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Time-domain Analysis of Nonlinear Circuits
Published in Jan Ogrodzki, Circuit Simulation Methods and Algorithms, 2018
Newton’s interpolation is an alternative to Lagrange interpolation where an lth order polynomial is fitted to l + 1 constraints xn,...xn−l at points tn,…, tn−l. The polynomial is calculated in a Newton form. To explain the Newton form of polynomial, let us consider an lth degree polynomial x(t). To look for its Newton form we divide it l times successively by t-tn, t − tn−1,… up to t − tn−l + 1. We denote residua from these divisions by rnk(k = 0,…, l).
Modeling of Thermal Systems
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
where the product sign Π denotes multiplication of the n factors obtained by varying j from 0 to n, excluding j = i, for the quantity within the parentheses. It is easy to see that this polynomial may be written in the general form of a polynomial, Equation (3.29), if needed. Lagrange interpolation is applicable to an arbitrary distribution of data points, and the determination of the coefficients of the polynomial does not require the solution of a system of equations, as was the case for the general polynomial. Because of the ease with which the method may be applied, Lagrange interpolation is extensively used for engineering applications.
Discretization of Physical Domains
Published in Guigen Zhang, Introduction to Integrative Engineering, 2017
The intuitive way of imagining the shape functions can actually be expressed by mathematical equations based on the Lagrange interpolation formula. In its original form, Lagrange interpolation uses a polynomial function to construct a smooth curve passing through a set of points. By passing through these points, the Lagrange interpolation formula reproduces the values of the ordinates of all the points.
Pseudo-spectral optimisation of smooth shift control strategy for a two-speed transmission for electric vehicles
Published in Vehicle System Dynamics, 2020
Weichen Wang, Junqiu Li, Fengchun Sun
Collocation point of the Radau pseudo-spectral method is LGR point, that is, the root of polynomial , where N is the order of the Lagrange Interpolation Polynomial. A total of N + 1 collocation points, denoted as , , where , . In this paper, at LGR points, the state variables X are discrete the control variable U are discrete The Lagrange Interpolation Polynomial approximation control variable and state variable based on collocation points where L is Lagrange Interpolation Polynomial Third step: State equation transform
Nonlinear vibration analysis of elastically supported multi-layer composite plates using efficient quadrature techniques
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
This technique is the classical DQM. The shape function for this method is Lagrange interpolation polynomial. The unknown u and its derivatives can be approximated as a weighted linear sum of nodal values, ui, (i = 1,N), as follows [43–49]: where u terms to N is the number of grid points. The weighting coefficients of the first and second order derivative can be determined by differentiating (17), as [43–49]:
Thermal bending response of functionally graded magneto-electric–elastic shell employing non-polynomial model
Published in Mechanics of Advanced Materials and Structures, 2023
The weighted coefficients for the and domain are considered as and respectively. The number of points for discretization in the and domain is interpreted as and The weighted coefficients can be calculated with different basis functions [18]. In this paper, Lagrange interpolation polynomial (LIP) [37] are employed as basis function: