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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 20
TRENDS IN ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis, B.H.V. Topping
Chapter 1
Isogeometric Analysis: Toward Unification of Computer Aided Design and Finite Element Analysis Y. Bazilevs^{1}, V.M. Calo^{1}, J.A. Cottrell^{1}, J. Evans^{1}, T.J.R. Hughes^{1}, S. Lipton^{1}, M.A. Scott^{1} and T.W. Sederberg^{2}
^{1}Computational and Applied Mathematics, University of Texas, Austin TX, United States of America Y. Bazilevs, V.M. Calo, J.A. Cottrell, J. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, T.W. Sederberg, "Isogeometric Analysis: Toward Unification of Computer Aided Design and Finite Element Analysis", in M. Papadrakakis, B.H.V. Topping, (Editors), "Trends in Engineering Computational Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 1, pp 116, 2008. doi:10.4203/csets.20.1
Keywords: isogeometric analysis, computer aided design, NURBS, Tsplines.
Summary
This chapter discusses the major challenges faced in engineering design and analysis. Going from an initial geometric design to a finite element mesh constitutes the most timeconsuming step of the overall design through analysis process. The concept of isogeometric analysis introduced in [1] has thus far addressed this issue by utilizing the NURBS functions commonly found in CAD packages as an isoparametric basis for both the geometry and the solution space. We extend the methodology to include Tsplines, a technology rapidly gaining favor in the geometry community due to its flexibility and efficiency in representing complex objects. This efficiency stems from its locally refinable basis, a feature ideally suited to finite element analysis where local refinement is frequently desirable.
Though NURBS have been, and continue to be widely used by designers, they have several drawbacks that we would like to avoid. One is that they achieve only C^{0}continuity across patch boundaries. Another is that the joining of two patches that were created separately can be problematic, frequently requiring the insertion of many knots from one patch into the other, and vice versa. This is a significant disadvantage of NURBS: knot insertion is a global operation. When we refine by inserting knots into the knot vectors of a surface, the knot lines extend throughout the entire domain.
Tsplines (see [2]) offer an interesting alternative to NURBS for both geometrical modeling and as a basis for isogeometric analysis. They will allow us to build bases that are complete up to a desired polynomial order, as smooth as an equivalent NURBS basis, and capable of being locally refined while keeping the original geometry and parameterization unchanged. The properties that make Tsplines useful for geometrical modeling also make them useful for finite element analysis. As mentioned, Tsplines even include NURBS as a special case, making their use in a finite element setting a natural extension of the existing isogeometric analysis technology.
Isogeometric analysis has been extended to include finite element analysis using Tsplines. Initial results suggest that the locally refinable Tspline basis may be wellsuited to analysis. Highly localized features can be captured without necessitating global refinements of the mesh, as was necessary in the case of NURBS.
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