To extend the proposed approach to handle more complex geometries, such as multiple layers or non-planar interfaces, several modifications and enhancements can be implemented.
Multiple Layers: For scenarios involving multiple layers, the Green's function formulation can be adapted to account for the additional interfaces and media. Each layer would introduce new boundary conditions and wave equations, requiring the development of corresponding integral equations. By iteratively solving these equations and incorporating the appropriate boundary conditions at each interface, the scattering behavior in multi-layered mediums can be accurately modeled.
Non-Planar Interfaces: When dealing with non-planar interfaces, the geometry of the boundary needs to be explicitly considered in the integral equation formulations. This may involve parametrizing the irregular boundary and adjusting the Green's function expressions to accommodate the non-planar surface. Techniques like boundary element methods can be employed to discretize the boundary and handle the integral equations efficiently.
Advanced Numerical Techniques: Utilizing advanced numerical techniques like adaptive mesh refinement, high-order quadrature methods, and parallel computing can enhance the computational efficiency and accuracy when dealing with complex geometries. These techniques can help in resolving intricate interfaces and capturing fine details in the scattering behavior.
By incorporating these strategies, the Nyström method can be extended to effectively tackle scattering problems in more intricate geometries, providing a comprehensive analysis of wave interactions in diverse environments.