Taking 3D acoustic analysis as an example, a semi-analytic algorithm is proposed which can be used to compute nearly singular integrals with high order element exactly. With analysis of geometry features of high order elements, approximate geometric parameters are constructed. Then, a kernel function of nearly singular integral is decomposed into two parts using subtraction method. One is regular part and the other is singular part. Integral of regular part is computed accurately using conventional Gauss quadrature. For integral of the singular part, semi-analytic algorithm gives exact result. Classical examples are given including 3D acoustic internal and external problems. Sound pressures at points near boundary are calculated with different methods. Comparisons of results demonstrate accuracy and effectiveness of the algorithm.

Boundary element method (BEM) is used to study propagation of crack under loading. First, all leading unknown coefficients in Williams series expansion and complete stress field for notched structures are calculated with BEM and former eigenanalysis for notch tip region. Then, with consideration of non-singular stress term and maximum circumferential stress criterion of brittle fracture, crack initiation extended direction from crack tip in a semicircular bending specimen is determined by BEM. Strategy for BEM tracking crack propagation is given. Numerical examples show that the method is correct and effective in simulating propagation of plane crack.