Trefftz finite element method. Th numerical approach is particularly effective for those problems with local effects and local singularity, in contrast with conventional finite element method. We intend to work on following aspects to promote this approach: 1)Development of efficient hybrid Trefftz (HT) finite element-boundary element method (FE-BEM) schemes for complex engineering structures and the related general purpose computer codes with preprocessing and postprocessing capabilities. 2) Generation of various special-purpose functions to effectively handle singularities attributable to local geometrical or load effects. As discussed previously, the special-purpose functions warrant that excellent results are obtained at minimal computational cost and without local mesh refinement. Extension of such functions could be applied to other cases such as the boundary layer effect between two materials, the interaction between fluid and structure in fluid-structure problems, and circular hole, corner and load singularities. 3) Development of hybrid Trefftz finite element approach in conjunction with a topology optimization scheme to contribute to microstructure design. 4) Development of efficient adaptive procedures including error estimation, h-extension element, higher order p-capabilities, and convergence studies. 5) Extensions of Trefftz finite element approach to soil mechanics, thermoelasticity, deep shell structure, fluid flow, piezoelectric materials, and rheology problems. 6) Indirect Trefftz method in conjunction with parallel processing to numerical models of continuous systems of science and engineering; Application of the indirect method of Trefftz to space-time problems, including parabolic (heat conduction), hyperbolic (wave propagation) transport (advection-diffusion) equations. 7) Development of dynamic Trefftz finite element approach using numerical integral transforms such as Laplace transform

Meshless method. The meshless numerical method has recently become an alternative to the finite element method (FEM) and the boundary element method (BEM), due to its advantages of avoiding meshing and remeshing, effective treatment of complicated load conditions, and avoidance of mesh distortion in large deformation problems. The major difficulty in this method is to find the Radial basis function to the related problem. So far this method is limited to Poisson problem. We intend to extend this method to more complex problem by developing new Radial basis functions.

Precise time-step integration method for dynamic analysis. This program explore effective second-order schemes of precise time-step integration (PTI) method for dynamic analysis with respect to long-term integration and transient responses while spatial discretization is realized with the differential quadrature method. Rather than transforming into first-order equations, a recursive scheme is developed for direct solution of the homogeneous part of second-order differential and algebraic equations. The sine and cosine matrices involved in the scheme are calculated using the so-called algorithm, and the corresponding particular solution is also presented where the excitation vector is approximated by the truncated Taylor series. The issue of spurious high-frequency responses resulting from spatial discretization for shock-excited structural dynamic analysis is also studied in the framework of the second-order PTI method.