An Adaptive Discontinuous Galerkin Method For Numerical Simulation Of Compressible Flows

Due to the rapid development of numerical methods and computational resources,computational fluid dynamics(CFD)methods have become important tools for engineering design and analysis in the aviation industry in recent years.Meanwhile,the accuracy and efficiency of numerical methods are always among the most critical issues in the field of CFD.Comparing with traditional lower order methods,from which the numerical simulations obtainer are usually not accurate enough,the discontinuous Galerkin(DG)finite element method now becomes one of the research highlights in the field of CFD for its high-order accuracy,great geometry flexibility,and simplicity for parallel computing.However,research on discontinuous Galerkin finite element method is still in the startup stage.Many crucial issues remain to be solved or improved,such as capturing strong discontinuity and obtaining higher-accuracy solution with lower mesh cost.In this thesis,an adaptive high-order discontinuous Galerkin method is developed for compressible flow simulations.Firstly,a robust two-dimensional high-order flow solver,which supports the mesh adaption as well,is developed.To ensure the stability of the computation,a high-order boundary modification approach and a curved mesh generation method are deployed on two-dimensional triangular mesh.For the spatial discretization of the governing equations,the LLF(Local Lax-Friedrichs)scheme is adopted for the convective numerical flux function,while the BR2 scheme is adopted for the viscous numerical flux function.An artificial viscosity approach is employed to capture shocks in the transonic flow simulations.As for the time integration,an explicit Runge-Kutta method and an implicit Newton's method are adopted,and a block Guass-Seidel iterative method is employed in order to solve the linear systems generated by Newton linearization efficiently.A mesh adaptive method is developed to enhance the solution accuracy and reduce the computation.The artificial viscous coefficient is regarded as the indicator of mesh adaption in the transonic flow simulations,whereas the vorticity is considered in the laminar flow simulations.A triangular element would be split into four smaller elements during the mesh refinement,while the mesh coarsening would do the opposite.In addition,the shape control points and the information of new elements are obtained by interpolating from the old elements to ensure the computational stability and solution accuracy.Secondly,a mesh adaption discontinuous Galerkin method for the three-dimensional flow simulations is developed by applying parallel computing.The spatial discretization and temporal discretization are similar to the two-dimensional case,and a high-order geometric approximation of curved boundaries is developed for high-order mesh generation as well.Compared with the two-dimensional mesh adaption,three-dimensional mesh adaption is more complicated.A tetrahedron element would be split into eight smaller elements during mesh refinement,and the mesh coarsening would again do the opposite.As the three-dimensional numerical simulation would demand huge amounts of grid and computation,a MPI-based parallel computing method is deployed and a dynamic mesh partitioning method is developed.Once the grids are in need of adaption,all processes are released and the mesh adaption procedure is executed in process 0.Then the mesh is re-partitioned and sent back to parallel computation.Since the load is balanced both before and after the adaption procedure,the method above is of high parallel efficiency.It should be noticed that the mesh adaption method for the discontinuous Galerkin method is far more complicated than it is in the traditional finite volume method case,as all the complex information about mapping and curved meshes must be handled very carefully.Finally,simulations of various two-dimensional and three-dimensional numerical cases are presented.Numerical solutions indicate that high accuracy solution can be obtained by increasing order and local mesh adaption on sparse initial mesh.Especially in the unsteady cases,highly accurate numerical results are obtained using the adaptive method at relatively low expense.