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Volumetric parameterization and trivariate B-spline fitting using harmonic functions. (English) Zbl 1205.65094
Summary: We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate B-spline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [T. J. R. Hughes et al., Comput. Methods Appl. Mech. Eng. 194, No. 39–41, 4135–4195 (2005; Zbl 1151.74419)]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate B-spline geometric and attribute representations are generated from the resulting parameterization, creating trivariate B-spline material property representations over the same parameterization in a way that is related to the first two authors’ work [Representation and extraction of volumetric attributes using trivariate splines. in: Symposium on Solid and Physical Modeling, 234–240 (2001)] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a B-spline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate B-spline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis.

65D17 Computer-aided design (modeling of curves and surfaces)
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[1] Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M., 2003. Anisotropic polygonal remeshing, vol. 22, pp. 485-493
[2] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J., 1938. Topics in the theory of algebraic curves · Zbl 0559.14017
[3] Axelsson, O., Iterative solution methods, (1994), Cambridge University Press Cambridge · Zbl 0795.65014
[4] Binford, T.O., 1971. Visual perception by computer. In: Proceedings of the IEEE Conference on Systems and Controls. Miami, Florida
[5] Chuang, J.-H.; Ahuja, N.; Lin, C.-C.; Tsai, C.-H.; Chen, C.-H., A potential-based generalized cylinder representation, Computers&graphics, 28, 6, 907-918, (2004)
[6] Cohen, E.; Riesenfeld, R.F.; Elber, G., Geometric modeling with splines: an introduction, (2001), A.K. Peters, Ltd. Natick, MA, USA · Zbl 0980.65016
[7] Davis, P.J., Circulant matrices, (1979), John Wiley & Sons, Inc. · Zbl 0418.15017
[8] Dong, S.; Kircher, S.; Garland, M., Harmonic functions for quadrilateral remeshing of arbitrary manifolds, Computer aided geometric design, 22, 5, 392-423, (2005) · Zbl 1205.65116
[9] Edelsbrunner, H., 1980. Dynamic data structures for orthogonal intersection queries. Technical Report F59, Inst. Informationsverarb., Tech. Univ. Graz
[10] Floater, M., Mean value coordinates, Computer-aided design, 20, 1, 19-27, (2003) · Zbl 1069.65553
[11] Floater, M.S.; Hormann, K., Surface parameterization: A tutorial and survey, (), 157-186 · Zbl 1065.65030
[12] Freitag, L., Plassmann, P., 1997. Local optimization-based simplicial mesh untangling and improvement. Technical report. Mathematics and Computer Science Division, Argonne National Laboratory · Zbl 0962.65098
[13] Grimm, C.M.; Hughes, J.F., Modeling surfaces of arbitrary topology using manifolds, (), 359-368
[14] Gu, X.; Yau, S.-T., Global conformal surface parameterization, (), 127-137
[15] Hormann, K.; Greiner, G., Quadrilateral remeshing, (), 153-162
[16] Hua, J.; He, Y.; Qin, H., Multiresolution heterogeneous solid modeling and visualization using trivariate simplex splines, (), 47-58
[17] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover
[18] Hughes, T.J.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: cad, finite elements, nurbs, exact geometry, and mesh refinement, Computer methods in applied mechanics and engineering, 194, 4135-4195, (2005) · Zbl 1151.74419
[19] Jaillet, F., Shariat, B., Vandorpe, D., 1997. Periodic b-spline surface skinning of anatomic shapes. In: 9th Canadian Conference in Computational Geometry
[20] Lazarus, F.; Verroust, A., Level set diagrams of polyhedral objects, (), 130-140
[21] Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H., 2007. Harmonic volumetric mapping for solid modeling applications. In: Symposium on Solid and Physical Modeling, pp. 109-120
[22] Loop, C., Smooth spline surfaces over irregular meshes, (), 303-310
[23] Lyche, T.; Morken, K., A data reduction strategy for splines with applications to the approximation of functions and data, IMA journal of numerical analysis, 8, 2, 185-208, (1988) · Zbl 0642.65008
[24] Martin, T., Cohen, E., Kirby, M., 2008. A comparison between isogeometric analysis versus fem applied to a femur. In preparation
[25] Martin, W., Cohen, E., 2001. Representation and extraction of volumetric attributes using trivariate splines. In: Symposium on Solid and Physical Modeling, pp. 234-240
[26] Milnor, J., Morse theory, Annual of mathematics studies, vol. 51, (1963), Princeton University Press Princeton, NJ
[27] Ni, X.; Garland, M.; Hart, J.C., Fair Morse functions for extracting the topological structure of a surface mesh, ACM transactions on graphics, 23, 3, 613-622, (2004)
[28] Schreiner, J.; Scheidegger, C.; Fleishman, S.; Silva, C., Direct (re)meshing for efficient surface processing, Proceedings of eurographics 2006, Computer graphics forum, 25, 3, 527-536, (2006)
[29] Sederberg, T.W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and t-nurccs, ACM transactions on graphics, 22, 3, 477-484, (2003)
[30] Sheffer, A.; Praun, E.; Rose, K., Mesh parameterization methods and their applications, Foundations and trends in computer graphics and vision, 2, 2, (2006)
[31] Shinagawa, Y.; Kunii, T.L.; Kergosien, Y.L., Surface coding based on Morse theory, IEEE computer graphics and applications, 11, 5, 66-78, (1991)
[32] Si, H., Tetgen: A quality tetrahedral mesh generator and three-dimensional Delaunay triangulator
[33] Tong, Y., Alliez, P., Cohen-Steiner, D., Desbrun, M., 2006. Designing quadrangulations with discrete harmonic forms. In: ACM/EG Symposium on Geometry Processing
[34] Verroust, A.; Lazarus, F., Extracting skeletal curves from 3d scattered data, The visual computer, 16, 1, 15-25, (2000) · Zbl 0955.68115
[35] Wang, Y., Gu, X., Thompson, P.M., Yau, S.-T., 2004. 3d harmonic mapping and tetrahedral meshing of brain imaging data. In: Proc. Medical Imaging Computing and Computer Assisted Intervention (MICCAI), St. Malo, France, September 26-30 2004
[36] Zhou, X.; Lu, J., Nurbs-based Galerkin method and application to skeletal muscle modeling, (), 71-78
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