Unstructured Mesh Generation and Adaptation

International audienceWe first describe the well established unstructured mesh generation methods as involved in the computational pipeline, from geometry definition to surface and volume mesh generation. These components are always a preliminary and required step to any numerical computations. From an historical point of view, the generation of fully unstructured mesh generation in 3D has been a real challenge so as to the design of robust and accurate second order schemes on such unstructured meshes. If the issue of generating volume meshes for geometries of any complexity is now mostly solved, the emergence of robust numerical schemes on unstructured meshes has paved the way to adaptivity. Indeed, unstructured meshes in contrast with structured or block structured grids have the necessary flexibility to control the discretization both in size and orientation. In the second part, we review the main components to perform adaptative computations: (i) anisotropic mesh prescription via a metric field tensor (ii) anisotropic error estimates, and (iii) anisotropic mesh generation. For each component, we focus on a particularly simple method to implement. In particular, we describe a simple but robust strategy for generating anisotropic meshes. Each adaptation entity, ie surface, volume or boundary layers, relies on a specific metric tensor field. The metric-based surface estimate is then 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. Several 3D examples issued from steady and unsteady simulations from systems of hyper-bolic laws are presented. In particular, we show that despite the simplicity of the introduced adaptive meshing scheme a high level of anisotropy can be reached. This includes the direct prediction of the sonic boom of an aircraft by computing the flow from the cruise altitude to the ground, the interaction between shock waves and boundary layer, or the prediction of complex unsteady phenomena in 3D

Recent Improvements on Cavity-Based Operators for RANS Mesh Adaptation

International audienceIf anisotropic mesh adaptation has been a reliable tool to predict inviscid flows, its use with viscous flows at high Reynolds number remains a tedious task. Indeed many issues tends to limit the efficiency of standard remeshing algorithms based on local modifications. First, the high Reynolds number require to handle a very high level of anisotropy O(1 : 10 6) near the geometry. In the range of anisotropy, interpolation of metric fields or the projection on geometry are typical components that may fail during an adaptive step. The need for high-resolution near the geometry imposes to use an accurate geometry description, and optimally, be linked to a continuous CAD geometries. However, the boundary layer sizing may become smaller than typical CAD tolerance. We present a simple hierarchical geometry approximation where the newly created points are projected linearly, then using a cubic approximation then the CAD data. Finally, the accuracy, speed of convergence of the flow solver highly depends on the topology of the grids. Typical quasi-structured grids are preferred in the boundary layer while this kind of grids are complicated to generate with typical anisotropic meshing algorithm. We discuss in this paper, new developments in metric-orthogonal approach where an advancing points techniques is used to propose new points. Then these newly created points are inserted by using the cavity operator

Anisotropic, Adaptive Finite Elements for a Thin 3D Plate

International audienceAn adaptive, anisotropic finite element algorithm is proposed to solve the 3D linear elasticity equations in a thin 3D plate. Numerical experiments show that adaptive computations can be performed in thin 3D domains having geometrical aspect ratio 1:1000

Parallel Anisotropic Unstructured Grid Adaptation

Computational Fluid Dynamics (CFD) has become critical to the design and analysis of aerospace vehicles. Parallel grid adaptation that resolves multiple scales with anisotropy is identified as one of the challenges in the CFD Vision 2030 Study to increase the capacity and capability of CFD simulation. The Study also cautions that computer architectures are undergoing a radical change and dramatic increases in algorithm concurrency will be required to exploit full performance. This paper reviews four different methods to parallel anisotropic grid generation. They cover both ends of the spectrum: (i) using existing state-of-the-art software optimized for a single core and modifying it for parallel platforms and (ii) designing and implementing scalable software with incomplete, but rapidly maturating functionality. A brief overview for each grid adaptation system is presented in the context of a telescopic approach for multilevel concurrency. These methods employ different approaches to enable parallel execution, which provides a unique opportunity to illustrate the relative behavior of each approach. Qualitative and quantitative metric evaluations are used to draw lessons for future developments in this critical area for parallel CFD simulation

Developments on the P2{P^2} cavity operator and Bézier Jacobian correction using the simplex algorithm.

International audienceThis paper describes developments on the ${P^2}$ cavity operator stemming from a new Bézier untangling algorithm. Both surface and volume are adapted to an anisotropic solution field with the cavity operator as the low-level driver handling all topological changes to the mesh. The ${P^2}$ extension of the cavity operator handles curvature through Riemannian curved edge length minimization in the volume and geometry projection on the surface. In particular, the anisotropy conserving log-euclidean metric interpolation scheme was extended to high-order elements to facilitate differentiating edge length in the metric field. As a step forward from previous iterations of the ${P^2}$ cavity operator, validity is now enforced through optimization of Jacobian coefficients using the simplex algorithm for linear programs. This is made possible by the fact that Jacobian control coefficients are linear with regards to each control point and enables the global optimization of the minimum of all control coefficients surrounding an edge at once. Numerical results illustrate the ability of metric-induced curving to relatively quickly curve 3D meshes with complex geometries involved in Computational Fluid Dynamics (CFD) using only local schemes. This framework allow us to curve highly anisotropic meshes with around 10 million elements within minutes

Shrimp User Guide. A Fast Mesh Renumbering and Domain Partionning Method

This technical note describes the main features of Shrimp, a software that renumbers mesh entities and splits mesh domain and handle the parallelization of adaptive mesh generators. The aim of the software, the input and the output files and the list of options are given in this document. Shrimp has been developed within the GAMMA research project at INRIA Paris-Rocquencourt. This document describes the features of the current version: release V1.0 (January 2009)

Continuous Mesh Model and Well-Posed Continuous Interpolation Error Estimation

Rapport de recherche INRIAIn the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density and stretching of anisotropic meshes. Such structures are used to compute lengths in adaptive mesh generators. In this report, a Riemannian metric space is shown to be more than a way to compute a distance. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be derived continuously for a continuous mesh. In its tangent space, a Riemannian metric space reduces to a constant metric tensor so that it simply spans a metric space. Metric tensors are then used to continuously model discrete elements. On this basis, geometric invariants have been extracted. They connect a metric tensor to the set of all the discrete elements which can be represented by this metric. As the behavior of a Riemannian metric space is obtained by patching together the behavior of each of its tangent spaces, the global mesh model arises from gathering together continuous element models. We complete the continuous-discrete analogy by providing a continuous interpolation error estimate and a well-posed definition of the continuous linear interpolate. The later is based on an exact relation connecting the discrete error to the continuous one. From one hand, this new continuous framework freed the analysis of the topological mesh constraints. On the other hand, powerful mathematical tools are available and well defined on the space of continuous meshes: calculus of variations, differentiation, optimization, ..., whereas these tools are not defined on the space of discrete meshes

P2 Cavity Operator with Metric-Based Volume and Surface Curvature

International audienceThis paper describes theoretical developments and algorithms involved in the design of a P2 cavity operator to generate anisotropic curved meshes. Both volume and surface are adapted. A high-level approach is chosen, such that the existing P1 cavity operator is used as-is to handle topology. The P2 extension performs the curving process and ensures geometric validity. Volume curvature is based on Riemannian edge length minimization, first requiring a description of the metric field along a Bézier edge: this leads to the proposed high-order extension of the log-Euclidean scheme and differentiation of geometrical quantities in this framework. Surface curvature is based on similar principles, with the added difficulty of CAD or CAD surrogate projection. Numerical results illustrating the P2 cavity operator’s ability to recover curvature, from surface geometry to boundary layers to metric fields are presented. Examples are based on 3D real-world geometries encountered in Computational Fluid Dynamics (CFD). This framework allow us to curve highly anisotropic meshes with around 10 million elements within minutes

Vizir: High-order mesh and solution visualization using OpenGL 4.0 graphic pipeline

International audienceOpenGL 4 with GLSL shading language have become a standard on many common archi-tectures (Mac, Linux, Windows, , ...) from a couple of years. In the mean time, high-order methods (for flow solution and for meshing algorithm) are emerging. Many of them have proven their abilities to provide accurate results on complex (3D) geometries. However, the assessment of a particular meshing algorithm or of a high-order numerical scheme strongly relies on the capacity to validate and inspect visually the current mesh/solution at hand. However, having at the same time, an accurate and interactive visualization process for high-order mesh/solution is still a challenge as complex process are usually involved in the graphic pipeline: non linear root finding, ray tracing, GPU programming,. .. . In this paper, we discuss the current status and issues of using the (raw) OpenGL 4 pipeline to render curved high-order entities, and almost pixel-exact solutions. We illustrate this process on meshes and solutions issued from high-order curved from CAD and with high-order interpolated solutions

Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D

International audienceIn this paper, we study the extension of anisotropic metric-based mesh adaptation to the case of very high-order solutions in 3D. This work is based on an extension of the continuous mesh framework and multi-scale mesh adaptation where the optimal metric is derived through a calculus of variation. Based on classical high order a priori error estimates, the point-wise leading term of the local error is a homogeneous polynomial of order k + 1. To derive the leading anisotropic direction and orientations, this polynomial is approximated by a quadratic positive definite form, taken to the power k+1 2. From a geometric point of view, this problem is equivalent to finding a maximal volume ellipsoid included in the level set one of the absolute value of the polynomial. This optimization problem is strongly non-linear both for the functional and the constraints. We first recast the continuous problem in a discrete setting in the metric-logarithm space. With this approximation, this problem becomes linear and is solved with the simplex algorithm. This optimal quadratic form in the Euclidean space is then found by iteratively solving a sequence of such log-simplex problems. From the field of the local quadratic forms that representing the high-order error, a calculus of variation is used to globally control the error in L p norm. A closed form of the optimal metric is then found. Anisotropic meshes are then generated with this metric based on the unit mesh concept. For the numerical experiments, we consider several analytical functions in 3D. Convergence rate and optimality of the meshes are then discussed for interpolation of orders 1 to 5