zbMATH — the first resource for mathematics

Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. (English) Zbl 1227.74109
Summary: Isogeometric analysis has been proposed as a methodology for bridging the gap between computer aided design (CAD) and finite element analysis (FEA). Although both the traditional and isogeometric pipelines rely upon the same conceptualization to solid model steps, they drastically differ in how they bring the solid model both to and through the analysis process. The isogeometric analysis process circumvents many of the meshing pitfalls experienced by the traditional pipeline by working directly within the approximation spaces used by the model representation. In this paper, we demonstrate that in a similar way as how mesh quality is used in traditional FEA to help characterize the impact of the mesh on analysis, an analogous concept of model quality exists within isogeometric analysis. The consequence of these observations is the need for a new area within modeling – analysis-aware modeling – in which model properties and parameters are selected to facilitate isogeometric analysis.

74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
65D17 Computer-aided design (modeling of curves and surfaces)
Full Text: DOI
[1] Hughes, T.J.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: cad, finite elements, NURBS, exact geometry, and mesh refinement, Comput. meth. appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[2] T. Dokken, V. Skytt, J. Haenisch, K. Bengtsson, Isogeometric representation and analysis - bridging the gap between CAD and analysis, in: 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, January 5-8, 2009.
[3] Sederberg, T.W.; Finnigan, G.T.; Li, X.; Lin, H.; Ipson, H., ACM trans. graphics, 27, 3, 79:1-79:8, (2008)
[4] Cohen, Elaine; Riesenfeld, Richard F.; Elber, Gershon, Geometric modeling with splines: an introduction, (2001), A.K. Peters, Ltd. Natick, MA, USA · Zbl 0980.65016
[5] Schwab, Ch., p- and hp-finite element methods: theory and applications to solid and fluid mechanics, (1999), Oxford University Press USA · Zbl 0910.73003
[6] Ciarlet, Philippe G., The finite element method for elliptic problems, (2002), Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 0999.65129
[7] Axelsson, Owe, Iterative solution methods, (1994), Cambridge University Press Cambridge · Zbl 0795.65014
[8] Deville, M.O.; Mund, E.H.; Fischer, P.F., High order methods for incompressible fluid flow, (2002), Cambridge University Press · Zbl 1007.76001
[9] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for CFD - second ed., (2005), Oxford University Press UK · Zbl 0857.76044
[10] Cohen, Elaine; Lyche, Tom; Riesenfeld, Richard F., Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics, Comput. graphics image process., 15, 2, 87-111, (1980)
[11] Cohen, Elaine; Lyche, Tom; Schumaker, L.L., Algorithms for degree-raising of splines, ACM trans. graphics, 4, 3, 171-181, (1986) · Zbl 0621.41009
[12] Morken, K., Products of splines as linear combinations of B-splines, Construct. approx., 7, 1, 195-208, (1991) · Zbl 0732.41004
[13] Casale, M.S.; Stanton, E.L., An overview of analytic solid modeling, IEEE comput. graphics appl., 45-56, (1985)
[14] Karen Lynn Paik, Trivariate B-splines. Master’s Thesis, Department of Computer Science, University of Utah, June 1992.
[15] William Martin, Elaine Cohen, Surface completion of an irregular boundary curve using a concentric mapping, in: Proceedings of the Fifth Conference on Curves and Surfaces, Nashboro Press, 2003, pp. 293-302. · Zbl 1043.65032
[16] Joel D. Daniels II, Elaine Cohen, Surface creation and curve deformations between two complex closed spatial spline curves, in: Springer-Verlag Lecture Notes in Computer Science 4077 (GMP 2006), 2006, pp. 221-234. · Zbl 1160.68618
[17] Cottrell, J.A.; Hughes, T.J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. meth. appl. mech. engrg., 196, 4160-4183, (2007) · Zbl 1173.74407
[18] Steven A. Coons, Surfaces for computer-aided design of space forms. Technical Report MAC-TR-41, MIT, 1967.
[19] Gordon, William J., Spline-blended surface interpolation through curve networks, J. math. mech., 18, 10, 931-952, (1969) · Zbl 0192.42201
[20] Piegl, Leslie; Tiller, Wayne, A menagerie of rational B-spline circles, IEEE comput. graphics appl., 9, 5, 48-56, (1989)
[21] Blinn, James, How many ways can you draw a circle?, IEEE comput. graphics appl., 7, 8, 39-44, (1987)
[22] Y. Zhang, Y. Bazilevs, S. Goswami, C.L. Bajaj, T.J.R. Hughes, Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow, in: Proceedings of the 15th International Meshing Roundtable, Springer, Berlin, 2006, pp. 73-92. · Zbl 1121.76076
[23] Bazilevs, Y.; Hughes, T.J.R., Nurbs-based isogeometric analysis for the computation of flows about rotating components, Comput. mech., 43, 143-150, (2008) · Zbl 1171.76043
[24] William Martin, Elaine Cohen, Representation and extraction of volumetric attributes using trivariate splines, in: Symposium on Solid and Physical Modeling, 2001, pp. 234-240.
[25] Tobias Martin, Elaine Cohen, Mike Kirby, Volumetric parameterization and trivariate B-spline fitting using harmonic functions, in: SPM ’08: Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, ACM, New York, NY, USA, 2008, pp. 269-280.
[26] Jonathan R. Shewchuk, What is a good linear element? interpolation, conditioning, and quality measures, in: 11th International Meshing Roundtable, 2002, pp. 115-126.
[27] Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O.C., Adaptive remeshing for compressible flow computations, J. comput. phys., 72, 2, 449-466, (1987) · Zbl 0631.76085
[28] M. Berzins, Mesh quality: a function of geometry, error estimates or both, in: Seventh International Meshing Roundtable, 1998, pp. 229-238.
[29] Venditti, David A.; Darmofal, David L., Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow, J. comput. phys., 164, 1, 204-227, (2000) · Zbl 0995.76057
[30] Sederberg, Thomas W.; Zheng, Jianmin; Bakenov, Almaz; Nasri, Ahmad, T-splines and T-NURCCS, ACM trans. graphics, 22, 3, 477-484, (2003)
[31] Butkov, Eugene, Mathematical physics, (1968), Addison-Wesley Publishing Company Reading, MA · Zbl 0189.33501
[32] Bazilevs, L.; amd Beirao da Veiga, Y.; Cottrell, J.A.; Hughes, T.J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. meth. models appl. sci., 16, 1031-1090, (2006) · Zbl 1103.65113
[33] Szabó, B.A.; Babuška, I., Finite element analysis, (1991), John Wiley and Sons New York
[34] Schumaker, Larry L., Spline functions: basic theory, (2007), Cambridge University Press · Zbl 1123.41008
[35] Pardhanani, Anand; Carey, Graham F., Optimization of computational grids, Numer. meth. partial diff. equat., 4, 2, 95-117, (1988) · Zbl 0645.65081
[36] Cottrell, J.A.; Reali, A.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric analysis of structural vibrations, Comput. meth. appl. mech. engrg., 195, 41-43, 5257-5296, (2006) · Zbl 1119.74024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.