A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call pliant mesh generation. It first constructs the constrained Delaunay triangulation of the domain, then iteratively smooths, refines, and retriangulates. On each iteration, a node is selected at random, it is repositioned according to attraction/repulsion with its neighbors, the neighborhood is retriangulated, and nodes are inserted or deleted as necessary. All operations are done relative to the metric tensor. This simple method generates high quality meshes whose elements conform well to the requested shape metric. The method appears particularly well suited to surface meshing and viscous flow simulations, where stretched triangles are desirable, and to time-dependent remeshing problems.

this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rate-of-strain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation.

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the "shape" of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three- dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case.

A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197--213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR-8921421 and under the Alan T. Waterman award, grant CCR-9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. We prove that if a polygon Q n has k reflex vertices, then any triangulation of Q n can be transformed to another triangulation of Q n with at most O(n + k 2 ) flips. We produce examples of polygons with two triangulations T and T such that to transform T to T requires O(n 2 ) flips. These results are then extended to triangulations of point sets. We also show that any triangulation of an n point set always has n - 4 2 edges that can be flipped. 1. Introduction Let P n = {v 1 , ..., v n } be a collection of points on the plane. A triangulation of P n is a partitioning of the convex hull Conv(P n ) of P n into a set of triangles T = {t 1 , ..., t m } with disjoint interiors in such a way that the vertices of each triangle t of T are points of P n . The elements of P n will be called the vertices of T and the edges of the triangles t 1 , ..., t m of T will be called the edges...

The edge-insertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.

A strategy is presented to find a set of points which yields a Conforming Delaunay tetrahedralization of a three-dimensional Piecewise-Linear Complex (PLC). This algorithm is novel because it imposes no angle restrictions on the input PLC. In the process, an algorithm is described that computes a planar conforming Delaunay triangulation of a Planar StraightLine Graph (PSLG) such that each triangle has a bounded circumradius, which may be of independent interest. 1 Introduction In many two- and three-dimensional geometric modeling problems, notably the numerical approximation of the solution to a Partial Differential Equation with a Finite-Element type method [SF73], it is very desirable to obtain a triangulation (tetrahedralization) that respects the domain of interest. The task of forming such decompositions, along with ensuring that the elements of the decompositions satisfy application-specific quality requirements, is sometimes referred to as unstructured mesh generation. Se...

An edge-flipping operation in a triangulation T of a set of points in the plane is a local restructuring that changes T into a triangulation that differs from T in exactly one edge. The edge-flipping distance between two triangulations of the same set of points is the minimum number of edge-flipping operations needed to convert one into the other. In the context of computing the rotation distance of binary trees Sleator, Tarjan, and Thurston [7] show an upper bound of 2n \Gamma 10 on the maximum edge-flipping distance between triangulations of convex polygons with n nodes, n ? 12. Using volumetric arguments in hyperbolic 3-space they prove that the bound is tight. In this paper we establish an upper bound on the edgeflipping distance between triangulations of a general set of points in the plane by showing that not more edge-flipping operations than the number of intersections between the edges of two triangulations are needed to transform these triangulations into another, and we pre...

Mesh coarsening and mesh enrichment are combined with an r-re nement scheme to produce a exible approach for mesh adaptation of time evolving domains. The robustness of this method depends heavily on maintaining mesh quality during each adaptation cycle. This in turn is inuenced by the ability to identify and remove badly shaped elements after the r-re nement stage. Measures of both element quality and element deformation can be de ned in terms of unitarily invariant matrix norms. The construction of these element deformation and quality measures is described, and details are provided of the three stages of the adaptation cycle.