On the Location of Zeroes of Polynomials from the Stability Analysis of Novel Strong-Form Meshless Random Differential Quadrature Method

摘要

In this paper, the stability characteristics of a novel strong-form meshless method, called the random differential quadrature (RDQ), are studied using the location of zeros or roots of its characteristic polynomials with respect to unit circle in complex plane by discretizing the domain with the uniform or random field nodes. This is achieved by carrying out the RDQ method stability analysis for the 1st-order wave, transient heat conduction and transverse beam deflection equations using both the analytical and numerical approaches. The RDQ method extends the applicability of the differential quadrature (DQ) method over irregular domain, discretized by randomly or uniformly distributed field nodes, by interpolating the function values based on the fixed reproducing kernel particle method (fixed RKPM). The stability analysis of the locally applied DQ and RDQ methods is carried out for the different single and multistep schemes by Von Neumann and Schur polynomials. The stable schemes are identified and their consistency analysis is carried out to obtain additional constraints on the temporal spacing. The analytical results from the stability and consistency analyses of the stable schemes are effectively verified by numerically implementing the RDQ method to solve the 1st-order wave, transient heat conduction and transverse beam deflection equations by discretizing domain using uniform and random field nodes. Thus, it is shown that the RDQ method is very well attuned to the stability analysis and provides stable results as compared with FEM and other meshless methods.