On the Convergence of Random Differential Quadrature (RDQ) Method and Its Application in Solving Nonlinear Differential Equations in Mechanics

Abstract

Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed RUM), and discretizing a governing differential equation by the locally applied DQ method. A superconvergence condition is developed for the RDQ method, which gives more than O(h(p+1)) function value convergence for the uniform as well as random field nodes scattered in the domain. The RDQ method convergence analysis is carried out, and the superconvergence condition is verified by solving several 1D, 2D and elasticity problems. The applicability of the RDQ method to solve the nonlinear governing differential equations is successfully demonstrated by solving the fixed-fixed and cantilever beams for deflection due to the nonlinear electrostatic loading. It is concluded that the RDQ method effectively handles the irregular or regular domains discretized by the uniform or random field nodes, with good convergence rates.