Abstract: In this paper the method of transformation of the boundaries for structure the admissible displacements with various boundary conditions is presented. What is called the method of transformation of the boundaries is that, first we transform the actual system into the basic system and additional boundary forces and displacements on the basis of the superposition principle, then apply variational principles to the basic system, finally find the permissible displacement of the actual system by means of the method of transformation of the series.In this paper, we also give mixed energy principles under v iriation of boundary conditions. The mixed energy principles as the potential and complementary energy principles under variation of boundary conditions are all the chief theoretical fundamental of the method of transformation of the boundaries.Applying the method of transformation of the boundaries, we form the permissible displacements of rectangular plates of plane stress and bending problems with various edge conditions.Because the method of transformation of the boundaries is in progress to follow the variational principles and definite program to form permissible displacements, the difficulty in supposing and piecing together permissible displacements in the Rayleigh-Ritz method will be overcome.

Key words: incompressible Navier-Stokes equation, numerical simulation, artificial compressibility, central and upwind difference scheme mixed algorithm, flow over a prolate spheroid, steady/unsteady flow