I. Babuska and A.K. Aziz. On the angle condition in the finite element method. SIAM J. Numer. Anal., 13(2):214--226, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....their extent and curvature. This is especially critical given that most problems in computational science are boundary value problems and require accurate boundary information to yield an accurate solution. Equally important is mesh quality, which affects the convergence rate and solution accuracy [7, 1, 6]. Finally, automatic mesh generation requires Correspondence to: Charles Boivin, Department of Mechanical Engineering, University of British Columbia, 2324 Main Mall, Vancouver, BC, Canada V6T 1Z4 Contract grant sponsor: Supported in part by a NSERC post graduate scholarship Contract grant ....

I. Babuska and A. Aziz. On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 13:214--226, 1976.

.... the stretched mesh regions, either simply as a result of the overall algorithm [11, 12] or by design [13, 14, 15, 16, 17] For example, it is well known that for a stretched two dimensional triangulation, obtuse triangles must be avoided due to the poor approximation accuracy which they afford [18]. Since the sum of the three angles in a triangle must be equal to 180 degrees, the only permissible stretched triangles are those with one small angle, and two angles which are close to 90 degrees. Hence, nearly right angle triangles are required in the highly stretched regions of the mesh. It is ....

I. Babushka and A. K. Aziz. On the angle condition in the finite-element method. SIAM Journal of Numerical Analysis, 13(6), 1976. 11

....element) # i The angle at vertex v i of a triangle. #min , #max The signed minimum and maximum angles of a triangle. # ij In a tetrahedron, the dihedral angle at the edge connecting vertices v i and v j . piecewise linear approximation of a function. The celebrated paper of Babuska and Aziz [2] demonstrates that the accuracy of finite element solutions on triangular meshes degrades seriously as angles are allowed to approach 180 # , but the same is not true as angles are allowed to approach 0 # , so long as the largest angles are not too large. In other words, small angles are not ....

....is scale invariant; the error bounds are not. 5 Related Work Error estimates and quality measures for finite elements have been the subject of much research. Only a tiny fraction can be mentioned here. Much of the work on error estimates (including the aforementioned paper of Babuska and Aziz [2]) is built on functional analysis and embedding theorems. Apel s habilitation [1, especially Section 10] includes an excellent summary. These results are asymptotic in nature, and ignore the constants associated with the error bounds. The premise of this paper is that small constants and ....

Ivo Babuska and A. K. Aziz. On the Angle Condition in the Finite Element Method. SIAM Journal on Numerical Analysis 13(2):214--226, April 1976.

....large or small angles can degrade the quality of the numerical solution to a finite element problem. In interpolation, triangles with large angles can cause large errors in the gradients of the interpolated surface. In the finite element method, large angles can cause a large discretization error [1]; the solution may be less accurate than the method would normally promise. Small angles can cause the coupled systems of algebraic equations that the finite element method yields to be ill conditioned [6] A lower bound on the smallest angle of a triangulation implicitly bounds the largest ....

Ivo Babuska and A. K. Aziz. On the Angle Condition in the Finite Element Method. SIAM Journal on Numerical Analysis 13(2):214--226, April 1976.

....associated with r circ . See Appendix A.2. 3 v v 2 3 4 2 3 r r in mc circumcircle incircle min containment circle Figure 1: Quantities associated with triangles and tetrahedra. 2 Element Size, Element Shape, and Interpolation Error The celebrated paper of Babuska and Aziz [4] demonstrates that the accuracy of finite element solutions on triangular meshes degrades seriously as angles are allowed to approach 180 # , but the same is not true as angles are allowed to approach 0 # , so long as the largest angles are not too large. In other words, small angles are not ....

....and have few enough elements to be usable. Bad Good Figure 4: The top four tetrahedron shapes incur little interpolation error. The bottom three tetrahedron shapes can cause to be unnecessarily large, and to grow without bound if the tetrahedra are flattened. The Babuska Aziz result [4] applies to the anisotropic case too, but it is an asymptotic result: as the largest angle approaches 180 # , the discretization error grows without bound. This result is often interpreted to mean large angles are bad. But this is only true if large is defined appropriately. There are ....

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Ivo Babuska and A. K. Aziz. On the Angle Condition in the Finite Element Method. SIAM Journal on Numerical Analysis 13(2):214--226, April 1976.

.... the reader con consult the classical book of Ciarlet [17] or the book of Strong ond Fix [47] or the paper of Bratable and Zlamal [14] The importonce of non obtuse triongulations is cited in [47] 7] 16] 29] The importonce of the large angle condition in general is cited in Babusk ond Aziz [1] for Part of this research was done while the author was employed by the Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. This research was supported in part by NSF grants CCR 88 03549 and CCR 91 0473 by DIMACS under NSF grant STC 88 09648, and by the Dutch ....

I. Babuska and A.K. Aziz, (1976), "On the angle condition in the finite element method",SIAM J. on Numerical Analysis, Vol. 13, No 2, pp. 214-226.

....[1] in a family of triangular meshes the minimum angle of elements should be bounded away from zero. Equivalently, the ratio of the maximum element edge to the diameter of the inscribed circle should be bounded [2] Slightly less restrictive is the Synge BabukaAziz maximum angle condition [3, 4], requiting that the maximum angle of triangular elements be bounded away from The situation is less clear in three dimensions, in particular for tetrahedral elements. Although quality criteria are abundant (e.g. 5] no theory is available to explain adequately which parameters are really ....

I. Babugka, A.K. Aziz, "On the angle condition in the finite element method," SIAMJ.. Numer. Anal., vol. 13, No. 2, pp.214-226, 1976.

....individual elements have quality shapes. Tests on Jacobian determinants have been essential to identify poorly performing elements, but when dealing with higher order displacement fields, the numerical performance of the mesh is also very sensitive to obtuse angles as shown by Babuska and Aziz [22]. Common quality indicators based on element geometry, and frequently used in pre and post processing mesh improvement schemes have thus been developed (Houman [21] Field [23] Despite the good properties of triangulation techniques, they still require much user preprocessing both to determine ....

Babuska I, Aziz AK. On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 1976; 13(2): 214-226.

....can be assessed. There are other criteria that constitute a good triangulation. For example, in applications like finiteelement methods, finding good triangulations involves either maximizing or minimizing the angles of the generated triangles. These methods for triangulations are discussed in [18, 19]. 3.1 An Analog Circuit for Minimum Weight Triangulation We now describe the construction of an analog neural circuit for solving the minimum weight triangulation problem. We describe the circuit in terms of the initial state, the external inputs and the interconnection weights and then prove ....

I. Babuska and A. K. Aziz, "On the angle condition in the finite element method," SIAM Journal on Numerical Analysis, vol. 13, no. 2, pp. 214--226, 1976.

....individual elements have quality shapes. Tests on Jacobian determinants have been essential to identify poorly performing elements, but when dealing with higher order displacement fields, the numerical performance of the mesh is also very sensitive to obtuse angles as shown by Babuska and Aziz [15]. Common quality indicators based on element geometry, and frequently used in pre and post processing mesh improvement schemes have thus been developed (Houman [16] Field [17] Despite the good properties of triangulation techniques, they still require much user preprocessing both to determine ....

Babuska I, Aziz AK. On the angle condition in the finite element method. SIAM Journal on Numerical Analysis, 1976; 13(2): 214-226.

....of Energy under contract DE AC04 76DP00789, and by the Applied Mathematical Sciences program, U.S. Department of Energy Research. Steiner triangulations whose triangles have bounded shape are important for numerical analysis, in particular for a mesh in a finite element method. Babuska and Aziz [1] shows that the convergence of a finite element method depends on the largest angle of the triangulation. Often one wishes to find a triangulation for a PSLG that is not a polygon. For example, a semiconductor may have two differently doped regions. Hence a description of the semiconductor would ....

I. Babuska and A. K. Aziz, On the angle condition in the finite element method, SIAM J. Num. Anal. 13:214--226, 1976.

....minimize the maximum edge length, and maximize the minimum hight are considered. For example, if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 10, 11]. Polynomial time algorithms have been developed in determining those triangulations [2, 7, 8, 15] In computational geometry another important research object is to compute the minimum weight triangulation. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the ....

I. Babuska and A. Aziz, "On the angle condition in the finite element method", SIAM Journal on Numerical Analysis 13, pp.214-226, 1976.

....that the tetrahedra have bounded aspect ratio. This means that the angle between any adjacent pair of edges of the tetrahedron, or between any edge and a 2 dimensional face not containing it, is bounded below by a constant. For information about aspect ratio bounds in numerical analysis, see Babuska and Aziz [1976]. Our algorithm generates a triangulation for a nonconvex bounded polyhedral domain with holes. In addition, the triangulation is optimal in two respects. In particular, the best possible aspect ratio is achieved for the tetrahedra, and the number of tetrahedra is within a constant factor of the ....

I. Babuska and A. K. Aziz [1976], On the angle condition in the finite element method, SIAM J. Num. Anal. 13:214--226.

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I. Babuska and A.K. Aziz. On the angle condition in the finite element method. SIAM J. Numer. Anal., 13(2):214--226, 1976.

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Babuska, I. and Aziz, A.K. - "On the Angle Condition in the Finite Element Method," SIAM J. Num. Anal., vol. 13, pp. 214-226, 1976.

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Babuska I. and Aziz A.K., "On the angle condition in the finite element method", SIAM Journal of Numerical Analysis, Vol.13, No.2, (1976), pp 214-226.

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BABUSKA, I., AND AZIZ, A. 1976. On the angle condition in the finite element method. SIAM Journal of Numerical Analysis, 13(2), 214--226.

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I. Babuska and A. Aziz. On the angle condition in the finite element method. SIAM Journal of Numerical Analysis, 13(2):214--226, 1976.

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I. Babuska and A. Aziz. On the angle condition in the Finite Element Method. Int. J. Numer. Meth. Eng., 12, 1978.

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I. Babuska and A. Aziz. On the angle condition in the Finite Element Method. Int. J. Numer. Meth. Eng., 12, 1978.

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I. Babuska and A. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal., Vol. 13, 1976, pp. 214-227.

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Babuska, I. and Aziz, A. K., On the angle condition in the finite element method. SIAM J. Numer. Anal. 13: 214-226, 1976.

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Babushka, I. and Aziz, A.K., On the Angle Condition in the Finite Element Method, SIAM Journal of Numerical Analysis, 13, 2, 1976.

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Babuska, I., and Aziz, A. K., "On the Angle Condition in the Finite Element Method", SIAM J. Numer. Anal., Vol. 13, No. 2, 1976.

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I. Babuska and A. Aziz, "On the angle condition in the finite element method", SIAM Journal on Numerical Analysis 13 (1976) 214-226. 15

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