An h-Adaptivity DG Method on Locally Curved Tetrahedral Mesh for Solving Compressible Flows

For the numerical simulation of compressible flows,normally different mesh sizes are expected in different regions.For example,smaller mesh sizes are required to improve the local numerical resolution in the regions where the physical variables vary violently(for example,near the shock waves or in the boundary layers)and larger elements are expected for the regions where the solution is smooth.h-adaptive mesh has been widely used for complex flows.However,there are two difficulties when employing h-adaptivity for high-order discontinuous Galerkin(DG)methods.First,locally curved elements are required to precisely match the solid boundary,which significantly increases the difficulty to conduct the"refining"and"coarsening"operations since the curved information has to be maintained.Second,h-adaptivity could break the partition balancing,which would significantly affect the efficiency of parallel computing.In this paper,a robust and automatic h-adaptive method is developed for high-order DG methods on locally curved tetrahedral mesh,for which the curved geometries are maintained during the h-adaptivity.Furthermore,the reallocating and rebalancing of the computational loads on parallel clusters are conducted to maintain the parallel efficiency.Numerical results indicate that the introduced h-adaptive method is able to generate more reasonable mesh according to the structure of flow-fields.

An h-adaptive Discontinuous Galerkin Method for Laminar Compressible Navier-Stokes Equations on Curved Mesh

An h-adaptive method is developed for high-order discontinuous Galerkin methods(DGM)to solve the laminar compressible Navier-Stokes(N-S)equations on unstructured mesh.The vorticity is regarded as the indicator of adaptivity.The elements where the vorticity is larger than a pre-defined upper limit are refined,and those where the vorticity is smaller than a pre-defined lower limit are coarsened if they have been refined.A high-order geometric approximation of curved boundaries is adopted to ensure the accuracy.Numerical results indicate that highly accurate numerical results can be obtained with the adaptive method at relatively low expense.

ADAPTIVE EIGENFREQUENCY ANALYSIS BY IMPROVED r-AND h-ADAPTIVE FINITE ELEMENT METHOD BASED ON PERTURBATION AND ELEMENT ENERGY RATIO

A new adaptive technique of r-and h-version for vibration problemsutilizing the matrix per- turbation theory and element energy ratiois proposed. In structural vibration analysis, through the r-conver-gence adaptvie finite element process, mesh optimization can berealized. In the light of the judgement on the changes in themagnitude of the element energy ratio, local refinement can beachieved in the process of h- convergence adaptive finite element sothat more accurate finite element solutions can be obtained with asfew meshes as possible. Many numerical examples are given and theproposed approach is shown to be feasible and effective.

An h-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media.In this paper,a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon.Towards the efficiency and accuracy of the numerical simulations,a predictor-corrector process is used for resolving the nonlinearity of the equation,and an h-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region,in which a heuristic a posteriori error indicator is designed for the purpose.In numerical simulations,a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method.The effectiveness of the h-adaptive method is also successfully demonstrated by numerical experiments.Finally the mechanism on generating fingers is discussed by numerically studying several examples.

An h-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In[35,36],we presented an h-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws.A tree data structure(binary tree in one dimension and quadtree in two dimensions)is used to aid storage and neighbor finding.Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled"children".Extensive numerical tests indicate that the proposed h-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities.In this paper,we apply this h-adaptive method to solve Hamilton-Jacobi equations,with an objective of enhancing the resolution near the discontinuities of the solution derivatives.One-and two-dimensional numerical examples are shown to illustrate the capability of the method.

A characteristic-featured shock wave indicator for simulating high-speed inviscid flows on 3D unstructured meshes

In this work,we extend the characteristic-featured shock wave indicator based on artificial neuron training to 3D high-speed flow simulation on unstructured meshes.The extension is achieved through dimension splitting.This leads to that the proposed indicator is capable of identifying regions of flow compression in any direction.With this capability,the indicator is further developed to combine with h-adaptivity of mesh refinement to improve resolution with less computational costs.The present indicator provides an attractive alternative for constructing high-resolution,high-efficiency shock-processing methods to simulate high-speed inviscid flows.

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

The implicit numerical methods have the advantages on preserving the physical properties of the quantum system when solving the time-dependent Kohn-Sham equation.However,the efficiency issue prevents the practical applications of those implicit methods.In this paper,an implicit solver based on a class of Runge-Kutta methods and the finite element method is proposed for the time-dependent Kohn-Sham equation.The efficiency issue is partially resolved by three approaches,i.e.,an h-adaptive mesh method is proposed to effectively restrain the size of the discretized problem,a complex-valued algebraic multigrid solver is developed for efficiently solving the derived linear system from the implicit discretization,as well as the OpenMP based parallelization of the algorithm.The numerical convergence,the ability on preserving the physical properties,and the efficiency of the proposed numerical method are demonstrated by a number of numerical experiments.