An equilibrium hybrid-Trefftz formulation based on the direct approximation of the stress and boundary displacement fields is presented. The general solution of the governing differential equations is used to approximate the stress field and the boundary displacements are represented by polynomial functions. When singular solutions are implemented to model local high stress gradients due to concentrated loads or to the presence of wedges or cracks, rational functions are used to approximate the boundary displacements in the neighbourhood of such singular stress points. The equilibrium conditions and the kinematic boundary conditions are locally satisfied. The remaining fundamental relations-the compatibility conditions, the static boundary conditions and the constitutive relations-are enforced in a weighted residual form so designed as to preserve the duality and constitutive reciprocity. The resulting governing system is symmetric and all intervening structural operators have boundary integral expressions. Numerical applications are presented to illustrate the performance of the formulation. the definitions given above for the intervening variables, to yield J J J J

A formulation based on the approximation of the stress field is used to compute directly the stress intensity factors in crack problems. The boundary displacements are independently approximated. In each finite element, the assumed stresses may model multipoint singularities of variable order. The differential equilibrium equations are locally satisfied as solutions of the governing differential system are used to build the stress approximation basis. The approximation on the boundary displacements is constrained to satisfy locally the kinematic boundary conditions. The remaining fundamental conditions, namely the differential compatibility equations, the constitutive relations and the static boundary conditions are enforced through weighted residual statements. The approximation criteria are so chosen as to ensure that the finite element model is described by a sparse, adaptive and symmetric governing system described by structural matrices with boundary integral expressions. Numerical applications are presented to show that accurate solutions can be obtained using structural discretizations based on coarse meshes of few but highly rich elements, each of which may have different geometries and alternative approximation laws.

4n efficient and simple infinite element for modelling the far field potential of water wave diffraction problems is presented. The shape functions in the radial direction comprise the first two terms of the asymptotic expansions of Hankel functions. The integrals with infinite limit for calculating the coefficient matrix have been worked out. Numerical tests on the diffraction by a surface-piercing circular cylinder give surprisingly accurate resultant forces even if the infinite elements are placed very near to the cylinder. Other typical three dimensional examples also show that satisfactory results can be obtained by the use of this simple infinite element. A computer program, WALOAD, has been developed for computing the wave forces on fixed two and three dimensional bodies.

In order to determine the，a test bench for the natural vibration characteristics of the bogie suspension was put forward. Firstly the natural vibration characteristics structure composition and the testing principle was introduced, the sine sweep incentive was simulated, and the inverse kinematics solution model and the real-time solution model were developed. The Comparison of the simulation result and the geometric theory results was carried out, proving the accuracy of the inverse kinematics solution model and the simulation model.

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