Email: jwteng@mail.iggcas.ac.cn and Xu, Tao, Email: xutao@mail.iggcas.ac.cn and CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101 and Badal, José, Email: badal@unizar.es},
abstractNote = {The masslumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonalmassmatrix spectral element method (SEM) in nontensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a pnormbased optimization algorithm to obtain higherorder cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Feketebased triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubaturebased TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a halfspace model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Feketebased TSEM. In particular, the accuracy of the 7thorder OTSEM is even higher than that of the 14thorder Feketebased TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a nonflat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Feketebased TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required.  Highlights: • Higherorder cubature points for degrees 7 to 9 are developed. • The effects of quadrature rule on the mass and stiffness matrices has been conducted. • The cubature points have always positive integration weights. • Freeing from the inversion of a wide bandwidth mass matrix. • The accuracy of the TSEM has been improved in about one order of magnitude.},
doi = {10.1016/J.JCP.2017.01.069},
journal = {Journal of Computational Physics},
number = ,
volume = 336,
place = {United States},
year = {Mon May 01 00:00:00 EDT 2017},
month = {Mon May 01 00:00:00 EDT 2017}
}

We implement a highorder finiteelement application, which performs the numerical simulation of seismic wave propagation resulting for instance from earthquakes at the scale of a continent or from active seismic acquisition experiments in the oil industry, on a large cluster of NVIDIA Tesla graphics cards using the CUDA programming environment and nonblocking message passing based on MPI. Contrary to many finiteelement implementations, ours is implemented successfully in single precision, maximizing the performance of current generation GPUs. We discuss the implementation and optimization of the code and compare it to an existing very optimized implementation in C language and MPI onmore »

Asynchronous communication in spectralelement and discontinuous Galerkin methods for atmospheric dynamics – a case study using the HighOrder Methods Modeling Environment (HOMMEhomme_dg_branch)
The scalability of computational applications on current and nextgeneration supercomputers is increasingly limited by the cost of interprocess communication. We implement nonblocking asynchronous communication in the HighOrder Methods Modeling Environment for the time integration of the hydrostatic fluid equations using both the spectralelement and discontinuous Galerkin methods. This allows the overlap of computation with communication, effectively hiding some of the costs of communication. A novel detail about our approach is that it provides some data movement to be performed during the asynchronous communication even in the absence of other computations. This method produces significant performance and scalability gains in largescalemore »Cited by 1 
Direct Vlasov solvers with highorder spectral element method.
This paper presents the development of parallel direct Vlasov solvers using the Spectral Element Method (SEM). Instead of the standard ParticleInCell (PIC) approach for kinetic space plasma simulation, i.e. solving the VlasovMaxwell equations, the direct method has been used in this paper. There are several benefits to solve the Vlasov equation directly, such as avoiding noise associated with the finite number of particles and the capability to capture the fine structure in the plasma, etc. The most challenging part of direct Vlasov solver comes from high dimension, as the computational cost increases as N{sup 2d}, where d is the dimensionmore » 
Variational multiscale turbulence modelling in a high order spectral element method
In the variational multiscale (VMS) approach to large eddy simulation (LES), the governing equations are projected onto an a priori scale partitioning of the solution space. This gives an alternative framework for designing and analyzing turbulence models. We describe the implementation of the VMS LES methodology in a high order spectral element method with a nodal basis, and discuss the properties of the proposed scale partitioning. The spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow. The performance of the turbulence model is then assessed by several coarse grid simulations ofmore » 
Boundary elementimage method approach to seismic modeling
The boundary elementimage method is proposed as a method for generating synthetic seismograms in a system of piecewise homogeneous layers. In contrast with many other methods (such as finite difference) in which the boundary conditions are represented simply by a change of material properties, the present approach uses the boundary conditions explicitly in order to develop a system of integral equations. As in the indirect boundary element method, the scattered seismic waves are presumed to result from the action of fictitious sources. However, as in the image method, these fictitious sources are located outside of the domain through which themore »