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Overview
Published in Dr Arzhang Angoshtari, Ali Gerami Matin, Finite Element Methods in Civil and Mechanical Engineering, 2020
Dr Arzhang Angoshtari, Ali Gerami Matin
We saw that once a set of basis functions {ψ1,…,ψN} is chosen, one can apply the Galerkin method to (1.2) to approximate a solution of (1.1). But how can we determine a suitable choice of basis functions? There are many different approaches for obtaining basis functions which yield different approximation methods such as finite element methods, finite-difference methods, spectral methods, etc. We will study finite element methods in this book. The main aspects of the finite element method for obtaining basis functions can be summarized as follows.
Differential Quadrature Method: A Robust Technique to Solve Differential Equations
Published in Mangey Ram, S. B. Singh, Mathematics Applied to Engineering and Management, 2020
The DQM has gone through a lot of development in terms of the degree of the polynomial, different forms of basis functions, implementation of boundary conditions, and the choice of grid points. Because of various developments in DQM, it is a now well-known numerical method for its admirable quality of computational efficiency and fast convergence. The initial phases of advancement of the DQM and its use in various applications to solve differential equations are reported by Shu [2]. The utmost requirement of DQM is the calculation of weighting coefficient whose formulation was further improved by Quan and Chang [3,4]. One of the key features of the DQM is the basis functions. The efficiency of the obtained numerical solution also depends on the choice of the basis function. Various kinds of basis functions, such as spline functions, Lagrange interpolation polynomials, and sinc function [5,6], for example, are successfully implemented to determine the weighting coefficients.
Computational mechanics
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2019
In the introduction of the finite element method in Chapter 7, we used basis functions to describe the approximate solutions. In order to approximate the solution inside the domain, the finite element technique uses a collection of low order polynomial basis functions. For a triangular element discretization of a two-dimensional domain, as shown in Figure 12.2, bilinear interpolation functions are commonly used in the form: u(x,y)=a+bx+cy. Here u represents any of the q, T or p physical solution quantities introduced in the past three sections.
PICAR: An Efficient Extendable Approach for Fitting Hierarchical Spatial Models
Published in Technometrics, 2022
In this section, we present our PICAR approach that is designed to efficiently fit hierarchical spatial models. In this framework, we represent the spatial random effects as a linear combination of basis functions:where is an n × p basis function matrix where each column contains a spatial basis function, are the reparameterized spatial random effects (or basis coefficients), and is the p × p covariance matrix for the coefficients. Basis functions can be interpreted as a set of distinct spatial patterns that can be used to construct a spatial random field, along with their coefficients. Basis representation has been a popular approach to model spatial data (see Higdon 1998; Haran, Hodges, and Carlin 2003; Griffith 2003; Christensen, Roberts, and Sköld 2006; Banerjee et al. 2008; Cressie and Johannesson 2008; Higdon et al. 2008; Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Hughes and Haran 2013; Nychka et al. 2015; Mak et al. 2018). Examples of basis functions include splines, wavelets, empirical orthogonal functions, combinations of sines and cosines, piece-wise linear functions, and many others. Basis representations tend to be computationally efficient as they help bypass large matrix operations, reduce the dimensions of the spatial random effects W, and as in our case, reduces correlation in W.
A numerical approach to hybrid nonlinear optimal control
Published in International Journal of Control, 2021
Esmaeil Sharifi, Christopher J. Damaren
Galerkin’s spectral method can be used to find a uniform approximation to the continuous-time HJB equation such that the approximate controls are still stable on a specified set (Beard, 1995). The successive Galerkin approximation (SGA), which simultaneously combines successive approximation and Galerkin approximation, introduces a design algorithm which systematically improves the closed-loop performance of arbitrary stabilising feedback control laws (Beard, 1995). The basic idea behind successive approximation is to compute the value function involved in the HJB equation and the associated optimal control iteratively, instead of calculating them simultaneously (Beard, 1995; Beard & McLain, 1998b). The essence of the SGA approach is first to reduce the continuous-time HJB equation, which is a nonlinear partial differential equation, to a sequence of linear first-order partial differential equations known as the Generalized Hamilton-Jacobi-Bellman (GHJB) equation. Galerkin’s approximation method is then utilised with basis functions defined globally on a compact set to approximate these partial differential equations (Beard, 1995). Starting with an arbitrary stabilising control law, the GHJB equation is first solved for the value function associated with this initial control law, and the optimal control in terms of the value function is then updated. When this process is iterated, the solution to the GHJB equation converges uniformly to the solution of the continuous-time HJB equation, which consequently solves the optimal control problem being considered (Beard, 1995).
A Galerkin discretisation-based identification for parameters in nonlinear mechanical systems
Published in International Journal of Systems Science, 2018
By applying the idea of Galerkin finite element, a novel discretisation-based method is established to identify linear and nonlinear parameters in mechanical systems. In the method, displacement vectors over time history are approximated by piecewise linear functions. The second-order terms in model equation are eliminated by integrating by parts. In this way, differential equation can be transformed into an algebraic one. Then, the lost function of integration form is derived to identify parameter vector. For the optimisation of the lost function, traditional least-squares algorithm is applied to the linear system and an iterative one to the nonlinear systems. The convergence of the method is also demonstrated and the proposed method ensures an exponential convergence in the neighbourhood of the fixed point. To conclude, highlights of the paper are listed as follows. The proposed method has a broad application in mechanical systems. For both linear and nonlinear systems, identification procedures are illustrated and the efficiency is verified in given examples.Basis functions used in the method are piecewise linear such that they are relatively simple compared to other basis functions. The failure of derivation of basis functions is overcome through integrating by parts.For the identification accuracy, the proposed method can show high performance in both linear and nonlinear systems. Particularly, under the condition of low sampling frequencies, the method still maintains high identification accuracy, which is verified in the simulations