E.F. D'Azzevedo and R.B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput., 10(6):1063--1076, 1989. 19  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that an optimal triangle s aspect ratio is # # # # , 14) where # are the eigenvalues of the Hessian. The Hessian of a function f(u,v)is the matrix H f uu f uv f vu f vv . 15) When det H 0, the aspect ratio of (14) is the unique optimum for all L p norms with p # 1[7, 18]. This case coincides with a positive Gaussian curvature, if we regard f as a surface in 3 D. When det H 0, the L 2 optimal aspect ratio is not unique; there is a one parameter family of solutions generated by stretching (14) along one of the directions of zero curvature [16, Eq. 3) The L# ....

....if we regard f as a surface in 3 D. When det H 0, the L 2 optimal aspect ratio is not unique; there is a one parameter family of solutions generated by stretching (14) along one of the directions of zero curvature [16, Eq. 3) The L# optimal aspect ratios differ from (14) by a small factor [7]. Long, thin sliver triangles can be bad in certain contexts; for instance, they can lead to large condition numbers in the matrices used for certain finite element simulations. Equilateral triangles are desirable in such contexts. But for our goal, deriving an approximation with minimal ....

E.F. D'Azevedo, R.B. Simpson, On optimal interpolation triangle incidences, SIAM J. Sci. Statist. Comput. 10 (6) (1989) 1063--1075.

....option on the maximum of two assets when r = 0.05, # S1 = 0.10, # S2 = 0.30, # = 0.70, T t # = 0.25 and K = 40. The solutions were computed with 50 timesteps using a positive coe#cient edge swapped mesh with 6724 nodes. curvature, the placement of the elements must be data dependent (see D Azevedo and Simpson, 1989; Rippa, 1992) In particular, the triangles should be short in directions where the curvature of U is high, and long in directions where it is low. Edge swapping to ensure positive coe#cients ignores the curvature of U , so that elements may be placed with their long side in the direction of high ....

D'Azevedo, E. F. and R. B. Simpson (1989). On optimal interpolation triangle incidences. SIAM Journal on Scientific and Statistical Computing 10, 1063--1075.

....[45] observed that equiangularity of a triangulation, which is the sorted list of its angles, increases lexicographically in this way. DT (S) thus maximizes the minimum angle. Coarseness of a triangulation is measured by the largest circumcircle that arises for its triangles. D Azevedo and Simpson [16] showed that DT (S) minimizes coarseness in this sense, and also if smallest enclosing circles are taken rather than circumcircles. The latter property unlike others generalizes to higherdimensional Delaunay triangulations; see Rajan [59] Similarly, fatness may be de ned as the sum of ....

E.F.D'Azevedo, R.B.Simpson: `On optimal interpolation triangle incidences', SIAM J. Sci. Statist. Comput. 10 (1989) pp. 1063-1075

....have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [20, 24] minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3], and minimizing the maximum edge length [5] The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight triangulation (MWT) in which the optimization criterion is the sum of the edge lengths. Indeed, this seems to ....

E.F. D'Azevedo and R.B. Simpson. On optimal interpolation triangle incidences. Report CS-88-17, Univ. Waterloo (1988).

....results are related to the Delaunay triangulation [Del34] It has been shown that among all triangulations of a given finite point set, the Delaunay triangulation optimizes various criteria. The Delaunay triangulation maximizes the minimum angle [Sib78] minimizes the maximum circumscribing circle [D AS89], and minimizes the maximum smallest enclosing circle [D AS89, Raj91] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide and conquer [ShHo75, GuSt85] geometric ....

....has been shown that among all triangulations of a given finite point set, the Delaunay triangulation optimizes various criteria. The Delaunay triangulation maximizes the minimum angle [Sib78] minimizes the maximum circumscribing circle [D AS89] and minimizes the maximum smallest enclosing circle [D AS89, Raj91]. Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide and conquer [ShHo75, GuSt85] geometric transformation [Brow79] plane sweep [For87] and randomized ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10 (1989), 1063--1075.

.... is the Delaunay triangulation [Dela34] It is dual to the socalled Voronoi diagram [Voro08] The popularity of the two dimensional Delaunay triangulation is partly due to the fact that it optimizes various quality measures, including the smallest angle [Sibs78] the largest circumscribed circle [D AS89], the largest minimum enclosing circle [D AS89, Raja91] and the integral of the gradient squares [Ripp90] Algorithms that construct the Delaunay triangulation of a given set of n points in the plane in time O(n log n) can be found in Guibas, Stolfi [GuSt85] Fortune [Fort87] Guibas et al. ....

.... dual to the socalled Voronoi diagram [Voro08] The popularity of the two dimensional Delaunay triangulation is partly due to the fact that it optimizes various quality measures, including the smallest angle [Sibs78] the largest circumscribed circle [D AS89] the largest minimum enclosing circle [D AS89, Raja91], and the integral of the gradient squares [Ripp90] Algorithms that construct the Delaunay triangulation of a given set of n points in the plane in time O(n log n) can be found in Guibas, Stolfi [GuSt85] Fortune [Fort87] Guibas et al. GKS90] and other publications in computational geometry ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10 (1989), 1063--1075.

....positive results are related to the Delaunay triangulation defined for finite point sets [Del34] It has been shown that among all triangulations of a given point set, the Delaunay triangulation optimizes various criteria. These include the maxmin angle [Sib78] the minmax circumscribed circle [D AS89], the minmax smallest enclosing circle [D AS89, Raj91] and the minimum integral of the gradient squared [Rip90] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] ....

....triangulation defined for finite point sets [Del34] It has been shown that among all triangulations of a given point set, the Delaunay triangulation optimizes various criteria. These include the maxmin angle [Sib78] the minmax circumscribed circle [D AS89] the minmax smallest enclosing circle [D AS89, Raj91], and the minimum integral of the gradient squared [Rip90] Efficient algorithms for constructing Delaunay triangulations are abundant in the literature and based on such diverse algorithmic paradigms as edge flipping [Laws72, Laws77] divide andconquer [ShHo75, GuSt85] geometric transformation ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10 (1989), 1063--1075.

.... plane sweep [Fort87] Furthermore, it can be computed in time O(n log n) with high probability by randomized incrementation [GKS92] Among all triangulations of a given point set S, the Delaunay triangulation optimizes criteria such as the max min angle [Sibs78] the min max circumscribed circle [D AS89], the min max smallest enclosing circle [D AS89, Raja91] and the minimum integral of the gradient squared [Ripp90] As a graph structure, the Delaunay triangulation of S, D(S) is the straight line dual of the so called Voronoi diagram [Voro07, Voro08, Aure91] Various subgraphs of D(S) have ....

.... in time O(n log n) with high probability by randomized incrementation [GKS92] Among all triangulations of a given point set S, the Delaunay triangulation optimizes criteria such as the max min angle [Sibs78] the min max circumscribed circle [D AS89] the min max smallest enclosing circle [D AS89, Raja91], and the minimum integral of the gradient squared [Ripp90] As a graph structure, the Delaunay triangulation of S, D(S) is the straight line dual of the so called Voronoi diagram [Voro07, Voro08, Aure91] Various subgraphs of D(S) have been studied in the literature. Three of those subgraphs, ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Statist. Comput. 10 (1989), 1063--1075.

....on the maximum of two assets when r = 0:05, oe S1 = 0:10, oe S2 = 0:30, ae = 0:70, T Gamma t = 0:25 and K = 40. The solutions were computed with Deltat = 0:005 using a positive coefficient edge swapped mesh with 6724 nodes. curvature, the placement of the elements must be data dependent (see [8, 22]) In particular, the triangles should be short in directions where the curvature of U is high, and long in directions where it is low. Edge swapping to ensure positive coefficients ignores the curvature of U , so that elements may be placed with their long side in the direction of high curvature. ....

E. F. D'Azevedo and R. B. Simpson. On Optimal Interpolation Triangle Incidences. SIAM Journal on Scientific and Statistical Computing, 10(6):1063--1075, 1989.

....generation problem in computational geometry. The most often used optimization criteria [4] include maximizing the minimum angle among all elements of the partition (solved by the wellknown Delaunay triangulation [48] minimizing the maximum angle [14] minimizing a maximum mincontainment ellipse [13], and minimizing total length (an outstanding open problem in the field [17] Variants of these problems allow one to add extra vertices, called Steiner points, in order to further improve the quality of the solution. Mesh generation is a great example of inter disciplinary research. Its ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. In SIAM Journal on Scientific and Statistical Computing, volume 10, pages 1063--1075, 1995.

....it states that if the mesh size h tends to zero in such a way that the maximum angle tends to , then it may well happen that the finite element solution fails to converge to the exact one. It is possible to circumvent this condition, however, if the mesh is adapted to the solution. More precisely [19, 20, 21], if the triangles are thin along the direction of maximum absolute value of the second directional derivative. Several recent results [22, 23, 24] have brought new attention to metric based stretching. They concerned compressible flow computations using several solvers on solution adapted meshes ....

....will leaving the refinement procedure as a fixed black box. This is certainly an attractive feature having automated engineering analysis in mind. Questionable aspects are not difficult to find. The mathematical justification of DStype methods is poor (though interpolation results are encouraging [19, 20, 21]) They are less robust than EE type methods in that, if the initial mesh is too coarse, some solution details may not be detected. Also, large angles and crudely refined meshes may cause trouble to many of the existing finite element solvers. On one hand, this is not always the case, as shown ....

E. D'Azevedo and R. Simpson, 'On optimal interpolation triangle incidences', SIAM j. sci. stat. comput. 10, 1063--1075 (1989).

....measured simply by maximum edge length [209] or by more complicated metrics. For some applications such as modeling diffusion in a nonisotropic medium size is appropriately measured by the area of the min containment circle , after the domain has been transformed by an appropriate affine map [55]. Sizes of elements are typically used to weight finite element residuals in a posteriori error estimates, in order to find regions of the mesh in need of refinement [6, 37] 2.2.1. Delaunay Triangulation and the Flip Algorithm A well known construction, called the Delaunay triangulation, ....

....algorithm also gives a number of useful optimality properties of the CDT. The min containment circle of a triangle t is the smallest circle containing t. Theorem 1. Of all triangulations of a PSLG, the CDT (1) minimizes the largest circumcircle; 2) minimizes the largest min containment circle [55, 176]; and (3) maximizes the minimum angle in the triangulation [128] Proof: Each of these properties is improved by flipping a reversed quadrilateral. The optimal triangulation cannot be improved, so it has no reversed quadrilaterals, and hence by Lemma 3 must be the CDT. Result (3) is sometimes ....

[Article contains additional citation context not shown here]

E.F. D'Azevedo and R.B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10 (1989) 1063--1075.

....be accomplished either by remeshing [4] or by mesh optimization [1] 2 HESSIAN RECOVERY: Error estimate The motivation of Hessian based methods comes from interpolation estimates. If K is a linear triangle, it can be shown that the Lagrange interpolant Pi hu of a quadratic function u satisfies [7] k u Gamma Pi hu k1;K C max 2 K fi fi fi fi fi X i;j 2 u x i x j Delta i Delta j fi fi fi fi fi (2) where Delta i is the i th component of the vector that joins both ends of edge , and the constant C is of order unity. This suggests to equilibrate the ....

E. D'Azevedo and R. Simpson, "On optimal interpolation triangle incidences", SIAM J. Sci. Stat. Comput. 10, 1063-1075 (1989).

....of Delaunay triangulations in mesh generation. Unfortunately, neither this property nor the min max circumcircle property generalizes to Delaunay triangulations in dimensions higher than two. The property of minimizing the largest min containment circle was first noted by D Azevedo and Simpson [19], and has been shown by Rajan [55] to hold for higher dimensional Delaunay triangulations. 2.1.2 Planar Straight Line Graphs and Constrained Delaunay Triangulations Given that the Delaunay triangulation of a set of vertices maximizes the minimum angle (in two dimensions) why isn t the problem ....

E. F. D'Azevedo and R. B. Simpson. On Optimal Interpolation Triangle Incidences. SIAM Journal on Scientific and Statistical Computing 10:1063--1075, 1989.

....by filtering points and later smoothing the mesh. The third and fourth approaches trade direct control over element shapes for ease of fitting complicated geometries. Nevertheless, one can achieve anisotropy with these approaches by computing the Delaunay triangulation within a stretched space [29, 36, 43]. For example, Bossen [29] uses a background triangulation to define local affine transformations; Delaunay flips (described below) are then made with respect to transformed circles. Stretched Delaunay triangulations have many more large angles than ordinary Delaunay triangulations, but this ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput., 10:1063--1075, 1989.

....et al. correctness follows from our analysis below. Minimum angle, however, is not the only measure of mesh quality. Various papers have provided theoretical justification for other measures including maximum angle [4] maximum edge length [32] minimum height [23] minimum containing circle [12], and most recently ratio of area to sum of squared edge lengths [6] Data dependent criteria [6, 16, 31] may be used in adaptive meshing, which uses the finite element method s output to improve the mesh for another run. In this paper, we study optimization based smoothing using quality ....

E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10, 1989, pp. 1063--1075.

....have applications to cartography, spatial data analysis, and finite element methods. Optimization criteria for which e#cient algorithms are known include maximizing the minimum angle [19, 23] minimizing the maximum angle [6] minimizing the minimum angle [7] minimizing the maximum aspect ratio [3], and minimizing the maximum edge length [5] The most longstanding open problem in computational geometry is the complexity of another optimal triangulation problem, the minimum weight triangulation (MWT) in which the optimization criterion is the sum of the edge lengths. Indeed, this seems to ....

E.F. D'Azevedo and R.B. Simpson. On optimal interpolation triangle incidences. Report CS-88-17, Univ. Waterloo (1988).

....aspect ratio is # = # # # # # 2 # 1 # # # # 1=2 (14) where f# i g are the eigenvalues of the Hessian. The Hessian of a function f#u; v# is the matrix H = # f uu f uv f vu f vv # (15) When det H # 0, the aspect ratio of (14) is the unique optimum for all L p norms with p # 1 [7, 18]. This case coincides with a positive Gaussian curvature, if we regard f as a surface in 3 D. When det H # 0,theL 2 optimal aspect ratio is not unique; there is a one parameter family of solutions generated by stretching (14) along one of the directions of zero curvature [16, eqn. 3) The L1 ....

....if we regard f as a surface in 3 D. When det H # 0,theL 2 optimal aspect ratio is not unique; there is a one parameter family of solutions generated by stretching (14) along one of the directions of zero curvature [16, eqn. 3) The L1 optimal aspect ratios differ from (14) by a small factor [7]. Long, thin sliver triangles can be bad in certain contexts; for instance, they can lead to large condition numbers in the matrices used for certain finite element simulations. Equilateral triangles are desirable in such contexts. But for our goal, deriving an approximation with minimal ....

Eduardo F. D'Azevedo and R. Bruce Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput., 10(6):1063--1075, 1989.

....Many other alternative definitions of optimality have been proposed. See [4, 13] for surveys. Amongst all triangulations, The Delaunay triangulation optimizes many triangulation measures. These include ffl maximizing the minimum angle [20] ffl minimizing the maximum circumscribed circle [5], ffl minimizing the maximum smallest enclosing circle 1 [5, 17] ffl minimizing the integral of the gradient squared [18, 16] Little [14] and Schumaker [19] have proposed that the triangles in a good triangulation should have large inradii. I will prove that the Delaunay triangulation ....

....proposed. See [4, 13] for surveys. Amongst all triangulations, The Delaunay triangulation optimizes many triangulation measures. These include ffl maximizing the minimum angle [20] ffl minimizing the maximum circumscribed circle [5] ffl minimizing the maximum smallest enclosing circle 1 [5, 17], ffl minimizing the integral of the gradient squared [18, 16] Little [14] and Schumaker [19] have proposed that the triangles in a good triangulation should have large inradii. I will prove that the Delaunay triangulation maximizes the sum of the inradii (and hence the arithmetic mean) 2 ....

E. F. Dazevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Statist. Comput., 10(6):1063--1075, 1989.

....that we only work with one variable denoted j. We are going to determine the metric tensor in order to equilibrate the interpolation error. Assume that a first solution has been computed over a given mesh and suppose that continuous piecewise linear approximation is applied for j, then (see [1] and [2] 1 For the sake of simplicity, we will consider in this section that Omega ae IR 2 . We obtain the same results if Omega defines a surface or if Omega ae IR 3 . In the latest case, triangles are replaced by tetrahedrons . the maximum interpolation error depends on the Hessian ....

E. F. d'Azevedo and R. B. Simpson, `On Optimal Interpolation Triangle Incidences '. SIAM'S Journal on Scientific and Statistical Computing , no. 6: pp. 1063-- 1075, (1989).

....or a lexicographically ordered vector of the measures of all triangles. The most commonly used triangulation is the Delaunay triangulation which is optimal according to the following criteria: it maximizes the minimum angle of its triangles [Law77, Pre85] it minimizes the maximum circumcircle [Daz89], and it minimizes the maximum containing circle [Raj91] Optimality criteria defined on the approximating surface have also been considered in the context of scattered data interpolation. Triangulations defined through such criteria are referred to as datadependent triangulations, since ....

D'Azavedo, E.F., Simpson, R.B., On optimal interpolation triangle incidences, SIAM Journal on Scientific and Statistic Computing, 20, 6, 1989, pp.1063-1075.

..... In the latest case, triangles are replaced by tetrahedrons . INRIA Anisotropic Grid Adaptation for Inviscid and Viscous Flows Simulations 3 Assume that a first solution has been computed over a given mesh and suppose that continuous piecewise linear approximation is applied for j, then (see [1] and [2] the maximum interpolation error depends on the Hessian matrix of j H = 2 j= x 2 2 j= x y 2 j= x y 2 j= y 2 = R 1 0 0 2 R Gamma1 ; 6) and a metric tensor M can be defined by M = R j 1 j 0 0 j 2 j R Gamma1 : 7) Thus, the error over a ....

E. F. d'Azevedo and R. B. Simpson, `On Optimal Interpolation Triangle Incidences'. SIAM'S Journal on Scientific and Statistical Computing , no. 6: pp. 1063--1075, (1989).

....c lies outside T , in which case E(T ) je T (x mid )j (4) where x mid is the midpoint of the side of T closest to x c . While this latter case is perhaps less geometrically obvious, it is established, along with the algebraic formulae that correspond to this geometry, in D Azevedo and Simpson, [6]. Figure 2.1 Error contours for interpolation on a triangle 2.3 Solving the model maximum efficiency mesh problem In the model maximum efficiency mesh problem, the maximum error is specified not to exceed , and we seek an appropriate triangle shape with which to mesh the plane. We note from ....

....to the local mesh by the isotropic Delaunay insertion, and these edges identified as the connections for the anisotropic mesh in the x coordinate plane. This technique for selecting incidences was shown to be optimal for the model problem of x 2 with a fixed vertex set by D Azevedo and Simpson in [6], and has been used as a heuristic for data fitting by Dyn et al. in [10] Calculations for the compressible Navier Stokes flow around a two element airfoil are reported in [14] for free stream Mach number of 0:5 and Reynold s number 5 2 10 3 . The adapted mesh covers recirculation zones, ....

E F D'Azevedo and R B Simpson. On optimal interpolation triangle incidences. SIAM J for Sci and Stat Comp, 10:1063--1075, 1989.

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E.F. D'Azzevedo and R.B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput., 10(6):1063--1076, 1989. 19

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E. F. D'Azevedo and R. B. Simpson, On optimal interpolation triangle incidences, Siam J. Sci. Stat Comput. Vol. 10, No. 6, (1989), pp. 1063-1075,

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