Two-dimensional Delaunay-based anisotropic mesh adaptation

Science and engineering applications often have anisotropic physics and therefore require anisotropic mesh adaptation. In common with previous researchers on this topic, we use metrics to specify the desired mesh. Where previous approaches are typically heuristic and sometimes require expensive optimization steps, our approach is an extension of isotropic Delaunay meshing methods and requires only occasional, relatively inexpensive optimization operations. We use a discrete metric formulation, with the metric defined at vertices. To map a local sub-mesh to the metric space, we compute metric lengths for edges, and use those lengths to construct a triangulation in the metric space. Based on the metric edge lengths, we define a quality measure in the metric space similar to the well-known shortest-edge to circumradius ratio for isotropic meshes. We extend the common mesh swapping, Delaunay insertion, and vertex removal primitives for use in the metric space. We give examples demonstrating our scheme’s ability to produce a mesh consistent with a discontinuous, anisotropic mesh metric and the use of our scheme in solution adaptive refinement.     

Adaptive anisotropic meshing for steady convection-dominated problems

Obtaining accurate solutions for convection–diffusion equations is challenging due to the presence of layers when convection dominates the diffusion. To solve this problem, we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution. Three main ingredients are used. First, the streamline upwind Petrov–Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal Voronoi tessellation algorithm. Our algorithm is tested on a variety of two-dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non-physical oscillations in the numerical approximation.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

An anisotropic unstructured triangular adaptive mesh algorithm based on error and error gradient information

In this paper, an algorithm based on unstructured triangular meshes using standard refinement patterns for anisotropic adaptive meshes is presented. It consists of three main actions: anisotropic refinement, solution-weighted smoothing and patch unrefinement. Moreover, a hierarchical mesh formulation is used. The main idea is to use the error and error gradient on each mesh element to locally control the anisotropy of the mesh. The proposed algorithm is tested on interpolation and boundary-value problems with a discontinuous solution.     

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.     

Obtuse triangle suppression in anisotropic meshes

Anisotropic triangle meshes are used for efficient approximation of surfaces and flow data in finite element analysis, and in these applications it is desirable to have as few obtuse triangles as possible to reduce the discretization error. We present a variational approach to suppressing obtuse triangles in anisotropic meshes. Specifically, we introduce a hexagonal Minkowski metric, which is sensitive to triangle orientation, to give a new formulation of the centroidal Voronoi tessellation (CVT) method. Furthermore, we prove several relevant properties of the CVT method with the newly introduced metric. Experiments show that our algorithm produces anisotropic meshes with much fewer obtuse triangles than using existing methods while maintaining mesh anisotropy.     

Generation of anisotropic mesh by ellipse packing over an unbounded domain

With the advance of the finite element method, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of anisotropic mesh of variable element size over an unbounded 2D domain by using the advancing front ellipse packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing ellipses is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as an ellipse is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new ellipse. Ellipses are packed closely and in contact with the existing ellipses by an iterative procedure according to the specified anisotropic metric tensor. The anisotropic meshes generated by ellipse packing can also be used through a mapping process to produce parametric surface meshes of various characteristics. The size and the orientation of the ellipses in the pack are controlled by the metric tensor as derived from the principal surface curvatures. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of ellipse packing, this reduces a lot of geometrical checks for intersection between frontal segments. Five examples are given to show the effectiveness and robustness of anisotropic mesh generation and the application of ellipse packing to mesh generation over various curved surfaces.     

基于黎曼度量的复杂参数曲面有限元网格生成方法

给出了三维空间的黎曼度量和曲面自身的黎曼度量相结合的三维复杂参数曲面自适应网格生成的改进波前推进算法.详细阐述了曲面参数域上任意一点的黎曼度量的计算和插值方法;采用可细化的栅格作为背景网格,在降低了程序实现的难度的同时提高了网格生成的速度;提出按层推进和按最短边推进相结合的方法,在保证边界网格质量的同时,提高曲面内部网格的质量.三维自适应黎曼度量的引入,提高了算法剖分复杂曲面的自适应性.算例表明,该算法对复杂曲面能够生成高质量的网格,而且整个算法具有很好的时间特性和可靠性.     

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.     

On Optimal Bilinear Quadrilateral Meshes

The novelty of this work is in presenting interesting error properties of two types of asymptotically ‘optimal’ quadrilateral meshes for bilinear approximation. The first type of mesh has an error equidistributing property, where the maximum interpolation error is asymptotically the same over all elements. The second type has faster than expected ‘super-convergence’ property for certain saddle-shaped data functions. The ‘super-convergent’ mesh may be an order of magnitude more accurate than the error equidistributing mesh. Both types of mesh are generated by a coordinate transformation of a regular mesh of squares. The coordinate transformation is derived by interpreting the Hessian matrix of a data function as a metric tensor. The insights in this work may have application in mesh design near known corner or point singularities.     

Anisotropic geodesics for live‐wire mesh segmentation

We present an interactive method for mesh segmentation that is inspired by the classical live‐wire interaction for image segmentation. The core contribution of the work is the definition and computation of wires on surfaces that are likely to lie at segment boundaries. We define wires as geodesics in a new tensor‐based anisotropic metric, which improves upon previous metrics in stability and feature‐awareness. We further introduce a simple but effective mesh embedding approach that allows geodesic paths in an anisotropic path to be computed efficiently using existing algorithms designed for Euclidean geodesics. Our tool is particularly suited for delineating segmentation boundaries that are aligned with features or curvature directions, and we demonstrate its use in creating artist‐guided segmentations.     

A nonconforming finite element method¶with anisotropic mesh grading¶for the incompressible Navier–Stokes equations¶in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫₁) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N _e is the number of elements of the mesh, we prove that the optimal order of convergence h∼ N _e ^{− 1/3}. Received:July 2001 / Accepted: July 2002     

全局各向异性四边形主导网格重建方法

四边形网格的结构特点要求网格单元满足全局一致性,难以取得网格质量与表达效率之间的平衡.为此,提出一种基于全局的各向异性四边形主导网格重建方法,可生成网格质量好且冗余程度低的四边形网格.重建过程以主曲率线为基本采样单元,首先计算模型表面的主曲率场并对主曲率场积分,得到密集的主曲率线采样;再根据贪心算法,利用几何形体自身的各向异性找出冗余度最高的主曲率线并予以删除;如此循环,直至达到理想的采样密度.该重建方法适用于任意拓扑网格模型,所得到的各向异性四边形主导网格在网格模型分辨率下降时,由于始终保留重要主曲率线,从而可以更好地保持模型特征.同时,在基于贪心算法的渐进式主曲率线删除过程中,可产生分辨率连续可调的四边形主导网格.     

Adaptive boundary layer meshing for viscous flow simulations

A procedure for anisotropic mesh adaptation accounting for mixed element types and boundary layer meshes is presented. The method allows to automatically construct meshes on domains of interest to accurately and efficiently compute key flow quantities, especially near wall quantities like wall shear stress. The new adaptive approach uses local mesh modification procedures in a manner that maintains layered and graded elements near the walls, which are popularly known as boundary layer or semi-structured meshes, with highly anisotropic elements of mixed topologies. The technique developed is well suited for viscous flow applications where exact knowledge of the mesh resolution over the computational domain required to accurately resolve flow features of interest is unknown a priori. We apply the method to two types of problem cases; the first type, which lies in the field of hemodynamics, involves pulsatile flow in blood vessels including a porcine aorta case with a stenosis bypassed by a graft whereas the other involves high-speed flow through a double throat nozzle encountered in the field of aerodynamics.     

Generic remeshing of 3D triangular meshes with metric-dependent discrete voronoi diagrams

In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user defined vertex budget. Our approach can deal with a wide range of applications, from high quality mesh generation to shape approximation. By using appropriate metric constraints the method generates isotropic or anisotropic elements. Based on point-sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of entirely discrete geometry processing . Besides the general described framework, we show experimental results using isotropic, quadric-enhanced isotropic and anisotropic metrics which prove the efficiency of our method on large meshes, for a low computational cost.     

On discrete boundaries and solution accuracy in anisotropic adaptive meshing

In the adaptive mesh generation, the space mesh should be adequate to the surface mesh. When the analytical surface representation is not known, additional surface information may be extracted from triangular surface meshes. We describe a new surface reconstruction method which uses approximate Hessian of a piecewise linear function representing the discrete surface. Efficiency of the proposed method is illustrated with two CFD applications.     

Anisotropic noise samples

We present a practical approach to generate stochastic anisotropic samples with Poisson-disk characteristic over a two-dimensional domain. In contrast to isotropic samples, we understand anisotropic samples as non-overlapping ellipses whose size and density match a given anisotropic metric. Anisotropic noise samples are useful for many visualization and graphics applications. The spot samples can be used as input for texture generation, e.g., line integral convolution (LIC), but can also be used directly for visualization. The definition of the spot samples using a metric tensor makes them especially suitable for the visualization of tensor fields that can be translated into a metric. Our work combines ideas from sampling theory and mesh generation. To generate these samples with the desired properties we construct a first set of non-overlapping ellipses whose distribution closely matches the underlying metric. This set of samples is used as input for a generalized anisotropic Lloyd relaxation to distribute noise samples more evenly. Instead of computing the Voronoi tessellation explicitly, we introduce a discrete approach which combines the Voronoi cell and centroid computation in one step. Our method supports automatic packing of the elliptical samples, resulting in textures similar to those generated by anisotropic reaction-diffusion methods. We use Fourier analysis tools for quality measurement of uniformly distributed samples. The resulting samples have nice sampling properties, for example, they satisfy a blue noise property where low frequencies in the power spectrum are reduced to a minimum.     

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

Anisotropic Porous Medium Equation (APME) is developed as an extension of the Porous Medium Equation (PME) for anisotropic porous media. A special analytical solution is derived for APME for time-independent diffusion. Anisotropic mesh adaptation for linear finite element solution of APME is discussed and numerical results for two dimensional examples are presented. The solution errors using anisotropic adaptive meshes show second order convergence.     

Statics Aware Grid Shells

We introduce a framework for the generation of polygonal gridshell architectural structures, whose topology is designed in order to excel in static performances. We start from the analysis of stress on the input surface and we use the resulting tensor field to induce an anisotropic nonEuclidean metric over it. This metric is derived by studying the relation between the stress tensor over a continuous shell and the optimal shape of polygons in a corresponding gridshell. Polygonal meshes with uniform density and isotropic cells under this metric exhibit variable density and anisotropy in Euclidean space, thus achieving a better distribution of the strain energy over their elements. Meshes are further optimized taking into account symmetry and regularity of cells to improve aesthetics. We experiment with quad meshes and hexdominant meshes, demonstrating that our gridshells achieve better static performances than stateoftheart gridshells.