Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 1069–1090 | Cite as

Position-based dynamic of a particle system: a configurable algorithm to describe complex behaviour of continuum material starting from swarm robotics

  • Ramiro dell’ErbaEmail author
Original Article


In a previous work, we considered a two-dimensional lattice of particles and calculated its time evolution by using an interaction law based on the spatial position of the particles themselves. The model reproduced the behaviour of deformable bodies both according to the standard Cauchy model and second gradient theory; this success led us to use this method in more complex cases. This work is intended as the natural evolution of the previous one in which we shall consider both energy aspects, coherence with the principle of Saint Venant and we start to manage a more general tool that can be adapted to different physical phenomena, supporting complex effects like lateral contraction, anisotropy or elastoplasticity.


Discrete mechanical systems Second gradient continua Fracture 


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  1. 1.
    Bender, J., Müller, M., Macklin, M.: Position-based simulation methods in computer graphics. In: Eurographics 2015 Tutorials, Eurographics Association, Zurich, Switzerland (2015)Google Scholar
  2. 2.
    Bender, J., Koschier, D., Charrier, P., Weber, D.: Position-based simulation of continuous materials. Comput. Graph. 44, 1–10 (2014)CrossRefGoogle Scholar
  3. 3.
    Rivers, A.R., James, D.: FastLSM: fast lattice shape matching for robust real-time deformation. ACM Trans. Graph. 26, 82 (2007). CrossRefGoogle Scholar
  4. 4.
    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, New York, NY, USA, pp. 237–246 (2011)Google Scholar
  5. 5.
    Macklin, M., Müller, M., Chentanez, N.: XPBD: position-based simulation of compliant constrained dynamics. In: Proceeding MIG ’16 Proceedings of the 9th International Conference on Motion in Games, pp. 49–54 (2016).
  6. 6.
    Battista, A., Rosa, L., dell’Erba, R., Greco, L.: Numerical investigation of a particle system compared with first and second gradient continua: deformation and fracture phenomena*. Math. Mech. Solids, p. 1081286516657889, lug (2016)Google Scholar
  7. 7.
    Della Corte, A., Battista, A., dell’Isola, F.: Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non Linear Mech. 80, 209–220 (2016)ADSCrossRefGoogle Scholar
  8. 8.
    Della Corte, A., Battista, A., dell’Isola, F., Giorgio, I.: Modeling deformable bodies using discrete systems with centroid-based propagating interaction: fracture and crack evolution. Mathematical Modelling in Solid Mechanics Volume 69 of the series Advanced Structured Materials, pp. 59–88 (2017)Google Scholar
  9. 9.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, vol. 159. Springer, Berlin (2013)zbMATHGoogle Scholar
  10. 10.
    Greco, L., Cuomo, M.: B-spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Greco, L., Cuomo, M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Contrafatto, L., Cuomo, M., Fazio, F.: An enriched finite element for crack opening and rebar slip in reinforced concrete members. Int. J. Fract. 178, 33–50 (2012)CrossRefGoogle Scholar
  13. 13.
    dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on the static response and wave propagation. Proc. Est. Acad. Sci. 64(3), 219 (2015)CrossRefGoogle Scholar
  14. 14.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21(5), 562–577 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. ZAMM J. Appl. Math. Mech. 96(10), 1220–1244 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bilotta, A., Turco, E.: A numerical study on the solution of the Cauchy problem in elasticity. Int. J. Solids Struct. 46(25), 4451–4477 (2009)CrossRefzbMATHGoogle Scholar
  17. 17.
    Placidi, L., Greve, R., Seddik, H., Faria, S.H.: Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor. Continuum Mech. Thermodyn. 22(3), 221–237 (2010)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17–21 March 2014 generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids 22(2), 144–157 (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM J. Appl. Math. Mech. 89(4), 242–256 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2012)zbMATHGoogle Scholar
  23. 23.
    Altenbach, H., Eremeyev, V.A.: Cosserat-type shells. In Generalized continua from the theory to engineering applications. Springer, Vienna, pp. 131–178 (2013)Google Scholar
  24. 24.
    Altenbach, H., Eremeyev, V.A. (eds.): Generalized Continua-from the Theory to Engineering Applications, vol. 541. Springer, Berlin (2012)Google Scholar
  25. 25.
    Altenbach, H., Bîrsan, M., Eremeyev, V.A.: Cosserat-type rods. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 541, pp. 179–248. Springer, Vienna (2013)CrossRefGoogle Scholar
  26. 26.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of anisotropic Cosserat continuum. Gen. Contin. Models Mater. 10 (2012)Google Scholar
  27. 27.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49(14), 1993–2005 (2012)CrossRefGoogle Scholar
  28. 28.
    Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)CrossRefGoogle Scholar
  29. 29.
    Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 05, 1–16 (2017)Google Scholar
  30. 30.
    Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. B Eng. 115, 423–448 (2017)CrossRefGoogle Scholar
  31. 31.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 85 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    dell’Isola, F., Madeo, A., Seppecher, P.: Cauchy tetrahedron argument applied to higher contact interactions. Arch. Ration. Mech. Anal. 219(3), 1305–1341 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471(2183), 20150415 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61, 21 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Alibert, J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Forest, S., Cordero, N.M., Busso, E.P.: First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput. Mater. Sci. 50(4), 1299–1304 (2011)Google Scholar
  37. 37.
    Placidi, L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. 27(4–5), 623 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rosi, G., Giorgio, I., Eremeyev, V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM J. Appl. Math. Mech. 93(12), 914–927 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola. Mech. Math. Solids (MMS) 20(8), 887–928. (Published online beforeprintFebruary 2, 2014) (2015)Google Scholar
  40. 40.
    Lanczos, C.: The Variational Principles of Mechanics. Courier Corporation, North Chelmsford (2012)zbMATHGoogle Scholar
  41. 41.
    Placidi, L., dell’Isola, F., Ianiro, N., Sciarra, G.: Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A Solids 27(4), 582–606 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    dell’Isola, F., Placidi, L.: Variational principles are a powerful tool also for formulating field theories. In Variational Models and Methods in Solid and Fluid Mechanics. Springer, Vienna, pp. 1–15 (2011)Google Scholar
  43. 43.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRefGoogle Scholar
  44. 44.
    dell’Isola, F., Gavrilyuk, S.L. (eds.): Variational Models and Methods in Solid and Fluid Mechanics, vol. 535. Springer, Berlin (2012)Google Scholar
  45. 45.
    dell’Isola, F., Auffray, N., Eremeyev, V.A., Madeo, A., Placidi, L., Rosi, G.: Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola’s point of view. In The Complete Works of Gabrio Piola, vol. I, pp. 606–694. Springer (2014)Google Scholar
  46. 46.
    Ladevèze, P.: Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation. Springer, Berlin (2012)Google Scholar
  47. 47.
    Steigmann, D.J.: Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity. J. Elast. 111(1), 91–107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Steigmann, D.J.: A concise derivation of membrane theory from three-dimensional nonlinear elasticity. J. Elast. 97(1), 97–101 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Steigmann, D.J.: Applications of polyconvexity and strong ellipticity to nonlinear elasticity and elastic plate theory. CISM Course Appl. Poly Quasi Rank One Convexity Appl. Mech. 516, 265–299 (2010)CrossRefGoogle Scholar
  50. 50.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 20150790 (2016)ADSCrossRefGoogle Scholar
  51. 51.
    Della Corte, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations. Math. Models Methods Appl. Sci. 27(08), 1391–1421 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Gabriele, S., Rizzi, N.L., Varano, V.: A one-dimensional nonlinear thin walled beam model derived from Koiter shell theory. In: Topping, B.H.V., Iványi, P. (eds.) Proceedings of the Twelfth International Conference on Computational Structures Technology, Civil-Comp Press, Stirlingshire, UK, Paper 156 (2014).
  53. 53.
    dell’Erba, R.: Determination of spatial configuration of an underwater swarm with minimum data. Int. J. Adv. Robot. Syst. 12, 97 (2015)CrossRefGoogle Scholar
  54. 54.
    Moriconi, C., dell’Erba, R.: The localization problem for harness: a multipurpose robotic swarm. In: SENSORCOMM 2012, The Sixth International Conference on Sensor Technologies and Applications, pp. 327–333 (2012)Google Scholar
  55. 55.
    Karaboga, D.: An idea based on honey bee swarm for numerical optimization. Technical report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005)Google Scholar
  56. 56.
    Passino, K.M., Seeley, T.D., Visscher, P.K.: Swarm cognition in honey bees. Behav. Ecol. Sociobiol. 62(3), 401–414 (2007)CrossRefGoogle Scholar
  57. 57.
    Janson, S., Middendorf, M., Beekman, M.: Honeybee swarms: how do scouts guide a swarm of uninformed bees? Anim. Behav. 70(2), 349–358 (2005)CrossRefGoogle Scholar
  58. 58.
    Khatib, O., Kumar, V., Rus, D.: Experimental Robotics: The 10th International Symposium on Experimental Robotics. Springer, Berlin (2008)CrossRefGoogle Scholar
  59. 59.
    Dos Reis, F., Ganghoffer, J.F.: Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput. Mater. Sci. 51(1), 314–321 (2012)CrossRefGoogle Scholar
  60. 60.
    Dos Reis, F., Ganghoffer, J.F.: Construction of micropolar continua from the asymptotic homogenization of beam lattices. Comput. Struct. 112, 354–363 (2012)CrossRefGoogle Scholar
  61. 61.
    Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Goda, I., Assidi, M., Ganghoffer, J.F.: Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization. J. Mech. Phys. Solids 61(12), 2537–2565 (2013)ADSCrossRefGoogle Scholar
  63. 63.
    Alibert, J.J., Della Corte, A., Giorgio, I., Battista, A.: Extensional Elastica in large deformation as\(\backslash \)Gamma-limit of a discrete 1D mechanical system. Zeitschrift für angewandte Mathematik und Physik 68(2), 42 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Alibert, J.J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für angewandte Mathematik und Physik 66(5), 2855–2870 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Keaveny, T.M., Morgan, E.F., Yeh, O.C.: Bone Mechanics. Biomedical Engineering and Design Handbook, pp. 221–243. McGraw-Hill, New York (2009)Google Scholar
  66. 66.
    Andreaus, U., Giorgio, I., Lekszycki, T.: A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time. ZAMM J. Appl. Math. Mech. 94(12), 978–1000 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Andreaus, U., Colloca, M., Iacoviello, D.: An optimal control procedure for bone adaptation under mechanical stimulus. Control Eng. Pract. 20(6), 575–583 (2012)CrossRefGoogle Scholar
  68. 68.
    Andreaus, U., Colloca, M., Toscano, A.: Mechanical behaviour of a prosthesized human femur: a comparative analysis between walking and stair climbing by using the finite element method. Biophys. Bioeng. Lett. 1(3), 1–15 (2008)Google Scholar
  69. 69.
    Lekszycki, T., dell’Isola, F.: A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM J. Appl. Math. Mech. 92(6), 426–444 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Giorgio, I., Andreaus, U., Scerrato, D., dell’Isola, F.: A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials. Biomech. Model. Mechanobiol. 15(5), 1325–1343 (2016)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ENEA Technical Unit Technologies for Energy and Industry – Robotics LaboratoryRomeItaly

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