Abstract
We review the fundamentals of the subdivision nite element method and introduce its extension to the analysis of thin shells with non-smooth and non-manifold features. The subdivision methods are a generalisation of splines and are able to generate, at least,C 1 continuous smooth surfaces on irregular meshes. In subdivision nite elements, subdivision surfaces are used for describing the reference geometry and the deformed geometry of a thin shell in accordance with the isogeometric analysis method.
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Bibliography
H. Biermann, A. Levin, and D. Zorin. Piecewise smooth subdivision surfaces with normal control. In SIGGRAPH 2000 Conference Proceedings, pages 113–120, July 2000.
N. Buechter and E. Ramm. Shell theory versus degeneration — A comparison in large rotation finite element analysis. International Journal for Numerical Methods in Engineering, 34:39–59, 1992.
E. Catmull and J. Clark. Recursively generated b-spline surfaces on arbitrary topological meshes. Computer-Aided Design, 10:350–355, 1978.
P.G. Ciarlet. An Introduction to Differential Geometry with Applications to Elasticity. Springer, 2005.
F. Cirak and M. Ortiz. Fully C 1-conforming subdivision elements for finite deformation thin-shell analysis. International Journal for Numerical Methods in Engineering, 51:813–833, 2001.
F. Cirak, M. Ortiz, and P. Schröder. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. International Journal for Numerical Methods in Engineering, 47:2039–2072, 2000.
F. Cirak, M.J. Scott, E.K. Antonsson, M. Ortiz, and P. Schröder. Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Computer-Aided Design, 34:137–148, 2002.
G. de Rham. Un peu de mathématiques à propos d’une courbe plane. Elemente der Mathematik, 2:73–76, 89–97, 1947.
D. Doo and M. Sabin. Behavior of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356–360, 1978.
F. Gruttmann, E. Stein, and P. Wriggers. Theory and numerics of thin elastic shells with finite rotations. Ingenieur Archiv, 59:54–67, 1989.
T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005.
A. Levin. Modified subdivision surfaces with continuous curvature. In SIGGRAPH 2006 Conference Proceedings, pages 1035–1040, 2006.
C.L. Liao and J. N. Reddy. Analysis of anisotropic, stiffened composite laminates using a continuum based shell element. Computers and Structures, 34:805–815, 1990.
Q. Long. Subdivision Finite Elements for Geometrically Complex Thin and Thick Shells. PhD thesis, University of Cambridge, 2010.
C.T. Loop. Smooth subdivision surfaces based on triangles. Master’s thesis, Department of Mathematics, University of Utah, 1987.
E. Oñate and F.G. Flores. Advances in the formulation of the rotation-free basic shell triangle. Computer Methods in Applied Mechanics and Engineering, 2005.
D. Peric and D.R.J. Owen. The Morley thin shell finite element for large deformation problems: simplicity versus sophistication. In Proceedings of the International Conference on Nonlinear Engineering Computations, pages 121–142, Swansea, 1991.
J. Peters and U. Reif. Subdivision Surfaces. Springer Series in Geometry and Computing. Springer, 2008.
R. Phaal and C.R. Calladine. A simple class of finite elements for plate and shell problems. II: An element for thin shells, with only translational degrees of freedom. International Journal for Numerical Methods in Engineering, 35:979–996, 1992.
L. Piegl and W. Tiller. The NURBS book. Monographs in visual communication. Springer, 1997.
S. Schaefer and R. Goldman. Non-uniform subdivision for b-splines of arbitrary degree. Computer Aided Geometric Design, 26:75–81, 2009.
T.W. Sederberg, J. Zheng, D. Sewell, and M. Sabin. Non-uniform recursive subdivision surfaces. In SIGGRAPH1998 Conference Proceedings, pages 387–394, 1998.
J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parameterization. Computer Methods in Applied Mechanics and Engineering, 72:267–304, 1989.
J. Stam. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In SIGGRAPH 1998 Conference Proceedings, pages 395–404, 1998.
J. Stam. On subdivision schemes generalizing uniform b-spline surfaces of arbitrary degree. Computer Aided Geometric Design, 18:383–396, 2001.
J. Warren and H. Weimer. Subdivision methods for geometric design: A constructive approach. Morgan Kaufmann, 2001.
D. Zorin and P. Schröder. Subdivision for modeling and animation. SIG-GRAPH 2000 Course Notes, 2000.
D. Zorin and P. Schröder. A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided Geometric Design, 18:429–454, 2001.
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Cirak, F., Long, Q. (2010). Advances in Subdivision Finite Elements for Thin Shells. In: De Mattos Pimenta, P., Wriggers, P. (eds) New Trends in Thin Structures: Formulation, Optimization and Coupled Problems. CISM International Centre for Mechanical Sciences, vol 519. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0231-2_8
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DOI: https://doi.org/10.1007/978-3-7091-0231-2_8
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