Marie-Gabrielle Vallet

Anisotropic mesh adaptation: towards user‐independent, mesh‐independent and solver‐independent CFD. Part I: general principles

The present paper is the lead article in a three-part series on anisotropic mesh adaptation and its applications to structured and unstructured meshes. A flexible approach is proposed and tested on two-dimensional, inviscid and viscous, finite volume and finite element flow solvers, over a wide range of speeds. The directional properties of an interpolation-based error estimate, extracted from the Hessian of the solution, are used to control the size and orientation of mesh edges. The approach is encapsulated into an edge-based anisotropic mesh optimization methodology (MOM), which uses a judicious sequence of four local operations: refinement, coarsening, edge swapping and point movement, to equi-distribute the error estimate along all edges, without any recourse to remeshing. The mesh adaptation convergence of the MOM loop is carefully studied for a wide variety of test cases

A DIRECTIONALLY ADAPTIVE METHODOLOGY USING AN EDGE-BASED ERROR ESTIMATE ON QUADRILATERAL GRIDS

The present paper describes a directionally adaptive finite element method for high-speed flows, using an edge-based error estimate on quadrilateral grids. The error of the numerical solution is estimated through its second derivatives and the resulting Hessian tensor is used to define a Riemannian metric. An improved mesh movement strategy, based on a spring analogy, but with no orthogonality constraints, is introduced to equidistribute the lengths of the edges of the elements in the defined metric. The grid adaptation procedure is validated on an analytical test case and the efficiency of the overall methodology is investigated on supersonic and hypersonic benchmarks.

Certifiable Computational Fluid Dynamics Through Mesh Optimization

The accuracy and reliability of computational fluid dynamics (CFD) are addressed by proposing a novel, efficient, and generic mesh optimization approach. By using an appropriate directional error estimator, coupled with an effective mesh adaptation technique that is tied closely to the solver, it can be demonstrated that, for each flow condition and geometry combination, a controllable error level and an optimal mesh can be obtained. It is further demonstrated that such an optimal mesh can be reached from almost any reasonable initial grid and, more astonishingly, that the order of accuracy of well-posed numerical algorithms has a considerably reduced impact on solution accuracy if the mesh is well adapted. Thus, the proposed approach can be considered a first step toward user-, mesh-, and solver-independent, and thus certifiable, CFD.

Anisotropic Mesh Adaptation: A Step Towards a Mesh-Independent and User-Independent CFD

This paper presents an anisotropic remeshing strategy to obtain accurate numerical solutions of problems showing distinct directional features. The adaptation criterion is measured on the mesh edges and is independent of the problem or the discretization scheme used. The strategy is shown to converge and is demonstrated for two-dimensional inviscid and viscous laminar flows, at a variety of speeds.

Two-dimensional metric tensor visualization using pseudo-meshes

Riemannian metric tensors are used to control the adaptation of meshes for finite element and finite volume computations. To study the numerous metric construction and manipulation techniques, a new method has been developed to visualize two-dimensional metrics without interference from an adaptation algorithm. This method traces a network of orthogonal tensor lines, tangent to the eigenvectors of the metric field, to form a pseudo-mesh visually close to a perfectly adapted mesh but without many of its constraints. Anisotropic metrics can be visualized directly using such pseudo-meshes but, for isotropic metrics, the eigensystem is degenerate and an anisotropic perturbation has to be used. This perturbation merely preserves directional information usually present during metric construction and is small enough, about 1% of the prescribed target element size, to be visually imperceptible. Both analytical and solution-based examples show the effectiveness and usefulness of the present method. As an example, pseudo-meshes are used to visualize the effect on metrics of Laplacian-like smoothing and gradation control techniques. Application to adaptive quadrilateral mesh generation is also discussed.

On ellipse intersection and union with application to anisotropic mesh adaptation

This paper presents ellipses as a convenient mean of representation for operating on several tensor fields. The first part of this paper discusses the representation of tensors as ellipses or ellipsoids. Two binary operators are defined inside the set of centred ellipses, intersection, and union, and their properties are studied. The second part of this paper presents an application of these operators to binary operations on metric tensor fields for the purpose of anisotropic mesh adaptation.

CGNS-Based Data Model for Turbine Blade Optimization

This paper presents a CFD data model specifically designed to address the needs of the OPALE project, whose goal is to automate the design optimization of hydraulic turbine blades. The purpose of this CFD model, based on the CGNS system, is to enable the integration of commercial and in-house analysis and design applications, and to allow the management and exchange of data in order to provide a support to MDO. The paper first presents the OPALE CFD data model structure based on the CGNS system. Some constraints in the use of CGNS are proposed in the context of the OPALE project, some interpretation problems concerning the CGNS system are resolved and the specific choices made are justified. The result is a complete CFD data model directly based on the CGNS standard specifications. The second part of the paper presents a flexible method, based on XML, proposed to drive the filtering process of the CGNS data files produced by commercials CFD solvers, with the aim of bringing them in the OPALE standardized CGNS form.