Abstract
A major bottleneck in numerical analysis is to represent a CAD model in a format appropriate for performing analysis. Often most of the analysis time is spent in this preprosessing step. In 2005, Professor Tom Hughes at the University of Texas introduced isogeometric analysis as a remedy for this problem. The idea is to use NURBS, well known from the representation of free form curves and surfaces in CAD models, as a solution space for Finite Element Analysis. This enables a unified representation for geometry and solution. Both professor Hughes and other authors have been using NURBS in numerical analysis with good results. A further claim is that this concept will bridge the gap between CAD and numerical analysis. Is this true? A CAD represented solid is described by its inner and outer hull, each hull is described as a patchwork of surfaces. The surfaces are either analytical ones or NURBS surfaces. Trimming is extensively used to achieve flexibility in the shape modelling as many analytical surfaces are infinite and NURBS surfaces are inherently 4-sided. Trimmed surfaces are normally described using approximate trimming curves as the exact closed form of a trimming curve cannot be represented by NURBS. This implies that the model is not watertight. However, the geometrical shape may be described with high accuracy and a high level of details. In Finite Element Analysis, the shape is represented by structures of finite elements where each element is a trivariate polynomial often of low degree. The FEA model will be watertight, but the shape is often a crude approximation of the shape of a corresponding CAD model. NURBS surfaces and a FEA models are both described by function spaces. The spaces have many similarities, but also differences: Both spaces have got compactly supported basis functions that sum up to unity. The traditional FE space have polynomial basis functions, the spline space have basis functions that are piecewise polynomials or rational
