A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem

Abstract

Owing to its particular characteristics, the direct discretization of the Dirac-delta function is not feasible when point discretization methods like the differential quadrature method (DQM) are applied. A way for overcoming this difficulty is to approximate (or regularize) the Dirac-delta function with simple mathematical 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. On the other hand, it is possible to combine the DQM with the integral quadrature method (IQM) to handle the Dirac-delta function. Alternatively, one may use another definition of the Dirac-delta function that the derivative of the Heaviside function, H(x), is the Dirac-delta function, δ(x), in the distribution sense, namely, dH(x)/dx = δ(x). This approach has been referred in the literature as the direct projection approach. It has been shown that although this approach yields highly oscillatory approximation of the Dirac-delta function, it can yield a non-oscillatory approximation of the solution. In this paper, we first present a modified direct projection approach that eliminates such difficulty (oscillatory approximation of the Dirac-delta function). We then demonstrate the applicability and reliability of the proposed method by applying it to some moving load problems of beams and rectangular plates.

Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers

“…However, for the second approximation, using the derivative of the ramp function, the solution is oscillatory especially near the singular point ( ). This effect has been reported earlier by Eftekhari [ 59 ] and Jung [ 60 ] in the case of the Dirac function.…”

An accurate differential quadrature procedure for the numerical solution of the moving load problem

“…The applications of the method are numerous and include the numerical solution of ordinary differential equations (ODEs), partial differential equations (PDEs), and integro-differential equations (IDEs) that arise in various engineering and applied mechanics problems [28,29]. More recently, the DQM has been applied to some type of moving load problem [31][32][33][34][35][36]. In the moving load problem, the position and movement of the load can be described by means of a time-dependent Dirac-delta function.…”

“…To overcome the difficulties in the discretization of the Dirac-delta function, in Refs. [31][32][33][34][35][36][37][38][39][40], various procedures have been proposed. Although the proposed approaches were shown to be accurate and reliable, as discussed in Refs.…”

“…Although the proposed approaches were shown to be accurate and reliable, as discussed in Refs. [32,33], they have some disadvantages and limitations. For instance, the procedures presented in Refs.…”

“…On the other hand, in Refs. [32,37,39,40], a projection technique has been proposed to solve the problem. In this technique, the Dirac-delta function is replaced by the derivative of the Heaviside function (i.e., δ(x) is replaced by dH(x)/dx, where δ(x) and H(x) are, respectively, the Diracdelta function and the Heaviside function).…”