Investigations on the hp-Cloud Method by solving Timoshenko beam problems

Abstract

 The hp-Cloud Method is a new and promising approximation technique which, without relying on a mesh, can be used in both finite and boundary element methods. In spite of its great success in solving problems with high accuracy and convergence rates, there are still a number of aspects to be qualitative and quantitatively investigated. Among these are: the sensitivity to the weight functions used in the construction of the Shepard functions, the sensitivity to the class of enrichment functions used in the p-adaptivity, the sensitivity to the cloud overlapping, and the variations of the condition number. This paper describes numerical experiments regarding some of the many choices allowed by the hp-Cloud methodology applied to Timoshenko beam problems. Since the Moving Least Squares Method is used to generate the partition of unity, some choices of weighting functions are studied and the results are compared to each other. In addition, convergence results are presented for successive h-refinements, when the number of clouds is increased, and for increasingly higher order approximation functions characterizing p-refinements. Since the new basis functions are in general not polynomials, an adaptive integration procedure is employed. The efficiency of several types of basis functions is verified. The rates of h and p convergence are determined as functions of other parameters. Also, examples of degradation of the stiffness matrix condition number is displayed and discussed.