Solution of Structural Mechanics Problems by the Differential Quadrature Finite Difference Method

2000 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Partial Differential Operator ◽  
Structural Mechanics ◽  
Plane Elasticity ◽  
Differential Quadrature ◽  
Difference Operators ◽  
Elasticity Problems ◽  
Nonuniform Plate

Abstract The differential quadrature finite difference method (DQFDM) has been proposed by the author. The finite difference operators are derived by the differential quadrature (DQ). They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of generic differential quadrature (GDQ). The derivation of higher order finite difference operators is also easy. By adopting the same order of approximation to all mathematical terms existing in the problem to be solved, excellent convergence can be obtained due to the consistent approximation. The DQFDM is effective for solving structural mechanics problems. The numerical simulations for solving anisotropic nonuniform plate problems and two-dimensional plane elasticity problems are carried out. Numerical results are presented. They demonstrate the DQFDM.

Determination of finite-difference weights using scaled binomial windows

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. W17-W26 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Pseudospectral Method ◽  
Difference Operators ◽  
Dispersion Properties ◽  
Taylor Series Expansions ◽  
The Finite Difference Method ◽  
Grid Nodes ◽  
A Minor

The finite-difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes. Conventional finite-difference weights can be calculated either from Taylor series expansions or by Lagrange interpolation polynomials. The finite-difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain. For this reason, we also can derive finite-difference operators by truncating the convolution series of the pseudospectral method. Various truncation windows can be employed for this purpose and they result in finite-difference operators with different dispersion properties. We found that there exists two families of scaled binomial windows that can be used to derive conventional finite-difference operators analytically. With a minor change, these scaled binomial windows can also be used to derive optimized finite-difference operators with enhanced dispersion properties.

scholarly journals Application of the finite difference method for modeling cantilever bar vibrations

2020 ◽  
Vol 224 ◽  
pp. 02002
Author(s):  
M I Volnikov
Keyword(s):  
Mathematical Modeling ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Structural Mechanics ◽  
Forced Vibrations ◽  
Cantilever Beams ◽  
System Parameters ◽  
Free And Forced Vibrations ◽  
The Finite Difference Method

The paper is devoted to mathematical modeling of cantilever bars using the finite difference method. This method is widely used in structural mechanics for solving static problems. The novelty lies in the application of the finite difference method to simulate the dynamics of free and forced vibrations of the cantilever. Models have been developed that allow calculating the static and dynamic deflections of the cantilevers during free and forced vibrations, as well as simulating the vibrations of cantilever beams with attached vibration dampers. The resulting models of cantilever structures make it easy to modify system parameters, external influences and damping elements. All calculations were performed using the finite difference approach when moving along geometric and temporal coordinates.

scholarly journals A mixed methods approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method

2019 ◽  
Vol 9 (2) ◽  
pp. 223-235 ◽  
Author(s):  
Ali Başhan
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Schrödinger Equation ◽  
Difference Method ◽  
Schrodinger Equation ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
B Spline ◽  
Wide Range

The present manuscript include, finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) to obtain the numerical solutions for the nonlinear Schr¨odinger (NLS) equation. For this purpose, firstly Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using special type of classical finite difference method namely, Crank-Nicolson scheme. After that, Rubin and Graves linearization techniques have been utilized and differential quadrature method has been applied. So, partial differential equation turn into algebraic equation system. Next, in order to be able to test the accuracy of the newly hybrid method, the error norms L2 and L? as well as the two lowest invariants I1 and I2 have been calculated. Besides those, the relative changes in those invariants have been given. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison has clearly indicated that the currently utilized method, namely FDM-DQM, is an effective and efficient numerical schemeand allowed us to propose to solve a wide range of nonlinear equations.

scholarly journals Solution of Finite Difference Method and Differential Quadrature Method in Burgers Equation

2019 ◽  
Vol 63 (3) ◽  
pp. 1-4
Author(s):  
Amiruddin Ab. Aziz ◽  
◽  
Noor Noor Syazana Ngarisan ◽  
Nur Afriza Baki ◽  
◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Burgers Equation ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Language Program ◽  
C Language

The Finite Difference Method and Differential Quadrature Method are used to solve the partial differential equation in Burgers equation. The different number of nodes is used in these methods to investigate the accuracy. The solutions of these methods are compared in terms of accuracy of the numerical solution. C language program have been developed based on the method in order to solve the Burgers equation. The results of this study are compared in terms of convergence as well as accuracy of the numerical solution. Generally, from the numerical results show that the Differential Quadrature Method is better than the Finite Different Method in terms of accuracy and convergence.

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