Residual-based adaptive refinement for meshless eigenvalue solvers

Abstract

The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimensional structures. Based on radial basis functions, eigenmodes are calculated in a collocation approach for the second-order wave equation. This type of meshless method promises highly accurate results with the simplicity of a node-based collocation approach. Thus, when changing the discrete representation of a physical model, only node locations have to be adapted, hence avoiding the numerical overhead of handling an explicit mesh topology. The accuracy of the method comes at a cost of dealing with poorly-conditioned matrices. This is circumvented by applying a leave-one-out-cross-validation optimization algorithm to get stable results. A node adaptivity algorithm is presented to efficiently refine an initially coarse discretization. The convergence is evaluated in two numerical examples with analytical solutions. The most relevant parameter of the adaptation algorithm is numerically investigated and its influence on the convergence rate examined.