Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.Comment: Submitted to JC

Development of a Data-Driven Wall Model for Separated flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior. However, they remain prohibitively expensive at high Reynolds numbers or actual complex geometries. Most of the cost is associated with the resolution of the boundary layer, and therefore, to save computational resources, wall-modeled LES (wmLES) has become a valued tool. However, wall models are not yet reliable in predicting the complex flow configurations occurring in turbomachinery passages. Most existing analytical wall models assume the flow to be fully turbulent, attached, flow aligned, and near-equilibrium. These assumptions no longer hold when different flow regimes and complex flow features coexist. Although significant progress has been made in recent years (e.g., non-equilibrium models using pressure gradients), they have not always brought a clear benefit for such realistic flows. This paper proposes an innovative data-driven wall model to treat separated flows. Among the many possibilities to solve this complex regression problem, deep neural networks have been selected for their universal approximation capabilities~\cite{hornik_approximation_1991}. In the present framework, the two-dimensional periodic hill problem is selected as a reference test case featuring the separation of a fully turbulent boundary layer. Gaussian Mixture Neural networks (GMN) and Convolutional Neural Networks (CNN) combined with a self-attention layer~\cite{Vaswani_sel_attention_2017} are trained to predict the wall-parallel components of the wall shear stress using instantaneous flow quantities and geometric parameters. The \textit{a priori} and \textit{a posteriori} validation of such data-driven wall models on the periodic hill problem will be presented.9. Industry, innovation and infrastructur

Machine Learning for wall modeling in LES of separating flows

Large Eddy Simulations (LES) are of increasing interest for turbomachinery design since they provide a more reliable prediction of flow physics and component behavior than standard RANS simulations. However, they remain prohibitively expensive at high Reynolds numbers or realistic geometries. The cost of resolving the near-wall region has justified the development of wall-modeled LES (wmLES), which uses a wall model to account for the effect of the energetic near-wall eddies. The classical assumptions of algebraic wall models do not hold for more complex flow patterns that frequently occur in turbomachinery passages (i.e., misalignment, separation). This work focuses on the extension of wall models to the separation phenomenon. Among possibilities to solve the complex regression problem (i.e., predicting the wall-parallel components of the shear stress from instantaneous flow data and geometrical parameters), neural networks have been selected for their universal approximation capabilities. Since DNS and LES perform well on academic and several industrial configurations, they are used to produce databases to train various neural networks. In the present work, we investigate the possibility of using neural networks to improve wall-shear stress models for flows featuring severe pressure gradients and separation. The database is composed of three building-blocks flows: (1) a flow aligned turbulent boundary layer at equilibrium; (2) a turbulent boundary layer subjected to a moderate pressure gradient; and (3) a turbulent boundary layer that separates and reattaches from a curved wall. These building blocks are referred to as a channel flow at a friction Reynolds number of 950 and the two walls (i.e., the flat upper surface and the curved lower one) of the two-dimensional periodic hill at a bulk Reynolds number of $10{,}595$, respectively. This work is constructed around three main questions: which input points should be considered for the data-driven wall model, how should one normalize the in- and output data to obtain a unified and consistent database, and which neural networks are considered

Efficient Runge-Kutta Discontinuous Galerkin Methods Applied to Aeroacoustics (Efficiënte Runge-Kutta discontinue Galerkin methoden voor aeroakoestische toepassingen)

The simulation of aeroacoustic problems sets demanding requirements on numerical methods, particularly in terms of accuracy. Runge-Kutta Discontinuous Galerkin (RKDG) schemes are increasingly popular for such applications, because they converge at an arbitrarily high order rate, they can deal with complex geometries, and they are amenable to parallel computing. However, they are still considered to be computationally costly. The work presented in this thesis aims at improving the computational efficiency of RKDG methods for linear aeroacoustic applications.The first part of the work is dedicated to the study of the stability and accuracy properties that affect the performance of RKDG methods applied to hyperbolic problems. An analysis technique inspired by the classical von Neumann method is used to determine the stability restrictions of the schemes, as well as their accuracy properties in terms of dissipation and dispersion. It is first used to investigate the influence of the element shape on CFL conditions with triangular grids, in order to improve the determination of the maximum allowable time step in practical simulations. Alternative methods to the CFL conditions are also devised for this purpose. Moreover, Runge-Kutta schemes specifically designed to maximize the computational efficiency of RKDG methods for wave propagation problems are derived.The second part of the work deals with the application of RKDG methods to linear aeroacoustics. RKDG formulations for the linearized Euler and Navier-Stokes equations are introduced, along with validation cases. Then, higher-order treatments of curved wall boundaries, needed to fully benefit from the efficiency of high-order RKDG methods in aeroacoustic propagation problems, are studied. Finally, the methods developed in this work are used in a hybrid approach to characterize the acoustic behaviour of orifices in plates under grazing flow. The results show a clear qualitative improvement over the existing analytical approaches.status: publishe

Optimal Runge-Kutta Schemes for Discontinuous Galerkin Spatial Discretizations Applied to Wave Propagation Problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. Stability and accuracy analysis techniques are used to assess the performance of the fully discrete scheme from the point of view of the user, who aims to choose the combination of mesh and numerical method that minimizes the computational time while fulfilling an accuracy requirement on a given frequency range. In this framework, two scenarios are defined. In the first one, the mesh size can be freely chosen by the user. The efficiency then depends on a trade-off between accuracy and stability: increasing the accuracy of the scheme enables the use of fewer, larger elements, while favouring the stability results in larger time steps. Thus, we define a cost metric involving both stability and accuracy, following the considerations in Ref. [Bernardini and Pirozzoli, J. Comput. Phys. 228(11):4182-4199, 2009]. In the second scenario, the elements are assumed to be constrained to a very small size by geometrical features of the computational domain. In this case, the accuracy can be disregarded, and the computational efficiency is considered to depend only on the stability. After reviewing relevant Runge-Kutta methods from the literature, we consider schemes of order q from 3 to 4, and number of stages up to q+4, for optimization. In the first scenario, a 8-stage, fourth-order Runge-Kutta scheme (named RKF84) is found to minimize the cost measure. In the second one, we derive one 7-stage, third-order scheme (RKC73) and one 8-stage, fourth-order scheme (RKC84) that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed. Depending on the scenario, they outperfom the Runge-Kutta methods found in the literature by 16% to 27%. Their benefits are also illustrated with examples. For each of the new Runge-Kutta schemes, we provide the coefficients of a 2N-storage implementation.status: publishe

Time Stepping with Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids

In order to optimize the time step determination in aeroacoustic simulations, the impact of element shape on the stability of a Runge-Kutta Discontinuous Galerkin method is systematically studied. The maximum time step for stability is calculated by comparing the eigenvalue spectrum of the semi-discrete scalar advection operator to the stability region of the Runge-Kutta integrator. To relate element shape to scheme stability, structured periodic grids made up of identical elements are considered. Stability bounds are computed for a set of different grids corresponding to a wide range of element shapes. Maximum Courant numbers based on three different measures of the element size are calculated from the stability results, for Carpenter's low-storage (4,5) Runge-Kutta scheme. For each element size measure, the accuracy of CFL conditions is evaluated, and the lowest values of the maximum Courant number, to be used in aeroacoustic simulations, are provided. Moreover, a simplified procedure for stability analysis is described, in order to compensate for the relative lack of reliability of CFL conditions. Time step calculations for an unstructured and a hybrid grid show that the procedure performs well compared to CFD conditions, and could be used for time step determination in practical aeroacoustic simulations.status: publishe

Optimal Runge-Kutta Schemes for Discontinuous Galerkin Space Discretizations Applied to Wave Propagation Problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q+4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge-Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge-Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.status: publishe

CFL Conditions for Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids

We study time step restrictions due to linear stability constraints of Runge-Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax-Friedrichs flux or the upwind flux, and integrated in time with various Runge-Kutta schemes designed for linear wave propagation problems or non-linear applications. Von-Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order $p$ ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax-Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.status: publishe

Time Stepping and Linear Stability of Runge-Kutta Discontinuous Galerkin Methods on Triangular Grids

The influence of element shape on the stability of a Runge-Kutta Discontinuous Galerkin method is systematically investigated, in order to improve the time step calculation in practical simulation. The maximum time step for stability is determined by comparing the eigenvalue spectrum of the semi-discrete scalar advection operator to the stability region of the Runge-Kutta integrator. Stability analyses are performed with a broad range of structured periodic triangular grids, all elements of each grid having the same shape, so that each element shape can be associated to a stability bound. Maximum Courant numbers are computed for Carpenter's low-storage (4,5) Runge-Kutta scheme, based on three different measures of the element size. Lower values of the maximum Courant number, to be used in practical simulations, are provided, and the accuracy of the CFL condition is assessed for each element size measure. In order to remedy the relative lack of reliability of CFL conditions, a simplified procedure for stability analysis is presented, that can be used for maximum time step calculation in practical simulations. It is shown in two examples involving respectively an unstructured and a hybrid grid, that it compares favorably to the CFL conditions.status: publishe

Efficient Computation of the Minimum of Shape Quality Measures on Curvilinear Finite Elements

We present a method for computing robust shape quality measures defined for any order of finite elements. All type of elements are considered, including pyramids. The measures are defined as the minimum of the pointwise quality of curved elements. The computation of the minimum, based on previous work presented by Johnen et al. (2013) [1] and [2], is very efficient. The key feature is to expand polynomial quantities into Bézier bases which allows to compute sharp bounds on the minimum of the pointwise quality measures