Convergence behaviour of inverse differential quadrature method for analysis of beam and plate structures

Abstract

Modelling of structures like beams and plates for advanced engineering applications is a tedious task considering the complexity of the kinematical assumptions governing their behaviour. A fundamental requirement in understanding the behaviour of these engineering structures is providing accurate numerical solutions to the high-order system of partial differential equations derived from the kinematic assumptions. To circumvent error accumulation arising from the numerical differentiation of high-order systems, the inverse differential quadrature method (iDQM) was recently proposed, and it is increasingly gaining attention due to its associated spectral convergence and numerical stability. To fully exploit the merits of iDQM for wider applications, this study establishes an understanding of sensitivity of static, buckling, and free vibration iDQM-based solutions of beam and plate structures to the choice of basis functions and grid distributions. Moreover, the computational complexity imposed by the full preconditioning required to transform the inherently underdetermined iDQM system to a square system suitable for eigenvalue analyses is addressed by proposing two novel subsystem preconditioning methods. From the ensuing comprehensive convergence analysis, it is shown that iDQM can be generalised for a specific class of polynomials that exhibit spectral convergence and negligible sensitivity to grid distributions. For other classes of polynomials that offer ill-conditioned coefficient matrices, iDQM can be applied to problems in which the desired accuracy can be achieved with few grid points. Finally, the study highlights the excellent computational gains offered by subsystem preconditioning methods for obtaining numerically stable eigenvalue solutions with high-order accuracy.