Evaluation of 2-D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

摘要

This paper presents the non-singular forms, in a global sense, of two-dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element-free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier-Legendre series, together with transforming the integration interval [a, b] to [-1, 1]; the series coefficients are thus to be determined. The hyper-singular integral, interpreted in the Hadamard finite-part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well-known Burton-Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non-uniqueness problem. A general non-singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made.