CAD-based collocation eigenanalysis of 2-D elastic structures C.G. Provatidis National Technical University of Athens, School of Mechanical Engineering, Mechanical Design and Control Systems Division, Heroon Polytechniou 9, 15780 Zografou, Greece article info Article history: Received 16 November 2015 Accepted 17 November 2016 Keywords: Coons-Gordon Bézier B-splines CAD/CAE Global collocation Eigenvalues abstract This paper investigates the performance of the global collocation method for the numerical eigenfre- quency extraction of 2-D elastic structures. The method is applied to CAD-based macroelements, starting from the older blending function Coons-Gordon interpolation (based on Lagrange polynomials) and extending to tensor product Bézier and B-splines. Numerical findings show equivalence between Lagrangian and Bézierian macroelements, while a mass lumping procedure is proposed for the former ones. Concerning B-splines, the influence of multiplicity of inner knots and the position of collocation points is thoroughly investigated. The theory is supported by 2-D numerical examples on rectangular and curvilinear structures of simple shape under plane stress conditions, in which the approximate solu- tion rapidly converges towards the exact solution faster than that of the conventional finite element of similar mesh density. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction The use of CAD-based global approximation for the numerical solution of partial differential equations (CAE: computer- aided-engineering) is as old as the theory of computer- aided-design (CAD) itself. It is well known that an industrial team (at General Motors) under Gordon’s leadership, in early 1970s, used blending function methods, based on the ideas put forward in [1], to produce some interesting element families [2,3]. Although this team presented the mathematical background for the common description between the geometric model and the unknown variable (CAD/CAE integration), unknown reasons (perhaps the high computational cost) prevented further dissemination of this excellent idea. One decade later, E1-Zafrany and Cookson [4,5] also used Coons’ and Barnhill’s ideas for quadrilateral and triangular patches, respectively, whereas Zhaobei and Zhiqiang proposed the use of Coons’ interpolation for the analysis of plates and shells [6]. Nevertheless computational results concerning CAD-based isoparametric macroelements (occupying a Coons patch ABCD) were presented for the first time by Kanarachos, Deriziotis and Provatidis [7,8] in 2D potential and elasticity (static and dynamic) problems, where the so-called ‘‘C”-elements were successfully compared with conventional finite elements and boundary ele- ments of similar mesh density. For a detailed review (of over 160 references) on the use of CAD-based macroelements the reader is referred to [9]. Summarizing some of the most important previous findings concerning macroelements that occupy a 2D quadrilateral patch ABCD or a 3D hexahedral block ABCDEFGH, in chronological corre- spondence with the progress in CAD-theory (Coons, Gordon, Bézier, B-splines and NURBS) (see, for instance, [10]), it has been reported that: (i) (Boundary-only) Coons interpolation is capable of creating a broad family of arbitrary-noded elements that may be equivalent to that of Serendipity type. For example, the conventional 4- up to 8-noded 2D elements, as well as the 8- and 20-noded 3D elements can be directly derived applying Coons interpolation [11–14]. (ii) Gordon-Coons (transfinite blending function) interpolation when applied to a structured macroelement of which the boundary and internal nodal points lay at the same normal- ized (n, g) positions, degenerates to the classical Lagrangian type finite element [14]. (iii) Coons-Gordon interpolation allows for dealing with a (relatively) unstructured mesh of internal nodes. Using a single quadrilateral macroelement, not only simple shapes such as a rectangular or a circle can be treated, but also it was possible to perform analysis until the complexity of a pi-shaped domain (see, [14–16], among others). For more complex shapes, domain decomposition using large macroelements becomes necessary. http://dx.doi.org/10.1016/j.compstruc.2016.11.007 0045-7949/Ó 2016 Elsevier Ltd. All rights reserved. E-mail address: cprovat@central.ntua.gr URL: http://users.ntua.gr/cprovat Computers and Structures 182 (2017) 55–73 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc