A simple strategy for generating anisotropic meshes is introduced. The approach belongs to the cl... more A simple strategy for generating anisotropic meshes is introduced. The approach belongs to the class of metric-based mesh adaptation procedures where a field of metric tensors governs the adaptation. This development is motivated by the need of generating anisotropic meshes for complex geometries and complex flows. The procedure may be used advantageously for cases where global remeshing techniques become either unfeasible or unreliable. Each of the local operations used is checked in a variety of ways by taking into account both the volume and the surface mesh. This strategy is illustrated with surface mesh adaptation and with the generation of meshes suited for boundary layers analysis. Two simple mesh operators are used to recursively modify the mesh: edge collapse and point insertion on edge. It is shown that using these operators jointly with a quality function allows to quickly produce an quality anisotropic mesh. Each adaptation entity, ie surface, volume or boundary layers, relies on a specific metric tensor field. The metric-based surface estimate is used to control the deviation to the surface and to adapt the surface mesh. The volume estimate aims at controlling the interpolation error of a specific field of the flow. The boundary layers metric-based estimate is deduced from a level-set distance function.
The aim of this work is to develop an algorithm that allows the common refinement of non-coincide... more The aim of this work is to develop an algorithm that allows the common refinement of non-coincident meshes composed of arbitrary 3D surfaces elements (triangles, quad elements). This study is motivated by computations in geologic applications which involve complex geometries with heterogeneous components and geologic faults. The resulting meshes are linked through a continuous bijection in order to ensure the accuracy and the conservativity of the data transferring through the surfaces. The strategy adopted consists, first, of the simultaneous convexification of the two surfaces by the mean of points connections. A projection according to given normals is then achieved. The mesh quality improved by applying a surface smoothing should accelerate the procedure convergence.
In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientati... more 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.
Efficiently parallelizing a whole set of meshing tools, as required by an automated mesh adaptati... more Efficiently parallelizing a whole set of meshing tools, as required by an automated mesh adaptation loop, relies strongly on data localization to avoid memory access contention. In this regard, renumbering mesh items through a space filling curve (SFC), like Hilbert or Peano, is of great help and proved to be quite versatile. This paper briefly introduces the Hilbert SFC renumbering technique and illustrates its use with two different approaches to parallelization: an out-of-core method and a shared-memory multi-threaded algorithm.
In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientati... more 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.
International Journal for Numerical Methods in Fluids, 2009
This paper describes the use of an a posteriori error estimator to control anisotropic mesh adapt... more This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h min is enforced across solution discontinuities. This h min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h min . We verify numerically that the effect of varying h min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. mesh adaptation techniques were initially based on a metric derived from a numerical approximation of the Hessian of the solutions with, in the background, the use of an a priori error estimator . The main idea behind the use of such metrics originates from the problem of finding the optimal mesh with a prescribed number of nodes, which minimizes the interpolation error of a given function . More recently, these a priori error estimators were used to drive unstructured mesh adaptation for 3D flows [9-11] and 3D phase change problems , just to name a few applications. For all these applications, sharp fronts, e.g. shocks in compressible flows or solidification interfaces, are meshed with high aspect ratio tetrahedra, allowing the computation of very accurate solutions with a minimal number of elements. These a priori error estimators successfully carry over to inviscid flows with shocks in spite of the fact that theoretically these estimates involve the Hessian of the exact solution, which is not regular. Practically, for these inviscid flows, flow solvers approximate regularized solutions (because of the artificial viscosity) with properly defined second derivatives that can be reconstructed with a recovery technique .
Mesh adaptation is considered here as the research of an optimum that minimizes the P1 interpolat... more Mesh adaptation is considered here as the research of an optimum that minimizes the P1 interpolation error of a function u of R n given a number of vertices. A continuous modeling is described by considering classes of equivalence between meshes which are analytically represented by a metric tensor field. Continuous metrics are exhibited for L p error model and mesh order of convergence are analyzed. Numerical examples are provided in two and three dimensions.
Key words: Anisotropic mesh adaptation, high-order method, high speed flows. Abstract. In this pa... more Key words: Anisotropic mesh adaptation, high-order method, high speed flows. Abstract. In this paper, we discuss the contribution of mesh adaptation to high-order convergence of high-speed flow simulations on complex geometries. At first, we analyse the ability of mesh adaptation to accurately capture a discontinuity. Then, we propose an anisotropic mesh adaptation strategy for achieving a global second-order mesh convergence for discontinuous numerical solutions in L p norm. This ability is illustrated and validated on 2D and 3D examples for supersonic flows.
This paper addresses classical issues that arise when applying anisotropic mesh adaptation to rea... more This paper addresses classical issues that arise when applying anisotropic mesh adaptation to real-life 3D problems as the loss of anisotropy or the necessity to truncate the minimal size when discontinuities are present in the solution. These problematics are due to the complex interaction between the components involved in the adaptive loop: the flow solver, the error estimate and the mesh generator. A solution based on a new continuous mesh framework is proposed to overcome these issues. We show that using this strategy allows an optimal level of anisotropy to be reached and thus enjoy the full benefit of unstructured anisotropic mesh adaptation: optimal distribution of the degrees of freedom, improvement of the ratio accuracy with respect to cpu time, ...
The shape optimization of a supersonic aircraft need a composite model combining a 3D CFD high-fi... more The shape optimization of a supersonic aircraft need a composite model combining a 3D CFD high-fidelity model and a simplified boom propagation model. The management of this complexity is studied in an optimization loop, with exact discrete adjoints of 3D flow and mesh deformation system. The introduction of a mesh adaptation algorithm is also considered. RÉSUMÉ. L'optimisation de la forme d'un avion supersonique nécessite un modèle composite comportant une composante 3D haute fidélité en mécanique des fluides et un modèle simplifié de propagation du bang. La prise en compte de cette complexité est étudiée dans le cadre d'une boucle d'optimisation, avec des adjoints discrets exacts de l'écoulement 3D et un système de déformation de maillage. L'introduction d'une méthode d'adaptation de maillage est aussi considérée.
A simple strategy for generating anisotropic meshes is introduced. The approach belongs to the cl... more A simple strategy for generating anisotropic meshes is introduced. The approach belongs to the class of metric-based mesh adaptation procedures where a field of metric tensors governs the adaptation. This development is motivated by the need of generating anisotropic meshes for complex geometries and complex flows. The procedure may be used advantageously for cases where global remeshing techniques become either unfeasible or unreliable. Each of the local operations used is checked in a variety of ways by taking into account both the volume and the surface mesh. This strategy is illustrated with surface mesh adaptation and with the generation of meshes suited for boundary layers analysis. Two simple mesh operators are used to recursively modify the mesh: edge collapse and point insertion on edge. It is shown that using these operators jointly with a quality function allows to quickly produce an quality anisotropic mesh. Each adaptation entity, ie surface, volume or boundary layers, relies on a specific metric tensor field. The metric-based surface estimate is used to control the deviation to the surface and to adapt the surface mesh. The volume estimate aims at controlling the interpolation error of a specific field of the flow. The boundary layers metric-based estimate is deduced from a level-set distance function.
The aim of this work is to develop an algorithm that allows the common refinement of non-coincide... more The aim of this work is to develop an algorithm that allows the common refinement of non-coincident meshes composed of arbitrary 3D surfaces elements (triangles, quad elements). This study is motivated by computations in geologic applications which involve complex geometries with heterogeneous components and geologic faults. The resulting meshes are linked through a continuous bijection in order to ensure the accuracy and the conservativity of the data transferring through the surfaces. The strategy adopted consists, first, of the simultaneous convexification of the two surfaces by the mean of points connections. A projection according to given normals is then achieved. The mesh quality improved by applying a surface smoothing should accelerate the procedure convergence.
In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientati... more 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.
Efficiently parallelizing a whole set of meshing tools, as required by an automated mesh adaptati... more Efficiently parallelizing a whole set of meshing tools, as required by an automated mesh adaptation loop, relies strongly on data localization to avoid memory access contention. In this regard, renumbering mesh items through a space filling curve (SFC), like Hilbert or Peano, is of great help and proved to be quite versatile. This paper briefly introduces the Hilbert SFC renumbering technique and illustrates its use with two different approaches to parallelization: an out-of-core method and a shared-memory multi-threaded algorithm.
In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientati... more 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.
International Journal for Numerical Methods in Fluids, 2009
This paper describes the use of an a posteriori error estimator to control anisotropic mesh adapt... more This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single-parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h min is enforced across solution discontinuities. This h min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h min . We verify numerically that the effect of varying h min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. mesh adaptation techniques were initially based on a metric derived from a numerical approximation of the Hessian of the solutions with, in the background, the use of an a priori error estimator . The main idea behind the use of such metrics originates from the problem of finding the optimal mesh with a prescribed number of nodes, which minimizes the interpolation error of a given function . More recently, these a priori error estimators were used to drive unstructured mesh adaptation for 3D flows [9-11] and 3D phase change problems , just to name a few applications. For all these applications, sharp fronts, e.g. shocks in compressible flows or solidification interfaces, are meshed with high aspect ratio tetrahedra, allowing the computation of very accurate solutions with a minimal number of elements. These a priori error estimators successfully carry over to inviscid flows with shocks in spite of the fact that theoretically these estimates involve the Hessian of the exact solution, which is not regular. Practically, for these inviscid flows, flow solvers approximate regularized solutions (because of the artificial viscosity) with properly defined second derivatives that can be reconstructed with a recovery technique .
Mesh adaptation is considered here as the research of an optimum that minimizes the P1 interpolat... more Mesh adaptation is considered here as the research of an optimum that minimizes the P1 interpolation error of a function u of R n given a number of vertices. A continuous modeling is described by considering classes of equivalence between meshes which are analytically represented by a metric tensor field. Continuous metrics are exhibited for L p error model and mesh order of convergence are analyzed. Numerical examples are provided in two and three dimensions.
Key words: Anisotropic mesh adaptation, high-order method, high speed flows. Abstract. In this pa... more Key words: Anisotropic mesh adaptation, high-order method, high speed flows. Abstract. In this paper, we discuss the contribution of mesh adaptation to high-order convergence of high-speed flow simulations on complex geometries. At first, we analyse the ability of mesh adaptation to accurately capture a discontinuity. Then, we propose an anisotropic mesh adaptation strategy for achieving a global second-order mesh convergence for discontinuous numerical solutions in L p norm. This ability is illustrated and validated on 2D and 3D examples for supersonic flows.
This paper addresses classical issues that arise when applying anisotropic mesh adaptation to rea... more This paper addresses classical issues that arise when applying anisotropic mesh adaptation to real-life 3D problems as the loss of anisotropy or the necessity to truncate the minimal size when discontinuities are present in the solution. These problematics are due to the complex interaction between the components involved in the adaptive loop: the flow solver, the error estimate and the mesh generator. A solution based on a new continuous mesh framework is proposed to overcome these issues. We show that using this strategy allows an optimal level of anisotropy to be reached and thus enjoy the full benefit of unstructured anisotropic mesh adaptation: optimal distribution of the degrees of freedom, improvement of the ratio accuracy with respect to cpu time, ...
The shape optimization of a supersonic aircraft need a composite model combining a 3D CFD high-fi... more The shape optimization of a supersonic aircraft need a composite model combining a 3D CFD high-fidelity model and a simplified boom propagation model. The management of this complexity is studied in an optimization loop, with exact discrete adjoints of 3D flow and mesh deformation system. The introduction of a mesh adaptation algorithm is also considered. RÉSUMÉ. L'optimisation de la forme d'un avion supersonique nécessite un modèle composite comportant une composante 3D haute fidélité en mécanique des fluides et un modèle simplifié de propagation du bang. La prise en compte de cette complexité est étudiée dans le cadre d'une boucle d'optimisation, avec des adjoints discrets exacts de l'écoulement 3D et un système de déformation de maillage. L'introduction d'une méthode d'adaptation de maillage est aussi considérée.
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Papers by Adrien Loseille