Static analysis of two-dimensional elastic structures using global collocation

Abstract

Based on previous findings concerning the numerical solution of one-dimensional elastodynamical problems [Provatidis in Arch Appl Mech 78(4):241-250, 2008] this paper extends the methodology to the static analysis of two-dimensional problems in quadrilateral domains. This target is achieved by replacing the Galerkin/Ritz procedure involved in Lagrangian (or Gordon-Coons) type finite elements by a global collocation scheme. In brief, the boundary conditions are fulfilled at all boundary nodes, while the governing equation is fulfilled at internal points. The theory is supported by four test cases concerning rectangular and curvilinear structures under plane-stress or plane-strain conditions, where the convergence rate is successfully compared with that of conventional bilinear finite elements with the same mesh density.

B-splines collocation eigenanalysis of 2D acoustic problems

“…Concerning mechanical analysis in problems of solids and structures including acoustics, it is well known that there are three main methodologies: the popular finite element method (FEM), the boundary element method (BEM), and the promising global collocation method ( [Provatidis 2008b;2009b;Provatidis and Ioannou 2010] and about 300 references therein). For the sake of brevity, finite volume, finite difference, mesh-less and mesh-free methodologies are not commented on.…”

“…For the first time, in 2005 the author expressed the above general idea of implementing a global collocation scheme in conjunction with CAD-based approximation [Provatidis 2006, p. 6704], while he continued with numerical applications in potential problems under Dirichlet [Provatidis 2008b] and Neumann [Provatidis 2009b] boundary conditions, as well as in plane stress elastostatics [Provatidis and Ioannou 2010]. Concerning eigenvalue and time response structural analysis, in 2008 he published a couple of papers [Provatidis 2008a;2008c] (1D problems), and also he supervised a thesis concerning 2D acoustics and 2D elastodynamics in conjunction with Lagrange polynomials [Filippatos 2010].…”