Development of a novel strong-form meshless technique : random differential quadrature (RDQ) method with applications for 2-D multiphysics simulation of pH-sensitive hydrogel

Abstract

Differential Quadrature (DQ) is one of the efficient techniques for derivative approximation, but it always requires a regular domain discretized with all the points distributed in a fixed pattern only along the straight lines. This severely restricts the DQ while solving problems with the irregular domain discretized by the random field nodes. This limitation of the DQ method is overcome by the presently proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. This method extends the applicability of the DQ technique over the irregular or regular domain discretized by the field nodes distributed randomly. In the RDQ method, the governing differential equation is discretized with the locally applied DQ method, and the value of function is interpolated approximately by the fixed reproducing kernel particle method. A superconvergence condition is developed first for the RDQ method, which gives more than convergence rate of the function for the uniform or random field nodes scattered in the domain, where is the highest order of the monomials used in the approximation of function. Approximate derivatives of the function, computed by the RDQ method, are then evaluated by the novel approaches, called the weighted derivative and improved weighted derivative. The convergence analysis of the RDQ method is then performed by solving several 1-D, 2-D, and the elasticity problems with locally high gradients of the field variable distributions. It is observed from the results that the approaches termed the weighted derivative and improved weighted derivative provide satisfactory rates of the derivative convergence for the field nodes distributed either in the uniform or random manner.