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2021, 3(2): 156-170

Published Date:2021-4-20 DOI: 10.1016/j.vrih.2021.01.002

A homogenization method for nonlinear inhomogeneous elastic materials

Abstract

Background
Fast simulation techniques are strongly favored in computer graphics, especially for the nonlinear inhomogeneous elastic materials. The homogenization theory is a perfect match to simulate inhomogeneous deformable objects with its coarse discretization, as it reveals how to extract information at a fine scale and to perform efficient computation with much less DOF. The existing homogenization method is not applicable for ubiquitous nonlinear materials with the limited input deformation displacements.
Methods
In this paper, we have proposed a homogenization method for the efficient simulation of nonlinear inhomogeneous elastic materials. Our approach allows for a faithful approximation of fine, heterogeneous nonlinear materials with very coarse discretization. Modal analysis provides the basis of a linear deformation space and modal derivatives extend the space to a nonlinear regime; based on this, we exploited modal derivatives as the input characteristic deformations for homogenization. We also present a simple elastic material model that is nonlinear and anisotropic to represent the homogenized materials. The nonlinearity of material deformations can be represented properly with this model. The material properties for the coarsened model were solved via a constrained optimization that minimizes the weighted sum of the strain energy deviations for all input deformation modes. An arbitrary number of bases can be used as inputs for homogenization, and greater weights are placed on the more important low-frequency modes.
Results
Based on the experimental results, this study illustrates that the homogenized material properties obtained from our method approximate the original nonlinear material behavior much better than the existing homogenization method with linear displacements, and saves orders of magnitude of computational time.
Conclusions
The proposed homogenization method for nonlinear inhomogeneous elastic materials is capable of capturing the nonlinear dynamics of the original dynamical system well.

Keyword

Physical-based simulation ; Homogenization theory ; Heterogeneous material ; Modal basis

Cite this article

Jing ZHAO, Fei ZHU, Liyou XU, Yong TANG, Sheng LI. A homogenization method for nonlinear inhomogeneous elastic materials. Virtual Reality & Intelligent Hardware, 2021, 3(2): 156-170 DOI:10.1016/j.vrih.2021.01.002

References

1. Terzopoulos D, Platt J, Barr A, Fleischer K. Elastically deformable models. ACM SIGGRAPH Computer Graphics, 1987, 21(4): 205–214 DOI:10.1145/37402.37427

2. Gloria A. Numerical homogenization: survey, new results, and perspectives. ESAIM: Proceedings, 2012, 37: 50–116 DOI:10.1051/proc/201237002

3. Kharevych L, Mullen P, Owhadi H, Desbrun M. Numerical coarsening of inhomogeneous elastic materials. In: ACM SIGGRAPH 2009 papers on-SIGGRAPH '09. New Orleans, Louisiana, New York, ACM Press, 2009, 28(3): 51:1–51:8 DOI:10.1145/1576246.1531357

4. Sifakis E, Barbic J. FEM simulation of 3D deformable solids: a practitioner's guide to theory, discretization and model reduction. In: ACM SIGGRAPH 2012 Posters on-SIGGRAPH '12. Los Angeles, California, New York, ACM Press, 2012, 1–50 DOI:10.1145/2343483.2343501

5. Barbič J, James D L. Real-Time subspace integration for St. Venant-Kirchhoff deformable models. SIGGRAPH '05: ACM SIGGRAPH 2005 Papers, 2005, 982–990 DOI:10.1145/1186822.1073300

6. Nealen A, Müller M, Keiser R, Boxerman E, Carlson M. Physically based deformable models in computer graphics. Computer Graphics Forum, 2006, 25(4): 809–836 DOI:10.1111/j.1467-8659.2006.01000.x

7. Gast T F, Schroeder C, Stomakhin A, Jiang C, Teran J M. Optimization integrator for large time steps. IEEE Transactions on Visualization and Computer Graphics, 2015, 21(10): 1103–1115 DOI:10.1109/tvcg.2015.2459687

8. Müller M, Dorsey J, McMillan L, Jagnow R, Cutler B. Stable real-time deformations. In: Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation-SCA'02. San Antonio, Texas, New York, ACM Press, 2002, 49–54 DOI:10.1145/545261.545269

9. Müller M, Gross M. Interactive virtual materials. Proceedings of Graphics Interface. London, 2004, 239–246

10. Georgii J, Westermann R. Corotated finite elements made fast and stable. Proceedings of the 5th Workshop on Virtual Reality Interactions and Pysical Simulations. VRIPHYS 2008, Grenoble, France, 2008,11–19

11. James D L, Pai D K. Real time simulation of multizone elastokinematic models. In: Proceedings 2002 IEEE International Conference on Robotics and Automation. Washington, DC, USA, IEEE, 2002, 927–932 DOI:10.1109/robot.2002.1013475

12. James D L, Pai D K. Multiresolution Green's function methods for interactive simulation of large-scale elastostatic objects. ACM Transactions on Graphics, 2003, 22(1): 47–82 DOI:10.1145/588272.588278

13. Capell S, Green S, Curless B, Duchamp T, Popović Z. Interactive skeleton-driven dynamic deformations. ACM Transactions on Graphics, 2002, 21(3): 586–593 DOI:10.1145/566654.566622

14. Teschner M, Heidelberger B, Muller M, Gross M. A versatile and robust model for geometrically complex deformable solids. In: Proceedings Computer Graphics International. Crete, Greece, IEEE, 2004, 312–319 DOI:10.1109/cgi.2004.1309227

15. Choi M G, Ko H S. Modal warping: real-time simulation of large rotational deformation and manipulation. IEEE Transactions on Visualization and Computer Graphics, 2005, 11(1): 91–101 DOI:10.1109/tvcg.2005.13

16. Barbič J, Zhao Y. Real-time large-deformation substructuring. Acm Transactions on Graphics, 2011, 30(4):91:1–91:8 DOI:10.1145/1964921.1964986

17. Hildebrandt K, Schulz C, von Tycowicz C, Polthier K. Interactive spacetime control of deformable objects. ACM Transactions on Graphics, 2012, 31(4): 71 DOI:10.1145/2185520.2185567

18. Harmon D, Zorin D. Subspace integration with local deformations. ACM Transactions on Graphics, 2013, 32(4): 1–10 DOI:10.1145/2461912.2461922

19. von Tycowicz C, Schulz C, Seidel H P, Hildebrandt K. An efficient construction of reduced deformable objects. ACM Transactions on Graphics, 2013, 32(6): 213 DOI:10.1145/2508363.2508392

20. Nesme M, Payan Y, Faure F. Animating shapes at arbitrary resolution with non-uniform stiffness. Proceedings of the 3rd Workshop in Virtual Reality Interaction and Physical Simulation. Aire-la-Ville: Eurographics Association Press, 2006,1–8

21. Nesme M, Kry P G, Jeřábková L, Faure F. Preserving topology and elasticity for embedded deformable models. ACM Transactions on Graphics, 2009, 28(3): 1–9 DOI:10.1145/1531326.1531358

22. Bickel B, Bächer M, Otaduy M A, Matusik W, Pfister H, Gross M. Capture and modeling of non-linear heterogeneous soft tissue. ACM Transactions on Graphics, 2009, 28(3): 1–9 DOI:10.1145/1531326.1531395

23. Faure F, Gilles B, Bousquet G, Pai D K. Sparse meshless models of complex deformable solids. In: ACM SIGGRAPH 2011 papers on-SIGGRAPH '11. Vancouver, British Columbia, Canada, New York, ACM Press, 2011, 30(4):76–79 DOI:10.1145/1964921.1964968

24. Chen D S, Levin D I W, Sueda S, Matusik W. Data-driven finite elements for geometry and material design. ACM Transactions on Graphics, 2015, 34(4): 1–10 DOI:10.1145/2766889

25. Chen D S, Levin D I W, Matusik W, Kaufman D M. Dynamics-aware numerical coarsening for fabrication design. ACM Transactions on Graphics, 2017, 36(4): 1–15 DOI:10.1145/3072959.3073669

26. Chen J, Bao H J, Wang T, Desbrun M, Huang J. Numerical coarsening using discontinuous shape functions. ACM Transactions on Graphics, 2018, 37(4): 1–12 DOI:10.1145/3197517.3201386

27. Chen J, Budninskiy M, Owhadi H, Bao H J, Huang J, Desbrun M. Material-adapted refinable basis functions for elasticity simulation. ACM Transactions on Graphics, 2019, 38(6): 161 DOI:10.1145/3355089.3356567

28. Johnson S G. The NLopt nonlinear-optimization package. http://ab-initio.mit.edu/nolpt

29. Stuart D A, Levine J A, Jones B, Bargteil A W. Automatic construction of coarse, high-quality tetrahedralizations that enclose and approximate surfaces for animation. In: Proceedings of Motion on Games-MIG '13. Dublin 2, Ireland, New York, ACM Press, 2019 DOI:10.1145/2522628.2522648

30. Barbič J, Sin F S, Schroeder D. VegaFEM Library, http://www.jernejbarbic.com/vega, 2012