用混合复变元无伽辽金法分析三维亥姆霍兹方程

IF 1.4 Q4 MATERIALS SCIENCE, MULTIDISCIPLINARY International Journal of Computational Materials Science and Engineering Pub Date : 2022-12-19 DOI:10.1142/s2047684123500057
Heng Cheng, Y. Liu, Dongqiong Liang
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引用次数: 2

摘要

在这项研究中,我们提出了求解三维亥姆霍兹方程的混合复无变元伽辽金(HCVEFG)方法。将分维法(DSM)引入到相应的控制方程中,对三维亥姆霍兹方程的问题域进行分维,得到一系列二维形式。对于每一个二维问题,采用改进的复变量移动最小二乘(ICVMLS)近似得到其形状函数,并采用惩罚法施加必要的边界条件,从而采用相应的伽辽金弱形式推导出二维问题的离散方程。利用有限差分法(FDM)在分维方向上对这两个方程进行耦合,从而得到三维亥姆霍兹方程的最终数值解。第四节给出了相对误差,并对其收敛性进行了数值分析。这些算例的数值结果表明,采用HCVEFG方法比采用改进的无单元伽辽金(IEFG)方法可以大大提高计算速度。
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Analyzing 3D Helmholtz equations by using the hybrid complex variable element-free Galerkin method
In this study, we present the hybrid complex variable element-free Galerkin (HCVEFG) method for solving 3D Helmholtz equations. The dimension splitting method (DSM) will be introduced into the corresponding governing equation, thus a series of 2D forms can be obtained by splitting the problem domain of 3D Helmholtz equation. For every 2D problem, the shape function can be obtained by using the improved complex variable moving least-squares (ICVMLS) approximation, and the essential boundary condition can be imposed by using the penalty method, thus the discretized equations of 2D problems can be derived by using the corresponding Galerkin weak form. These equations can be coupled by using the finite difference method (FDM) in the dimension splitting direction, thus final formulae of the numerical solution for 3D Helmholtz equation can be obtained. In Sec. 4, the relative errors are given, and the convergence is analyzed numerically. The numerical result of these examples illustrates that the calculation speed can be improved greatly when the HCVEFG method is used rather than the improved element-free Galerkin (IEFG) method.
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CiteScore
2.10
自引率
15.40%
发文量
27
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