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.

Thermo-elastic singularities of V-notches in orthotropic materials are studied. By introducing series asymptotic expansions of physics fields near notch tip, stress and flux equilibrium equations are transformed into characteristic ordinary differential equations with respect to singularity orders. Singularity orders and corresponding characteristic angular functions can be derived synchronously as interpolating matrix method is introduced to solve characteristice quations. The method evaluates stress and flux singularity orders at the same time. Numerical results show that it has high accuracy and strong adaptability.

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.

Based on asymptotic extension of displacement field at a composite notch tip, equilibrium equation for a notch subjected to anti-plane loading is transformed into a characteristic differential equation respects to notch singularity orders. A transformation is applied to convert the equation into a set of characteristic linear ordinary differential equations. Interpolate matrix method is introduced to solve the equations for getting notch singularity orders. A single material notch, a bi-material notch and a notch terminated at bimaterial interface are studied successively. Examples indicate that the method provides all stress singularity orders synchronously. Though singular stress state is not shown with regard to non-singular orders, non-singular stress orders are indispensable parameters as evaluating complete stress field at notch tip region.

An algorithm is developed to calculate stresses at the interior points near boundary in boundary element method of thermoelasticity. A series of transformations is manipulated on conventional derivative boundary integral equations(BIE). It leads to a new natural BIE in thermoelasticity problem named thermal stress natural boundary integral equation(NBIE). Hyper-singularity and strong-singularity in conventional BIE are converted into strong-singularity in NBIE. Nearly strong singular integrals are evaluated in NBIE by the regularization algorithm. Thermal stresses at points near boundary are calculated by NBIE. Numerical examples illustrate efficiency of the method.

The fundamental theory of interpolating matrix method is eatablished for solving mixed order systems of nonlinear multipoint boundary value problems of ODEs.A general purpose solver IVMMS of ODEs is written based on this know ledge,and which can support the finite element method of lines for solid mechanics.

Interpolating matrix method is a numerical method for solving multi-point boundary value problems. This paper provides the error analysis of the method, proving the same precision for the solutions y(x), y'(x), ..., y(^m)(x)obtained by the method. Author gives the convergence order estimate and proves good stability property of the method the second order scalar equations.