The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.11 - November (2011 vol.17)
pp: 1663-1675
Christian Dick , Technische Universität München, Garching
Joachim Georgii , Fraunhofer MEVIS, Bremen
Rüdiger Westermann , Technische Universität München, Garching
ABSTRACT
We present a hexahedral finite element method for simulating cuts in deformable bodies using the corotational formulation of strain at high computational efficiency. Key to our approach is a novel embedding of adaptive element refinements and topological changes of the simulation grid into a geometric multigrid solver. Starting with a coarse hexahedral simulation grid, this grid is adaptively refined at the surface of a cutting tool until a finest resolution level, and the cut is modeled by separating elements along the cell faces at this level. To represent the induced discontinuities on successive multigrid levels, the affected coarse grid cells are duplicated and the resulting connectivity components are distributed to either side of the cut. Drawing upon recent work on octree and multigrid schemes for the numerical solution of partial differential equations, we develop efficient algorithms for updating the systems of equations of the adaptive finite element discretization and the multigrid hierarchy. To construct a surface that accurately aligns with the cuts, we adapt the splitting cubes algorithm to the specific linked voxel representation of the simulation domain we use. The paper is completed by a convergence analysis of the finite element solver and a performance comparison to alternative numerical solution methods. These investigations show that our approach offers high computational efficiency and physical accuracy, and that it enables cutting of deformable bodies at very high resolutions.
INDEX TERMS
Deformable objects, cutting, finite elements, multigrid, octree meshes.
CITATION
Christian Dick, Joachim Georgii, Rüdiger Westermann, "A Hexahedral Multigrid Approach for Simulating Cuts in Deformable Objects", IEEE Transactions on Visualization & Computer Graphics, vol.17, no. 11, pp. 1663-1675, November 2011, doi:10.1109/TVCG.2010.268
REFERENCES
[1] A. Brandt, “Multi-Level Adaptive Solutions to Boundary-Value Problems,” Math. of Computation, vol. 31, no. 138, pp. 333-390, 1977.
[2] W. Hackbusch, Multi-Grid Methods and Applications: Springer Series in Computational Mathematics. Springer, 1985.
[3] W.L. Briggs, V.E. Henson, and S.F. McCormick, A Multigrid Tutorial, second ed. SIAM, 2000.
[4] D. Bielser, V.A. Maiwald, and M.H. Gross, “Interactive Cuts through 3-Dimensional Soft Tissue,” Computer Graphics Forum, vol. 18, no. 3, pp. 31-38, 1999.
[5] D. Bielser and M.H. Gross, “Interactive Simulation of Surgical Cuts,” Proc. Pacific Graphics, pp. 116-125, 2000.
[6] A.B. Mor and T. Kanade, “Modifying Soft Tissue Models: Progressive Cutting with Minimal New Element Creation,” Proc. Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 598-608, 2000.
[7] M. Wicke, M. Botsch, and M. Gross, “A Finite Element Method on Convex Polyhedra,” Computer Graphics Forum, vol. 26, no. 3, pp. 355-364, 2007.
[8] I. Babuška and J.M. Melenk, “The Partition of Unity Method,” Int'l J. Numerical Methods in Eng., vol. 40, no. 4, pp. 727-758, 1997.
[9] T. Strouboulis, K. Copps, and I. Babuška, “The Generalized Finite Element Method,” Computer Methods in Applied Mechanics and Eng., vol. 190, nos. 32/33, pp. 4081-4193, 2001.
[10] T. Belytschko and T. Black, “Elastic Crack Growth in Finite Elements with Minimal Remeshing,” Int'l J. Numerical Methods in Eng., vol. 45, no. 5, pp. 601-620, 1999.
[11] N. Moës, J. Dolbow, and T. Belytschko, “A Finite Element Method for Crack Growth without Remeshing,” Int'l J. Numerical Methods in Eng., vol. 46, no. 1, pp. 131-150, 1999.
[12] N. Sukumar, N. Moës, B. Moran, and T. Belytschko, “Extended Finite Element Method for Three-dimensional Crack Modelling,” Int'l J. Numerical Methods in Eng., vol. 48, no. 11, pp. 1549-1570, 2000.
[13] Y. Abdelaziz and A. Hamouine, “A Survey of the Extended Finite Element,” Computers and Structures, vol. 86, nos. 11/12, pp. 1141-1151, 2008.
[14] L. Jeřábková and T. Kuhlen, “Stable Cutting of Deformable Objects in Virtual Environments Using XFEM,” IEEE Computer Graphics and Applications, vol. 29, no. 2, pp. 61-71, Mar./Apr. 2009.
[15] P. Kaufmann, S. Martin, M. Botsch, E. Grinspun, and M. Gross, “Enrichment Textures for Detailed Cutting of Shells,” ACM Trans. Graphics, vol. 28, no. 3, pp. 50:1-50:10, 2009.
[16] G. Debunne, M. Desbrun, A.H. Barr, and M.-P. Cani, “Interactive Multiresolution Animation of Deformable Models,” Proc. Eurographics Workshop Computer Animation and Simulation, pp. 133-144, 1999.
[17] M. Müller and M. Gross, “Interactive Virtual Materials,” Proc. Graphics Interface, pp. 239-246, 2004.
[18] J. Georgii and R. Westermann, “Interactive Simulation and Rendering of Heterogeneous Deformable Bodies,” Proc. Vision, Modeling and Visualization, pp. 383-390, 2005.
[19] M. Botsch, M. Pauly, M. Wicke, and M. Gross, “Adaptive Space Deformations Based on Rigid Cells,” Computer Graphics Forum, vol. 26, no. 3, pp. 339-347, 2007.
[20] N. Pietroni, F. Ganovelli, P. Cignoni, and R. Scopigno, “Splitting Cubes: A Fast and Robust Technique for Virtual Cutting,” The Visual Computer, vol. 25, no. 3, pp. 227-239, 2009.
[21] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer, “Elastically Deformable Models,” Proc. ACM SIGGRAPH, pp. 205-214, 1987.
[22] D. Terzopoulos and K. Fleischer, “Modeling Inelastic Deformation: Viscoelasticity, Plasticity, Fracture,” Proc. ACM SIGGRAPH, pp. 269-278, 1988.
[23] A. Nealen, M. Müller, R. Keiser, E. Boxerman, and M. Carlson, “Physically Based Deformable Models in Computer Graphics,” Computer Graphics Forum, vol. 25, no. 4, pp. 809-836, 2006.
[24] D.L. James and D.K. Pai, “ArtDefo: Accurate Real Time Deformable Objects,” Proc. ACM SIGGRAPH, pp. 65-72, 1999.
[25] G. Debunne, M. Desbrun, M.-P. Cani, and A.H. Barr, “Dynamic Real-Time Deformations Using Space and Time Adaptive Sampling,” Proc. ACM SIGGRAPH, pp. 31-36, 2001.
[26] S. Capell, S. Green, B. Curless, T. Duchamp, and Z. Popović, “A Multiresolution Framework for Dynamic Deformations,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 41-47, 2002.
[27] E. Grinspun, P. Krysl, and P. Schröder, “CHARMS: A Simple Framework for Adaptive Simulation,” ACM Trans. Graphics, vol. 21, no. 3, pp. 281-290, 2002.
[28] M. Müller, B. Heidelberger, M. Teschner, and M. Gross, “Meshless Deformations Based on Shape Matching,” ACM Trans. Graphics, vol. 24, no. 3, pp. 471-478, 2005.
[29] E. Sifakis, T. Shinar, G. Irving, and R. Fedkiw, “Hybrid Simulation of Deformable Solids,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 81-90, 2007.
[30] M. Bro-Nielsen and S. Cotin, “Real-Time Volumetric Deformable Models for Surgery Simulation Using Finite Elements and Condensation,” Computer Graphics Forum, vol. 15, no. 3, pp. 57-66, 1996.
[31] X. Wu, M.S. Downes, T. Goktekin, and F. Tendick, “Adaptive Nonlinear Finite Elements for Deformable Body Simulation Using Dynamic Progressive Meshes,” Computer Graphics Forum, vol. 20, no. 3, pp. 349-358, 2001.
[32] J.F. O'Brien and J.K. Hodgins, “Graphical Modeling and Animation of Brittle Fracture,” Proc. ACM SIGGRAPH, pp. 137-146, 1999.
[33] J.F. O'Brien, A.W. Bargteil, and J.K. Hodgins, “Graphical Modeling and Animation of Ductile Fracture,” ACM Trans. Graphics, vol. 21, no. 3, pp. 291-294, 2002.
[34] M. Nesme, P.G. Kry, L. Jeřábková, and F. Faure, “Preserving Topology and Elasticity for Embedded Deformable Models,” ACM Trans. Graphics, vol. 28, no. 3, pp. 52:1-52:9, 2009.
[35] H.-W. Nienhuys and A.F. van der Stappen, “Combining Finite Element Deformation with Cutting for Surgery Simulations,” Proc. Eurographics—Short Presentations, pp. 43-52, 2000.
[36] S. Cotin, H. Delingette, and N. Ayache, “A Hybrid Elastic Model for Real-Time Cutting, Deformations, and Force Feedback for Surgery Training and Simulation,” The Visual Computer, vol. 16, no. 8, pp. 437-452, 2000.
[37] C. Forest, H. Delingette, and N. Ayache, “Removing Tetrahedra from a Manifold Mesh,” Proc. Computer Animation, pp. 225-229, 2002.
[38] H.-W. Nienhuys and A.F. van der Stappen, “A Surgery Simulation Supporting Cuts and Finite Element Deformation,” Proc. Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 145-152, 2001.
[39] D. Serby, M. Harders, and G. Székely, “A New Approach to Cutting into Finite Element Models,” Proc. Int'l Conf. Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 425-433, 2001.
[40] D. Steinemann, M.A. Otaduy, and M. Gross, “Fast Arbitrary Splitting of Deforming Objects,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 63-72, 2006.
[41] D. Bielser, P. Glardon, M. Teschner, and M. Gross, “A State Machine for Real-Time Cutting of Tetrahedral Meshes,” Proc. Pacific Graphics, pp. 377-386, 2003.
[42] F. Ganovelli, P. Cignoni, C. Montani, and R. Scopigno, “A Multiresolution Model for Soft Objects Supporting Interactive Cuts and Lacerations,” Computer Graphics Forum, vol. 19, no. 3, pp. 271-281, 2000.
[43] N. Molino, Z. Bao, and R. Fedkiw, “A Virtual Node Algorithm for Changing Mesh Topology During Simulation,” ACM Trans. Graphics, vol. 23, no. 3, pp. 385-392, 2004.
[44] S. Martin, P. Kaufmann, M. Botsch, M. Wicke, and M. Gross, “Polyhedral Finite Elements Using Harmonic Basis Functions,” Computer Graphics Forum, vol. 27, no. 5, pp. 1521-1529, 2008.
[45] E. Sifakis, K.G. Der, and R. Fedkiw, “Arbitrary Cutting of Deformable Tetrahedralized Objects,” Proc. ACM SIGGRAPH/Eurographics Symp. Computer Animation, pp. 73-80, 2007.
[46] S. Popinet, “Gerris: A Tree-Based Adaptive Solver for the Incompressible Euler Equations in Complex Geometries,” J. Computational Physics, vol. 190, no. 2, pp. 572-600, 2003.
[47] L. Shi and Y. Yu, “Visual Smoke Simulation with Adaptive Octree Refinement,” Proc. Computer Graphics and Imaging, pp. 13-19, 2004.
[48] F. Losasso, F. Gibou, and R. Fedkiw, “Simulating Water and Smoke with an Octree Data Structure,” ACM Trans. Graphics, vol. 23, no. 3, pp. 457-462, 2004.
[49] E. Haber and S. Heldmann, “An Octree Multigrid Method for Quasi-Static Maxwell's Equations with Highly Discontinuous Coefficients,” J. Computational Physics, vol. 223, no. 2, pp. 783-796, 2007.
[50] R.S. Sampath and G. Biros, “A Parallel Geometric Multigrid Method for Finite Elements on Octree Meshes,” SIAM J. Scientific Computing, vol. 32, no. 3, pp. 1361-1392, 2010.
[51] J. Bolz, I. Farmer, E. Grinspun, and P. Schröder, “Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid,” ACM Trans. Graphics, vol. 22, no. 3, pp. 917-924, 2003.
[52] X. Wu and F. Tendick, “Multigrid Integration for Interactive Deformable Body Simulation,” Proc. Int'l Symp. Medical Simulation, pp. 92-104, 2004.
[53] J. Georgii and R. Westermann, “A Multigrid Framework for Real-Time Simulation of Deformable Bodies,” Computer and Graphics, vol. 30, no. 3, pp. 408-415, 2006.
[54] L. Shi, Y. Yu, N. Bell, and W.-W. Feng, “A Fast Multigrid Algorithm for Mesh Deformation,” ACM Trans. Graphics, vol. 25, no. 3, pp. 1108-1117, 2006.
[55] M. Kazhdan and H. Hoppe, “Streaming Multigrid for Gradient-Domain Operations on Large Images,” ACM Trans. Graphics, vol. 27, no. 3, pp. 21:1-21:10, 2008.
[56] X. Jin, S. Chen, and X. Mao, “Computer-Generated Marbling Textures: A GPU-Based Design System,” IEEE Computer Graphics and Applications, vol. 27, no. 2, pp. 78-84, Mar./Apr. 2007.
[57] C. Dick, J. Georgii, R. Burgkart, and R. Westermann, “Computational Steering for Patient-Specific Implant Planning in Orthopedics,” Proc. Eurographics Workshop Visual Computing for Biomedicine, pp. 83-92, 2008.
[58] Y. Zhu, E. Sifakis, J. Teran, and A. Brandt, “An Efficient Multigrid Method for the Simulation of High-Resolution Elastic Solids,” ACM Trans. Graphics, vol. 29, no. 2, pp. 16:1-16:18, 2010.
[59] S.A. Sauter and R. Warnke, “Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients,” Computing, vol. 77, no. 1, pp. 29-55, 2006.
[60] T. Preusser, M. Rumpf, and L.O. Schwen, “Finite Element Simulation of Bone Microstructures,” Proc. 14th Workshop the Finite Element Method in Biomedical Eng., Biomechanics and Related Fields, pp. 52-66, 2007.
[61] F. Liehr, T. Preusser, M. Rumpf, S. Sauter, and L.O. Schwen, “Composite Finite Elements for 3D Image Based Computing,” Computing and Visualization in Science, vol. 12, no. 4, pp. 171-188, 2009.
[62] S.F. Frisken-Gibson, “Using Linked Volumes to Model Object Collisions, Deformation, Cutting, Carving, and Joining,” IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 4, pp. 333-348, Oct.-Dec. 1999.
[63] K.-J. Bathe, Finite Element Procedures. Prentice Hall, 2002.
[64] C.C. Rankin and F.A. Brogan, “An Element Independent Corotational Procedure for the Treatment of Large Rotations,” ASME J. Pressure Vessel Technology, vol. 108, no. 2, pp. 165-174, 1986.
[65] J. Georgii and R. Westermann, “Corotated Finite Elements Made Fast and Stable,” Proc. Workshop Virtual Reality Interactions and Physical Simulation, pp. 11-19, 2008.
[66] A. Lorusso, D.W. Eggert, and R.B. Fisher, “A Comparison of Four Algorithms for Estimating 3-D Rigid Transformations,” Proc. British Conf. Machine Vision, pp. 237-246, 1995.
[67] M.J. Aftosmis, M.J. Berger, and G. Adomavicius, “A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries, AIAA 2000-0808,” Proc. 38th AIAA Aerospace Sciences Meeting and Exhibit, 2000.
[68] L. Jeřábková, T. Kuhlen, T.P. Wolter, and N. Pallua, “A Voxel Based Multiresolution Technique for Soft Tissue Deformation,” Proc. ACM Symp. Virtual Reality Software and Technology, pp. 158-161, 2004.
[69] W. Wang, “Special Bilinear Quadrilateral Elements for Locally Refined Finite Element Grids,” SIAM J. Scientific Computing, vol. 22, no. 6, pp. 2029-2050, 2001.
[70] S. Toledo, D. Chen, V. Rotkin, and O. Meshar, “TAUCS: A Library of Sparse Linear Solvers,” http://www.tau.ac.il/~stoledotaucs, 2003.
[71] L. Kharevych, P. Mullen, H. Owhadi, and M. Desbrun, “Numerical Coarsening of Inhomogeneous Elastic Materials,” ACM Trans. Graphics, vol. 28, no. 3, pp. 51:1-51:8, 2009.
[72] M. Teschner, S. Kimmerle, B. Heidelberger, G. Zachmann, L. Raghupathi, A. Fuhrmann, M.-P. Cani, F. Faure, N. Magnenat-Thalmann, W. Strasser, and P. Volino, “Collision Detection for Deformable Objects,” Computer Graphics Forum, vol. 24, no. 1, pp. 61-81, 2005.
9 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool