A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $$\Delta t$$ Δ t satisfy $$h\cong C\Delta t$$ h ≅ C Δ t , with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $$L^\infty$$ L ∞ and $$L^2$$ L 2 -norms is order $$p+1$$ p + 1 for polynomial orders $$p=1$$ p = 1 and $$p=3$$ p = 3 and order p for polynomial order $$p=2$$ p = 2 .

Assessment of a Discontinuous Galerkin Method for the Simulation of the Turbulent Flow around the DrivAer Car Model

The turbulent flow over the DrivAer fastback model is here investigated with an order-adaptive discontinuous Galerkin (DG) method. The growing need of high-fidelity flow simulations for the accurate determination of problems, e.g., vehicle aerodynamics, promoted research on models and methods to improve the computational efficiency and to bring the practice of Scale Resolving Simulations (SRS), like the large-eddy simulation (LES), to an industrial level. An appealing choice for SRS is the Implicit LES (ILES) via a high-order DG method, where the favourable numerical dissipation of the space discretization scheme plays directly the role of a subgrid-scale model. Implicit time integration and the p-adaptive algorithm reduce the computational cost allowing a high-fidelity description of the physical phenomenon with very coarse mesh and moderate number of degrees of freedom. Two different models have been considered: (i) a simplified DrivAer fastback model, without the rear-view mirrors and the wheels, and a smooth underbody; (ii) the DrivAer fastback model, without rear-view mirrors and a smooth underbody. The predicted results have been compared with experimental data and CFD reference results, showing a good agreement.

FINITE ELEMENT ANALYSIS FOR TEXTILE COMPOSITES USING FIBER-BUNDLES/MATRIX-RESIN SEPARATED MESH

This paper proposes a finite element modeling method for textile composite, in which fiber bundle and matrix resin are separately meshed, and they are connected by using discontinuous Galerkin (DG) method. The fiber bundle geometry is often complex in the textile composite. In the conventional FEM, it causes small, distorted resin elements surrounded by the fiber bundles, because the resin must be meshed along the fiber bundle geometry. These distorted elements result in the increased effort to meshing, computing cost, and degraded accuracy. In the proposed method, we apply the DG method to the 3-dimensional analysis of the textile composite. DG method is a method which can connect two separately divided meshes in the FEM. The method proposed in this paper has a distinct advantage, because matrix resin has not to be meshed along the fiber bundle geometry. Moreover, regular cubic grid mesh can be used for matrix resin. In the present paper, the formulation of the DG method is presented first. The method and results of the microscopic stress analysis for textile composite is then described. The results agree well with those of conventional FEM, and validity of the proposed method is demonstrated.

Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial differential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unified form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first-order hyperbolic PDEs.