Higher order meshless approximation applied to Finite Difference and Finite Element methods in selected thermomechanical problems

Abstract

This paper is focused on the application of higher order meshless schemes in a numerical analysis of selected thermomechanical problems. Numerical investigation is based upon meshless Finite Difference method as well as its combinations with Finite Element method, performed at two different levels of analysis. The first variant assumes conjugation of element and meshless approximation schemes, while in the second one, the problem domain is divided into several disjoint subdomains, with a parallel and independent operating of coupled methods in each subdomain.The most important advantage of the applied approximation technique is no requirements of modification or enhancement of the existing discretization model. Therefore, high approximation orders may be assumed without providing new nodes, elements and degrees of freedom, maintaining the entire numerical model as simple as possible. This approach is especially convenient in coupled multi-field 2D and 3D problems, for instance stationary and non-stationary thermoelastic ones. In those problems, standard higher order approximation techniques may lead to complex numerical models caused by the rapid growth of a number of degrees of freedom and ill-conditioned schemes.The proposed approach is derived for three dimensional non-stationary thermoelastic problems. Moreover, it is examined on variety of 2D and 3D benchmark examples and engineering applications. Both solution accuracy and convergence rate are taken into account. Obtained results are very promising as they reflect the competitiveness of the approach comparing to other commonly applied higher order approaches.