Abstract
Mass-spring systems (MSS) simulating elastic materials obey constraints known in elasticity as the Cauchy relations, restricting the Poisson ratio of isotropic systems to be exactly \(\nu =1/4\). We remind that this limitation is intrinsic to centrosymmetric spring systems (where each node is a center of symmetry), forbidding them for instance to simulate incompressible materials (with \(\nu =1/2\)). To overcome this restriction, we propose to supplement the spring deformation energy with an energy depending on the volume only, insensitive to change of shape, permitting MSS to simulate any real isotropic materials. In addition, the freedom in choosing the spring constants realizing a given elastic behavior allows to manage instabilities. The proposed hybrid model is evaluated by comparing its response to various deformation geometries with analytical model and/or finite element model. The results show that the hybrid MSS model allows to simulate any compressible isotropic elastic material and in particular the nearly incompressible (Poisson ratio \(\nu \simeq 1/2\)) biological soft tissues to which it is dedicated.
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Arnab, S., Raja, V.: Chapter 4: Simulating a deformable object using a surface mass spring system. In: 2008 3rd international conference on geometric modeling and imaging, 21–26 (2008). https://doi.org/10.1109/GMAI.2008.24
Baraff, D., Witkin, A.: Large steps in cloth simulation. In: Proceedings of the 25th Annual conference on computer graphics and interactive techniques, SIGGRAPH ’98, pp. 43–54. ACM, New York (1998). https://doi.org/10.1145/280814.280821
Baudet, V., Beuve, M., Jaillet, F., Shariat, B., Zara, F.: Integrating tensile parameters in Hexahedral mass-spring system for simulation. In: International Conference on Computer Graphics, Visualization and Computer Vision’2009, WSCG’2009, Plzen, Czech Republic, pp. 145–152 (2009)
Bender, J., Müller, M., Otaduy M. A., Teschner, M.: Position-based methods for the simulation of solid objects in computer graphics. In: STAR Proceedings of Eurographics, (2013)
Born, M., Huang, K.: The Dynamical Theory of Crystal Lattices. Oxford University Press, London (1954)
Bourguignon, D., Cani, M.P.: Controlling Anisotropy in Mass-Spring Systems, pp. 113–123. Springer, Vienna (2000)
Bridson, R., Fedkiw, R., Anderson, J.: Robust treatment of collisions, contact and friction for cloth animation. ACM Trans. Graph. 21(3), 594–603 (2002)
Cauchy, A.L.: De la pression ou tension dans un système de points matériels. Œuvres complètes vol. série 2, tome 8, 1882–1974
Cauchy, A.L.: Sur l’équilibre et le mouvement d’un système de points matériels sollicités par des forces d’attraction ou de répuslion mutuelle. Exercices de Mathématiques 3, 1882–1974 (1828). https://gallica.bnf.fr/ark:/12148/bpt6k90200c/f226
Chen, Y., Zhu, Q., Kaufman, A.: Physically-based animation of volumetric objects. In: Proceedings of IEEE computer animation ’98, pp. 154–160 (1998)
de Saint-Venant, A.J.C.B.: De la torsion des prismes, avec des considérations sur leur flexion ainsi que sur l’équililbre des solides élastiques en général et des formules pratiques pour le calcul de leur résistance à divers efforts s’exerçant simultanément. No. 14 in Mémoires présentés par divers savants à l’Académie des Sciences de l’Institut Impérial de France. Académie des Sciences de l’Institut Impérial de France (1855). https://gallica.bnf.fr/ark:/12148/bpt6k99739z/f1.image
Diziol, R., Bender, J., Bayer, D.: Robust real-time deformation of incompressible surface meshes. In: Proceedings of the 2011 ACM SIGGRAPH/Eurographics symposium on computer animation, SCA ’11, pp. 237–246. ACM, New York (2011)
Duan, Y., Huang, W., Chang, H., Chen, W., Zhou, J., Teo, S.K., Su, Y., Chui, C., Chang, S.: Volume preserved mass-spring model with novel constraints for soft tissue deformation. IEEE J. Biomed. Health Inform. 2194(c), 1–12 (2014). https://doi.org/10.1109/JBHI.2014.2370059
Elcoro, L., Etxebarria, J.: Common misconceptions about the dynamical theory of crystal lattices: Cauchy relations, lattice potentials and infinite crystals. Eur. J. Phys. 32, 25–35 (2011). https://doi.org/10.1088/0143-0807/32/1/003
Gelder, A.V.: Approximate simulation of elastic membranes by triangulated spring meshes. J. Graph. Tools 3(2), 21–41 (1998). https://doi.org/10.1080/10867651.1998.10487490
Hallquist, J.O.: LS-DYNA Theory manual (2006) http://www.lstc.com/pdf/ls-dyna_theory_manual_2006.pdf
Jarrousse, O., Fritz, T., Dössel, O.: Implicit Time Integration in a Volumetric Mass-Spring System for Modeling Myocardial Elastomechanics, pp. 876–879. Springer, Berlin (2010)
Keating, P.N.: Effect of invariance requirements on the elastic strain energy of crystals with applications to the diamond structure. Phys. Rev. 145, 637–645 (1966). https://doi.org/10.1103/PhysRev.145.637
Keating, P.N.: Relationship between the macroscopic and microscopic theory of crystal elasticity I. Primitive crystals. Phys. Rev. 152, 774–779 (1966). https://doi.org/10.1103/PhysRev.152.774
Kirkwood, J.G.: The Skeletal modes of vibration of long chain molecules. J. Chem. Phys. 7, 506–509 (1939). https://doi.org/10.1063/1.1750479
Kot, M., Nagahashi, H.: Second degree of freedom of elastic objects - adjustable Poisson’s ratio for mass spring models. In: GRAPP 2015 - Proceedings, Berlin, Germany, 11–14 March, 2015, pp. 138–142. SciTePress (2015). https://doi.org/10.5220/0005303601380142
Kot, M., Nagahashi, H.: Mass spring models with adjustable Poisson’s ratio. Vis. Comput. 33(3), 283–291 (2017). https://doi.org/10.1007/s00371-015-1194-8
Kot, M., Nagahashi, H., Szymczak, P.: Elastic moduli of simple mass spring models. Vis. Comput. 31(10), 1339–1350 (2015). https://doi.org/10.1007/s00371-014-1015-5
Lifshitz, E., Kosevich, A., Pitaevskii, L.: Theory of Elasticity, 3rd edn. Butterworth-Heinemann, Oxford (1986)
Lloyd, B.A., Székely, G., Harders, M.: Identification of spring parameters for deformable object simulation. IEEE Trans. Vis. Comput. Graph. 13(1), 1081–1093 (2007). https://doi.org/10.1109/TVCG.2007.1055
Marchal, M.: Soft tissue modeling for computer assisted medical interventions. Theses, Université Joseph-Fourier - Grenoble I (2006). https://tel.archives-ouvertes.fr/tel-00129430
Mollemans, W., Schutyser, F., Van Cleynenbreugel, J., Suetens, P.: Tetrahedral Mass Spring Model for Fast Soft Tissue Deformation, pp. 145–154. Springer, Berlin (2003)
Natsupakpong, S., Çavuşoğlu, M.: Cenk: Determination of elasticity parameters in lumped element (mass-spring) models of deformable objects. Graph. Models 72(6), 61–73 (2010). https://doi.org/10.1016/j.gmod.2010.10.001
Niiranen, J.: Fast and accurate symmetric Euler algorithm for electromechanical simulations. IMACS 1, 71–78 (1999)
Ostoja-Starzewski, M., Sheng, P.Y., Alzebdeh, K.: Spring network models in elasticity and fracture of composites and polycrystals. Comput. Mater. Sci. 7, 82–93 (1996)
Sahimi, M., Arbabi, S.: Mechanics of disordered solids. II. Percolation on elastic networks with bond-bending forces. Phys. Rev. B 47, 703–712 (1993). https://doi.org/10.1103/PhysRevB.47.703
San-Vicente, G., Aguinaga, I., Tomás Celigüeta, J.: Cubical mass-spring model design based on a tensile deformation test and nonlinear material model. IEEE Trans. Vis. Comput. Graph. 18(2), 228–41 (2012). https://doi.org/10.1109/TVCG.2011.32
Terzopoulos, D., Platt, J., Fleischer, K.: Heating and melting deformable models. J. Vis. Comput. Animat. 2(2), 68–73 (1991)
Todhunter, I.: A History of the Theory of Elasticity and of the Strength of Materials: From Galilei to the Present Time. Cambridge Library Collection - Mathematics. Cambridge University Press, Cambridge (2014). https://doi.org/10.1017/CBO9781107280076
Vincente-Otamendi, G.S.: Designing deformable models of soft tissue for virtual surgery planning and simulation using the Mass-Spring Model. Ph.D. thesis, Universidad de Navarra (2011)
Acknowledgements
This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The authors would like to thank for support of E. Flechon for providing the original version of TopoSim (software implementing the LCC+MSS model).
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Appendix A. Theoretical notions of linear elasticity
Appendix A. Theoretical notions of linear elasticity
1.1 The multi-constant theory of elasticity
Consider a deformation that brings a material point from its rest position \(\mathbf {x}\in \mathbb {R}^3\) to a new position \(\mathbf {x}+\mathbf {u}(\mathbf {x})\). Obviously, a homogeneous displacement \(\mathbf {u}(\mathbf {x})=cst\) amounts to a uniform translation, generating no stress in the material, so the lowest-order quantity of interest in elasticity is the displacement gradient \({\varvec{\nabla }}\mathbf {u}\), the symmetric part of which is called the strain tensor defined by:
In linear elasticity, the deformation energy is a quadratic function of the deformation tensor, its volume density being written
where the fourth-rank tensor C is the stiffness tensor. [We use the implicit summation convention of repeated Cartesian indices i, j, k and l over the values 1, 2 and 3.] The antisymmetric part of \({\varvec{\nabla }}\mathbf {u}\), which corresponds to solid-body rotations, does not enter expression (7.2) since the deformation energy is invariant with respect to such rotations. Moreover, since \(\varepsilon _{ij}=\varepsilon _{ji}\), one has
The stress tensor \({\varvec{\sigma }}\) derives from the energy:
the component \(\sigma _{ij}\) being the elastic force along the i-direction transmitted across a unit area of a surface normal to the j-direction. This proportionality relation between stress and strain is Hooke’s law.
1.2 Relations for an isotropic material
Among the \(3^4=81\) components of the stiffness tensor C, only 21 are independent in the general case due to relations (7.3) [24]. Symmetries of the material further reduce this number; the most symmetric materials being isotropic, in which case \(C^{(iso)}\) is independent of the orientation of the reference axes and can be written
in terms of the two Lamé constants \(\lambda \) and \(G\), where \(G\) is named the shear modulus (the relation between stiffness tensor and material symmetry is derived in [24].) The deformation energy of an isotropic material then reads
and the stress tensor is
Besides, an alternative set of elastic constants, widely used to wholly describe an isotropic elastic material, is the Young modulus E and the Poisson ratio \(\nu \) which are related to Lamé parameters by:
In particular, they are most useful in uniaxial stress situations: in a stress state such that the only nonzero stress component is \(\sigma _{xx}^{(iso)}\), the nonzero strain components are \(\varepsilon _{xx}=\sigma ^{(iso)}_{xx}/E\) and \(\varepsilon _{zz}=\varepsilon _{yy}=-\nu \,\varepsilon _{xx}\).
A last elastic constant of interest is the bulk modulus
such that, in an uniform dilation \(\varepsilon _{ij}=\frac{1}{3}\delta _{ij}\varDelta V/V\) changing volume V by \(\varDelta V=V\varepsilon _{kk}\), the stress is \(\sigma ^{(iso)}_{ij}=K\varDelta V/V\delta _{ij}=K\varepsilon _{kk}\delta _{ij}\).
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Golec, K., Palierne, JF., Zara, F. et al. Hybrid 3D mass-spring system for simulation of isotropic materials with any Poisson’s ratio. Vis Comput 36, 809–825 (2020). https://doi.org/10.1007/s00371-019-01663-0
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DOI: https://doi.org/10.1007/s00371-019-01663-0