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Abstract

We continue our research on the performance of CAD-based global approximation to the analysis of 2D acoustic problems. In addition to previous "boundary-only" Coons and transfinite Gordon-Coons interpolations, we now investigate the quality of the solution when utilizing "tensor product B-splines" interpolation. For the latter, we propose a global collocation method that is successfully compared with the well known Galerkin-Ritz formulation. Particular attention is paid to the handling of Neumann boundary conditions as well as to the role of multiplicity of internal knots. The theory is supported by two numerical examples, one for a rectangular and the other for a circular acoustic cavity in which the approximate solution rapidly converges towards the exact solution.259 260 CHRISTOPHER G. PROVATIDIS literature therein. Our conclusion in that article is that rapid CAD-based global collocation methods have to be applied instead of time-consuming domain Galerkin-Ritz methods.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]. It is worth mentioning that isogeometric collocation methods were recently presented by others [Auricchio et al. 2010].It is well known that B-splines collocation methods were initially developed by mathematicians, well before engineers understood that CAD-based Galerkin-Ritz methods require high computing effort, a fact that motivated them to investigate their replacement by isogeometric collocation schemes . For a detailed review of 273 papers covering the period 1934-1989 we refer to [Fairweather and Meade 1989], whereas recent works are cited in a survey [Bialecki et al. 2011]. A fair comparison between B-spline collocation and Galerkin methods on the basis of the same bandwidth (and not the same degree of shape functions) in one spatial dimension is [Kwok et al. 2001]. However, most works are limited to one-dimensional problems [de Boor 2001, Chapter XV], where, for cubic B-splines, the key point is to use two collocation points between any two breakpoints, thus resulting in as many equations as the number of unknowns; this is strictly related to using double knots (de Boor, personal communication, 2007). On the other hand, excellent results have been obtained in some particular examples using for collocation the images of either Greville or Demko abscissae [Auricchio et al. 2010].It is remarkable that de Boor's software, aiming at the solution of one-dimen...