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Victor Sofonea - One of the best experts on this subject based on the ideXlab platform.
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application of mixed Quadrature lattice boltzmann models for the simulation of poiseuille flow at non negligible values of the knudsen number
Journal of Computational Science, 2016Co-Authors: Victor E Ambrus, Victor SofoneaAbstract:Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian Quadratures, where the Quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range Quadratures outperform the full-range Gauss–Hermite Quadratures for the direction perpendicular to the diffuse-reflecting plates, while the Quadrature on the periodic direction along the flow is the full-range Gauss–Hermite Quadrature. Our results are validated against numerical results available in the literature.
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lattice boltzmann models based on half range gauss hermite Quadratures
Journal of Computational Physics, 2016Co-Authors: Victor E Ambrus, Victor SofoneaAbstract:We discuss general features of thermal lattice Boltzmann models based on half-range Gauss Quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the Quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different Quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less Quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.
Victor E Ambrus - One of the best experts on this subject based on the ideXlab platform.
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application of mixed Quadrature lattice boltzmann models for the simulation of poiseuille flow at non negligible values of the knudsen number
Journal of Computational Science, 2016Co-Authors: Victor E Ambrus, Victor SofoneaAbstract:Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian Quadratures, where the Quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range Quadratures outperform the full-range Gauss–Hermite Quadratures for the direction perpendicular to the diffuse-reflecting plates, while the Quadrature on the periodic direction along the flow is the full-range Gauss–Hermite Quadrature. Our results are validated against numerical results available in the literature.
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lattice boltzmann models based on half range gauss hermite Quadratures
Journal of Computational Physics, 2016Co-Authors: Victor E Ambrus, Victor SofoneaAbstract:We discuss general features of thermal lattice Boltzmann models based on half-range Gauss Quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the Quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different Quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less Quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.
N Sukumar - One of the best experts on this subject based on the ideXlab platform.
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numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons
Computational Mechanics, 2011Co-Authors: S E Mousavi, N SukumarAbstract:We construct efficient Quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The Quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous Quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous Quadratures is based on Lasserre's method for the integration of homogeneous functions over convex polytopes. We also construct Quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.
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generalized gaussian Quadrature rules for discontinuities and crack singularities in the extended finite element method
Computer Methods in Applied Mechanics and Engineering, 2010Co-Authors: S E Mousavi, N SukumarAbstract:New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like Quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the Quadratures, which ensures that the final Quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.
Gradimir V Milovanovic - One of the best experts on this subject based on the ideXlab platform.
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symbolic numeric computation of orthogonal polynomials and gaussian Quadratures with respect to the cardinal b spline
Numerical Algorithms, 2017Co-Authors: Gradimir V MilovanovicAbstract:The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ m as the weight functions on [0, m] (m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian Quadrature up to 60 nodes. The efficiency of these Gaussian Quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.
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numerical Quadratures and orthogonal polynomials
2011Co-Authors: Gradimir V MilovanovicAbstract:Orthogonal polynomials of dierent kinds as the basic tools play very important role in construction and analysis of Quadrature formulas of maximal and nearly maximal algebraic degree of exactness. In this survey paper we give an account on some important connections between orthogonal polynomials and Gaussian Quadratures, as well as several types of generalized orthogonal polynomials and corresponding types of Quadratures with simple and multiple nodes. Also, we give some new results on a direct connection of generalized Birkho-Young Quadratures for analytic functions in the complex plane with multiple orthogonal polynomials.
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on the remainder term of gauss radau Quadratures for analytic functions
Journal of Computational and Applied Mathematics, 2008Co-Authors: Gradimir V Milovanovic, Miodrag M Spalevic, Miroslav S PranicAbstract:For analytic functions the remainder term of Gauss-Radau Quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours with foci at the points +/-1 and a sum of semi-axes @r>1 for the Chebyshev weight function of the second kind. Starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined. The corresponding Gautschi's conjecture from [On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau Quadratures, Rocky Mountain J. Math. 21 (1991), 209-226] is proved.
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Quadratures with multiple nodes power orthogonality and moment preserving spline approximation
Journal of Computational and Applied Mathematics, 2001Co-Authors: Gradimir V MilovanovicAbstract:Quadrature formulas with multiple nodes, power orthogonality, and some applications of such Quadratures to moment-preserving approximation by defective splines are considered. An account on power orthogonality (s- and σ-orthogonal polynomials) and generalized Gaussian Quadratures with multiple nodes, including stable algorithms for numerical construction of the corresponding polynomials and Cotes numbers, are given. In particular, the important case of Chebyshev weight is analyzed. Finally, some applications in moment-preserving approximation of functions by defective splines are discussed.
Bernie D Shizgal - One of the best experts on this subject based on the ideXlab platform.
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the computation of radial integrals with nonclassical Quadratures for quantum chemistry and other applications
Journal of Mathematical Chemistry, 2017Co-Authors: Bernie D Shizgal, Xingwei YangAbstract:The computation of radial integrals on the semi-infinite axis is an important computationally intensive feature of quantum chemistry computer codes with additional applications to physics and engineering. There have been numerous algorithms proposed to evaluate these integrals efficiently. Many of these approaches involve the transformation of the semi-infinite axis, \(r \in [0,\infty )\), to the finite interval, \(x \in [-1,1]\), and the use of the Gauss–Legendre or Gauss–Chebyshev Quadratures to evaluate the integrals. These mappings redistribute the Quadrature points in many different ways. The approach in this paper is to compute the radial integrals with the Gauss–Maxwell nonclassical Quadrature defined by the weight function \(w(r)=r^2e^{-r^2}\) appropriate for the semi-infinite interval and to also use scaling of the Quadrature points. We carry out numerical experiments with simple model radial integrands and compare with the results of previous workers.
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an efficient nonclassical Quadrature for the calculation of nonresonant nuclear fusion reaction rate coefficients from cross section data
Computer Physics Communications, 2016Co-Authors: Bernie D ShizgalAbstract:Abstract Nonclassical Quadratures based on a new set of half-range polynomials, T n ( x ) , orthogonal with respect to w ( x ) = e − x − b / x for x ∈ [ 0 , ∞ ) are employed in the efficient calculation of the nuclear fusion reaction rate coefficients from cross section data. The parameter b = B / k B T in the weight function is temperature dependent and B is the Gamow factor. The polynomials T n ( x ) satisfy a three term recurrence relation defined by two sets of recurrence coefficients, α n and β n . These recurrence coefficients define in turn the tridiagonal Jacobi matrix whose eigenvalues are the Quadrature points and the weights are calculated from the first components of the eigenfunctions. For nonresonant nuclear reactions for which the astrophysical function can be expressed as a lower order polynomial in the relative energy, the convergence of the thermal average of the reactive cross section with this nonclassical Quadrature is extremely rapid requiring in many cases 2–4 Quadrature points. The results are compared with other libraries of nuclear reaction rate coefficient data reported in the literature.
