A new numerical method to solve three-dimensional wave equations in media with arbitrarily-shaped interfaces on a Cartesian grid is proposed. The present method aims to achieve two objectives to simulate wave propagation through realistic geometries: (1) handling wave interaction at the interface with high ratios of acoustic material properties and (2) treating complex geometries involving both smooth and non-smooth interfaces. To achieve the first objective, the present method extends the solution smoothly across the interface in the direction normal to the interface. A cell layer of ghost points on each side of the interface is used to enforce interface conditions, which support not only reflection but also transmission of incident waves. Ghost-point values are determined by applying a local coordinate-transform and a weighted least squares error method, which suppress numerical instabilities. To achieve the second objective, the interface geometry is approximated using an unstructured surface mesh, which does not require analytic information about the interface geometry. Finally, the accuracy and effectiveness of the present method are validated and demonstrated for wave propagation over or through several two-dimensional and three-dimensional obstacles.

提出了一种求解笛卡尔网格上任意形状界面介质中三维波动方程的数值方法。本方法旨在实现两个目的，以模拟通过实际几何形状的波传播：（1）以高比率的声学材料特性处理界面处的波相互作用，以及（2）处理涉及光滑和非光滑界面的复杂几何形状。为了实现第一个目的，本方法在垂直于界面的方向上将解决方案平滑地扩展到界面上。界面两侧的虚点的单元层用于强制执行界面条件，该条件不仅支持反射而且还支持入射波的传输。重影点值是通过应用局部坐标变换和加权最小二乘误差方法确定的，可以抑制数值不稳定性。为了实现第二个目的，可以使用非结构化表面网格来近似界面几何形状，该网格不需要关于界面几何形状的分析信息。最后，本方法的准确性和有效性得到了验证，并证明了通过或穿过几个二维和三维障碍物的波传播。