High-order Nodal Discontinuous Galerkin Research Articles

Toward a robust detection of viscous and turbulent flow regions using unsupervised machine learning

We propose an invariant feature space for the detection of viscous-dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of the strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous-dominated, rotational region (a boundary layer and a wake region) and an inviscid, irrotational region (an outer flow region). We have tested the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at Re = 40 and Re = 3900 and a laminar flow around an airfoil at Re=1×105. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method. The results obtained are analyzed to show that Gaussian mixture clustering provides an effective identification method of viscous-dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors.

A high-order nodal discontinuous Galerkin method for simulation of three-dimensional non-cavitating/cavitating flows

In this study, the nodal discontinuous Galerkin method is formulated in three-dimensions and applied to simulate three-dimensional non-cavitating/cavitating flows. For this aim, the three-dimensional preconditioned Navier-Stokes equations based on the artificial compressibility approach considering appropriate source terms to model cavitating phenomena are used. The spatial derivative terms in the resulting equations are discretized by utilizing the nodal discontinuous Galerkin method on tetrahedral elements and the derivative of the solution vector with respect to the artificial time is discretized by applying an explicit time integration method. An artificial viscosity method is formulated in three-dimensions and applied to properly capture the discontinuities in the cavitating flow-field. Here, different non-cavitating/cavitating flow problems including non-cavitating flow in a 3:1:1 driven cavity, non-cavitating flow between two concentric rotating spheres, cavitating flow over a sphere and cavitating flow over a wavy cylinder are solved to examine the accuracy and robustness of the proposed solution method. These flow problems are selected to properly assess the numerical method in the presence of straight and curved boundaries and for different Reynolds and cavitation numbers. From linear to cubic polynomial degrees are used in the simulations and the numerical results are compared with the available analytical and numerical results. Indications are that the present numerical method based on the NDGM formulated in three-dimensions can accurately simulate the three-dimensional non-cavitating/cavitating flows. • Formulating a nodal discontinuous Galerkin method (NDGM) in 3D to simulate noncavitating/cavitating flows. • Implementing an artificial viscosity approach in 3D for capturing the steep discontinuities in the cavity interface. • Examining the accuracy and robustness of the solution method by computing different 3D noncavitating/cavitating flow problems. • Considering the 3D NDGM proposed to assess other solution methods for computing noncavitating/cavitating flows.

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

Numerical simulation of the interaction of wave phase conjugation with bubble clouds

The interaction of the Wave Phase Conjugation (WPC) process with bubble cloud dynamics is investigated via numerical modeling in application to bubble size measurements. The WPC simulator by Modarreszadeh and Timofeev (Computers and Fluids 197, 104353) is used as a starting point. In the simulator, a modified version of the high-order Nodal Discontinuous Galerkin method is used for modeling of acoustic wave propagation in the active WPC conjugator and the surrounding linear or non-linear fluid in 2D/axisymmetric domains. The effects of bubble dynamics are incorporated with the help of induced volume fractions. The modified Keller–Miksis model is employed to simulate vibrations of bubbles subjected to acoustic waves. After thorough verification and validation of the numerical scheme and the physical model, the interaction of various bubble clouds, including poly-disperse ones, with the WPC process is examined with the help of some existing and newly suggested descriptive indicators, e.g., the amplification factor and the absolute and relative signal intensities. The results show that signals scattered from bubble-plane clouds, in which the bubbles are non-physically synced, are strong enough to affect the modulation process if the stimulation frequency is close to the natural frequency of bubbles. For more realistic bubble clouds, the modulation process is almost intact. However, conjugate waves do carry some signatures of bubble-cloud dynamics, e.g., signals with frequencies higher than the natural frequency of bubbles are significantly diminished. The study demonstrates that WPC-based techniques have a promising potential to be used for measuring bubble dimensions.

Algebraic Multigrid Schemes for High-Order Nodal Discontinuous Galerkin Methods

We present algebraic multigrid (AMG) methods for the efficient solution of the linear system of equations stemming from high-order discontinuous Galerkin (DG) discretizations of second-order elliptic problems. For DG methods, standard multigrid approaches cannot be employed because of redundancy of the degrees of freedom associated to the same grid point. We present new aggregation procedures and test them in extensive two-dimensional numerical experiments to demonstrate that the proposed AMG method is uniformly convergent with respect to all of the discretization parameters, namely the mesh-size and the polynomial approximation degree.

An implicit dual-time stepping high-order nodal discontinuous Galerkin method for solving incompressible flows on triangle elements

In this work, a high-order nodal discontinuous Galerkin method (NDGM) is developed and assessed for the simulation of 2D incompressible flows on triangle elements. The governing equations are the 2D incompressible Navier–Stokes equations with the artificial compressibility method. The discretization of the spatial derivatives in the resulting system of equations is made by the NDGM and the time integration is performed by applying the implicit dual-time stepping method. Three numerical fluxes, namely, the local Lax–Friedrich, Roe and AUSM+-up are formulated and applied to assess and compare their accuracy and performance in the simulation of incompressible flows using the NDGM. Several steady and unsteady incompressible flow problems are simulated to examine the accuracy and robustness of the proposed solution methodology and they are the Kovasznay, backward facing step, NACA0012 airfoil, circular cylinder and two side-by-side circular cylinders. Indications are that the NDGM applied for solving the incompressible Navier–Stokes equations with the artificial compressibility approach and the implicit dual-time stepping method is accurate and robust for the simulation of steady and unsteady incompressible flow problems.

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. (2017) [46] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around N=7). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings.

A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations

This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor–Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers' turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax–Friedrichs and Roe-type Riemann solvers. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.

A high-order nodal discontinuous Galerkin method for solution of compressible non-cavitating and cavitating flows

In this work, a high-order nodal discontinuous Galerkin method is applied and assessed for the simulation of compressible non-cavitating and cavitating flows. The one-fluid approach with the thermal effects is used to properly model the cavitation phenomenon. Here, the spatial and temporal derivatives in the system of governing equations are discretized using the nodal discontinuous Galerkin method and the third-order TVD Runge–Kutta method, respectively. Various numerical fluxes such as the Roe, Rusanov, HLL, HLLC and AUSM+-up and two discontinuity capturing methods, namely, the generalized MUSCL limiter and a generalized exponential filter are implemented in the solution algorithm. At first, the sinusoidal density wave problem which has a smooth solution is simulated and the effects of the numerical fluxes on the accuracy and performance of the nodal discontinuous Galerkin method are studied. Two problems, namely, the shock–density interaction (non-cavitating flow) and the two symmetric expansion waves (cavitating flow) are then computed and the effects of the numerical fluxes and the discontinuity capturing methods on the accuracy and computational cost of the solution are investigated. For non-cavitating flows, the high-pressure water–water shock tube and the low-pressure water–water shock tube are also simulated. Then, three cavitating flow problems, namely, the two symmetric expansion waves, the shock-condensation tube and the collapsing cavitation bubble are simulated to assess the accuracy and robustness of the solution algorithm. Results show that the solution methodology based on the high-order NDGM is accurate and robust for simulating the compressible non-cavitating and cavitating flows.

A hierarchical uniformly high order DG-IMEX scheme for the 1D BGK equation

A class of high order nodal discontinuous Galerkin implicit–explicit (DG-IMEX) schemes with asymptotic preserving (AP) property has been developed for the one-dimensional (1D) BGK equation in Xiong et al. (2015) [40], based on a micro–macro reformulation. The schemes are globally stiffly accurate and asymptotically consistent, and as the Knudsen number becomes small or goes to zero, they recover first the compressible Navier–Stokes (CNS) and then the Euler limit. Motivated by the recent work of Filbet and Rey (2015) [27] and the references therein, in this paper, we propose a hierarchical high order AP method, namely kinetic, CNS and Euler solvers are automatically applied in regions where their corresponding models are appropriate. The numerical solvers for different regimes are coupled naturally by interface conditions. To the best of our knowledge, the resulting scheme is the very first hierarchical one being proposed in the literature, that enjoys AP property as well as uniform high order accuracy. Numerical experiments demonstrate the efficiency and effectiveness of the proposed approach. As time evolves, three different regimes are dynamically identified and naturally coupled, leading to significant CPU time savings (more than 80% for some of our test problems).

A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn–Hilliard equation

In this paper, we develop and analyze a nodal discontinuous Galerkin method for the linearized fractional Cahn–Hilliard equation containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative. The linearized fractional Cahn–Hilliard problem has been expressed as a system of low order differential/integral equations. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization using a high-order nodal basis set of orthonormal Lagrange–Legendre polynomials of arbitrary order in space on each element of computational domain. Moreover, we prove the stability and optimal order of convergence N+1 for the linearized fractional Cahn–Hilliard problem when polynomials of degree N are used. Numerical experiments are displayed to verify the theoretical results.

Numerical Solution of Multidimensional Compressible Flow by High Order Nodal Discontinuous Galerkin Method

In this paper we present a robust, high order method for numerical solution of multidimensional compressible inviscid flow equations. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method utilizes the favorable features of Finite Volume Method (FVM) and Finite Element Method (FEM). In this method, space discretization is carried out by finite element discontinuous approximations. The resulting semi discrete differential equations were solved using explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. In this article, we demonstrate the use of exponential filter to remove Gibbs oscillations near the shock waves. Numerical predictions for two dimensional compressible fluid flows are presented here. The solution was obtained with overall order of accuracy of 3. The numerical results obtained are compared with experimental and finite volume method results.

Numerical Solution of Compressible Euler Equations by High Order Nodal Discontinuous Galerkin Method

In this paper we present a robust, high order method for numerical solution of compressible Euler Equations of the gas dynamics. Euler equations are hyperbolic in nature. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method combines mainly two key ideas which are based on the finite volume and finite element methods. In this method, we employ Discontinuous Galerkin (DG) technique for finite element space discretization by discontinuous approximations. Whereas, for temporal discretization, we used explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. We used filter to remove spurious oscillations near the shock waves. Numerical predictions for Shock tube problem (SOD) are presented and compared with exact solution at different polynomial order and mesh sizes. Results show the suitability of DG method for modeling gas dynamics equations and effectiveness of high order approximations.

The use of hybrid meshes to improve the efficiency of a discontinuous Galerkin method for the solution of Maxwell’s equations

This paper investigates the efficiency of a high-order nodal discontinuous Galerkin method for the numerical solution of Maxwell’s equations using hybrid meshes. An unstructured triangular or tetrahedral mesh is used near curved boundaries and a structured Cartesian mesh is used to fill the remainder of the domain. A quadrature-free implementation is employed for the regular quadrilateral and hexahedral elements which, together with the reduction in the number of internal faces, leads to a reduction in the cpu time requirements. Numerical examples in two and three dimensions are used to illustrate the benefits of using hybrid meshes.

DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations

We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated and absorbed in the interior of the computational domain using a flexible relaxation technique applied on the free surface variables.

High-order nodal discontinuous Galerkin particle-in-cell method on unstructured grids

We present a high-order particle-in-cell (PIC) algorithm for the simulation of kinetic plasmas dynamics. The core of the algorithm utilizes an unstructured grid discontinuous Galerkin Maxwell field solver combining high-order accuracy with geometric flexibility. We introduce algorithms in the Lagrangian framework that preserve the favorable properties of the field solver in the PIC solver. Fast full-order interpolation and effective search algorithms are used for tracking individual particles on the general grid and smooth particle shape functions are introduced to ensure low noise in the charge and current density. A pre-computed levelset distance function is employed to represent the geometry and facilitates complex particle–boundary interaction. To enforce charge conservation we consider two different techniques, one based on projection and one on hyperbolic cleaning. Both are found to work well, although the latter is found be too expensive when used with explicit time integration. Examples of simple plasma phenomena, e.g., plasma waves, instabilities, and Landau damping are shown to agree well with theoretical predictions and/or results found by other computational methods. We also discuss generic well known problems such as numerical Cherenkov radiation and grid heating before presenting a few two-dimensional tests, showing the potential of the current method to handle fully relativistic plasma dynamics in complex geometries.