A partially orthonormal basis is constructed with better conditioning properties for tetrahedral •H(curl)-conforming Nédélec elements. The shape functions are classified into several categories with respect to their topological entities on the reference 3-simplex. The basis functions in each category are constructed to achieve maKimum orthogonality. The numerical study on the matrix conditioning shows that for the mass and quasi-stiffness matrices, and in a logarithmic scale the condition number grows linearly vs. order of approximation up to order three. For each order of approximation, the condition number of the quasi-stiffness matrix is about one order less than the corresponding one for the mass matrix. Also, up to order six of approximation the conditioning of the mass and quasistiffness matrices with the proposed basis is better than the corresponding one with the Ainsworth-Coyle basis Internat. J. Numer. Methods. Engrg., 58:2103-2130, 2003. except for order four with the quasi-stiffness matrix. Moreover, with the new basis the composite matrix ¡JLM + 5 has better conditioning than the Ainsworth-Coyle basis for a wide range of the parameter ß.Mathematics subject classification: 65N30, 65F35, 65F15.On the Construction of Well-Conditioned Hierarchical Bases 527 Nédélec's foundational work [20] and following Webb [28], many researchers had constructed various hierarchical bases for several commonly known elements in 2D and 3D [1,[3][4][5]15,16,21,25,27], From the perspective of differential forms, Hiptmair [13] laid a general framework for canonical construction of 7{(curl)-and ?i(div)-conforming finite elements. In this respect, the reader is referred to the works [14,[22][23][24] and the monograph [9].One problem with hierarchical bases is the matrix ill-conditioning when higher-order bases are applied [2,28,31,32], For a hierarchical basis to be useful, the issue of ill-conditioning has to be resolved. Using Gram-Schmidt orthogonalization procedure Webb [28] gave the explicit formulas of the basis functions up to order three for triangular and tetrahedral elements. Following the same line of development [28], i.e., decomposing the basis functions into rotational and irrotational groups. Sun and collaborators [27] investigated the conditioning issue more carefully and also gave the basis functions up to the third order. Ainsworth and Coy le [3] studied both the dispersive and conditioning issues for the hierarchical basis on the hybrid quadrilateral/triangular meshes. With the aid of Jacobi polynomials, the interior bubble functions are made orthogonal over an equilateral reference triangle [3]. With this partial orthogonality it was shown that the condition numbers of both the mass matrix and the stiffness matrix could be reduced significantly [3]. Using Legendre polynomials J0rgensen et al. constructed a nearorthogonal basis for the quadrilaterals and suggested that the same procedure could be applied for the triangles with the help of collapsed coordinate system [17]. Partially addressing the conditionin...
Hierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.
A novel interpolation algorithm, fuzzy interpolation, is presented and compared with other popular interpolation methods widely implemented in industrial robots calibrations and manufacturing applications. Different interpolation algorithms have been developed, reported, and implemented in many industrial robot calibrations and manufacturing processes in recent years. Most of them are based on looking for the optimal interpolation trajectories based on some known values on given points around a workspace. However, it is rare to build an optimal interpolation results based on some random noises, and this is one of the most popular topics in industrial testing and measurement applications. The fuzzy interpolation algorithm (FIA) reported in this paper provides a convenient and simple way to solve this problem and offers more accurate interpolation results based on given position or orientation errors that are randomly distributed in real time. This method can be implemented in many industrial applications, such as manipulators measurements and calibrations, industrial automations, and semiconductor manufacturing processes.
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