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# Introduction To Finite And Spectral Element Methods Using Matlab Pdf

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Published: 13.12.2020  Incorporating new topics and original material, Introduction to Finite and Spectral Element Methods Using MATLAB enables readers to quickly understand the theoretical foundation and practical implementation of the finite element method and its companion spectral element method. Utilizing the book as a user guide, readers can immediately run the codes and graphically display solutions to a variety of elementary and advanced problems. Suitable for self-study or as a textbook in various science and engineering courses, this self-contained book introduces the fundamentals on a need-to-know basis and emphasizes the development of algorithms and the computer implementation of essential procedures.

Metrics details. In this paper, we study the Legendre spectral element method for solving the sine-Gordon equation in one dimension. Firstly, we discretize the equation by Legendre spectral element in space and then discretize the time by the second-order leap-frog method. We study the stability and convergence of the method and show the convergence of our method.

## Stochastic time domain spectral element analysis of beam structures

Metrics details. In this paper, we study the Legendre spectral element method for solving the sine-Gordon equation in one dimension. Firstly, we discretize the equation by Legendre spectral element in space and then discretize the time by the second-order leap-frog method. We study the stability and convergence of the method and show the convergence of our method.

Finally, we show the results with numerical examples. A spectral element method combines the high accuracy of spectral methods and flexibility of finite element method, and the approximate result of this method provides high accuracy and spectral convergence. In this method the solution is approximated on each element using spectral methods. One of the advantages of this method is the high accuracy and stable solving algorithm with a small number of elements under a wide range of conditions [ 1 ].

Finite element method was proposed for the first time in by Courant [ 2 ]. He solved the Poisson equation based on minimizing piecewise linear approximations on finite subdomains. The spectral method is a conventional method for solving partial differential equations, which was first introduced by Navier for elastic sheet problems in In spectral method the solution is approximated on one general domain. In , Patera applied a spectral method to a greater number of subdomains by a division of domains.

He proposed the spectral element method by combining the spectral method and the finite element method [ 3 ]. In his innovative method, Patera uses the Chebyshev polynomials as the interpolation basis functions. The use of the Lagrangian interpolation conjugate with the Gauss—Legendre—Lobatto quadrature leads to a matrix of mass with diameter structure [ 5 ]. The Legendre spectral element method is widely used in solving partial differential equations.

Chen et al. The aim of [ 9 ] is the Lagrange—Galerkin spectral element method for solving two-dimensional shallow water equations. These basis functions are constructed so that the axial conditions along a plane or axis of symmetry are satisfied identically. Zhuang and Chen [ 13 ] used this method to solve biharmonic equations. In [ 14 ], the authors used the spectral element method with least-square formulation for parabolic interface problems.

Ai et al. The sine-Gordon equation is one of the most important partial differential equations, which applies to many scientific fields such as the motion of a rigid pendula attached to a stretched wire [ 17 ], solid state physics, nonlinear optics, and the stability of fluid motions.

Different numerical methods are presented for Eq. Dehghan and Shokri [ 18 ] solved a one-dimensional sine-Gordon equation using collocation points and approximating the solution using thin plate splines radial basis function.

Dehghan and Mirzaei [ 19 ] used a numerical method of the boundary integral equation to approximate the solution of one-dimensional equation 1. Mohebbi and Dehghan [ 20 ] have also used the finite difference method for numerical solution of equation 1. In [ 21 ] the authors present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. Yousif an Mahmood [ 22 ] used the variational homotopy perturbation method for solving the Klein—Gordon and sine-Gordon equations.

In [ 23 ] a new scheme, which has energy-preserving property, is proposed for solving the sine-Gordon equation with periodic boundary conditions. This method is obtained by the Fourier pseudo-spectral method and the fourth-order average vector field method.

Baccouch [ 24 ] presented superconvergence results for the local discontinuous Galerkin method for the sine-Gordon nonlinear hyperbolic equation in one space dimension. Other numerical methods have also been used to solve the sine-Gordon equation such as Chebyshev tau meshless method [ 25 ], meshless method of lines [ 26 ], high-accuracy multiquadric quasi-interpolation [ 27 ], reduced differential transform method [ 28 ], pseudo-spectral method [ 29 ], modified cubic B-spline differential quadrature method [ 30 ], modified cubic B-spline collocation method [ 31 ], etc.

In this paper, we study the Legendre spectral element method for solving Eq. First, using the Legendre spectral element method, we obtain a semi-discrete spatial form of Eq. We bring theorems on stability and convergence, and, finally, we show the results by a numerical example. This paper is organized as follows: In Sect. In Sect. Finally, in Sect. In this section, we explain the Legendre spectral element method and spatial discretization of Eq.

Basis functions are considered as the Lagrangian interpolation polynomials defined at Gauss—Lobatto integration points on each element. The stiffness [ 33 ] and mass matrices [ 34 ] on each element are calculated as follows:.

Using the Gauss quadrature, we obtain [ 35 ]. We obtain the weak form of Eq. The second integral on the left-hand side is obtained by integration by parts. Now, taking the k th Lagrange function of order N as the test function v and using Eq. We obtain the right-hand side of Eq. The matrix form of the semidiscrete form of Eq. So Eq. For full discretization of Eq.

Now, using the leap-frog method, we obtain the full discrete form of Eq. After simplifying, Eq. Because the mass matrix M is diagonal, solving Eqs. In this section, we analyze the stability of leap-frog method and the convergence of the spectral element method presented in the previous sections.

Equation 9 can be written as. Equation 10 is stable under the following condition :. We must show that. According to [ 37 ], we have that.

According to Eq. In [ 38 ] the convergence theorem is presented for the spectral element method for acoustic waves. In this section, we consider a numerical example to validate the proposed scheme. Absolute error for sine-Gordon equation for Example 5. In the following figures 4 , 5 , and 6 we have used a logarithmic scale for both axes. In Fig. In this example, we obtain the numerical solutions of Eq. The analytical solution is given in [ 39 ] as. The boundary conditions are obtained from the exact solution.

The obtained results are compared with the results in [ 18 , 30 , 31 ]. The exact solution [ 31 ] is given as. The boundary conditions can be obtained from the exact solution.

The numerical solution for Example 5. Computed results are compared with the results obtained in [ 30 , 31 , 40 ]. The spectral polynomials are useful tools for solving ordinary and partial differential equations.

Also, the incorporation of the finite element method with spectral polynomials, that is, the use of spectral polynomials as new shape functions in the finite element method is very efficient for obtaining a numerical algorithm with high accuracy.

In this paper, we constructed a Legendre spectral element method for the solution of the one-dimensional sine-Gordon equation. We used the Legendre spectral element method for discretizing the spatial space.

We presented theorems on the stability and convergence. Finally, using one test problem, we demonstrated that the algorithm is efficient for obtaining approximation solutions of the sine-Gordon equation. Vosse, F. Courant, R. Patera, A. Maday, Y. Google Scholar. Bathe, K. Prentice Hall International, Englewood Cliffs Priolo, E. Chen, Y. SIAM J. Zeng, F. Shanghai Univ. Giraldo, F. Zampieri, E. VanOs, R. Xu, C. Zhuang, Q. Khan, A. Methods Appl. Ai, Q. ## Introduction to Finite and Spectral Element Methods Using MATLAB

Finite element method postprocessing free download - SourceForge. Finite element method - Wikipedia. Finite element 3d matlab free download - SourceForge. Application of the Trefftz method in an original variational. Programing the Finite Element Method with Matlab. Finite Element Solution of the Two-dimensional Incompressible. For simplicity, we will use the relaxation-based algorithm rather than the Newton method to couple. ## We apologize for the inconvenience...

Fea Using Python. WordPress Site using Python. A Basic Guide. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions. Finite element approximation of initial boundary value problems.

Further Applications in One Dimension. Quadratic and Spectral Elements in Two Dimensions. Applications in Mechanics.

### Legendre spectral element method for solving sine-Gordon equation

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We propose a new spectral element model for finite rectangular plate elements with arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting method used in our previous study so that the four corner nodes of a finite rectangular plate element become active. Thus, the new spectral element model can be applied to any finite rectangular plate element with arbitrary boundary conditions, while the spectral element model introduced in the our previous study is valid only for finite rectangular plate elements with four fixed corner nodes. The new spectral element model can be used as a generic finite element model because it can be assembled in any plate direction. The accuracy and computational efficiency of the new spectral element model are validated by a comparison with exact solutions, solutions obtained by the standard finite element method, and solutions from the commercial finite element analysis package ANSYS. The plate is a representative structural element that is widely used in many engineering fields such as mechanical, civil, aerospace, shipbuilding, and structural engineering.

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Incorporating new topics and original material, Introduction to Finite and Spectral Element Methods Using MATLAB, Second Edition enables.

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In this work, a stochastic time domain spectral element method STSEM is proposed for stochastic modeling and uncertainty quantification of engineering structures.

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