Abstract
The differential quadrature method (DQM) is one of the most elegant and efficient methods for the numerical solution of partial differential equations arising in engineering and applied sciences. It is simple to use and also straightforward to implement. However, the DQM is well-known to have some difficulty when applied to partial differential equations involving singular functions like the Dirac-delta function. This is caused by the fact that the Dirac-delta function cannot be directly discretized by the DQM. To overcome this difficulty, this paper presents a simple differential quadrature procedure in which the Dirac-delta function is replaced by regularized smooth functions. By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. To demonstrate the applicability and reliability of the proposed method, it is applied here to solve some moving load problems of beams and rectangular plates, where the location of the moving load is described by a time-dependent Dirac-delta function. The results generated by the proposed method are compared with analytical and numerical results available in the literature. Numerical results reveal that the proposed method can be used as an efficient tool for dynamic analysis of beam- and plate-type structures traversed by moving dynamic loads.
Highlights
Most practical problems in engineering are governed by partial differential equations with proper boundary conditions
To demonstrate its applicability and reliability, the method is applied here to solve some moving load problems of beams and rectangular plates, where the location of the moving load is described by a time-dependent Dirac-delta function
The first and second examples demonstrate the applications of the proposed method to static analysis of supported beams and rectangular plates subjected to concentrated point loads
Summary
Most practical problems in engineering are governed by partial differential equations with proper boundary conditions. It is well known that the weak-form-based methods such as the Ritz and finite element methods can handle the problems involving the Dirac-delta function, since they directly integrate the governing differential equation of the. The techniques considered in their study were the regularization using the Gaussian smooth function, the points source approach, the direct projection technique, and the domain decomposition method They discretized the differential equation using the finite difference and pseudo-spectral methods and concluded that the domain decomposition method is the best among these methods at highest accuracy for solving sineGordon equation involving the Dirac-delta function. A differential quadrature procedure based on the regularization of the Dirac-delta function is presented for the numerical solution of the moving load problem. The numerical results prove that the proposed method is reliable and accurate and can be used as an efficient tool for handling the moving load problem
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