To this end, we consider metric tensors to specify different sizes in different directions. Metric tensors for anisotropic mesh generation request pdf. On anisotropic mesh generation and quality control in complex flow problems, proceedings, 10th international meshing roundtable, sandia national laboratories, pp. Anisotropy, mesh generation, delaunay triangulation, voronoi diagram. In section 2, we recall the main stages of the mesh adaptation scheme, i. Anisotropic mesh adaptation for image representation.
In general relativity, the metric tensor in this context often abbreviated to simply the metric is the fundamental object of study. Mesh generation, mesh adaptation, metric, anisotropy, control space abstract. After recalling the framework of metric based mesh generation,5,8 and the metric orthogonal process,7,10 this section details our continuous approach of size gradation control. Selberherr institute for microelectronics technical university of vienna gusshausstr. Metric tensors for anisotropic mesh generation sciencedirect. Organised according to the geometry and dimension of the problem domains, it develops from the basic meshing algorithms to the most advanced schemes to deal with problems. The anisotropic mesh generation problem generate anisotropic mesh. A study on using hierarchical basis error estimates in. A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. Optimal 3d highly anisotropic mesh adaptation based on the continuous mesh framework adrien loseille1, 2and fr.
Anisotropic voronoi diagrams and guaranteedquality. Computational fluid dynamics simulations typically require triangles stretched in the direction of the flow. We develop a dimensionindependent, delaunaybased anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. The key to the approach is to choose a proper metric tensor.
A local sampling approach to anisotropic metricbased mesh. Optimal 3d highly anisotropic mesh adaptation based on the. May 12, 2005 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. Thus, it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation. 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. Numerical results show that this approach can be successfully applied to deal with poisson and steady convectiondominated problems. Citeseerx metric tensors for anisotropic mesh generation. These meshes are obtained when most of the edges are aligned with the eigenvectors of a provided input metric. Anisotropic mesh generation we briefly detail our approach to anisotropic mesh generation based on the notion of unit mesh. Despite this progress, the basic mathematical principles in anisotropic.
The advancing front method can remesh a domain anisotropically by positioning new nodes according to the metric 16. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex two. In this basis, the terms of the diagonal matrices associated to the metric. Adaptive control space structure for anisotropic mesh generation tomasz jurczyk and barbara g lut agh university of science and technology institute of computer science, 30059 krak. A general formulation of anisotropic mesh gen eration or adaptation in 2d or 3d is to define a d. Our extension of the original mesh metric duality for piecewise linear functions by loseille et al. In the field of mesh generation, the metric tensor, mx, is commonly used such that an anisotropic mesh is generated as a quasiuniform mesh in the metric space determined by mx. In this paper, we develop a general formula for the.
Then we explain the details of the bubble packing and mesh generation process in sections 4 and 5, followed by experimental results in section 6 and conclusions. Pdf mesh adaptation is studied from the mesh control point of view. On 3d anisotropic local remeshing for surface, volume and. The approach uses an advancingpoint type point placement that aligns the elements with the underlying metric field. A metric tensor is introduced to quantify the stretching. Pdf mathematical principles of anisotropic mesh adaptation. Many mesh generation researchers were first users of meshes. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. An elegant formulation for related quasiconformal maps based on beltrami factors described in 36.
We use a metric tensor developed in 20, mainly because. In this paper, we develop a general formula for the metric ten. Metric orthogonal anisotropic mesh generation starting from a provided input 3d valid volume mesh and metric, the generation of an adapted metric orthogonal mesh is a combination of the previous operators with a frontal algorithm to propose the points to be inserted. Unstructured mesh generation and adaptation 5 figure 4. Aligned metricbased anisotropic solution adaptive mesh generation david marcum. Anisotropic mesh generation, metric, nash embedding theorem, isometric. Metric orthogonal anisotropic mesh generation adrien loseillea ainria parisrocquencourt, 78153 le chesnay, france abstract we introduce a procedure to generate metric orthogonal anisotropic volume meshes. Riemannian metric tensors and introduced the anisotropic delaunay kernel, their. We show that the inverse of the anisotropic diffusion matrix of the anisotropic elliptic problem is a natural metric tensor for the anisotropic mesh generation in three aspects.
Metric based mesh adaptation and generation provides this mathematical foundation. In practice, an anisotropic mesh is commonly generated as a quasiuniform mesh in the metric determined by a tensor specifying the shape, size, orientation of elements. Hybrid anisotropic mesh adaptation using metricorthogonal. It theoretically allowsto transformany unstructured uniform mesh generator into an anisotropic one. We introduce a procedure to generate metricorthogonal anisotropic volume. The approach uses an advancingpoint type point placement that aligns the elements with the underlying metric. Guaranteedquality anisotropic mesh generation for domains. Bossen and heckbert used a 2x2 metric tensor to generate an anisotropic triangular mesh using a system of interacting.
The remainder of this paper is structured as follows. Metric tensors for anisotropic mesh generation core. Anisotropic mesh generation there are two main issues in anisotropic mesh generation. Instead, it proceeds by local optimizations and uses an anisotropic shape criterion to fit a metric field. Very high order anisotropic metricbased mesh adaptation in 3d olivier coulaud a, adrien loseille ainria saclayledefrance, batiment. Aligned metricbased anisotropic solution adaptive mesh. In this paper, we are concerned with the natural metric tensor for use in anisotropic mesh generation for anisotropic. Robust, guaranteedquality anisotropic mesh generation by. The idea to generate anisotropic meshes by computing a unit mesh in a given riemannian metric space rather. Sep 29, 20 anisotropic meshes are known to be wellsuited for problems which exhibit anisotropic solution features. Anisotropic centroidal voronoi tessellations and their applications qiang duyand desheng wangz abstract.
Metricorthogonal anisotropic mesh generation adrien loseillea ainria parisrocquencourt, 78153 le chesnay, france abstract we introduce a procedure to generate metricorthogonal anisotropic volume meshes. Anisotropic mesh adaptation for cfd computations laboratoire. This new approach to mesh generation appears quite promising. The optimization will determine changes to the current. Numerical results show that mid mesh with manual adaptation around fractures can. The metricbased surface estimate is used to control the deviation to the surface and to adapt the surface mesh. Robust metricaligned quaddominant meshing using lp. Section 2 presents some background material on anisotropic mesh generation and metric based mesh adaptation.
A scalar metric tensor will lead to an isotropic mesh while a full metric tensor will generally result in an anisotropic mesh. Anisotropic mesh adaptation for evolving triangulated surfaces. At this point, it can be of practical interest to introduce a metric tensor in the. Pdf new metric tensors for anisotropic mesh generation. Generating smooth mixed blockunstructured meshes is a promising way to reduce mesh sizes for outer aerodynamic flow analysis. Although mainly concerned with anisotropic mesh generation, for comparison purpose we also record the results for the isotropic case. There is an infinite variety of geometry found in nature and manmade objects. A smooth and wellgraded metric field makes the generation of the anisotropic mesh generation easier and generally improves the final quality. Metric tensors for anisotropic mesh generation weizhang huang.
Defining an appropriate metric tensor and designing an efficient algorithm for anisotropic mesh generation are two important aspects of the anisotropic mesh methodology. From left to right, cad of a torus with 2 edges and one face, 2d mesh in the parametric. The interpolation and stiffness properties of anisotropic linear triangles in. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex twodimensional simulations. The metric field provided has a direct, albeit complex, impact on the quality of the resulting mesh. Metricbased mesh adaptation and generation provides this mathematical foundation. The generation of anisotropic meshes requires the specification of a mesh size in each direction at each point of the domain. Metricorthogonal anisotropic mesh generation sciencedirect.
Finite elements in analysis and design 25, 12, 85109. In our adaptive algorithm, we use a metric tensor to serve as a guide to align the mesh with the anisotropy of the computed solution. Anisotropic meshes are known to be wellsuited for problems which exhibit anisotropic solution features. Adaptive anisotropic meshing for steady convectiondominated. Finite element mesh generation provides a concise and comprehensive guide to the application of finite element mesh generation over 2d domains, curved surfaces, and 3d space. Metric tensor, anisotropic mesh, adaptive finite element, node placement and local mesh generation. The first theoretically guaranteed anisotropic mesh generation algorithm.
Conclusions future work should work in practice in 3d though the theoretical properties dont all follow. On the imbedding of systems of compacta in simplicial complexes. A convenient means to specify this information is in terms of a 3. Anisotropic geometr yconforming d simplicial meshing via. Alan turing, 91120 palaiseau, france abstract in this paper, we study the extension of anisotropic metricbased mesh adaptation to the case of very highorder solutions in 3d. Bamg generates a mesh given a metric eld, which is speci ed at nodes of a background mesh the current mesh in an adaptive setting. Both the monitor function denoted by the letter m and metric tensor denoted by the calligraphy letter m. In this sense, the mesh generation procedure is the same for both isotropic and anisotropic mesh generation.
The optimization approach was implemented in the cubit mesh generation code and tested on several problems. Anisotropic voronoi diagrams offer an elegant and fast way to define anisotropic delaunay triangulations. Pdf metricorthogonal anisotropic mesh generation researchgate. Anisotropic delaunay mesh adaptation for unsteady simulations 5 to achieve this result, the matrix n m. Most of presen t mesh generation algorithms are structured in the follo wing w a y. Adaptive anisotropic meshing for steady convection.
The matrix n is diagonalizable in r and admits a basis of eigenvectors orthogonal. Pdf anisotropic mesh generation methods based on acvt. A new aligned metric based anisotropic mesh generation procedure is presented. Levelsets and anisotropic mesh adaptation alexandra claisse. The triangulation procedure is changed to generate delaunay meshes in the riemannian space defined by the metric. Consequently, the quest of anisotropic mesh generation. Metricorthogonal anisotropic mesh generation semantic scholar. Anisotropic delaunay mesh adaptation for unsteady simulations. New metric tensors for anisotropic mesh generation. New metric tensors for anisotropic mesh generation core. In this paper we present a 3d tetrahedral, unstructured and anisotropic mesh generator that is not based on the delaunay, frontal or octree method.
Unstructured mesh generation and adaptation sciencedirect. Achieving finite element mesh quality via optimization of. In section 3, we present a new adaptation scheme based on a transient. Metric tensors that are useful for anisotropic g rid generation are discussed in, e. Anisotropic mesh adaptation for 3d anisotropic diffusion. May 12, 2005 generation of anisotropic mesh by ellipse packing over an unbounded domain generation of anisotropic mesh by ellipse packing over an unbounded domain lo, s wang, w. Fidkowski department of aerospace engineering, university of michigan, ann arbor, mi 48109, usa this work presents an outputbased anisotropic mesh optimization algorithm that builds on previous work of yano. This is done by computing the distance in a riemannian space instead of the classical euclidean metric space. Numerical examples are given to demonstrate the features of the linear nite element method for anisotropic meshes generated with the metric tensors. Adaptive anisotropic tetrahedral mesh generator request pdf. A local sampling approach to anisotropic metric based mesh optimization krzysztof j. For highquality anisotropic adaptive mesh generation, three basic approaches exist to achieve mesh re. Pdf we introduce a procedure to generate metricorthogonal anisotropic volume meshes. Generation of anisotropic mesh by ellipse packing over an.
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