zbMATH — the first resource for mathematics

From geometric design to numerical analysis: a direct approach using the finite cell method on constructive solid geometry. (English) Zbl 1410.65460
Authors’ abstract: During the last ten years, increasing efforts were made to improve and simplify the process from computer aided design (CAD) modeling to a numerical simulation. It has been shown that the transition from one model to another, i.e., the meshing, is a bottleneck. Several approaches have been developed to overcome this time-consuming step, e.g. isogeometric analysis (IGA), which applies the shape functions used for the geometry description (typically B-splines and NURBS) directly to the numerical analysis. In contrast to IGA, which deals with boundary represented models (B-Rep), our approach focuses on parametric volumetric models such as constructive solid geometries (CSG). These models have several advantages, as their geometry description is inherently watertight and they provide a description of the models’ interior. To be able to use the explicit mathematical description of these models, we employ the finite cell method (FCM). Herein, the only necessary input is a reliable statement whether an (integration-) point lies inside or outside the geometric model. This paper mainly discusses such point-in-membership tests on various geometric objects like sweeps and lofts, as well as several geometric operations such as filleting or chamfering. We demonstrate that, based on the information of the construction method of these objects, the point-in-membership-test can be carried out efficiently and robustly.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D17 Computer-aided design (modeling of curves and surfaces)
68U07 Computer science aspects of computer-aided design
PDF BibTeX Cite
Full Text: DOI
[1] Foley, J. D.; Dam, A. V.; Feiner, S. K.; Hughes, J. F.; Phillips, R. L., Introduction to computer graphics, (1997), Addison-Wesley
[2] Gomes, A. J.P.; Teixeira, J. G., Form feature modelling in a hybrid CSG/brep scheme, Comput. Graph., 15, 2, 217-229, (1991)
[3] Shah, J. J.; Mäntylä, M., Parametric and feature-based CAD/CAM: concepts, techniques, and applications, (1995), John Wiley & Sons
[4] Cottrell, J.; Hughes, T. J.; Bazilevs, Y., Isogeometric analysis: toward integration of CAD and FEA, (2009), Wiley and Sons New York · Zbl 1378.65009
[5] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[6] Piegl, L.; Tiller, W., (The NURBS Book, Monographs in Visual Communication, (1997), Springer Berlin Heidelberg Berlin, Heidelberg)
[7] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J.R.; Lipton, S.; Scott, M. A.; Sederberg, T. W., Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Engrg., 199, 229-263, (2010) · Zbl 1227.74123
[8] Massarwi, F.; Elber, G., A B-spline based framework for volumetric object modeling, Comput. Aided Des., 78, 36-47, (2016)
[9] Zuo, B.-Q.; Huang, Z.-D.; Wang, Y.-W.; Wu, Z.-J., Isogeometric analysis for CSG models, Comput. Methods Appl. Mech. Engrg., 285, 102-124, (2015) · Zbl 1425.65192
[10] A. Patera, Nonconforming mortar elements methods: Application to spectral dicretizations, 1988. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890002965.pdf.
[11] Natekar, D.; Zhang, X.; Subbarayan, G., Constructive solid analysis: A hierarchical, geometry-based meshless analysis procedure for integrated design and analysis, Comput. Aided Des., 36, 473-486, (2004)
[12] Schillinger, D.; Dedè, L.; Scott, M. A.; Evans, J. A.; Borden, M. J.; Rank, E.; Hughes, T. J., An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Engrg., 249-252, 116-150, (2012) · Zbl 1348.65055
[13] Parvizian, J.; Düster, A.; Rank, E., Topology optimization using the finite cell method, Optim. Eng., 13, 57-78, (2011) · Zbl 1293.74357
[14] Rank, E.; Ruess, M.; Kollmannsberger, S.; Schillinger, D.; Düster, A., Geometric modeling, isogeometric analysis and the finite cell method, Comput. Methods Appl. Mech. Engrg., 249-252, 104-115, (2012) · Zbl 1348.74340
[15] Bungartz, H.-J.; Griebel, M.; Zenger, C., Introduction to computer graphics, (2004), Charles River Media
[16] A.A.G. Requicha, H.B. Voelker, Constructive solid geometry, TM-25. Production Automation Project, University of Rochester, 1977.
[17] Lorensen, W. E.; Cline, H. E., Marching cubes: A high resolution 3D surface construction algorithm, (Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, (1987), ACM Press New York, NY), 163-169
[18] Düster, A.; Parvizian, J.; Yang, Z.; Rank, E., The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Engrg., 197, 3768-3782, (2008) · Zbl 1194.74517
[19] Hughes, T. J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Mineola, NY · Zbl 1191.74002
[20] Reddy, B. D., (Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements, Texts in Applied Mathematics, No. 27, (1997), Springer New York)
[21] Hughes, T. J.R., A simple scheme for developing ‘upwind’ finite elements, Internat. J. Numer. Methods Engrg., 12, 1359-1365, (1978), 00258 · Zbl 0393.65044
[22] Parvizian, J.; Düster, A.; Rank, E., Finite cell method, Comput. Mech., 41, 121-133, (2007) · Zbl 1162.74506
[23] Dauge, M.; Düster, A.; Rank, E., Theoretical and numerical investigation of the finite cell method, J. Sci. Comput., 65, 1039-1064, (2015) · Zbl 1331.65160
[24] Szabó, B. A.; Düster, A.; Rank, E., The p-version of the finite element method, (Stein, E., Encyclopedia of Computational Mechanics, (2004), John Wiley & Sons Chichester, West Sussex)
[25] Schillinger, D.; Rank, E., An unfitted \(h p\)-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry, Comput. Methods Appl. Mech. Engrg., 200, 3358-3380, (2011) · Zbl 1230.74197
[26] Schillinger, D.; Ruess, M.; Zander, N.; Bazilevs, Y.; Düster, A.; Rank, E., Small and large deformation analysis with the \(p\)- and B-spline versions of the finite cell method, Comput. Mech., 50, 445-478, (2012) · Zbl 1398.74401
[27] S. Duczek, M. Joulaian, A. Düster, U. Gabbert, Simulation of Lamb waves using the spectral cell method, pp. 86951U-86951U-11, Apr. 2013.
[28] Joulaian, M.; Düster, A., Local enrichment of the finite cell method for problems with material interfaces, Comput. Mech., 52, 741-762, (2013) · Zbl 1311.74123
[29] Abedian, A.; Parvizian, J.; Düster, A.; Khademyzadeh, H.; Rank, E., Performance of different integration schemes in facing discontinuities in the finite cell method, Int. J. Comput. Methods, 10, 1350002, (2013), 00027 · Zbl 1359.65245
[30] Kudela, L.; Zander, N.; Kollmannsberger, S.; Rank, E., Smart octrees: accurately integrating discontinuous functions in 3D, Comput. Methods Appl. Mech. Engrg., 306, 406-426, (2016)
[31] Kollmannsberger, S.; Özcan, A.; Baiges, J.; Ruess, M.; Rank, E.; Reali, A., Parameter-free, weak imposition of Dirichlet boundary conditions and coupling of trimmed and non-conforming patches, Internat. J. Numer. Methods Engrg., 101, 670-699, (2015), 00001 · Zbl 1352.65520
[32] Ruess, M.; Schillinger, D.; Bazilevs, Y.; Varduhn, V.; Rank, E., Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method, Internat. J. Numer. Methods Engrg., 95, 811-846, (2013) · Zbl 1352.65643
[33] Whitney, H., On regular closed curves in the plane, Compos. Math., 4, 276-284, (1937) · JFM 63.0647.01
[34] Machchhar, J.; Elber, G., Revisiting the problem of zeros of univariate scalar Béziers, Comput. Aided Geom. Design, 43, 16-26, (2016) · Zbl 1417.65087
[35] Zander, N.; Bog, T.; Kollmannsberger, S.; Schillinger, D.; Rank, E., Multi-level \(h p\)-adaptivity: high-order mesh adaptivity without the difficulties of constraining hanging nodes, Comput. Mech., 55, 499-517, (2015) · Zbl 1311.74133
[36] J. Schöberl, NETGEN, 2003. http://www.hpfem.jku.at/netgen/.
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.