Thomas Toulorge

Robust untangling of curvilinear meshes

This paper presents a technique that allows to untangle high-order/curvilinear meshes. The technique makes use of unconstrained optimization where element Jacobians are constrained to lie in a prescribed range through moving log-barriers. The untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded Jacobians. Bounds on Jacobians are computed using the results of Johnen et al. (2012, 2013) [1,2]. The technique is applicable to any kind of polynomial element, for surface, volume, hybrid or boundary layer meshes. A series of examples demonstrate both the robustness and the efficiency of the technique. The final example, involving a time explicit computation, shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure.

Curved Boundary Treatments for the Discontinuous Galerkin Method Applied to Aeroacoustic Propagation

A discontinuous Galerkin method is applied to unstructured grids to simulate aeroacoustic propagation, modeled by the linearized Euler equations. On triangular and tetrahedral elements with straight edges, the quadrature-free form of the discontinuous Galerkin method is used. In addition to the classical linear treatment of wall boundaries, two treatments involving a second-order representation of the geometry are presented. The simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order, and demonstrates how the boundary treatment involving curved elements overcomes this restriction. The benefits of higher-order treatments are also assessed for more realistic geometries, namely a high-lift airfoil and an elliptical muffler.

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.

Transition Prediction and Impact on a Three-Dimensional High-Lift-Wing Configuration

The evolution of maximum lift coefficient of a transport aircraft as a function of Reynolds number can be linked to modifications of the laminar-turbulent transition process. In the framework of European project EUROLIFT (I), a task was dedicated to the physical understanding and the numerical modeling of the transition process in high-lift configurations. Then, in the follow-on project EUROLIFT II, a major step is the integration of transition prediction tools within Reynolds-averaged Navier-Stokes (RANS) solvers in order to estimate the impact of transition on performance. This paper presents an overview of the different activities dealing with transition in the EUROLIFT II project.

An efficient parallel implementation of explicit multirate Runge-Kutta schemes for discontinuous Galerkin computations

Although explicit time integration schemes require small computational efforts per time step, their efficiency is severely restricted by their stability limits. Indeed, the multi-scale nature of some physical processes combined with highly unstructured meshes can lead some elements to impose a severely small stable time step for a global problem. Multirate methods offer a way to increase the global efficiency by gathering grid cells in appropriate groups under local stability conditions. These methods are well suited to the discontinuous Galerkin framework. The parallelization of the multirate strategy is challenging because grid cells have different workloads. The computational cost is different for each sub-time step depending on the elements involved and a classical partitioning strategy is not adequate any more. In this paper, we propose a solution that makes use of multi-constraint mesh partitioning. It tends to minimize the inter-processor communications, while ensuring that the workload is almost equally shared by every computer core at every stage of the algorithm. Particular attention is given to the simplicity of the parallel multirate algorithm while minimizing computational and communication overheads. Our implementation makes use of the MeTiS library for mesh partitioning and the Message Passing Interface for inter-processor communication. Performance analyses for two and three-dimensional practical applications confirm that multirate methods preserve important computational advantages of explicit methods up to a significant number of processors.

Automatic Transition Prediction for RANS Computations Applied to a Generic High-Lift Wing

With the objective to integrate the transition prediction within RANS codes with a minimal user interaction, cooperative actions are under way in the frame of the European EUROLIFT 2 Project, especially involved with the improvement of the knowledge of the flow developed around the modeling of high lift configurations. Transition criteria and simplified stability analysis are considered for insertion or automatic coupling to 3D RANS codes, possibly involving a coupling to a 2.5D or 3D boundary layer code. Recent developments from INTA and ONERA are presented, with comparisons to a common validation case, the AFV three-element wing in high lift configuration.

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.

Transition Prediction and Impact on 3D High-Lift Wing Configuration

The evolution of maximum lift coefficient of a transport aircraft as a function of Reynolds number can be linked to modifications of the laminar-turbulent transition process. In the framework of European project EUROLIFT (I), a task was dedicated to the physical understanding and the numerical modeling of the transition process in high-lift configurations. Then, in the follow-on project EUROLIFT II, a major step is the integration of transition prediction tools within Reynolds-averaged Navier-Stokes (RANS) solvers in order to estimate the impact of transition on performance. This paper presents an overview of the different activities dealing with transition in the EUROLIFT II project.

The Generation of Valid Curvilinear Meshes

It is now well-known that a curvilinear discretization of the geometry is most often required to benefit from the computational efficiency of high-order numerical schemes in simulations. In this article, we explain how appropriate curvilinear meshes can be generated. We pay particular attention to the problem of invalid (tangled) mesh parts created by curving the domain boundaries. An efficient technique that computes provable bounds on the element Jacobian determinant is used to characterize the mesh validity, and we describe fast and robust techniques to regularize the mesh. The methods presented in this article are thoroughly discussed in Ref. [1, 2], and implemented in the free mesh generation software Gmsh [4, 12].

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 Hausdorff 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.