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Application of Variational Methods and Galerkin Method in Solving Engineering Problems Represented by Ordinary Differential Equations

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Application of Variational Methods and Galerkin Method in Solving Engineering Problems Represented by Ordinary Differential Equations International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 4, Issue-4, Apr.-2016 APPLICATION OF VARIATIONAL METHODS AND GALERKIN METHOD IN SOLVING ENGINEERING PROBLEMS REPRESENTED BY ORDINARY DIFFERENTIAL EQUATIONS 1B.V. SIVA PRASAD REDDY, 2K. RAJESH BABU 1,2Department of Mechanical Engineering, Sri Venkateswara University college of Engineering, Tirupati, India E-mail: [email protected], [email protected]; Abstract – Nowadays the accuracy of problem solving is very important. In olden days the Variational methods were used to solve all engineering problems like structural, heat transfer and fluid mechanics problems. With the emergence of Finite Element Method (FEM) those methods are become less important, although FEM is also an approximate method of numerical technique. The concept of variational methods is inducted to solve majority of engineering problems, which gives more accurate results than any other type of approximate methods. The engineering problems like uniform bar, beams, heat transfer and fluid flow problems are used in our daily life and they play an important role in the development of our society. To achieve drastic development in the society, it is a must to focus on adopting approximation methods that improve the accuracy of engineering solution. Of all the methods, Galerkin method is emerging as an alternative and more accurate method than those of Ritz, Rayleigh – Ritz methods. Any physical problem in nature can be transformed into an equivalent mathematical model by idealization process and describing its behavior by a suitable governing equation with associated boundary conditions. Against this backdrop, the present work focuses on application of different variational methods in solving ordinary differential equations. The reason behind choosing second order differential equation (ODE) is that most of the structural and heat transfer problems can well be represented by an ODE. As an illustration the work herein reported highlights the utility of above cited methods with a simple bar problem. Furthermore the numerical part of this work is carried out on a MATLAB platform. Keywords— Variational methods, Second order differential equation, elastic bar, Ritz method, Rayleigh – Ritz method, Galerkin method and MATLAB I. INTRODUCTION problems. The Robert D. Cook and David S. Malkus [2] are introduced variational methods on The objective of this research is to evaluate and engineering problems. J. N. Reddy [1], The Robert examine the variational methods like Ritz, Rayleigh D. Cook and David S. Malkus [2], S. S. Rao [3], O.C – Ritz and weighted residual methods like Galerkin Zienkiewicz, R.L. Taylor & J. Z. Zhu [4] are used methods based on MATLAB. these principles to Finite Element Method in Solutions for field problems are widely used engineering applications. Recently the Sanjay mathematical tools in engineering analysis. These Govindjee [5] introduced the variational methods methods are applied in such areas as the analysis of solving with the MATLAB software. solids and structures, heat transfer, fluids and almost Any physical problem in nature can be transformed any other areas of engineering analysis. The into an equivalent mathematical model by variational methods are introduced to solve the idealization process and describing its behavior by a Engineering problems around 1820. The variational suitable governing equation with associated method was first used by Lord Rayleigh in 1870. boundary conditions. Some real engineering However, the approach did not receive much problems are shown in the following figures. recognition by the scientific community. Nearly 40 years later, due to the publication of two papers by Ritz, the method came to be called the Ritz method. To recognize the contributions of both men, the theory was later renamed the Rayleigh – Ritz method. The Ritz method proposed by the Swiss mathematician Walther Ritz in between 1878 to Fig 1 Beam having transverse load 1909. After that the Galerkin method is proposed by Russion mathematician “Boris Galerkin” in 1915. The Galerkin method is one of best method of weighted residual method. In the Galerkin method, it only requires that the residual of the differential Fig 2 A cantilever beam clamped at one end equation be orthogonal to each term of the series that satisfy the boundary conditions. These methods are In the present study we have chosen a simple second discussed in much research paper to apply on order differential equation, whose solution is sought different engineering problems. The J. N. Reddy [1] by different variational methods which include Ritz, was applied on his book to solve bar, beam Rayleigh – Ritz and Galerkin method. All of these Application of Variational Methods and Galerkin Method in Solving Engineering Problems Represented by Ordinary Differential Equations 75 International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 4, Issue-4, Apr.-2016 methods seek an approximate solution in the form of III. ANALYSIS OF 2nd ORDER ODE BY a linear combination of suitable approximate APPROXIMATE METHODS function, preferably power series. The parameters or coefficients are determined such that the approximate The given second order ODE has been analyzed by solution satisfies weak or variational form or adopting Ritz method, Rayleigh – Ritz method and minimizes the quadratic functional of the equation Galerkin method for one and two – parameter (as in Rayleigh – Ritz method) under study. Various approximation. The corresponding results were methods differ from each other in the choice of the tabulated and illustrated as detailed below. And also approximate functions. the comparisons of those results with exact results In addition an elastic bar subjected to uniformly were shown below. distributed load is also analyzed and solved for displacement field by Galerkin method. The 3.1 Solution by Ritz Method approximate solution so obtained is validated by a Governing equation suitable numerical data. The following sections will − − + = 0 For 0 < x < 1 briefly highlight the various variational methods to be adopted in solving the selected governing Boundary conditions u (0) = 0, u (1) = 0 equation and an elastic bar problem. The given governing equation is in strong form. In II. CHOSEN SECOND ORDER GOVERNING Ritz method first of all strong form is converted into EQUATIONS weak form. Then the governing equation written as ∫ ∗ [− − + ] = 0 The following governing equation is selected for its − ∗ + ∫ ∗ − ∫ ∗ + analysis by different variational methods like Ritz, Rayleigh – Ritz method and Galerkin methods. ∫ ∗ = 0 A. − − + = 0 For 0 < x < 1 SET: 1 Boundary conditions u (0) = 0, u (1) = 0 The above weak form can be expressed as leaner and ( ) bilinear forms (, ) = (). Bilinear form of SET: 2 Boundary conditions 0 = 0, = 1 above equation B (w, u) = ∫ ∗ − Exact solution of second order ODE for set:1 boundary condition ∫ ∗ and linear form is L (w) 2 sin(1 − x) + sin x () = + ( − 2) = − ∫ + ∗ sin 1 Exact solution of second order ODE for set:2 To get solution in Ritz method select approximate boundary condition equation with satisfy above boundary conditions 2 cos(1 − x) − sin x conditions ɸ = (1 − ) , which is the simplest () = + ( − 2) cos 1 function satisfying the boundary conditions The following governing equation is selected for its analysis by Galerkin method = C(1 − ) + C (1 − ) involving two different approaches that include + … … … . +C (1 − ) formulation by variational method and weighted residual method. This equation also governs an By substituting approximate function (w, u) elastic bar element subject to uniformly distributed becomes load and end load as well. ɸi ɸ , ɸ = ∗ ( C ɸ ) L P − ɸ ( C ɸ) A, E q = Cx ɸ ɸ = ∑ C ∫ ∗ − ɸ ∗ ɸ this is also Fig 3 Uniform elastic bar, loaded by axial tip force called stiffness matrix D and linear term ɸ = The above physical modal is converted into mathematical governing equation as bellow − ∫ ɸ this is a force matrix F. B. + = 0 For 0 < x < L The algebraic equations can be expressed in matrix Boundary conditions are u (0) = 0, = from [D]{C} = {F} by solving this the coefficients of Exact solution of 1-D elastic bar approximate function (,,……., ) are obtained.. 3 − + 6 () = 6 Application of Variational Methods and Galerkin Method in Solving Engineering Problems Represented by Ordinary Differential Equations 76 International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 4, Issue-4, Apr.-2016 Table 1 Ritz coefficients values for one and two − ∗ + ∫ ∗ − ∫ ∗ + parameter approximation ∫ ∗ = 0 The above weak form can be expressed as leaner and bilinear forms B (w, u) = ∫ ∗ − ∫ ∗ L (w) = − ∫ + [] ∗ 1 The functional is () = (, ) + () These coefficients are finally submitted in approximate function, one can get the desired () = ∫ − − 2 + (1) approximate solution To get solution in Rayleigh – Ritz method, select an approximate function which will satisfy above Table 2 Ritz field variable’ u’ at various points for boundary conditions ɸ = , which is the simplest one and two parameter approximation function satisfying the boundary conditions. = C + C + … … … . +C Substituting in functional The necessary condition for the minimizing of () functional “I” is that Apply = 0 , = ɸ ɸ ∫ 2 ∑ − 2ɸ ∑ ɸ − 2 ɸ + (1)
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