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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
Lagrange polynomialsLn(r): Lagrange polynomials are widely used for interpolation. Ln(r) (n = 1,.., N) are polynomials of order N−1 obtained from data prescribed at a set of N points ri with the property Ln(ri)={1fori=n0fori≠n
Computational Numerical Methods
Published in Timothy Bower, ®, 2023
Lagrange polynomial interpolation finds a polynomial that goes through a set of data points. The Lagrange polynomial, P(x), has the property that P(xi)=yi for all given (xi,yi) data points. We first find a Lagrange basis polynomial, Pi(x), for each of given xi value. The basis polynomials are products of factors such that Pi(xi)=1 and Pi(xj)=0 when i≠j. Then the Lagrange polynomial is a sum of products between the basis polynomials and the given yi values. The properties of the basis polynomials ensure that the Lagrange polynomial passes through each point.
Characterizing Tradeoffs in Memory, Accuracy, and Speed for Chemistry Tabulation Techniques
Published in Combustion Science and Technology, 2023
Elizabeth Armstrong, John C. Hewson, James C. Sutherland
B-splines were marginally more accurate than Lagrange polynomials and provide more flexibility, such as choosing knot locations, in obtaining accurate solutions. However, Lagrange polynomials were the fastest to evaluate for low dimensionality and/or low degrees and their simple construction allows for query optimization that can lead to significant further speedup. The larger ANNs showed comparable evaluation times in four or more dimensions for general table queries and were faster than higher-degree interpolants at this dimensionality. However, in the context of table queries involving extra conversions, including a nonlinear solve, to compute interpolated table inputs from simulation variables, such as demonstrated with both three- and four-dimensional nonadiabatic flamelet queries, the larger ANNs became faster than cubic Lagrange interpolated tables. In general, as dimensionality increases and/or more interpolated table input variables need to be determined with a nonlinear solve, larger-architecture ANN evaluation times will be favorable. ANNs also often have lower memory requirements than interpolants, which is advantageous for high-dimensional problems relevant to turbulent combustion.
Efficient CUF-based FEM analysis of thin-wall structures with Lagrange polynomial expansion
Published in Mechanics of Advanced Materials and Structures, 2022
Xiangyang Xu, Nasim Fallahi, Hao Yang
In current study three types of beam element - two nodes (B2), three nodes (B3) and four nodes (B4) - are provided for a linear (Figure 2), a quadratic and a cubic interpolation of the displacement variable along the beam axis, respectively. Choice of and M are arbitrary which can be defined with a different base function such as polynomials, harmonics and exponential of any order for modeling the displacement field of the beam over the cross section. For structural analysis, different types of polynomials can be used such as Taylor, Legendre and Lagrange polynomials for approximation function for numerical problems. In the present study, function is assumed to be Lagrange-like expansion to describe the deformation over the thin-walled open cross-section of the beams. Thus, the cross-section of the beam structure can be discretized by various types of Lagrange elements (LEs) for instance linear three-point (L3), bilinear four-point (L4), quadratic nine-point (L9) and cubic sixteen-point (L16) elements, among which in this study L9 elements are used over the cross-section. The Lagrange polynomial expansion for an L9 quadratic nine-point element can be written in the natural coordinate system (ξ, η);
One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects
Published in Mechanics of Advanced Materials and Structures, 2019
B. Umesh, A. Rajagopal, J. N. Reddy
Given n + 1 data points (x0, f0), (x1, f1), …(xj, fj), …(xn, fn), such that subsequent points are not repeated, then an degree Lagrange polynomial is constructed as a linear combination of Lagrange basis polynomials. where Lagrange basis polynomials are obtained as with the following Kronecker delta property which makes sure that the polynomial, L(xk), interpolates all the dataFigure 1, Figure 2, Figure 3points A Lagrange basis polynomial plotted for degree 1, 2 and 3 is as shown in Figure 1. The derivatives of Lagrange basis polynomials are obtained as follows: