Discontinuous Galerkin Finite Element Method Of Liao And Jin And Power And Wave Propagation Problems In Elastic-plastic Solids And Saturated Porous Media Analysis

Discontinuous Galerkin finite element method (DGFEM) has been attracted more attentions and studies to solve time-dependent problems such as structural dynamics and wave propagation problems. The essential features of the method can be summarized in the following three aspects. The finite element discretizations are used in both space and time simultaneously. The assumed nodal primary unknown vector and its derivative with respect to time for the semi-discrete field equation are independently interpolated by piecewise polynomial functions in time domain. In addition, both of them are permitted to be discontinuous at the discrete time levels.The traditional (continuous) Galerkin finite element method (CGFEM) to solve time-dependent problems is mainly characterized by its semi-discrete procedure. The field equation is discretized using finite elements in spatial domain and the semi-discrete field equation in turn is discretized using finite difference methods in time domain such as the Newmark method. CGFEM usually provides successful results to the time dependent problems in which the low frequency response dominates. However, it generally fails to properly capture discontinuities or sharp gradients of the solution due to propagating impulsive waves in space. It is indicated that CGFEM is incapable of filtering out the effects of spurious high modes and controlling spurious numerical oscillation. In contrast with CGFEM, DGFEM possesses the possibility to filter the spurious oscillations and provides much more accurate solutions than does CGFEM using the Newmark method as the same time step size is used. A time-discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation analysis in one and two phase elstoplastic continua is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3-P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, are particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non-linear problems, as compared with that required to the existing DGFEM. Both the implicit and explicit algorithms to solve the derived formulations for the materially non-linear problems are developed. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and providing with much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in time domain.