Explicit Evaluation of Hypersingular Boundary Integral Equation for 3-D Helmholtz Equation Discretized with Constant Triangular Element
Explicit Evaluation of Hypersingular Boundary Integral Equation for 3-D Helmholtz Equation Discretized with Constant Triangular Element
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DOI：
10.1299/jcst.4.194
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Toshiro Matsumoto;C. Zheng;S. Harada;Toru Takahashi
中科院分区：
文献类型：
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作者：
Toshiro Matsumoto;C. Zheng;S. Harada;Toru Takahashi
众所周知，当基于常规边界积分方程（CBIE）时，在相关内部问题的本征中违反了由Helmholtz方程所控制的外部声学问题的解决方案。治疗以解决它。为了解决这个问题，使用CBIE及其正常衍生物（NDBIE）的线性组合的Burton-Miller配方作为有效且有效的公式出现，如果耦合常数的想象部分，则证明可以为所有频率产生独特的解决方案两个方程式中的非零。实施Burton-Miller公式的最困难的部分是NDBIE是一种超源类型，并且通常通过使用Laplace方程的基本解决方案进行正规化。但是，文献中的各种正规化程序都会引起积分，这些积分仍然很困难和/或一般耗时。但是，当使用恒定的三角元素来离散边界时，可以在没有任何困难的情况下明确评估所有强度的和超级积分的积分，并且比任何其他单身性辅助技术都更有效。因此，在本文中，这些奇异积分被严格评估三角形常数元素作为发散积分的有限部分，通过取消明确显示在限制过程中的发散项。通过一些数值测试示例也证明了公式的正确性。
It is well known that the solution of an exterior acoustic problem governed by the Helmholtz equation is violated at the eigenfrequencies of the associated interior problem when the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) is applied without any special treatment to solve it. To tackle this problem, the Burton-Miller formulation using a linear combination of the CBIE and its normal derivative (NDBIE) emerges as an effective and efficient formula which is proved to yield a unique solution for all frequencies if the imaginary part of the coupling constant of the two equations is nonzero. The most difficult part in implementing the Burton-Miller formulation is that the NDBIE is a hypersingular type, and it is often regularized by using the fundamental solution of the Laplace's equation. But various regularization procedures in the literature give rise to integrals which are still difficult and/or extremely time consuming to evaluate in general. However, when constant triangular elements are used to discretize the boundary, all the strongly-singular and hypersingular integrals can be evaluated in finite-part sense explicitly without any difficulty, and the numerical computation becomes more efficient than any other singularity-subtraction technique. Therefore, in this paper, these singular integrals are evaluated rigorously for triangular constant element as finite parts of the divergent integrals by canceling out the divergent terms which appears in the limiting process explicitly. The correctness of the formulation is also demonstrated through some numerical test examples.