A global collocation method for two-dimensional rectangular domains

Abstract

This paper proposes the use of a global collocation procedure in conjunction with a previously developed functional set suitable for the numerical solution of Poisson's equation in rectangular domains. We propose to expand the unknown variable in a bivariate series of monomials x i y j that exist in Pascal's triangle. We also propose the use of the bivariate Gordon-Coons interpolation, apart from previous intuitive choices of the aforementioned monomials. The theory is sustained by two numerical examples of Dirichlet boundary conditions, in which we find that the approximate solution monotonically converges towards the exact solution.

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].…”

Static analysis of two-dimensional elastic structures using global collocation

“…Coons-Gordon as in the previous Galerkin/Ritz scheme). To this direction, pilot studies in one-dimensional problems revealed an excellent overall quality of the numerical solution [14,15]; also, another pilot study in rectangular domains governed by Poisson's equation (field problem) and Dirichlet boundary conditions led to encouraging results when a special set of Cartesian basis functions was used in the form x i y j [16].…”

“…Moreover, instead of the aforementioned special set of basis functions restricted to rectangular domains only [16], this paper proposes the use of cardinal global shape functions thus becoming capable of treating any structure of arbitrary curvilinear quadrilateral shape provided it is adequately smooth.…”