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Multigrid Solver Algorithms for DG Methods and Applications to Aerodynamic Flows

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Abstract

In this chapter we collect results obtained within the IDIHOM project on the development of Discontinuous Galerkin (DG) methods and their application to aerodynamic flows. In particular, we present an application of multigrid algorithms to a higher order DG discretization of the Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Spalart-Allmaras as well as the Wilcox-kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting solver algorithms are characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to IDIHOM test cases, namely theL1T2 high lift configuration and the deltawing of the second Vortex Flow Experiment (VFE-2) with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of number of nonlinear iterations and computing time. Furthermore, higher order DG methods are combined with adaptive mesh refinement, in particular, with residual-based and adjoint-based mesh refinement. These adaptive methods are applied to a subsonic and transonic flow around the VFE-2 delta wing.

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... Here, κ controls the strictness of the line search algorithm, and κ = 1.2 performs well for both viscous and inviscid flows [17]. In addition, the vector entries corresponding to the turbulence working variable are omitted in the evaluation of the norms in Equation (4.6) to enhance convergence [77]. Finally, the CFL number is updated at each iteration according to the value of ω: ...
... For example, Shahbazi et al. [65] reported that the linear multigrid method can be ten times faster than conventional single grid algorithms. On the other hand, Diosady and Darmofal [20], and Wallraff et al. [77] reported cases for which they were not able to achieve a significant speedup factor. ...
Thesis
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High-order accurate numerical discretization methods are attractive for their potential to significantly reduce the computational costs compared to the traditional second-order methods. Among the various unstructured higher-order discretization schemes, the k-exact reconstruction finite volume method is of interest for its straightforward mathematical formulation, and its compatibility with the current lower-order industrial solvers. However, current three-dimensional finite volume solvers are limited to the solution of inviscid and laminar viscous flow problems. Since three-dimensional turbulent flows appear in many industrial applications, the current thesis takes the first step towards the development of a three-dimensional higher-order finite volume solver for the solution of both inviscid and viscous turbulent steady-state flow problems. The k-exact finite volume formulation of the governing equations is rederived in a dimension-independent manner, where the negative Spalart-Allmaras turbulence model is employed. This one-equation model is reasonably accurate for many flow conditions, and its simplicity makes it a good starting point for the development of numerical algorithms. Then, the three-dimensional mesh preprocessing steps for a finite volume simulation are presented, including higher-order accurate numerical quadrature, and capturing the boundary curvature in highly anisotropic meshes. Also, the issues of k-exact reconstruction in handling highly anisotropic meshes are reviewed and addressed. Since three-dimensional problems can require much more memory than their two-dimensional counterparts, solution methods that work in two dimensions might not be feasible in three dimensions anymore. As an attempt to overcome this issue, a practical and parallel scalable method for the solution of the discretized system of nonlinear equations is presented. Finally, the solution of four three-dimensional test problems are studied: Poisson's equation in a cubic domain, inviscid flow over a sphere, turbulent flow over a flat plate, and turbulent flow over an extruded NACA 0012 airfoil. The solution is verified, and the resource consumption of the flow solver is measured. The results demonstrate the benefit and practicality of using higher-order methods for obtaining a certain level of accuracy.
... Here, κ LS controls the strictness of the line search algorithm, and κ LS = 1.2 performs well for both viscous and inviscid flows [23]. In addition, the vector entries corresponding to the turbulence working variable are omitted in the evaluation of the norms in Equation (21) to enhance convergence [25]. Finally, the CFL number is updated at each iteration according to the value of ω: ...
... Nevertheless, the linear system of Equation (25) can be as large as the original linear system Ax = b, and its solution cannot be carried out exactly. Instead of seeking the exact solution, the proposal is to use further inner ILU preconditioned GMRES iterations to find an approximate solution for Equation (25). The resulting preconditioner will be referred to as GMRES-LO-ILU in this work, and its performance will be examined in Section V.A. ...
Conference Paper
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We present a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the k-exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the GMRES method. We propose a preconditioning algorithm based on inner GMRES iterations and lines of control volumes with strong coupling, and show that it can enhance the speed and reduce the memory cost of the solver. Finally, we verify the developed finite volume algorithm by solving a set of test problems, where we attain optimal solution convergence with mesh refinement.
... The issue of developing optimal solvers for Composite discontinuous Galerkin Methods, first developed and analyzed Antonietti et al. [10] was considered by Antonietti et al. [11,12]. More recently Antonietti et al. [13] analysed multigrid strategies for Interior Penalty dG discretizations over agglomerated elements meshes, while Wallraff and Leicht [14] and Wallraff et al. [15] applied an agglomeration based h-multigrid solver to dG discretizations of the compressible Reynolds Averaged Navier-Stokes (RANS) equations. ...
Preprint
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... In the presence of turbulence, the residual of the turbulence working variable is usually ill-scaled and may accordingly affect the efficiency of the line search algorithm [28]. To enhance convergence, one potential approach is to leave out the vector entries corresponding to the turbulence working variable [29] in the evaluation of the norms in Equation (18). This we shall explore in Section V.A. ...
Article
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This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the k-exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.
... Fidkowski et al. [18] proposed to use the element line Jacobi smoother to improve the convergence of P -multigrid methods for high Reynolds number flow simulation with stretched grids. To further accelerate convergence, P -multigrid methods can be combined with the geometric multigrid methods [19,50]. We also note that P -multigrid methods can serve as preconditioners for Newton-Krylov methods [39,51]. ...
Preprint
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We develop a P -multigrid solver to simulate locally preconditioned unsteady compressible Navier-Stokes equations at low Mach numbers with implicit high-order methods. Specifically, the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method is employed for spatial discretization and the high-order time integration is conducted by means of the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) method. Local preconditioning is used to alleviate the stiffness of the compressible Navier-Stokes equations at low Mach numbers and is only conducted in pseudo transient continuation to ensure the high-order accuracy of ESDIRK methods. We employ the element Jacobi smoother to update the solutions at different P -levels in the P -multigrid solver. High-order spatiotemporal accuracy of the new solver for low-Mach-number flow simulation is verified with the isentropic vortex propagation when the Mach (Ma) number of the free stream is 0.005. The impact of the hierarchy of polynomial degrees on the convergence speed of the P -multigrid method is studied via several numerical experiments, including two dimensional (2D) inviscid and viscous flows over a NACA0012 airfoil at Ma=0.001\text{Ma} = 0.001, and a three dimensional (3D) inviscid flow over a sphere at Ma=0.001\text{Ma} = 0.001. The P -multigrid solver is then applied to coarse resolution simulation of the transitional flows over an SD7003 wing at 8 8^\circ angle of attack when the Reynolds number is 60000 and the Mach number is 0.1 or 0.01.
... Those authors focused on the analysis of a p-multigrid (p-MG) non-linear solver, proving convergence properties and performance in the context of compressible flows using element-or line-Jacobi smoothing. Several authors also considered multigrid operators built on agglomerated coarse grids, such as h-multigrid, see for example [15,16,17]. The possibil-2 ity of using multigrid operators as a preconditioner was also explored in the context of steady compressible flows, see for example [18,19]. ...
Preprint
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In recent years several research efforts focused on the development of high-order discontinuous Galerkin (dG) methods for scale resolving simulations of turbulent flows. Nevertheless, in the context of incompressible flow computations, the computational expense of solving large scale equation systems characterized by indefinite Jacobian matrices has often prevented from dealing with industrially-relevant computations. In this work we seek to improve the efficiency of Rosenbrock-type linearly-implicit Runge-Kutta methods by devising robust, scalable and memory-lean solution strategies. In particular, we combined p-multigrid preconditioners with matrix-free Krylov iterative solvers: the p-multigrid preconditioner relies on specifically crafted rescaled-inherited coarse operators and cheap block-diagonal smoother's preconditioners to obtain satisfactory convergence rates and a low memory occupation. Extensive numerical validation is performed. The rescaled-inherited p-multigrid algorithm for the BR2 dG discretization is firstly validated solving Poisson problems. The Rosenbrock formulation is then applied to test cases of growing complexity: the laminar unsteady flow around a two-dimensional cylinder at Re=200 and around a sphere at Re=300, and finally the transitional T3L1 flow problem of the ERCOFTAC test case suite with different levels of free-stream turbulence. For the latter good agreement with experimental data is documented, moreover, strong memory savings and execution time gains with respect to state-of-the art preconditioners are reported.
... The issue of developing optimal solvers for Composite discontinuous Galerkin Methods, first developed and analyzed Antonietti et al. [10] was considered by Antonietti et al. [11,12]. More recently Antonietti et al. [13] analysed multigrid strategies for Interior Penalty dG discretizations over agglomerated elements meshes, while Wallraff and Leicht [14] and Wallraff et al. [15] applied an agglomeration based h-multigrid solver to dG discretizations of the compressible Reynolds Averaged Navier-Stokes (RANS) equations. ...
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... DG methods are particularly attractive for the following features: (1) a great geometrical flexibility without spoiling the high-order accuracy, see Bassi et al. (2012b), Luo, Baum, and Lohner (2008), and Bassi et al. (2010), (ii) a straightforward implementation of h/p adaptive techniques, see Hartmann et al. (2010), and Wang and Mavriplis (2009), (iii) a compact stencil, suited to exploit massively parallel computers platform. The higher accuracy comes at an increased computational cost with respect to standard finite volume (FV) methods, see Sørensen et al. (2015), preventing a widespread application in industry, even if a considerable research effort has been recently devoted to devise more efficient computational strategies, see , Ghidoni (2014), Crivellini and Bassi (2011), Wallraff, Leicht, and Lange-Hegermann (2013), and Wallraff, Hartmann, and Leicht (2015). The objective of this work is to assess the ability of high-order DG methods to simulate complex threedimensional (3D) turbulent turbomachinery flows characterised by the mixing of main and secondary flows, leakage and blockage phenomena and the possible presence of shock waves and transitional boundary layers. ...
Article
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The only implicit smoothing method implemented in the DLR Flow Solver TAU is the LU-SGS method. It was chosen several years ago because of its low memory requirements and low operation counts. Since in the past for many examples a severe restriction of the CFL number and loss of robustness was observed, it is the goal of this paper to revisit the LU-SGS implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. Starting point is a full implicit multistage Runge-Kutta method. Based on this method we develop and suggest several additional features and simplifications such that the implicit method is applicable to high Reynolds number viscous flows, that is the required matrices fit into the fast memory of our cluster hardware and the arising linear systems can be approximately solved efficiently. To this end we focus on simplifications of the Jacobian as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods we take care of grid anisotropy for both approximately solving the linear systems and agglomeration strategy. The procedure creating coarse grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse grid meshes in the way that an overall better reliability of multigrid can be observed. Furthermore we exploit grid information within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.
Conference Paper
A comparison of nonlinear and linear p- and h-multigrid algorithms will be presented with respect to both algorithmic convergence properties and run-time behavior. In addi- tion to that a comparison with a single-level solver, namely a Backward-Euler method, will be presented as well. The algorithms will be used to solve the RANS-equations in combination with a kω-ω{closed} turbulence models for CFD applications in the area of compressible aerodynamic ows with high Reynolds numbers. The h-multigrid algorithms are formu- lated on agglomerated unstructured meshes. Results will be presented on both structured and unstructured meshes.
Article
We present an eigen‐decomposition of the quasi‐linear convective flux formulation of the completely coupled Reynolds‐averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p‐multigrid algorithm. To this end, we formulate a framework with a backward‐Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix‐free implementations and memory‐lean line‐Jacobi preconditioners and compare the effects of some parameter choices. In particular, p‐multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.
Article
In this work we exploit the flexibility associated to discontinuous Galerkin methods to perform high-order discretizations of the Euler and Navier–Stokes equations on very general meshes obtained by means of agglomeration techniques. Agglomeration is here considered as an effective mean to decouple the geometry representation from the solution approximation being alternative to standard isoparametric or breakthrough isogeometric discretizations. The mesh elements can be composed of a set of standard sub-elements belonging to a fine low-order mesh whose cardinality can be freely chosen according to the domain discretization capabilities. The number of mesh sub-elements still have an impact on the cost of numerical integration. Since the agglomerated elements are arbitrarily shaped, physical space orthonormal basis functions are here considered as a key ingredient to build a suitable discrete dG space. As a result we are allowed to perform high-order discretizations on top of the set of (possibly) subparametric sub-cells composing an agglomerated element. Our approach is validated on challenging viscous and inviscid test cases. We demonstrate the use of low-order meshes as a starting point to obtain high-order accurate solutions on coarse meshes.
Article
A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: (1) conventional κ-ε and κ-ω+2$/ formulations generally are inaccurate for boundary layers in adverse pressure gradient; (2) using 'wall functions' tends to mask the shortcomings of such models; and (3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows.
Article
A simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented. The approach consists of solving for the natural logarithm of the turbulence variables, which are known to be strictly positive. The approach is valid for any numerical scheme, be it a finite difference, a finite volume, or a finite element method. The work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method. The turbulence equations in logarithmic variables are presented for the standard k-ε model. Error estimation and mesh adaptation procedures are described. The formulation is validated on a shear layer case for which an analytical solution is available. This provides a framework for rigorous comparison of the proposed approach with the standard solution technique, which makes use of k and ε as dependent variables. The approach is then applied to solve turbulent flow over a NACA 0012 airfoil for which experimental measurements are available.
Article
Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high-order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p-multigrid solution strategy has been developed, which is based on a semi-implicit Runge–Kutta smoother for high-order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p-multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 John Wiley & Sons, Ltd.
Article
In this article we present the extension of the a posteriori error estimation and goal-oriented mesh refinement approach from laminar to turbulent flows, which are governed by the Reynolds-averaged Navier-Stokes and k-ω turbulence model (RANS-kω) equations. In particular, we consider a discontinuous Galerkin discretization of the RANS-kω equations and use it within an adjoint-based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients. The accuracy of the error estimation and the performance of the goal-oriented mesh refinement algorithm is demonstrated for various test cases, including a two-dimensional turbulent flow around a three-element high lift configuration and a three-dimensional turbulent flow around a wing-body configuration.
Article
Over the last years, the discontinuous Galerkin method (DGM) has demonstrated its excellence in accurate, high-order numerical simulations for a wide range of applications in computational physics. However, the development of practical, computationally efficient flow solvers for industrial applications is still in the focus of active research. This paper deals with solving the Navier-Stokes equations describing the motion of three-dimensional, viscous compressible fluids. We present details of the PADGE code under development at the German Aerospace Center (DLR) that is aimed at large-scale applications in aerospace engineering. The discussion covers several advanced aspects like the solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, a curved boundary representation, anisotropic mesh adaptation for reducing output error and techniques for solving the nonlinear algebraic equations. The performance of the solver is assessed for a set of test cases.
Conference Paper
The objectives of the International Vortex Flow Experiment 2 (VFE-2) are to perform new wind-tunnel tests on a delta wing by using modern measurement techniques and to compare these data with results of numerical state-of-the-art codes. In the present paper results of the first part of the VFE-2 experiments carried out at DLR Göttingen on a NASA wind tunnel model are described. These investigations comprise pressure distribution measurements by means of the Pressure Sensitive Paint (PSP) technique which captures the complete model surface giving much more insights in details of the flow topology than would be possible by discrete pressure taps. The delta wing of 65° sweep angle was equipped with sharp as well as with rounded leading edges. The pressure distributions are analyzed for two Mach numbers M = 0.4 and 0.8 and two Reynolds numbers 2 and 3 million. Dependent on the angle of attack (10º -25º) the aerodynamics are described with emphasis to secondary vortex formation and vortex breakdown. For the rounded leading edge the aerodynamics are also described with regard to attached flow, onset of outer primary vortex and particularly the formation of an inner primary vortex.
Article
In this study we present a solution method for the compressible Navier–Stokes equations as well as the Reynolds-averaged Navier–Stokes equations (RANS) based on a discontinuous Galerkin (DG) space discretisation. For the turbulent computations we use the standard Wilcox k–ω or the Spalart–Allmaras model in order to close the RANS system. We currently apply either a local discontinuous Galerkin (LDG) or one of the Bassi–Rebay formulations (BR2) for the discretisation of second-order viscous terms. Both approaches (LDG and BR2) can be advanced explicitly as well as implicitly in time by classical integration methods. The boundary is approximated in a continously differentiable fashion by curved elements not to spoil the high-order of accuracy in the interior of the flow field.Computations are performed for the circular cylinder, the flat plate and classical airfoil sections like NACA0012. We compare our obtained results with experimental and computational data as well as analytical (boundary layer) predictions for the flat plate. The excellent parallelisation characteristics of the scheme are demonstrated, achieved by hiding communication latency behind computation.
Article
A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the compressible Euler equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The developed method is used to compute a variety of both steady-state and time-accurate flow problems on arbitrary grids. The numerical results demonstrated the superior accuracy of this discontinuous Galerkin method in comparison with a second order finite volume method and a third-order WENO method, indicating its promise and potential to become not just a competitive but simply a superior approach than its finite volume and ENO/WENO counterparts for solving flow problems of scientific and industrial interest.
Article
Discontinuous Galerkin methods, originally developed in the advective case, have been successively extended to advection–diffusion problems, and are now used in very diverse applications. We here consider the numerical solution of the compressible Reynolds-averaged Navier–Stokes and k–ω turbulence model equations by means of DG space discretization and implicit time integration. Detailed description of the DG discretization of the viscous part of the equations and of several implementation details of the k–ω turbulence model are given. To assess the performance of the proposed methodology we present the results obtained in the computation of the turbulent flow over a flat plate and of the turbulent unsteady wake developing behind a turbine blade.
Article
We present a p-multigrid solution algorithm for a high-order discontinuous Galerkin finite element discretization of the compressible Navier–Stokes equations. The algorithm employs an element line Jacobi smoother in which lines of elements are formed using coupling based on a p = 0 discretization of the scalar convection–diffusion equation. Fourier analysis of the two-level p-multigrid algorithm for convection–diffusion shows that element line Jacobi presents a significant improvement over element Jacobi especially for high Reynolds number flows and stretched grids. Results from inviscid and viscous test cases demonstrate optimal hp + 1 order of accuracy as well as p-independent multigrid convergence rates, at least up to p = 3. In addition, for the smooth problems considered, p-refinement outperforms h-refinement in terms of the time required to reach a desired high accuracy level.
Article
In this work we show that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of arbitrarily shaped elements. Specifically, we propose and investigate a new h-adaptive technique based on agglomeration coarsening of a fine mesh. The main building block of our dG method consists of defining discrete polyno- mial spaces on arbitrarily shaped elements. For this purpose we orthonormalize with respect to the L2-product a set of monomials relocated in a specific element frame. This procedure provides high-order hierarchical physical space basis functions that are also optimal from the point of view of conservation property. To complete the dG formulation for second order problems, two extensions of the BRMPS scheme to arbitrary polyhedral grids, including a sharp estimate of the stabilization parameter ensuring the coercivity property, are here proposed. The freedom in defining the mesh topology leads to a new, agglomeration-based, mesh adaptivity approach, which is validated on a Poisson problem. The possibility to enhance the error distribution over the computational domain is investigated with the goal of obtaining a mesh independent discretization. The grid is considered as a degree of freedom of the computation and the nodes connectivity is decided on the fly as is usually done in mesh-free implementations. Finally, we propose an easy way to reduce the cost related to numerical integration on agglomerated meshes.
A high-order accurate discontinuous Finite Element method for inviscid and viscous turbomachinery flows
  • F Bassi
  • S Rebay
  • G Mariotti
  • S Pedinotti
  • M Savini
Analysis of PSP results obtained for the VFE-2 65° delta wing configuration at sub-and transonic speeds
  • R Konrath
  • C Klein
  • R Engler
  • D Otter
3D application of higher order multigrid algorithms for a RANS-kω DG-solver
  • M Wallraff
  • T Leicht
  • R Abgrall
  • H Beaugendre
  • P M Congedo
  • C Dobrzynski
  • V Perrier
  • R Hartmann
  • A Dillmann
  • G Heller
  • H.-P Kreplin
  • W Nitsche