In this paper, a half-space fast multipole BEM is developed for the simulation of three-dimensional acoustic problems above an infinite impedance plane. The half-space impedance Green’s function involving a complex line source is used, so that both mass-like and spring-like impedance boundary conditions on the infinite plane can be explicitly satisfied and the infinite plane is not required to be discretized. The Burton–Miller method is employed to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Image relations of the multipole expansion coefficients are used and the half-space impedance Green’s function is modified to apply such relations to avoid calculating, translating and saving the multipole/local expansion coefficients in the image domain. An automatic integrator with adaptive interval subdivision is further adopted to calculate the line integral contained in the M2L translation formula accurately and efficiently. Numerical examples are given to show the validity and potential of the method.

中文翻译：

---

使用快速多极边界元法模拟无限阻抗平面上的声音传播

在本文中，开发了一种半空间快速多极边界元法，用于模拟无限阻抗平面上的三维声学问题。使用涉及复线源的半空间阻抗格林函数，可以明确满足无限平面上的类质量和类弹簧阻抗边界条件，并且不需要对无限平面进行离散化。Burton-Miller 方法用于解决传统边界积分方程方法中涉及的虚构特征频率问题。使用多极展开系数的图像关系并修改半空间阻抗格林函数以应用这种关系以避免在图像域中计算、转换和保存多极/局部展开系数。进一步采用自适应区间细分的自动积分器，准确、高效地计算M2L平移公式中包含的线积分。给出了数值例子来说明该方法的有效性和潜力。