Discontinuous Galerkin Method And Its Application In Powder Forming

The High Velocity Compaction (HVC) technology is an important molding technique in powder metallurgy field, This thesis has applied the Discontinuous Galerkin method into the research of powder high velocity compaction process, and analyzed the spatial basic functions structures, the usual time discretization methods and the common numerical flux forms, the numerical solutions and error analysis of powder high velocity compaction stress-wave equations are given out in the end.Firstly, the basic theories of Galerkin method are summarized, the spatial discretization are considered with three types——the general multinomial basic functions, the orthogonal multinomial basic functions and the non-multinomial basic functions, in which the process of structuring general multinomial basic function, the Chebyshev orthogonal multinomial and the commonly non-multinomial basic function are given; the time discretization are considered with two types——Runge-Kutta method and the low-storage explicit Runge-Kutta method that overcome the shortness of Runge-Kutta method with high request in storage; Meanwhile the numerical flux of Discontinuous Galerkin method are also considered with two commonly types——Up-wind form and the Lax-friedrich form.Next, one dimension wave equation and one dimension wave equations are given, both applied with general multinomial basic functions and the low-storage explicit Runge-Kutta method, respectively applied numerical flux with the Up-wind form and the Lax-friedrich form; according to the one dimension wave equation has analyzed the calculation errors under the multinomial basic functions with different orders, which explained the Runge phenomenon in numerical method.Finally, based on simplifying stress-wave equations of power high velocity compaction, the equations’weak form of Discontinuous Galerkin method are deduced out, and the error estimation on the Cartesian space has been discussed, then taking suitable Lax-friedrich form as the numerical flux, using the Matlab7.1programming computation, the numerical solutions are obtained in the end.