In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more than a way to compute a length. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be evaluated continuously on a Riemannian metric space.From one hand, this new continuous framework proves that prescribing a Riemannian metric field is equivalent to the local control in L 1 norm of the interpolation error. This proves the consistency of classical metric-based mesh adaptation procedures. On the other hand, powerful mathematical tools are available and well defined on Riemannian metric spaces: calculus of variations, differentiation, optimization, . . . whereas these tools are not defined on discrete meshes.

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In the context of steady CFD computations, some numerical experiments point out that only a global mesh convergence order of one is numerically reached on a sequence of uniformly refined meshes although the considered numerical scheme is second order. This is due to the presence of genuine discontinuities or sharp gradients in the modelled flow. In order to address this issue, a continuous mesh adaptation framework is proposed based on the metric notion. It relies on a L p control of the interpolation error for twice differentiable functions. This theory provides an optimal bound of the interpolation error involving the Hessian of the solution. From this estimate, an optimal metric is exhibited to govern the adapted mesh generation. As regards steady flow computations with discontinuities, a global second order mesh convergence should be obtained. To this end, a higher order smooth approximation of the solution is reconstructed providing an accurate and reliable Hessian evaluation. Several numerical examples in two and three dimensions illustrate that the global convergence order is recovered using this mesh adaptation strategy.

Please cite this article as: Alauzet F, Loseille A. A decade of progress on anisotropic mesh adaptation for computational fluid dynamics. Computer-Aided Design (2015), http://dx. AbstractIn the context of scientific computing, the mesh is used as a discrete support for the considered numerical methods. As a consequence, the mesh greatly impacts the e ciency, the stability and the accuracy of numerical methods. The goal of anisotropic mesh adaptation is to generate a mesh which fits the application and the numerical scheme in order to achieve the best possible solution. It is thus an active field of research which is progressing continuously. This review article proposes a synthesis of the research activity of the INRIA Gamma3 team in the field of anisotropic mesh adaptation applied to inviscid flows in computational fluid dynamics since 2000. It shows the evolution of the theoretical and numerical results during this period. Finally, challenges for the next decade are discussed.

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