三维Helmholtz问题的间接规则化边界积分方程

Regularized Boundary Integral Equation with Indirect Unknowns for 3D Helmholtz Problems

进行了三维Helmholtz方程边值问题的间接变量规则化边界元法研究。使用位势问题的基本解逼近Helmholtz问题的基本解,将Helmholtz问题的边界积分方程的规则化问题转化为位势问题边界积分方程的规则化问题。应用已有的三维位势问题的基本解积分恒等式及规则化边界积分方程的极限定理,建立三维Helmholtz方程边值问题的间接变量规则化边界积分方程。此外,本文提出一般参数表示的几何边界上的精确单元的数值实施方案,即边界几何采用精确单元描述,单元上的边界量用二次8节点不连续插值函数逼近。数值算例表明:本文方法在低频下可获得很高的计算精度和效率。

The indirect boundary element method（ IBEM） for 3D Helmholtz equation problems was investigated by using the fundamental solution of the potential problem approaching to the fundamental solution of the Helmholtz equation problem. The boundary integral equation for Helmholtz problemwas transformed into the boundary integral equation for potential problem, so, we applied the fundamental solution to 3D potential problem and the limit theorem of the boundary integral equation to establish the indirect boundary element method（ IBEM） for 3D Helmholtz equation problems. In addition,the numerical implementation of the exact element with the geometric boundary was described by general parametric. Specifically,the geometric boundary was described by the exact element,and the boundary of the element was approximated by the two times discontinuous interpolation function with 8 nodes. Numerical examples with low wave number were investigated,demonstrating the feasibility and efficiency of this method.