Abstract
In this paper we review several aspects of older and contemporary attempts to integrate computer-aided design (CAD: geometric model) and computer-aided engineering (CAE: finite elements, boundary elements, etc.). After a short review on formulas for the description of CAD surfaces, a systematic mechanism for creating several types of corresponding isoparametric macroelements is presented. Gordon-Coons is initially applied in conjunction with piecewise linear, Lagrange polynomials and natural B-splines. Then, it is extended to more basis and blending functions. In addition to the well-known 'Lagrangian'-type elements, equivalent 'Bezierian'-type elements are introduced. Tensor product B-splines and aspects of NURBS isogeometric formulation are given. In addition to quadrilaterals, triangular macroelements based on Barnhill's interpolation are presented for the first time. The review covers applications of CAD-based macroelements in conjunction with the Galerkin-Ritz formulation, the Boundary Element Method, as well as recent Global Collocation procedures. A numerical example on a vibrating membrane elucidates the performance of the CAD-based global interpolation and depicts its superiority over the conventional finite element method.
Highlights
Computer-Aided Design (CAD) is the use of computer systems to assist in the creation, modification, analysis, or optimization of a design
It was shown that all five computer-aided design (CAD) interpolations, i.e. Coons, Gordon, Bézier, B-splines and non uniform rational B-splines (NURBS) are capable of producing global shape functions that influence the entire patch of volume, constructing reliable macroelements that can deal with a large portion of a structure
The great amount of relevant work performed by the finite element (FEM)/boundary element (BEM) group at National Technical University of Athens (NTUA) started in 1982 but it was published for the first time in 1989 [62]
Summary
Computer-Aided Design (CAD) is the use of computer systems to assist in the creation, modification, analysis, or optimization of a design. At MIT’s Electronic Systems Laboratory he investigated the mathematical formulation for these patches, and in 1967 published one of the most significant contributions to the area of geometric design, a treatise which has become known as “The Little Red Book” [1]. His “Coons Patch” was a formulation that presented the notation, mathematical foundation, and intuitive interpretation of an idea that would become the foundation for other surface descriptions that are commonly used today, such as B-spline surfaces, NURB surfaces, etc. The interested reader may find more details in [2]
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