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H T Rathod - One of the best experts on this subject based on the ideXlab platform.
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synthetic division based integration of rational functions of Bivariate Polynomial numerators with linear denominators over a unit triangle 0 ξ η 1 ξ η 1 in the local parametric space ξ η
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: H T Rathod, M Shajedul D KarimAbstract:The domain of real problems in mechanics often contains curved boundaries. Curved boundaries are often more accurately modelled by curved finite elements than by straight edged elements, as straight sides are perfectly satisfactory if the domain has a polygonal boundary. Because fewer curved elements are required, the effort needed to obtain a solution is usually reduced. If some parts of the boundary are curved, however, elements with at least one curved side are desirable. Our aim in this paper is to consider the triangular element with two straight sides and one curved side. This paper is concerned with explicit formulae for evaluating integrals of rational functions of Bivariate Polynomial numerators with linear denominators over a unit triangle {0⩽ξ,η⩽1,ξ+η⩽1} in the local parametric two dimensional space (ξ,η). These integrals arise in finite element formulations of second order linear partial differential equations by use of triangular element with two straight sides and one curved side of quadratic variation which often require relatively large numerical effort to integrate. The curved elements considered here are the four node, six node and ten node triangular elements with one curved side of quadratic variation and the other two sides have straight edges under the isoparametric and subparametric transformations, respectively. We have shown that by use of a method similar to synthetic division, the rational integrals of nth order Bivariate Polynomial numerator with a linear denominator having (n+1)(n+2)/2 integrals can be reduced to rational integrals of nth order Polynomial numerator in one variate with the same linear denominator having (n+1) integrals and a simple integral of Bivariate Polynomial expression containing n(n+1)/2 terms (which is free from denominator), and this amounts to a substantial reduction in the numerical effort for such integrals. The explicit analytical integration formulae (obtained) up to sextic Polynomial numerator in Bivariates ξ,η due to different element geometry are, for clarity and reference, summarised in tables. Finally three application examples are also considered. For the first example we have used integration formulae derived in all theorems of this paper and explained the detailed computational scheme. For the other two examples computational scheme follows in a similar way. It is observed in the solution of all problems that the displacements and torsional constants are satisfactory when components of all element matrices are calculated by analytical integration formulae derived in this paper. It is also observed that in the calculation of components of element matrices by using numerical integration formulae (e.g., 7-point and 13-point rule) much discrepancy occurs if the element geometry is coarse with a concave curve side. But it is found that the analytical integration technique is always consistent. Therefore the symbolic integration formulae presented in this paper are reliable and may lead to an easy and systematic incorporation of element matrices required in the finite element solution procedure.
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integration of rational functions of Bivariate Polynomial numerators with linear denominators over a 1 1 square in a local parametric two dimensional space
Computer Methods in Applied Mechanics and Engineering, 1998Co-Authors: H T Rathod, Md Shafiqul IslamAbstract:Abstract This paper is concerned with explicit formulas and algorithms for computing integral of rational function of Bivariate Polynomial numerators with linear denominators over a (−1,1) square in the local parametric space. These integrals arise in finite element formulations of second-order partial differential equations. Explicit evaluation of these integrals produce analytical finite element relations, provided that the original element geometry is restricted to a linear convex quadrilateral. We have also presented two algorithms, either of these can be used to compute n(n + 1) 2 integrals of 0th to nth order Bivariate Polynomial numerator whenever (n + 1) such integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulas. The analytical quadrature formulas from cubic to quintic monomial numerator in one of the variâtes due to different element geometry are, for clarity and reference, summarized in tabular forms. All the explicit formulas are constructed with three simple functions of element nodal values. They can be easily coded and save much computation time besides their inherent merit on numerical accuracy associated with analytical integration. We have also derived some integration formulas for the product of global derivatives which has applications to second-order linear partial differential equations in a variety of disciplines. Finally, an application example to compute the torsional constant K for an equilateral triangular cross-section is also considered for which we have explained the detailed computational scheme.
Wei Hong - One of the best experts on this subject based on the ideXlab platform.
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efficient two dimensional direction finding via auxiliary variable manifold separation technique for arbitrary array structure
Mathematical Problems in Engineering, 2015Co-Authors: Guang Hua, Hou-xing Zhou, Xicheng Zhu, Wei HongAbstract:A Polynomial rooting direction of arrival (DOA) algorithm for multiple plane waves incident on an arbitrary array structure that combines the multiPolynomial resultants and matrix computations is proposed in this paper. Firstly, a new auxiliary-variable manifold separation technique (AV-MST) is used to model the steering vector of arbitrary array structure as the product of a sampling matrix (dependent only on the array structure) and two Vandermonde-structured wavefield coefficient vectors (dependent on the wavefield). Then the propagator operator is calculated and used to form a system of Bivariate Polynomial equations. Finally, the automatically paired azimuth and elevation estimates are derived by Polynomial rooting. The presented algorithm employs the concept of auxiliary-variable manifold separation technique which requires no sector by sector array interpolation and thus does not suffer from any mapping errors. In addition, the new algorithm does not need any eigenvalue decomposition of the covariance matrix and exhausted search over the two-dimensional parameter space. Moreover, the algorithm gives automatically paired estimates, thus avoiding the complex pairing procedure. Therefore, the proposed algorithm shows low computational complexity and high robustness performance. Simulation results are shown to validate the effectiveness of the proposed method.
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Efficient two dimensional direction finding via auxiliary-variable manifold separation technique for arbitrary array structure
2014 IEEE International Conference on Communiction Problem-solving, 2014Co-Authors: Jiu-dong Wu, Hou-xing Zhou, Wei HongAbstract:A Polynomial rooting Direction of Arrival (DOA) algorithm for multiple plane waves incident on an arbitrary array structure that combines the multiPolynomial resultants and matrix computations is presented in this paper. Firstly, a new auxiliary-variable manifold separation technique (AV-MST) is proposed to modal the steering vector of arbitrary array structure as the product of a sampling matrix (dependent only on the array structure) and two Vandermonde-structured wavefield coefficient vectors (dependent on the wavefield). Then the propagator operator is calculated and used to form a system of Bivariate Polynomial equations. Finally, the automatically paired azimuth and elevation estimates are derived by Polynomial rooting. The presented algorithm employs the concept of auxiliary-variable manifold separation technique which requires no sector by sector array interpolation and thus does not suffer from any mapping errors. In addition, the new algorithm does not need any eigenvalue decomposition of the covariance matrix and exhausted search over the two dimensional parameter space. Moreover, the algorithm gives automatically paired estimates, thus avoiding the complex pairing procedure. Therefore, the proposed algorithm shows low computational complexity and high robustness performance. Simulation results are shown to validate the effectiveness of the proposed method.
Jinsan Cheng - One of the best experts on this subject based on the ideXlab platform.
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certified numerical real root isolation for Bivariate Polynomial systems
International Symposium on Symbolic and Algebraic Computation, 2019Co-Authors: Jinsan Cheng, Junyi WenAbstract:In this paper, we present a new method for isolating real roots of a Bivariate Polynomial system. Our method is a subdivision method which is based on real root isolation of univariate Polynomials and analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple real zero of the system in the box. We implement our method to isolate the real zeros of a given Bivariate Polynomial system. The experiments show the effectivity and efficiency of our method, especially for systems with high degrees and sparse terms. Our method also works for non-Polynomial systems.
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a generic position based method for real root isolation of zero dimensional Polynomial systems
Journal of Symbolic Computation, 2015Co-Authors: Jinsan Cheng, Kai JinAbstract:We improve the local generic position method for isolating the real roots of a zero-dimensional Bivariate Polynomial system with two Polynomials and extend the method to general zero-dimensional Polynomial systems. The method mainly involves resultant computation and real root isolation of univariate Polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is O ? B ( N 10 ) for the Bivariate case, where N = max ? ( d , ? ) , d resp., ? is an upper bound on the degree, resp., the maximal coefficient bitsize of the input Polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for Bivariate Polynomial systems.
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a generic position based method for real root isolation of zero dimensional Polynomial systems
arXiv: Symbolic Computation, 2013Co-Authors: Jinsan Cheng, Kai JinAbstract:We improve the local generic position method for isolating the real roots of a zero-dimensional Bivariate Polynomial system with two Polynomials and extend the method to general zero-dimensional Polynomial systems. The method mainly involves resultant computation and real root isolation of univariate Polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the Bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input Polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for Bivariate Polynomial systems.
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root isolation for Bivariate Polynomial systems with local generic position method
International Symposium on Symbolic and Algebraic Computation, 2009Co-Authors: Jinsan Cheng, Xiaoshan GaoAbstract:A local generic position method is proposed to isolate the real roots of a Bivariate Polynomial system ∑={f(x,y),g(x,y)}. In this method, the roots of the system are represented as linear combinations of the roots of two univariate Polynomial equations t(x)=0 and T(X)=0: {x = α, y = β -- α/s | α e V(t(x)), β e V(T(X)), ||β -- α|
Md Shafiqul Islam - One of the best experts on this subject based on the ideXlab platform.
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integration of rational functions of Bivariate Polynomial numerators with linear denominators over a 1 1 square in a local parametric two dimensional space
Computer Methods in Applied Mechanics and Engineering, 1998Co-Authors: H T Rathod, Md Shafiqul IslamAbstract:Abstract This paper is concerned with explicit formulas and algorithms for computing integral of rational function of Bivariate Polynomial numerators with linear denominators over a (−1,1) square in the local parametric space. These integrals arise in finite element formulations of second-order partial differential equations. Explicit evaluation of these integrals produce analytical finite element relations, provided that the original element geometry is restricted to a linear convex quadrilateral. We have also presented two algorithms, either of these can be used to compute n(n + 1) 2 integrals of 0th to nth order Bivariate Polynomial numerator whenever (n + 1) such integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulas. The analytical quadrature formulas from cubic to quintic monomial numerator in one of the variâtes due to different element geometry are, for clarity and reference, summarized in tabular forms. All the explicit formulas are constructed with three simple functions of element nodal values. They can be easily coded and save much computation time besides their inherent merit on numerical accuracy associated with analytical integration. We have also derived some integration formulas for the product of global derivatives which has applications to second-order linear partial differential equations in a variety of disciplines. Finally, an application example to compute the torsional constant K for an equilateral triangular cross-section is also considered for which we have explained the detailed computational scheme.
Irina Voiculescu - One of the best experts on this subject based on the ideXlab platform.
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affine arithmetic in matrix form for Polynomial evaluation and algebraic curve drawing
Progress in Natural Science, 2002Co-Authors: Irina VoiculescuAbstract:This paper shows how tight bounds for the range of a Bivariate Polynomial can be found using a matrix method based on affine arithmetic. Then, this method is applied to drawing an algebraic curve with a hierarchical algorithm, which demonstrates that more accurate answers can be obtained more rapidly than using conventional interval arithmetic.