# Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

## Abstract

The mass-lumped 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 diagonal-mass-matrix spectral element method (SEM) in non-tensor 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 p-norm-based optimization algorithm to obtain higher-order 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 Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based 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, E-mail: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, E-mail: jwteng@mail.iggcas.ac.cn, Xu, Tao, E-mail: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, and Badal, José, E-mail: badal@unizar.es.
```*Higher-order 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, E-mail: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, E-mail: jwteng@mail.iggcas.ac.cn, Xu, Tao, E-mail: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, & Badal, José, E-mail: badal@unizar.es.
```*Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling*. United States. doi:10.1016/J.JCP.2017.01.069.

```
Liu, Youshan, E-mail: ysliu@mail.iggcas.ac.cn, Teng, Jiwen, E-mail: jwteng@mail.iggcas.ac.cn, Xu, Tao, E-mail: xutao@mail.iggcas.ac.cn, CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101, and Badal, José, E-mail: badal@unizar.es. Mon .
"Higher-order 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 = {Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling},

author = {Liu, Youshan, E-mail: ysliu@mail.iggcas.ac.cn and Teng, Jiwen, E-mail: jwteng@mail.iggcas.ac.cn and Xu, Tao, E-mail: xutao@mail.iggcas.ac.cn and CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing, 100101 and Badal, José, E-mail: badal@unizar.es},

abstractNote = {The mass-lumped 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 diagonal-mass-matrix spectral element method (SEM) in non-tensor 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 p-norm-based optimization algorithm to obtain higher-order 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 Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based 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 non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required. - Highlights: • Higher-order 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}

}