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P.v. Jeyakarthikeyan - One of the best experts on this subject based on the ideXlab platform.
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Fast numerical algorithms with universal matrices for finding element matrices of quadrilateral and hexahedral elements
Ain Shams Engineering Journal, 2020Co-Authors: P.v. Jeyakarthikeyan, M. Radha Jeyakarthikeyan, Hamza Sulayman AbdullahiAbstract:Abstract In this paper, a new closed-form formulation with universal matrices strongly recommends that Weighted Richardson Extrapolation (WRE) with robust and hourglass controlled one-point Quadrature can absolutely replace the conventional Gauss Quadrature in terms of efficiency, accuracy, and speed to find element stiffness matrices of quadrilateral and hexahedral elements. Linearizing the geometric transformation and averaging the material property over an element, using sampling point at the origin (0,0) of a standard mapped 2-square ( ξ , η ) plane for two-dimensional analysis and origin (0,0,0) of a standard mapped 2-cube ( ξ , η , ζ ) system in the hexahedral element (mid-point rule), helps to obtain element stiffness matrix quickly and explicitly through constant universal matrices. This technique is used independently to each of the eight sub-cubes of the mapped 2-cube ( ξ , η , ζ ) of the element for three-dimensional problems and four sub-squares of the mapped 2-square of the element for two-dimensional problems. These matrices are assembled appropriately for a second and better approximation. A weighted addition of the two approximations produces a stiffness matrix as accurate as from conventional Gauss Quadrature. However, finding the stiffness matrix in this way, due to explicit integrations, demands only a third of the time needed for Gauss Quadrature.
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Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application
American Journal of Applied Sciences, 2018Co-Authors: Subramanian G., P.v. JeyakarthikeyanAbstract:A weighted integration route with robust one-point integration (hourglass-controlled) is proposed as efficient, time saving alternative to Gauss Quadrature for stiffness matrix of bilinear quadrilaterals. One-point rule relies on sampling at the center of the element to linearize the geometric transformation and average the material property over it. This enables, for a given element, explicit integration of stiffness matrix yielding a first approximation. For a second and better approximation, this procedure is applied independently to each of the four sub-squares of the mapped 2-square of the element and the matrices are assembled. A weighted addition of the two approximations produces a stiffness matrix as accurate as from 3-point Gauss-Quadrature (G9P). Whereas, due to explicit integrations, obtaining stiffness matrix in this way demands less than a third of the time needed for 2-point Gauss-Quadrature (G4P). On both counts (speed and accuracy) this approach outperforms Gauss-Quadrature. Sampling (material and geometry) at 5-points makes this element superior to G4P for Functionally Graded Material (FGM) applications. Bench mark examples by this approach are validated with Gauss Quadrature and analytical solutions.
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An alternate stable midpoint Quadrature to improve the element stiffness matrix of quadrilaterals for application of functionally graded materials (FGM)
Computers & Structures, 2017Co-Authors: P.v. Jeyakarthikeyan, G. Subramanian, R. YogeshwaranAbstract:A weighted integration route with robust stable one-point integration is recommended.Its an alternative to Gauss Quadrature to obtain stiffness matrix of quadrilaterals.A weighted addition of the two approximations produces a far better stiffness matrix.Due to many simple sampling points, the method is most suited for FGM applications. An efficient, stable and accurate quadrilateral element and its improved stiffness matrix on the midpoint Quadrature concept is proposed in this research study. As a first approximation, the integrating point is considered as midpoint of the element of the mapped 2-square in the (, ) plane (same as one-point Gauss-Quadrature). As a second approximation or stabilizing function, integrating points are assumed to be either at the midpoint of the four quadrants or four element edges of the mapped 2-square element in the (, ) plane and these interpolated data are assembled. An appropriate weighted addition of the two approximations is found to result in a better and stable stiffness matrix than equivalent time delayed value of Gauss Quadrature.
R. Yogeshwaran - One of the best experts on this subject based on the ideXlab platform.
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An alternate stable midpoint Quadrature to improve the element stiffness matrix of quadrilaterals for application of functionally graded materials (FGM)
Computers & Structures, 2017Co-Authors: P.v. Jeyakarthikeyan, G. Subramanian, R. YogeshwaranAbstract:A weighted integration route with robust stable one-point integration is recommended.Its an alternative to Gauss Quadrature to obtain stiffness matrix of quadrilaterals.A weighted addition of the two approximations produces a far better stiffness matrix.Due to many simple sampling points, the method is most suited for FGM applications. An efficient, stable and accurate quadrilateral element and its improved stiffness matrix on the midpoint Quadrature concept is proposed in this research study. As a first approximation, the integrating point is considered as midpoint of the element of the mapped 2-square in the (, ) plane (same as one-point Gauss-Quadrature). As a second approximation or stabilizing function, integrating points are assumed to be either at the midpoint of the four quadrants or four element edges of the mapped 2-square element in the (, ) plane and these interpolated data are assembled. An appropriate weighted addition of the two approximations is found to result in a better and stable stiffness matrix than equivalent time delayed value of Gauss Quadrature.
Lothar Reichel - One of the best experts on this subject based on the ideXlab platform.
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Generalized block anti-Gauss Quadrature rules
Numerische Mathematik, 2019Co-Authors: Hessah Alqahtani, Lothar ReichelAbstract:Golub and Meurant describe how pairs of Gauss and Gauss–Radau Quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real-valued matrix functionals defined by a symmetric matrix. However, there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds. In this situation, it may be possible to determine upper and lower bounds by evaluating pairs of Gauss and anti-Gauss rules. Unfortunately, it is difficult to ascertain whether the values determined by Gauss and anti-Gauss rules bracket the value of the given real-valued matrix functional. Therefore, generalizations of anti-Gauss rules have recently been described, such that pairs of Gauss and generalized anti-Gauss rules may determine upper and lower bounds for real-valued matrix functionals also when pairs of Gauss and (standard) anti-Gauss rules do not. The available generalization requires the matrix that defines the functional to be real and symmetric. The present paper reviews available anti-Gauss and generalized anti-Gauss rules and extends them in several ways that allow applications in new situations. In particular, the genarlized anti-Gauss rules for a real-valued non-negative measure described in Pranić and Reichel (J Comput Appl Math 284:235–243, 2015 ) are extended to allow the estimation of the error in matrix functionals defined by a non-symmetric matrix, as well as to matrix-valued matrix functions. Modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described.
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Simplified anti-Gauss Quadrature rules with applications in linear algebra
Numerical Algorithms, 2017Co-Authors: Hessah Alqahtani, Lothar ReichelAbstract:The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w T f(A)v with \(A\in {\mathbb {R}}^{n\times n}\) a large matrix, f a function, and \(v,w\in {\mathbb {R}}^{n}\) arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v T f(A)v by a pair of Gauss and Gauss-Radau Quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss Quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A T . The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss Quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.
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Truncated generalized averaged Gauss Quadrature rules
Journal of Computational and Applied Mathematics, 2016Co-Authors: Dušan Lj. Djukić, Lothar Reichel, Miodrag M. SpalevićAbstract:Generalized averaged Gaussian Quadrature formulas may yield higher accuracy than Gauss Quadrature formulas that use the same moment information. This makes them attractive to use when moments or modified moments are cumbersome to evaluate. However, generalized averaged Gaussian Quadrature formulas may have nodes outside the convex hull of the support of the measure defining the associated Gauss rules. It may therefore not be possible to use generalized averaged Gaussian Quadrature formulas with integrands that only are defined on the convex hull of the support of the measure. Generalized averaged Gaussian Quadrature formulas are determined by symmetric tridiagonal matrices. This paper investigates whether removing some of the last rows and columns of these matrices gives Quadrature rules whose nodes live in the convex hull of the support of the measure.
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Generalized averaged Gauss Quadrature rules for the approximation of matrix functionals
BIT Numerical Mathematics, 2015Co-Authors: Lothar Reichel, Miodrag M. Spalević, Tunan TangAbstract:The need to compute expressions of the form \(u^*f(A)v\), where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss Quadrature formulas.
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Generalized anti-Gauss Quadrature rules
Journal of Computational and Applied Mathematics, 2015Co-Authors: Miroslav S. Pranić, Lothar ReichelAbstract:Gauss Quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis. Laurie proposed an ( n + 1 ) -point Quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n -point Gauss Quadrature rule for all polynomials of degree up to 2 n + 1 . This rule is referred to as an anti-Gauss rule. It is useful for the estimation of the error in the approximation of the desired integral furnished by the n -point Gauss rule. This paper describes a modification of the ( n + 1 ) -point anti-Gauss rule, that has n + k nodes and gives an error of the same magnitude and of opposite sign as the associated n -point Gauss Quadrature rule for all polynomials of degree up to 2 n + 2 k - 1 for some k 1 . We refer to this rule as a generalized anti-Gauss rule. An application to error estimation of matrix functionals is presented.
Gustaaf Jacobs - One of the best experts on this subject based on the ideXlab platform.
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An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations
Computers & Fluids, 2020Co-Authors: Hareshram Natarajan, Gustaaf JacobsAbstract:Abstract An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss Quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss Quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss Quadrature points from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.
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An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations.
arXiv: Numerical Analysis, 2019Co-Authors: Hareshram Natarajan, Gustaaf JacobsAbstract:An explicit high order semi-Lagrangian method is developed for solving Lagrangian transport equations in Eulerian-Lagrangian formulations. To ensure a semi-Lagrangian approximation that is consistent with an explicit Eulerian, discontinuous spectral element method (DSEM) discretization used for the Eulerian formulation, Lagrangian particles are seeded at Gauss Quadrature collocation nodes within an element. The particles are integrated explicitly in time to obtain an advected polynomial solution at the advected Gauss Quadrature locations. This approximation is mapped back in a semi-Lagrangian fashion to the Gauss Quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. An explicit time integration is used for the semi-Lagrangian approximation that is consistent with the grid based DSEM solver, which ensures that particles seeded at the Gauss Quadrature points do not leave the element's bounds. The method is hence local and parallel and facilitates the solution of the Lagrangian formulation without the grid complexity, and parallelization challenges of a particle solver in particle-mesh methods. Numerical tests with one and two dimensional advection equation are carried out. The method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the accuracy of the time integration.
G Lombardi - One of the best experts on this subject based on the ideXlab platform.
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machine precision evaluation of singular and nearly singular potential integrals by use of Gauss Quadrature formulas for rational functions
IEEE Transactions on Antennas and Propagation, 2008Co-Authors: R D Graglia, G LombardiAbstract:A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical Quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss Quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provided.