The multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general convex domains are discussed in this paper. A multiscale asymptotic analysis formulation for this problem is presented by constructing properly the boundary layer. A multiscale numerical algorithm and a postprocessing technique are given. Finally, numerical results show that the method presented in this paper is effective and reliable.

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In this paper, improved shape functions for one-dimensional hierarchal finite element are suggested such that the condition numbers of the corresponding stiffness matrices for both uniform and nonuniform bars can greatly be reduced as compared with the traditional shape functions. Therefore, it is beneficial to avoiding numerical trouble during calculation when high order elements are used. Furthermore, a simple but effective algorithm is proposed for extending the application of hierarchal finite element to structural natural vibration problems, and a sample example is given.

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In this paper the analytical solutions of the impact of a particle on Timoshenko beams with four kinds of different boundary conditions are obtained according to Navier's idea, which is further developed. The initial values of the impact forces are exactly determined by the momentum conservation law.. The propagation of the longitudinal and transverse waves along the beam, especially, the effects of boundary conditions on the characteristics of the reflected waves, are investigated in detail. Some results are compared with those by MSC/NASTRAN.Recently the strategy by Navier was realized and developed in Refs. [7,8], and this paper will developed the strategy further, where the impactor and the target are treated as an integrated vibration system, and only the impactor has a distributive velocity. Therefore, the problems of elastic impact between structures are transformed into traditional vibration problems with initial values, and the impact loads are the internal forces on the contact area, not the external forces acting on the integrated system. The analysis method of impact problems based on this strategy might as well be called as Direct Mode Superposition Method (DMSM).Among the methods mentioned above, the direct solution of differential equations is greatly limited because of the difficulty of getting the closed form solutions. IMSM is not suitable for the analysis of locking problems of mechanisms, and the time step size and the number of modes used influence seriously the obtained results. In comparison with IMSM, DMSM is a general method of solving the elastic problems of linear impact between structures and the locking problems of mechanisms. According to DMSM, the software systems such as NASTRAN can be directly applied to the analysis of elastic impact problems.About the applications of Timoshenko theory, much research work [3~5,9~12] has been done, where Anderson [9] gave the responses of finite Timoshenko beam with known loads by mode superposition method; Boley et alfl0] obtained the dynamical responses of semiinfinite Timoshenko beam by Laplace transform, but the loads applied to the beam are assumed to be given as simple pulses; Yamamoto[ 11] carried out a numerical analysis of impact problems with contact deformation of a ball on Timoshenko beam by IMSM; Shueei et al. [ 12] obtained analytical solutions of loaded Timoshenko beam with general elastic boundary conditions, the loads are known and no additional mass is attached to the beam, and the method is not suitable for the analysis of impact problems of structures in which the force boundary condition is defined directly by the acceleration of the impactor and there are initial conditions, and the analysis of impacts between structures such as a rod impacting against a beam and so on, But the complicated impact problems mentioned above can be easily solved by DMSM.