一种高效评估曲面元素BEM奇异积分的方法及其在声学分析中的应用
An efficient method for evaluating BEM singular integrals on curved elements with application in acoustic analysis
AI总结：
本文提出结合共形变换和S型变换的高效方法，用于计算高阶BEM中弱、强和超奇异积分，减少积分点数量，提升计算效率和稳定性。
摘要：
极坐标变换（PCT）方法已被广泛用于处理边界元法（BEM）中的各种奇异积分。然而，当（1）元素的长宽比较大或（2）场点接近元素边界时，PCT后的被积函数趋于近奇异，因此需要大量积分点以获得相对较高的精度。本文通过使用共形变换解决第一个问题，保持曲面物理元素的几何形状；通过使用S型变换缓解第二个问题，使积分点更集中在近奇异区域。通过将所提的两种变换与Guiggiani等人在[1]中提出的方法结合，获得一种高效且稳健的数值方法，用于计算高阶BEM中曲面元素的弱、强和超奇异积分。数值积分结果表明，与原始PCT相比，本文方法可以显著减少积分点数量，以达到给定的精度。为进一步验证，该方法被整合到一个2阶Nyström BEM代码中，用于求解声学Burton-Miller边界积分方程。结果表明，该方法在使用远少于现有PCT的积分点的情况下，仍能保持BEM的收敛率。该方法用C语言实现并免费提供。
英文摘要：
The polar coordinate transformation (PCT) method has been extensively used to treat various singular integrals in the boundary element method (BEM). However, the resultant integrands of the PCT tend to become nearly singular when (1) the aspect ratio of the element is large or (2) the field point is closed to the element boundary; thus a large number of quadrature points are needed to achieve a relatively high accuracy. In this paper, the first problem is circumvented by using a conformal transformation so that the geometry of the curved physical element is preserved in the transformed domain. The second problem is alleviated by using a sigmoidal transformation, which makes the quadrature points more concentrated around the near singularity. By combining the proposed two transformations with the Guiggiani's method in [M. Guiggiani, \emph{et al}. A general algorithm for the numerical solution of hypersingular boundary integral equations. \emph{ASME Journal of Applied Mechanics}, 59(1992), 604-614], one obtains an efficient and robust numerical method for computing the weakly-, strongly- and hyper-singular integrals in high-order BEM with curved elements. Numerical integration results show that, compared with the original PCT, the present method can reduce the number of quadrature points considerably, for given accuracy. For further verification, the method is incorporated into a 2-order Nyström BEM code for solving acoustic Burton-Miller boundary integral equation. It is shown that the method can retain the convergence rate of the BEM with much less quadrature points than the existing PCT. The method is implemented in C language and freely available.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
