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An Introduction to the Finite Element Method
Published in Santanu Saha Ray, Numerical Analysis with Algorithms and Programming, 2018
The Rayleigh–Ritz method is a variational technique that involves minimization of functional for the solution of boundary value problems. Therefore, it has a disadvantage regarding existence of functional, which is not always possible to obtain. In order to overcome the above-mentioned difficulty, Galerkin methods may be used. In numerical analysis, Galerkin methods are a class of methods for converting a differential equation to a discrete problem. This method belongs to wider classes of methods, called the weighted residual methods, and uses the trial functions (approximating functions) that satisfy the boundary conditions of the problem. The trail function is substituted in the given differential equation, results in the residual. The integral of the weighted residual, taken over a domain, is then set to zero. For further details about this method, see Section 7.6.4. The common thread between all three approaches, that is, Rayleigh–Ritz, Galerkin, and collocation, is that the solution is approximated by a linear combination of trial functions, and the coefficients are obtained by solving a system of equations.
Variational and Related Methods
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
In this chapter, we have presented an overview of variational methods. Because of its relation with the Galerkin method, the related principles of virtual work, virtual displacement, and virtual traction are introduced. The more general Hamilton principle, Veubeke-Hu-Washizu principle, and Hellinger-Reissner principle are reviewed in view oftheir fundamental importance in numerical analysis, such as the finite element method. Approximate techniques related to the variational principle are introduced in Sections 14.8−14.10, including the Rayleigh-Ritz method, weighted residue method, and Galerkin method. A semi-Galerkin type method known as Kantorovich’s method, a term coined by Reiss (1965), is introduced using plate bending problems as an example in Section 14.11. In essence, for problems governed by partial differential equations, the Galerkin method is used to approximate one of the variables whereas the other one is solved analytically. Therefore, it is a semi-Galerkin type or semi-analytic technique. Functional formulation for plate bending is considered in Section 14.12 before the vibrations of circular plates are considered.
Introduction
Published in Juan C. Heinrich, Darrell W. Pepper, Intermediate Finite Element Method, 2017
Juan C. Heinrich, Darrell W. Pepper
There is a major difference between the Rayleigh-Ritz method and the Galerkin method. The Rayleigh-Ritz method finds the unknown coefficients through an energy minimization process; this process requires a minimum principle. The Galerkin method is based on making the projection of the error in the approximating functions φi vanish in the finite dimensional space spanned by the functions in order to determine the unknown coefficients ai. This approach allows the Galerkin method to be used in situations when minimum principles do not exist. Such cases occur when convection is the dominant transport mechanism in a fluid system. The Galerkin method is therefore the method of choice in problems involving fluid flow.
Nonlinear bending of FG metal/graphene sandwich microplates with metal foam core resting on nonlinear elastic foundations via a new plate theory
Published in Mechanics Based Design of Structures and Machines, 2023
Mohammed Sobhy, Fatimah Alsaleh
The equations of motion are often partial differential equations. The accurate solutions can be only obtained for some special cases of partial differential equations. So the researchers mostly transform partial differential equations into ordinary ones by Galerkin method, Weighted Residual method, Rayleigh-Ritz method, Boundary Element method, Finite Element method (FEM), etc, and then solve the obtained ordinary differential equations. Safaei, Onyibo, and Hurdoganoglu (2022) employed the FEM to investigate the bending and mechanical buckling analyses of sandwich beams with carbon foam core and composite face layers subjected to thermal load. Based on the FEM and a new third-order shear deformation theory, Ton-That (2021) illustrated the nonlinear free vibration of FG plates. Galerkin method represents the simplest method to convert a system of partial differential equations into a system of ordinary differential equations or a system of algebraic equations. Therefore, Galerkin method is widely used by many researchers to solve the governing equations such as Duc et al. (2021); Fallah and Aghdam (2011); Zhang and Li (2019); Li et al. (2018); Zeng, Wang, and Wang (2019); Liu, Chen, et al. (2021); Niu, Yao, and Wu (2022); She and Ding (2023).
A novel weighted local averaging for the Galerkin method with application to elastic buckling of Euler column
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
Anh Tay Nguyen, Nguyen Cao Thang, Tran Tuan Long, N. D. Anh, P. M. Thang, Nguyen Xuan Thanh
Since the Galerkin method is general in scope and can be used for both conservative and nonconservative, both linear and nonlinear systems, this method constitutes a general effective tool for the numerical solution for approximate analytical investigations in engineering and applied science. One of the applications of the Galerkin method is to analyze nonlinear vibration problems by converting nonlinear partial differential equations to ordinary ones. Barati and Zenkour [6] studied free vibrational behavior of porous nanocomposite shells reinforced with graphene platelets. The reinforced shell was modeled via first order shear deformation theory and the Galerkin method was implemented to obtain vibration frequencies. Sofiyev [7] derived the equations of motion of the non-linear free vibration of functionally graded material (FGM) orthotropic cylindrical shells considering the shear stresses using non-linear shell theory. The superposition and the Galerkin methods were then adopted to obtain a non-linear ordinary differential equation to be solved using the homotopy perturbation method.
Size dependent buckling analysis of nano sandwich beams by two schemes
Published in Mechanics of Advanced Materials and Structures, 2020
M. Rezaiee-Pajand, M. Mokhtari
The Galerkin method is a simple tool which can be applied to solve a great variety of boundary value problems. First, the unknown displacement functions must be assumed in such a way that the essential boundary conditions are satisfied. Hence, the following asymptotic solutions for and are considered: where n refers to the number of selected terms, and are the shape functions proposed for CC and SS boundary conditions in Table 1. In the current study, exact shape functions are chosen based on [78]. In these functions, and are unknown constants to be determined. Substitution of Eqs. (55) and (56) into Eqs. (38) and (39) yields the following equalities: