Modal Discontinuous Galerkin Research Articles

A modal discontinuous Galerkin method for simulating dusty and granular gas flows in thermal non-equilibrium in the Eulerian framework

A modal discontinuous Galerkin method was developed for computing compressible rarefied gaseous flows interacting with rigid particles and granular medium. In contrast to previous particle-based models that were developed to handle rarefied flows or solid phase particles, the present computational method employs full continuum-based models. This work is one of the first attempts to apply the modal discontinuous Galerkin method to a two-fluid model framework, which covers a wide range of gas and solid phase regimes, from a continuum to non-equilibrium gas, and from dusty to collisional regimes. The rarefaction effects were taken into account by applying the second-order Boltzmann-Curtiss-based constitutive relationship in a two-fluid system of equations. For the dust phase, computational models were developed based on the kinetic theory of the granular flows. Due to the orthogonal property of the basis functions in the method, no specific treatment of the source terms, commonly necessary in the conventional finite volume method, was required. Moreover, a high-fidelity approach was selected to treat the non-strictly hyperbolic equations of a dusty gas. This allows the same inviscid numerical flux functions to be applied to both the gaseous Euler and solid pressureless-Euler system of equations. Further, we observed that, for the discretization of the viscous fluxes in multiphase cases, the local discontinuous Galerkin is superior to the first method by Bassi and Rebay. After a verification and validation study, the new computational model was used to simulate the impingement of an underexpanded jet on a dusty surface in a rarefied condition. A surface erosion model based on viscous erosion associated with aerodynamic entrainment was implemented at a solid surface. Simulation cases in the near-field of the nozzle flow were tested to evaluate the capabilities of the present computational model in handling the challenging problems of multi-scale multiphase flows.

Effect of Strong Electric Fields on Material Responses: The Bloch Oscillation Resonance in High Field Conductivities.

In this paper, we investigated the effect of strong electric fields on material responses and the Bloch oscillation resonance in high field conductivities. For this purpose, a high-order accurate explicit modal discontinuous Galerkin (DG) solver is employed for solving the quantum Boltzmann transport equation (BTE) in the context of electron transport at nanoscales under strongly out-of-equilibrium conditions. Here, we study the transient behavior and the convergence of a steady-state response to an external oscillating electric field switched on at time zero. We first benchmark our numerical results with known analytic steady-state responses at low fields. The computational results show that the present DG scheme is in excellent agreement with analytic solutions over the whole range of parameters and to an extremely high precision, allowing us to achieve good agreement even for the fifth-order response at low fields. We then extend the method to strong electric fields and show how the responses are deviated from the low-field ones and the transition to a dampened Bloch oscillation regime. Most importantly, we report the observation of a new regime induced by the resonance between the standard low-field response and Bloch oscillations.

Supersonic turbulent flow simulation using a scalable parallel modal discontinuous Galerkin numerical method

The scalability and efficiency of numerical methods on parallel computer architectures is of prime importance as we march towards exascale computing. Classical methods like finite difference schemes and finite volume methods have inherent roadblocks in their mathematical construction to achieve good scalability. These methods are popularly used to solve the Navier-Stokes equations for fluid flow simulations. The discontinuous Galerkin family of methods for solving continuum partial differential equations has shown promise in realizing parallel efficiency and scalability when approaching petascale computations. In this paper an explicit modal discontinuous Galerkin (DG) method utilizing Implicit Large Eddy Simulation (ILES) is proposed for unsteady turbulent flow simulations involving the three-dimensional Navier-Stokes equations. A study of the method was performed for the Taylor-Green vortex case at a Reynolds number ranging from 100 to 1600. The polynomial order P = 2 (third order accurate) was found to closely match the Direct Navier-Stokes (DNS) results for all Reynolds numbers tested outside of Re = 1600, which had a normalized RMS error of 3.43 × 10−4 in the dissipation rate for a 603 element mesh. The scalability and performance study of the method was then conducted for a Reynolds number of 1600 for polynomials orders from P = 2 to P = 6. The highest order polynomial that was tested (P = 6) was found to have the most efficient scalability using both the MPI and OpenMP implementations.

A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for solid dynamics

We present a new multidimensional high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher-order elements. We also present new one-dimensional test problems that have an analytic solution corresponding to a hyperelastic–plastic wave. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated high-order Taylor series polynomials. The stress fields are represented using nodal quantities. The constitutive models used to calculate the deviatoric stress are either a hypoelastic–plastic, infinitesimal strain hyperelastic–plastic, or finite strain hyperelastic–plastic model. These constitutive models require new methods for calculating high-order polynomials for the velocity gradient and deformation gradient in an element. The plasticity associated with the strength model is determined using a radial return method with a J2 yield criterion and perfect plasticity. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. A diverse suite of 1D and 2D test problems are calculated. The new 1D piston test problems, which have analytic solutions for each elastic–plastic model, are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new Lagrangian DG method. 2D test problems are calculated to demonstrate the stability and robustness of the new Lagrangian DG method on multidimensional problems with high-order elements, which have faces that can bend.

A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme

We present a Lagrangian discontinuous Galerkin (DG) hydrodynamic method that is up to third-order accurate using subcell mesh stabilization (SMS) for compressible flows on quadratic meshes in two-dimensional (2D) Cartesian coordinates. Similar to the second-order accurate Lagrangian DG method with linear meshes, the physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with Taylor series polynomials. The Riemann velocity at the vertices of a curvilinear cell, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem. Curvilinear cells (e.g., quadratic quadrilateral meshes in this work) have many deformational degrees of freedom, and with these cells, they can deform in unphysical ways. Likewise, the Riemann solution at an edge vertex differs from the one at the corner of a cell. With SMS, each quadratic quadrilateral cell is decomposed into four quadrilateral subcells, that move in a Lagrangian manner. The edge vertex is surrounded by four subcells so that it is similar to the vertex at the cell corner. SMS can detect inconsistent density fields between the cell and subcells. The difference between these two density fields is used to correct the stress (pressure) input to the Riemann solver. This SMS scheme enables stable mesh motion and accurate solutions in the context of a Lagrangian high-order DG method that is up to third-order with quadratic cells. We also present effective limiting strategies that ensure monotonicity of the primitive variables with the high-order DG method. This Lagrangian DG hydrodynamic method with SMS conserves mass, momentum, and total energy. A suite of test problems are calculated to demonstrate the designed order of accuracy (up to third-order accurate) of this method, and that the Lagrangian DG method using SMS preserves cylindrical symmetry on 1D radial flow problems with an equal-angle polar quadratic mesh.

혼합 Modal 불연속 Galerkin 기법을 이용한 Lid-Driven Cavity 내의 희박유동 전산 연구

With the advancement of fabrication technology and miniaturization, fluid flows at micro- and nanoscales has received considerable attention. Flow characteristics in these systems significantly vary from those of macro- scale devices, due to geometric restrictions. In such cases, Navier-Stokes-Fourier(NSF) equations with no-slip condition may not be valid for studying gas flows inside a micro- cavity. This article therefore investigates the cavity flows using modified NSF equations with velocity slip and temperature jump conditions. In the present case, a monatomic gas is considered for modelling gas flows. To accurately predict the flow physics, mixed modal discontinuous Galerkin(DG) method is being developed. The flow characteristics of monatomic gas is studied by varying the Reynolds number(Re) and Knudsen number(Kn), respectively. Results obtained are compared with the previous results and are found to be in good agreement with them.

A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for elastic–plastic flows

We present a new high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas and solid dynamics. The evolution equations for specific volume, momentum, and total energy are discretized using the modal DG approach. The specific volume, velocity, and specific total energy fields are approximated with up to quadratic Taylor series polynomials. The specific internal energy, pressure, and stress deviators are nodal quantities. The stress deviators are evolved forward in time using a hypoelastic–plastic approach, which requires a velocity gradient. A new method is presented for calculating a high-order polynomial for the velocity gradient in an element. Plasticity is handled by applying a radial return model to the stress deviators. Limiting approaches are presented for modal and nodal fields. The TVD RK time integration method is used to temporally advance all governing evolution equations. Generalized Lagrangian DG equations are derived but test problems are calculated for 1D Cartesian coordinates. A suite of gas and solid dynamics test problem results are calculated to demonstrate the stability and formal accuracy of the new Lagrangian DG method.

Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates

We present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem in 2D Cartesian geometry. This area-weighted approach, in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. A suite of test problems are calculated to demonstrate stable mesh motion, the expected second order accuracy of these methods, and that the new area-weighted DG method preserves spherical symmetry on 1D radial flow problems with equal-angle polar meshes.

Off-fault plasticity in three-dimensional dynamic rupture simulations using a modal Discontinuous Galerkin method on unstructured meshes: implementation, verification and application

The dynamics and potential size of earthquakes depend crucially on rupture transfers between adjacent fault segments. To accurately describe earthquake source dynamics, numerical models can account for realistic fault geometries and rheologies such as nonlinear inelastic processes off the slip interface. We present implementation, verification and application of off-fault Drucker–Prager plasticity in the open source software SeisSol (www.seissol.org). SeisSol is based on an arbitrary high-order derivative modal Discontinuous Galerkin method using unstructured, tetrahedral meshes specifically suited for complex geometries. Two implementation approaches are detailed, modelling plastic failure either employing subelemental quadrature points or switching to nodal basis coefficients. At fine fault discretizations, the nodal basis approach is up to six times more efficient in terms of computational costs while yielding comparable accuracy. Both methods are verified in community benchmark problems and by 3-D numerical h- and p-refinement studies with heterogeneous initial stresses. We observe no spectral convergence for on-fault quantities with respect to a given reference solution, but rather discuss a limitation to low-order convergence for heterogeneous 3-D dynamic rupture problems. For simulations including plasticity, a high fault resolution may be less crucial than commonly assumed, due to the regularization of peak slip rate and an increase of the minimum cohesive zone width. In large-scale dynamic rupture simulations based on the 1992 Landers earthquake, we observe high rupture complexity including reverse slip, direct branching and dynamic triggering. The spatiotemporal distribution of rupture transfers are altered distinctively by plastic energy absorption, correlated with locations of geometrical fault complexity. Computational cost increases by 7 per cent when accounting for off-fault plasticity in the demonstrating application. Our results imply that the combination of fully 3-D dynamic modelling, complex fault geometries and off-fault plastic yielding is important to realistically capture dynamic rupture transfers in natural fault systems.

Implicit-Modal Discontinuous Galerkin Scheme for Two-Phase Flow with Discontinuous Capillary Pressure

In this paper, we propose an implicit-modal discontinuous Galerkin scheme for solving an incompressible and immiscible oil-water flow through a heterogeneous reservoir with different rock types. The scheme uses the modal discontinuous Galerkin method for spatial discretization of both pressure and saturation equations and the backward Euler method for temporal discretization. In the modal discontinuous Galerkin method, we apply the well-localized bases that are constructed by both the nodal and modal basis functions. To find the approximate solution for the saturation equation, we propose the upwind technique. In the stability analysis of the scheme, we obtain the constant of the trace inequality for the approximate function and its first derivatives using the modal basis functions. In this way, we give suitable ranges for penalty terms. To guarantee the convergence of the approximate solution, we find suitable ranges of the time step. To demonstrate the applicability of the proposed scheme, we apply it to the dense nonaqueous phase liquids and the Versuchseinrichtung zur Grundwasser und Altlastensanierung infiltration problems. Numerical results are presented.

Non-equilibrium effects of diatomic and polyatomic gases on the shock-vortex interaction based on the second-order constitutive model of the Boltzmann-Curtiss equation

The rotational mode of molecules plays a critical role in the behavior of diatomic and polyatomic gases away from equilibrium. In order to investigate the essence of the non-equilibrium effects, the shock-vortex interaction problem was investigated by employing an explicit modal discontinuous Galerkin method. In particular, the first- and second-order constitutive models for diatomic and polyatomic gases derived rigorously from the Boltzmann-Curtiss kinetic equation were solved in conjunction with the physical conservation laws. As compared with a monatomic gas, the non-equilibrium effects result in a substantial change in flow fields in both macroscale and microscale shock-vortex interactions. Specifically, the computational results showed three major effects of diatomic and polyatomic gases on the shock-vortex interaction: (i) the generation of the third sound waves and additional reflected shock waves with strong and enlarged expansion, (ii) the dominance of viscous vorticity generation, and (iii) an increase in enstrophy with increasing bulk viscosity, related to the rotational mode of gas molecules. Moreover, it was shown that there is a significant discrepancy in flow fields between the microscale and macroscale shock-vortex interactions in diatomic and polyatomic gases. The quadrupolar acoustic wave source structures, which are typically observed in macroscale shock-vortex interactions, were not found in any microscale shock-vortex interactions. The physics of the shock-vortex interaction was also investigated in detail to examine vortex deformation and evolution dynamics over an incident shock wave. A comparative study of first- and second-order constitutive models was also conducted for the enstrophy and dissipation rate. Finally, the study was extended to the shock-vortex pair interaction case to examine the effects of pair interaction on vortex deformation and evolution dynamics.

Three-dimensional plasma actuation for faster transition to turbulence

We demonstrate that a 3D non-linear plasma actuation method creates secondary instabilities by forming lambda vortices for a spatially developing turbulent boundary layer flow over a flat plate. Both bypass transition and controlled transition processes are numerically investigated using wall resolved modal discontinuous Galerkin based implicit large eddy simulation. The largest momentum thickness based Reynolds numbers tested are 1250 and 1100 for the bypass transition and the controlled transition, respectively. The 3D actuation method is based on a square serpentine plasma actuator (Durscher and Roy 2012 J. Phys. D: Appl. Phys. 45 035202). The transition is achieved via oblique wave transition by perturbing the flow at a frequency of 1 kHz with amplitude of 10% of the freestream velocity. Although the flow is perturbed at a single frequency, the instabilities arising due to the nonlinear interaction between the consecutive lambda vortices, creates subharmonic lambda vortices (half of the fundamental frequency), which finally break down into fully turbulent flow. These interactions have been thoroughly studied and discussed. Since the actuation creates oblique wave transition it will allow faster transition compared to the standard secondary instability mechanism with similar disturbance amplitude reducing the amount of energy input required for flow control.

Development of a multiscale LES model in the context of a modal discontinuous Galerkin method

This paper introduces a variational multiscale simulation (VMS) approach in the context of high-order discontinuous Galerkin (DG) discretizations. First, a new calibration of the Smagorinsky model parameter (CSΔ) is proposed in terms of the DG discretization (grid size, h, and polynomial order, p) and the selected VMS scale partition. Second, an efficient and simple implementation of the VMS version of the Smagorinsky model is presented. The dissipation properties of the DG discretizations for different values of h and p are then analysed based on Taylor–Green vortex computations in the limit of inviscid flow. The low levels of dissipation found for high-order DG discretizations emphasize the suitability of this type of discretization for LES. The performance of the presented approach is demonstrated by comparing DG/VMS computations of the Taylor–Green vortex at Re=3000 with a reference spectral DNS.

Study on flow and heat transfer of small scale gas flow for air cooling system

The small scale pipe plays an important role in the heat and mass transfer of industrial application. For the study of the flow characteristics in small scale pipe, a modal mixed Discontinuous Galerkin (DG) method was proposed for solving axial-symmetrical Navier-Stokes (NS) equations with solid slip boundary conditions. A new limiter of modal DG was proposed. And, the special treatments of velocity slip and temperature jump in wall solid boundary were given in the numerical scheme. Extensive numerical studies for the flow characteristics of small scale pipe were conducted. For validation, the comparison between the present study and experiment data were performed. The heat transfer coefficients and frictional factor are found to be increased with the diameter decreasing. Furthermore, it is observed that the frictional factor of small scale pipe is larger than that of macro scale pipe. And, the proposed DG-NS method provides a new way for the investigation of the flow characteristics in small scale flow.

A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases

The discontinuous Galerkin (DG) method has been popular as a numerical technique for solving the conservation laws of gas dynamics. In the present study, we develop an explicit modal DG scheme for multi-dimensional conservation laws on unstructured triangular meshes in conjunction with non-Newtonian implicit nonlinear coupled constitutive relations (NCCR). Special attention is given to how to treat the complex non-Newtonian type constitutive relations arising from the high degree of thermal nonequilibrium in multi-dimensional gas flows within the Galerkin framework. The Langmuir velocity slip and temperature jump conditions are also implemented into the two-dimensional DG scheme for high Knudsen number flows. As a canonical scalar case, Newtonian and non-Newtonian convection–diffusion Burgers equations are studied to develop the basic building blocks for the scheme. In order to verify and validate the scheme, we applied the scheme to a stiff problem of the shock wave structure for all Mach numbers and to the two-dimensional hypersonic rarefied and low-speed microscale gas flows past a circular cylinder. The computational results show that the NCCR model yields the solutions in better agreement with the direct simulation Monte Carlo (DSMC) data than the Newtonian linear Navier–Stokes–Fourier (NSF) results in all cases of the problem studied.

Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier–Stokes equations

We compare different block preconditioners in the context of parallel time adaptive higher order implicit time integration using Jacobian-free Newton–Krylov (JFNK) solvers for discontinuous Galerkin (DG) discretizations of the three dimensional time dependent Navier–Stokes equations. A special emphasis of this work is the performance for a relative high number of processors, i.e. with a low number of elements on the processor. For high order DG discretizations, a particular problem that needs to be addressed is the size of the blocks in the Jacobian. Thus, we propose a new class of preconditioners that exploits the hierarchy of modal basis functions and introduces a flexible order of the off-diagonal Jacobian blocks. While the standard preconditioners ‘block Jacobi’ (no off-blocks) and full symmetric Gauss–Seidel (full off-blocks) are included as special cases, the reduction of the off-block order results in the new scheme ROBO-SGS. This allows us to investigate the impact of the preconditioner’s sparsity pattern with respect to the computational performance. Since the number of iterations is not well suited to judge the efficiency of a preconditioner, we additionally consider CPU time for the comparisons. We found that both block Jacobi and ROBO-SGS have good overall performance and good strong parallel scaling behavior.

A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods. Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.

Polymorphic nodal elements and their application in discontinuous Galerkin methods

In this work, we discuss two different but related aspects of the development of efficient discontinuous Galerkin methods on hybrid element grids for the computational modeling of gas dynamics in complex geometries or with adapted grids. In the first part, a recursive construction of different nodal sets for hp finite elements is presented. They share the property that the nodes along the sides of the two-dimensional elements and along the edges of the three-dimensional elements are the Legendre–Gauss–Lobatto points. The different nodal elements are evaluated by computing the Lebesgue constants of the corresponding Vandermonde matrix. In the second part, these nodal elements are applied within the modal discontinuous Galerkin framework. We still use a modal based formulation, but introduce a nodal based integration technique to reduce computational cost in the spirit of pseudospectral methods. We illustrate the performance of the scheme on several large scale applications and discuss its use in a recently developed space-time expansion discontinuous Galerkin scheme.