Babu ska I., Aziz A.K. "On the angle condition in the finite element method." SIAM J. Numer. Anal., vol. 13, no. 2, 214--226, 1976  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for Re = 40 to 10 7 [6] In the finite element method, it is well known that poor element quality can have a deleterious effect on the solution. Angles near 0 can lead to ill conditioning of the discrete operator [13] while angles near can lead to large approximation errors in the H 1 norm [4]. When the geometry undergoes large changes during optimization, it becomes very difficult, if not impossible, to construct a structured mesh scheme that maintains element quality for a wide range of shapes. An unstructured mesh is capable of resolving complicated, deforming geometries with ....

I. Babu ska and A. Aziz, On the angle condition in the finite element method, SIAM Journal on Numerical Analysis, 13 (1976), pp. 214--227.

....including numerical methods such as the finite element method, computer graphics and geographic information system. To ensure accurate result, the simplices of the mesh must be well shaped, in the sense that they have small aspect ratios, i.e. the smallest angles are bounded from below [1, 15]. To get accurate numerical simulation, we also expect to have small element size. On the other hand, we prefer to have larger element size due to the time complexity of the simulation. Hence elements should have properly chosen size and shape that adapt to the complex geometry and solution ....

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

....method, but its correctness is not directly justified by the work of Matou sek 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 ....

....things that there may be many local optima instead of one global optimum. Indeed, it seems likely that the height and perimeter criteria mentioned above do not lead to good element shapes. However there is evidence that the maximum angle is an appropriate quality measure for finite element meshes [4], so we now discuss methods for optimizing this measure. Our results should be seen as preliminary and unready for practical implementation. THEOREM 6.1. We can find the placement of a Steiner point in a star shaped polygon, minimizing the maximum angle, in time O(n log c n) for some constant ....

I. Babu ska and A. Aziz. On the angle condition in the finite element method. SIAM J. Num. Anal. 13, 1976, pp. 214--227.

....approximation being very far from the continuous solution. A final requirement is that all angles in the mesh be bounded away from 0 and . The latter condition is necessary because the discretization error in a finite element approximation has been shown to grow as the maximum angle approaches [1]. We would like to avoid small angles because the condition number of the matrices arising from mesh elements has been shown to grow as O( 1 min ) where min is the smallest angle in the mesh [5] 2.1. Related Work. A number of mesh refinement algorithms have been shown to maintain the mesh ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....being very far from the continuous solution. A final requirement is that all angles in the mesh be bounded away from 0 and . The latter condition is necessary because the discretization error in a finite element approximation has been shown to grow as the maximum angle approaches [1]. We would like to avoid small angles because the condition number of the matrices arising from mesh elements has been shown to grow as O( 1 min ) where min is the smallest angle in the mesh [4] 2.1. Related Work. A number of mesh refinement algorithms have been shown to maintain the ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....mesh on the right. Fig. 6. The mesh on the left is a conforming mesh. The mesh on the right is nonconforming. Note the midpoint on the left side of the rightmost triangle. To ensure the quality of a mesh during refinement, it is desirable that no very large or very small angles are generated. In [1], Babuska and Aziz show that the accuracy of the finite element approximation degrades as the maximum angle approaches . Small angles should be avoided because the condition number of the matrices that arise from the finite element discretization grows as O( 1 min ) where min is the ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....in the solution of PDEs typically depends on the number and location of the elements vertices and the shape of the elements. For example, in the two dimensional case it is known that the accuracy of the finite element approximation degrades as the maximum interior angle of an element approaches [2]; further, the condition number of the resulting matrices grow as O( 1 min ) where min is the minimum interior angle of the mesh [11] The same basic principles apply in three dimensions; however, more complex measures of element quality are required as discussed in [19] Adaptive ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....graded mesh, adjacent triangles should not differ dramatically in area. Finally, all angles in the mesh must be bounded away from 0 and . The latter requirement is necessary because the discretization error in a finite element approximation has been shown to grow as the maximum angle approaches [1]. Small angles are to be avoided because the condition number of the matrices arising from mesh elements has been shown to grow as O( 1 min ) where min is the smallest angle in the mesh [5] 2.1. A Parallel Bisection Algorithm. The bisection algorithm bisects triangles across the largest ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....gap cases. # Figure 8. Kite decomposition of four sided gap with one side on domain boundary: add small circles along boundary edge making three sided gaps. 6 No large angles The maximum angle of any triangle has been shown to be one of the more important indicators of triangular mesh quality [1] and it is believed that the maximum angle is similarly important in quadrilateral meshes. For triangular meshes, a maximum angle of 90 # can be achieved [5] but for quadrilaterals this would imply that all elements are rectangles, which can only be achieved when the domain has axis parallel ....

I. Babu ska and A. Aziz. On the angle condition in the finite element method. SIAM J. Numerical Analysis 13:214--227, 1976.

....adjacent triangles should not differ dramatically in area. Finally, we require that all angles in the mesh be bounded away from 0 and . The latter requirement is necessary because the discretization error in a finite element approximation has been shown to grow as the maximum angle approaches [1]. We would like to avoid small angles because the condition number of the matrices arising from mesh elements has been shown to grow as O( 1 min ) where min is the smallest angle in the mesh [7] 3.1. A parallel bisection algorithm. The bisection algorithm bisects triangles across the ....

I. Babu ska and A. K. Aziz, On the angle condition in the finite element method, SIAM Journal of Numerical Analysis, 13 (1976), pp. 214--226.

....) 0; T6 oe(T h ) oe; for all h with oe independent of h. Condition T6 is satisfied if the minimal angle of all triangles is bounded from below independently of h. For a discussion of this minimal angle condition and its replacement by the weaker maximal angle condition see Babuska and Aziz [2]. With angles larger than 90 degrees, however, we loose the M matrix feature of the stiffness matrix. 2.5 Algebraic structure When the topology and geometry of a mesh is determined, the physical and mathematical problem description is used to derive the algebraic structure of the finite element ....

I. Babu ska and A. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal., 13 (1976), pp. 214--226.

.... two and three dimensions is provided by Bern and Eppstein [3] The real difficulty, though, is that a mesh generator should avoid producing elements with large aspect ratios skinny triangles and tetrahedra, so to speak because angles close to 180 ffi cause a large discretization error [1], and very small angles cause the stiffness matrix K to be ill conditioned [6] This constraint is difficult to reconcile with the correct meshing of arbitrary geometries. Archimedes includes a two dimensional mesh generator that triangulates complex geometries with high quality elements, and ....

BABU SKA, I., AND AZIZ, A. K. On the Angle Condition in the Finite Element Method. SIAM Journal on Numerical Analysis 13, 2 (Apr. 1976), 214--226.

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Babu ska I., Aziz A.K. "On the angle condition in the finite element method." SIAM J. Numer. Anal., vol. 13, no. 2, 214--226, 1976

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