Module helmholtz 

Helmholtz Green’s function and derivatives

The 3D Helmholtz Green’s function is:

G(x, y) = exp(ik|x-y|) / (4π|x-y|)

This module provides computation of G and its normal derivatives, which form the kernels for BEM integration.

Functions

all_kernels_3d

Compute all four BEM kernels at once for efficiency

distance

Distance between two points

greens_function_2d

2D Helmholtz Green’s function G = (i/4) H_0^(1)(kr)

greens_function_3d

3D Helmholtz Green’s function G = exp(ikr)/(4πr)

greens_function_adjoint_derivative_3d

Adjoint double layer kernel ∂G/∂n_x

greens_function_gradient_3d

Gradient of 3D Green’s function ∇_y G

greens_function_hypersingular_3d

Hypersingular kernel ∂²G/(∂n_x ∂n_y)

greens_function_normal_derivative_3d

Normal derivative of 3D Green’s function ∂G/∂n_y

laplace_greens_function_2d

2D Laplace Green’s function: G = -ln(r)/(2π)

laplace_greens_function_3d

Laplace Green’s function (k=0 limit): G = 1/(4πr)