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Numerical Techniques
Published in Gianni Comini, Stefano Del Giudice, Carlo Nonino, Finite Element Analysis in Heat Transfer, 2018
Gianni Comini, Stefano Del Giudice, Carlo Nonino
For purposes of finite element analysis, the best suited numerical integration technique is the Gaussian quadrature in which the integration rules can be easily expressed in terms of summations. The one-dimensional Gaussian quadrature formula is written as () I1=∫−11g(ξ)dξ=∑i=1pg(ξi)wi
Computationally efficient estimation of the probability density function for the load bearing capacity of concrete columns exposed to fire
Published in Jaap Bakker, Dan M. Frangopol, Klaas van Breugel, Life-Cycle of Engineering Systems, 2017
R. Van Coile, R. Caspeele, P. Criel, L. Wang, G.P. Balomenos, M.D. Pandey, A. Strauss
The kth moment for the lth cut function can be approximated by Gaussian quadrature. In its most basic form, Gaussian quadrature approximates the integration of a function g(z) over the entire domain of a standard normally distributed variable Z by a weighted sum of a limited number of well-chosen evaluation points zj, as mathematically given by Equation (24) with ϕ the standard normal PDF, L the number of Gauss integration points (for most cases 5 integration points is sufficiently accurate), and wj the associated Gauss weights (Zhang, 2013). ∫−∞∞g(z)ϕ(z)dz≈∑j=1Lwjg(zj)
Numerical Differentiation and Integration
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
Suppose that the (m + 1)-point quadrature formula is exact for polynomials of degree ≤ m. A quadrature formula is a Gaussian quadrature formula (exact for polynomials of degree ≤ 2m + 1) if and only if the sample points x0, x1, …, xm are the zeros of the orthogonal polynomial pm+1(x), and the coefficients α0, α1, …, αmcan be expressed as in (6.34).
Estimation of contact stresses in EN31 rolling contact bearings for screw compressor using Gauss quadrature & statistical analysis
Published in Australian Journal of Mechanical Engineering, 2023
S. H. Gawande, K. Balashowry, K. A. Raykar, K. H. Munde
Gaussian quadrature method is a numerical method to evaluate smooth definite integrals which are difficult to evaluate. This method involves using abscissae and corresponding weights in a tabular form to convert the integral into the summation of a set of linear equations. As shown in Eq.(22), the definite integral I is expressed as a summation of product of set of n weights and set of functions of n abscissae.
On spatio-temporal dynamics of sine-Gordon soliton in nonlinear non-homogeneous media using fully implicit spectral element scheme
Published in Applicable Analysis, 2021
Note that, density at each node points. Equation (14) can be written in matrix form as where product approximation is used for term, see Argyris et al. [27]. Mass matrix M, diffusion matrix D, density matrix and Neumann boundary term are given as To maintain the accuracy and efficiency of the space discretization, exact evaluation of integrals (especially integrals with nonlinear terms) involved in the weak formulation is important. Thus, applying quadrature formulae with insufficient accuracy can lead to aliasing error which degrades the accuracy of the solution and can cause numerical instability, see Kirby & Sherwin [60]. To eliminate aliasing error, Kirby & Karniadakis [61] investigated the super-collocation as well as over-integration for nonlinear terms on non-uniform grid, due to which the computational cost increases and the mass matrix loses its diagonal structure. Moreover, as reported by Lomtev et al. [62] over-integration may also lead to instabilities. Gaussian quadrature rules are widely used methods, both Gauss-Legendre (GL) and GLL quadratures are accurate and efficient. Using N quadrature points GL quadrature evaluates 2N−1 or less-order polynomial exactly, whereas GLL quadrature evaluates 2N−3 or less-order polynomial exactly. Thus, there is no unique quadrature rule that can be used to evaluate all integrals exactly. For higher order scheme (polynomial order greater than 3), loss of order of accuracy is out weighted by the computational efficiency and ease of implementation, Nair [63], see [44] for a brief discussion. In this paper, GLL quadrature is used to integrate all the integrals involved in weak formulation which results in diagonal mass matrix.