Volume 20, Issue 5
An Element Decomposition Method for the Helmholtz Equation

Gang Wang, Xiangyang Cui & Guangyao Li

Commun. Comput. Phys., 20 (2016), pp. 1258-1282.

Published online: 2018-04

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  • Abstract

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the "pollution error" caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weak form. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

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@Article{CiCP-20-1258, author = {}, title = {An Element Decomposition Method for the Helmholtz Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {5}, pages = {1258--1282}, abstract = {

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the "pollution error" caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weak form. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.110415.240316a}, url = {http://global-sci.org/intro/article_detail/cicp/11189.html} }
TY - JOUR T1 - An Element Decomposition Method for the Helmholtz Equation JO - Communications in Computational Physics VL - 5 SP - 1258 EP - 1282 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.110415.240316a UR - https://global-sci.org/intro/article_detail/cicp/11189.html KW - AB -

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the "pollution error" caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weak form. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

Gang Wang, Xiangyang Cui & Guangyao Li. (2020). An Element Decomposition Method for the Helmholtz Equation. Communications in Computational Physics. 20 (5). 1258-1282. doi:10.4208/cicp.110415.240316a
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