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