Meshless local discontinuous Petrov-Galerkin method with application to blasting problems

Abstract

Abstract：A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily.Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes. Keywords：meshless method; discontinuous Galerkin method; meshless local discontinuous PetrovGalerkin (MLDPG) method; finite-volume particle method; convection-dominated flow Conventional computational fluid dynamics (CFD) methods need a priori definition of the connectivity of nodes, i.e., they rely on a mesh, which leads to complications for certain classes of problems. The generation of good quality non-overlapping meshes presents fateful difficulties in the analysis of engineering systems (especially in the multi-dimensional problems). In recent years, more and more attention has been paid to the socalled meshless method(MM), as the method can solve partial differential equations only based on a scatter set of nodes without the need for an additional mesh, which eliminates the disadvantages of mesh-based method essentially. Nevertheless, MM is mostly applied to the solid mechanics problems, only very few work is reported using MM for convection-dominated flows.In the numerical simulation of blast phenomenon, the existence of the convection term makes that the solution will develop strong discontinuity as well as shock waves. In order to handle this kind of weak solutions numerically, one needs to introduce a special treatment to stabilize the numerical approximation. Some of the meshless schemes have been incorporated into the stabilization algorithms in the mesh-based method to obtain a good accuracy for convection-dominated problems, e.g., the smoothed particle hydrodynamics (SPH) [1,2] employs artificial viscosity in the formulations to improve the stability in the simulation of shock phenomenon, and a few of sophisticated MMs introduce the upwind algorithm for the calculation of convection-dominated problems including the finite particle method (FPM) [3] , reproducing kernel particle method (RKPM) [4] and meshless local Petrov-Galerkin (MLP...

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