Discontinuous Galerkin Framework Research Articles

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of accuracy. Our lowest order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations, and show that our scheme is positivity preserving. In the presence of shocks additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

Accelerating high order discontinuous Galerkin solvers using neural networks: Wall bounded flows

High order solvers are accurate but computationally expensive as they require small time steps to advance the solution in time. In this work we include a corrective forcing to a low order solution to improve the accuracy while advancing in time with larger time steps, and achieve fast computations. The work uses a discontinuous Galerkin framework, where the polynomial order, inside each mesh element, can be varied to provide low or high accuracy. The corrective forcing is included for each high order Gauss nodal point in the mesh. This work is a continuation of [1, 2], where we extend the methodology to wall bounded flows. Namely, we adapt the methodology to a turbulent channel at Reτ = 182. In this case, we use three neural networks to correct different regions of the flow, which are distinguished by their y+ distance to the wall. The methodology is able to correct the low resolution simulation to attain flow statistics that are comparable to high order simulations. We include comparisons for the mean, Reynolds stresses and shear stress on the wall. We achieve good predictions using the corrected low order solution, in mean velocity and its corresponded fluctuations, as well as the shear stress on the wall.

Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.

Conservative cut finite element methods using macroelements

We develop a conservative cut finite element method for an elliptic coupled bulk-interface problem. The method is based on a discontinuous Galerkin framework where stabilization is added in such a way that we retain conservation on macroelements containing one element with a large intersection with the domain and possibly a number of elements with small intersections. We derive error estimates and present confirming numerical results.

A Comparative Study of Different Sets of Variables in a Discontinuous Galerkin Method with Entropy Balance Enforcement

This paper investigates the effectiveness of different sets of variables in solving the compressible Euler equations using a modal Discontinuous Galerkin framework. Alongside the commonly used conservative and primitive variables, the entropy and logarithmic sets are considered to enforce entropy conservation/stability and positivity preservation of the thermodynamic state, respectively. An explicit correction to enforce entropy conservation/stability at the discrete level is also considered, with a significant increase in robustness for some of the solution strategies. Several two-dimensional inviscid test cases are computed to compare the performance of the different sets of variables, adding a directional shock-capturing term to the discretised equations when necessary. The entropy and logarithmic sets proved to be the most robust, completing simulations of an astrophysical jet at Mach number 2000 up to polynomial degree seven.

Benchmark Solutions for Radiative Transfer with a Moving Mesh and Exact Uncollided Source Treatments

The set of benchmark solutions used in the thermal radiative transfer community suffers some coverage gaps, in particular, nonlinear, nonequilibrium problems. Also, there are no nonequilibrium, optically thick benchmarks. These shortcomings motivated the development of a numerical method free from the requirement of linearity and easily able to converge on smooth optically thick problems, i.e., a moving mesh Discontinuous Galerkin framework that utilizes an uncollided source treatment. Having already proven this method on time-dependent scattering transport problems, we present here solutions to nonequilibrium thermal radiative transfer problems for familiar linearized systems together with more physical nonlinear systems in both optically thin and thick regimes, including both the full transport and the / solution. Geometric convergence is observed for smooth sources at all times and some nonsmooth sources at late times when there is local equilibrium. Also, accurate solutions are achieved for step sources when the solution is not smooth.

Wall-distance free transition model based on the laminar kinetic energy

The next fluid dynamics solvers will be based on innovative numerical schemes and models to increase the fidelity and decrease the computational cost. The higher accuracy and geometrical flexibility guaranteed by discontinuous Galerkin spatial discretization methods in solving Reynolds-Averaged Navier–Stokes equations could represent an appealing solution in comparison with finite volume solvers for real-life simulations. In this context, numerical models able to accurately predict transitional flows are mandatory to overcome the limits of turbulence models and the costs of high-fidelity approaches, e.g., Direct Numerical Simulations and Large Eddy Simulations, for the efficient design of many industrial applications, e.g., aerospace, turbomachinery, maritime, automotive, and cooling applications. Among the transition models proposed in the literature, the local and phenomenological formulation seems to guarantee better robustness, fidelity, and easiness of implementation in all the solvers. All the transition models are based also on the wall-distance, to define some local terms or parameters and model the transition phenomenon. The calculation of the wall-distance can be critical in the discontinuous Galerkin framework for the high-order representation of the boundaries, which can become very expensive and high-memory consuming. To alleviate this problem, a wall-distance free version of a transition model based on the laminar kinetic energy is proposed and implemented in a high-order discontinuous Galerkin solver, and the robustness and fidelity are assessed by computing flows with bypass and separation-induced transition and different Reynolds number, turbulent intensity, and pressure gradient on flat plates. The wall-distance free formulation proves robustness and fidelity in all the cases, in comparison with the original formulation and an ad hoc modified formulation for the separation-induced transition cases.

Assessment of solving the RANS equations with two-equation eddy-viscosity models using high-order accurate discretization

A challenge that faces high-order methods for industrial applications generally is turbulence modeling at high Reynolds numbers. Large eddy simulation is studied extensively for high-order methods, nevertheless, its computational cost is enormous for industrial applications. The hybrid LES/RANS compromises the computational cost and the modeling error, however, solving the Reynolds-averaged Navier-Stokes equations is a resilient task for high-order methods, due to the non-smooth profiles of the turbulence quantities. Taking into account the complexity with high-order methods and the fairly large modeling errors of the RANS modeling, low-order methods has proved to be more pragmatic. For instance, in the discontinuous Galerkin framework, the polynomial approximation for these quantities leads to large oscillations that obstructs the non-linear solver. To use the high-order methods for industrial cases it is essential to have reliable implementations of two-equation turbulence models in RANS formulations. In this paper, a RANS discretization based on hybridizable discontinuous Galerkin is presented for the standard, TNT, BSL and SST versions of the k−ω model for applications at Reynolds numbers up to 109. A particular focus is given to the treatment of the specific rate of turbulence dissipation ω in the high-order framework. The complexity increases with these types of models as the value of ω goes to infinity at solid walls. Additionally, with minor modifications to the numerical flux definition, the turbulence model formulation can be solved by discontinuous Galerkin method as well. The results show remarkable improvements regarding the error magnitudes and non-linear convergence rate (iterative error) compared to second-order finite volume based solvers.

: A high-order discontinuous Galerkin solver for flow simulations and multi-physics applications

We present the latest developments of our High-Order Spectral Element Solver (▪), an open source high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including compressible flows (with or without shocks), incompressible flows, various RANS and LES turbulence models, particle dynamics, multiphase flows, and aeroacoustics. We provide an overview of the high-order spatial discretisation (including energy/entropy stable schemes) and anisotropic p-adaptation capabilities. The solver is parallelised using MPI and OpenMP showing good scalability for up to 1000 processors. Temporal discretisations include explicit, implicit, multigrid, and dual time-stepping schemes with efficient preconditioners. Additionally, we facilitate meshing and simulating complex geometries through a mesh-free immersed boundary technique. We detail the available documentation and the test cases included in the GitHub repository. Program summaryProgram Title:▪CPC Library link to program files:https://doi.org/10.17632/738py2shk4.1Developer's repository link:https://github.com/loganoz/horses3d.Licensing provisions: MITProgramming language: Fortran 2008External routines/libraries: METIS, MPI, HDF5, MKL, PETSc (all are optional).Nature of problem:▪ is a high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including compressible flows (with or without shocks), incompressible flows, various RANS and LES turbulence models, particle dynamics, multiphase flows, and aeroacoustics.Solution method: high-order discontinuous Galerkin Spectral Element (DGSEM) and explicit/implicit time-steppers.Additional comments including restrictions and unusual features:▪ is a multiphysics environment where the compressible Navier-Stokes equations, the incompressible Navier–Stokes equations, the Cahn–Hilliard equation and entropy–stable variants are solved. Arbitrary high–order, p–anisotropic discretisations are used, including static and dynamic p–adaptation methods (feature-based and truncation error-based). Explicit and implicit time-steppers for steady and time-marching solutions are available, including efficient multigrid and preconditioners. Numerical and analytical Jacobian computations with a colouring algorithm have been implemented. Multiphase flows are solved using a diffuse interface model: Navier–Stokes/Cahn–Hilliard. Turbulent models implemented include RANS: Spalart-Allmaras and LES: Smagorinsky, Wale, Vreman; including wall models. Immersed boundary methods can be used, to avoid creating body fitted meshes. Acoustic propagation can be computed using Ffowcs-Williams and Hawkings models. ▪ supports curvilinear, hexahedral, conforming meshes in GMSH, HDF5 and SpecMesh/HOHQMesh format. A hybrid CPU-based parallelisation strategy (shared and distributed memory) with OpenMP and MPI is followed.

VARIATIONALLY DERIVED DISCONTINUOUS GALERKIN METHOD: APPLICATION TO DYNAMIC THERMOELASTICITY

This paper presents a variational approach for deriving discontinuous Galerkin (DG) method for coupled field problems. Starting from the variational multiscale discontinuous Galerkin (VMDG) framework that is applied to the mechanical and thermal fields across embedded interfaces, an interface DG method for the coupled multifield problems is developed. Extending the interface DG to all inter-element boundaries naturally leads to a full DG method. An important aspect of the proposed methods is that time dependency appears naturally in the analytical expressions for the Lagrange multipliers that enforce the continuity of the fields and their fluxes. Unique attributes of the analytical expressions are that they comprise material and geometric parameters that automatically embed the concepts of stress averaging and area averaging in the method. In addition, the interface terms also provide an avenue to variationally account for interfacial kinetic and kinematic models for a robust representation of interfacial physics in dynamic thermomechanical problems. The interface DG method where discontinuity in the fields exists only at the embedded interfaces while continuous formulation is employed in the rest of the domain is well suited for bimaterial interface problems as well as for obtaining computationally economic solutions to the general class of mathematically nonsmooth thermomechanical problems. Several benchmark test cases are investigated that highlight the enhanced stability and variational consistency of the proposed VMDG formulations.

Accurate solutions to time dependent transport problems with a moving mesh and exact uncollided source treatments

To find benchmark quality solutions to time dependent Sntransport problems, we develop a numerical method in a Discontinuous Galerkin (DG) framework that utilizes time dependent cell edges, called a moving mesh, and an uncollided source treatment. The moving mesh and uncollided source treatment is devised to circumvent discontinuities in the solution and better realize the potential of the DG method to converge on smooth problems. The resulting method achieves spectral convergence on smooth problems. When applied to problems with discontinuity-inducing sources, our combined method returns a significantly better solution than the standard DG method. On smooth problems, we observe spectral convergence even in problems with wave fronts. In problems where the angular flux is inherently non-smooth, we do not observe high-order of convergence when compared with static meshes, but there is a quantitative reduction in error of nearly three orders of magnitude.

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier continuation (FC) is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions. These methods have been used in partial differential equation (PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving the stability and the convergence. Here we propose the use of FC in forming a new basis for the DG framework.

Assessment of an adaptive time integration strategy for a high‐order discretization of the unsteady RANS equations

SummaryMany industrial applications, for example, aeronautics, aeroacoustics, and turbomachinery, are characterized by complex turbulent flow problems, whose numerical studies are mostly based on statistical models. These models rely on the Navier–Stokes equations averaged in time, namely the Reynolds‐averaged Navier–Stokes (RANS) equations. RANS unsteady solutions require accurate and efficient time integration strategies, and the choice of the time‐step can have a strong impact on the robustness and efficiency of the simulation. Adaptive time step algorithms can improve considerably the effectiveness of time integration, but in literature little information is available on their application to unsteady RANS equations for high‐Reynolds number turbulent flows. This work aims at reducing this gap, presenting a numerical investigation of the performance of different adaptive temporal strategies applied to linearly implicit Rosenbrock‐type schemes within a high‐order discontinuous Galerkin framework. Several test cases of increasing stiffness and difficulty are considered for both compressible and incompressible turbulent flows.

High order conservative positivity-preserving discontinuous Galerkin method for stationary hyperbolic equations

This is a follow-up work of Yuan et al. (2016) [19] and Ling et al. (2018) [13] that further investigates the positivity-preserving discontinuous Galerkin (DG) methods for stationary hyperbolic equations. In 2016, Yuan et al. proposed a high order positivity-preserving DG method for stationary hyperbolic equations with constant coefficients, but the scheme has to be used in combination with a non-conservative rotational limiter introduced in case of negative cell averages. Ling et al. (2018) improved the results in one dimensional space by rigorously proving the positivity of cell averages of the unmodulated DG scheme, which allows the conservative scaling limiter in Zhang and Shu (2010) [22] to be used to maintain positivity without affecting accuracy, but extension to two space dimensions requires an augmentation of the DG space and works only in the second order case. Considering that the aforementioned works only address stationary hyperbolic equations with constant coefficients and higher than second order conservative methods are still unavailable in two and three space dimensions, we propose high order conservative positivity-preserving DG methods for variable coefficient and nonlinear stationary hyperbolic equations in one dimension and constant coefficient stationary hyperbolic equations in two and three dimensions, via a suitable quadrature in the DG framework. We show the good performance of the algorithms by ample numerical experiments, including their applications in time-dependent problems.

A generalized Eulerian-Lagrangian discontinuous Galerkin method for transport problems

We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in Cai et al. (2021) [5], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties.

Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods

A high-order accurate implicit-mesh discontinuous Galerkin framework for wave propagation in single-phase and bi-phase solids is presented. The framework belongs to the embedded-boundary techniques and its novelty regards the spatial discretization, which enables boundary and interface conditions to be enforced with high-order accuracy on curved embedded geometries. High-order accuracy is achieved via high-order quadrature rules for implicitly-defined domains and boundaries, whilst a cell-merging strategy addresses the presence of small cut cells. The framework is used to discretize the governing equations of elastodynamics, written using a first-order hyperbolic momentum-strain formulation, and an exact Riemann solver is employed to compute the numerical flux at the interface between dissimilar materials with general anisotropic properties. The space-discretized equations are then advanced in time using explicit high-order Runge–Kutta algorithms. Several two- and three-dimensional numerical tests including dynamic adaptive mesh refinement are presented to demonstrate the high-order accuracy and the capability of the method in the elastodynamic analysis of single- and bi-phases solids containing complex geometries.

A three-dimensional modal discontinuous Galerkin method for the second-order Boltzmann-Curtiss-based constitutive model of rarefied and microscale gas flows

A three-dimensional mixed modal discontinuous Galerkin (DG) method based on tetrahedral meshes was developed for simulating all flow regimes from subsonic to hypersonic rarefied and microscale gas flows within a single framework. The mixed modal DG scheme was used for solving conservation laws in conjunction with the second-order Boltzmann-Curtiss-based constitutive model of diatomic and polyatomic gases in strong thermal nonequilibrium. A decomposition algorithm based on the compression-expansion and velocity shear sub-problems was presented for solving the multi-dimensional second-order constitutive model. The Langmuir and Maxwell-Smoluchowski velocity-slip and temperature-jump boundary conditions were also implemented into the DG framework. To assess the ability of the new computational model to capture correct physical phenomena, we applied the new model to various gas flows in a wide range of continuum-rarefied and microscale regimes. The computational results in the rarefied and microscale flow regimes showed that the second-order constitutive model yielded solutions that were in better agreement with the direct simulation Monte Carlo and experimental data than the first-order constitutive model.

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

Zonal direct-hybrid aeroacoustic simulation of trailing edge noise using a high-order discontinuous Galerkin spectral element method

We present the extension of a discontinuous Galerkin framework to zonal direct-hybrid aeroacoustic simulations. This extension provides the ability to simultaneously perform a zonal large eddy simulation (LES), solving the compressible Navier–Stokes equations, and an acoustic propagation simulation, solving the acoustic perturbation equations. In doing so, the acoustic sources are exchanged without using the file system, and the bottleneck of I/O operations is avoided. This approach is well suited for large-scale simulations done in high-performance computing. The zonal LES uses the recently introduced recycling rescaling anisotropic linear forcing as a turbulent inflow method. We present a methodology to model the required Reynolds stresses based on the distribution of the turbulent kinetic energy obtained from solving the Reynolds-averaged Navier–Stokes equations. We show at the example of a turbulent flow over a flat plate and a NACA 64418 trailing edge simulation that the chosen model of the Reynolds stresses is valid. Direct-hybrid simulation results of a NACA 0012 airfoil, including tonal self-noise and a NACA 64418 trailing edge, demonstrate the presented approach’s applicability. This zonal direct-hybrid simulation approach shows great potential for efficient hybrid computational aeroacoustic simulations in high-performance computing.

Discontinuous Galerkin Methods for a Class of Nonvariational Problems

We extend the finite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2): 786–801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the “finite element Hessian” as an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et al. (SIAM J. Numer. Anal. 39(5): 1749–1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the finite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting.