Research Article
A Hamiltonian particle method for non-linear elastodynamics
Yukihito Suzuki,
Seiichi Koshizuka,
Corresponding Author
Yukihito Suzuki
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, JapanSearch for more papers by this authorSeiichi Koshizuka
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Search for more papers by this authorYukihito Suzuki,
Seiichi Koshizuka,
Corresponding Author
Yukihito Suzuki
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, JapanSearch for more papers by this authorSeiichi Koshizuka
Department of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Search for more papers by this authorAbstract
Particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles. They have a great degree of flexibility, for instance, in dealing with complex large deformations or the fragmentation of solids. In this paper, a particle method for non-linear elastodynamics of compressible and incompressible materials is developed based on a discretization of the Lagrangian from which the governing equations of elastodynamics are derived using the principle of least action. The discretized Lagrangian leads to a finite-dimensional Hamiltonian system via the Legendre transformation. If the material is incompressible, the Hamiltonian system is accompanied by holonomic constraints. Depending on whether the material is compressible or incompressible, the symplectic scheme adopted for numerical time integration is either the Störmer/Verlet scheme or the RATTLE method, respectively. The resulting particle method inherits the symplectic structure possessed by the governing equations of elastodynamics. In the case of incompressible materials, incompressibility is strictly enforced at each time step. Some numerical tests indicate the excellence of the method for conservation of mechanical energy besides that of linear and angular momenta. Copyright © 2007 John Wiley & Sons, Ltd.
REFERENCES
- 1 Lucy LB. A numerical approach to the testing of fission hypothesis. The Astronomical Journal 1977; 82: 1013–1024.
- 2 Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society 1977; 181: 1013–1024.
10.1093/mnras/181.3.375 Google Scholar
- 3 Koshizuka S, Tamako H, Oka Y. A particle method for incompressible viscous flow with fluid fragmentation. Computational Fluid Dynamics Journal 1995; 4: 29–46.
- 4 Koshizuka S, Oka Y. Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nuclear Science and Engineering 1996; 123: 421–434.
- 5 Yserentant H. A new class of particle methods. Numerische Mathematik 1997; 76: 87–109.
- 6 Gauger C, Leinen P, Yserentant H. The finite mass method. SIAM Journal on Numerical Analysis 2000; 37: 1768–1799.
- 7 Belytschko T, Lu YY, Gu L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering 1994; 37: 229–256.
- 8 Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids 1995; 20: 1081–1106.
- 9 Liu WK, Li S, Belytschko T. Moving least-square reproducing kernel methods (I). Methodology and convergence. Computer Methods in Applied Mechanics and Engineering 1997; 143: 1081–1106.
- 10 Duarte CA, Oden JT. An h–p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering 1996; 139: 237–262.
- 11 Melenk JM, Babuska I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139: 289–314.
- 12 Li S, Liu WK. Meshfree Particle Methods. Springer: Berlin, 2004.
- 13 Libersky LD, Petschek AG, Carney TC, Hipp JR, Allahdadi FA. High strain Lagrangian hydrodynamics, a three-dimensional SPH code for dynamic material response. Journal of Computational Physics 1993; 109: 67–75.
- 14 Song MS, Koshizuka S, Oka Y. A particle method for dynamic simulation of elastic solids. Proceedings of the 6th World Congress on Computational Mechanics (WCCM VI), Beijing, 5–10 September 2004.
- 15 Marsden JE, Hughes TJR. Mathematical Foundations of Elasticity. Prentice Hall: Englewood Cliffs, NJ, 1983.
- 16 Beris AN, Edwards BJ. Thermodynamics of Flowing Systems. Oxford University Press: New York, 1994.
- 17 Gingold RA, Monaghan JJ. Kernel estimates as a basis for general particle methods in hydrodynamics. Journal of Computational Physics 1982; 46: 429–453.
- 18 Rasio FA. Particle methods in astrophysical fluid dynamics. In Proceedings of the 5th International Conference on Computing in Physics (ICCP5), Kanazawa, Japan, 11–13 October 1999, Hiwatari Y et al. (eds). Progress of Theoretical Physics, Supplement No. 138, 2000; 609–621.
- 19 Inutsuka S, Imaeda Y. Reformulation of smoothed particle hydrodynamics for astrophysical fluid dynamics. Computational Fluid Dynamics Journal 2001; 9: 316–325.
- 20 Monaghan JJ, Price DJ. Variational principles for relativistic smoothed particle hydrodynamics. Monthly Notices of the Royal Astronomical Society 2001; 328: 381–392.
- 21 Bonet J, Kulasegaram S, Rodriguez-Paz MX, Profit M. Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems. Computer Methods in Applied Mechanics and Engineering 2004; 193: 1245–1256.
- 22 Koshizuka S, Chikazawa Y, Oka Y. Particle method for fluid and solid dynamics. Computational Fluid and Solid Mechanics 2001; 2: 1269–1271. Proceedings of the 1st MIT Conference on Computational Fluid and Solid Mechanics, Boston, 12–15 June 2001.
10.1016/B978-008043944-0/50893-3 Google Scholar
- 23 Suzuki Y, Koshizuka S, Oka Y. A Hamiltonian particle method for incompressible fluid flows. Proceedings of the 6th World Congress on Computational Mechanics (WCCM VI), Beijing, 5–10 September 2004.
- 24 Suzuki Y, Koshizuka S, Oka Y. Hamiltonian moving-particle semi-implicit (HMPS) method for incompressible fluid flows. Computer Methods in Applied Mechanics and Engineering 2007; 196: 2876–2894.
- 25 Sanz-Serna JM, Carvo MP. Numerical Hamiltonian Problems. Chapman & Hall: London, 1994.
10.1007/978-1-4899-3093-4 Google Scholar
- 26 Iserles A, Munthe-Kaas HZ, Nørsett SP, Zanna A. Lie-group methods. Acta Numerica 2000; 9: 215–365.
10.1017/S0962492900002154 Google Scholar
- 27 Marsden JE, West M. Discrete mechanics and variational integrators. Acta Numerica 2001; 10: 357–514.
10.1017/S096249290100006X Google Scholar
- 28 Hairer E, Lubich C, Wanner G. Geometric Numerical Integration. Springer: Berlin, 2002.
10.1007/978-3-662-05018-7 Google Scholar
- 29 Leimkuhler B, Reich S. Simulating Hamiltonian Dynamics. Cambridge University Press: Cambridge, 2004.
- 30 Frank J, Gottwald G, Reich S. A Hamiltonian particle-mesh method for the rotating shallow water equations. In Meshfree Methods for Partial Differential Equations, M Griebel, MA Schweitzer (eds). Lecture Notes in Computational Science and Engineering, vol. 26. Springer: Berlin, 2003; 131–142.
10.1007/978-3-642-56103-0_10 Google Scholar
- 31 Frank J, Reich S. Conservation properties of smoothed particle hydrodynamics applied to the shallow-water equations. BIT 2003; 43: 40–54.
- 32 Leimukuhler B, Skeel RD. Symplectic numerical integrators in constrained Hamiltonian systems. Journal of Computational Physics 1994; 112: 117–125.
- 33 Simo JC, Tarnow N. The discrete energy–momentum method. Conserving algorithms for nonlinear elastodynamics. Zeitschrift für Angewandte Mathematik und Physik 1992; 43: 757–793.
- 34 Simo JC, Tarnow N. A new energy and momentum conserving algorithm for the non-linear dynamics of shells. International Journal for Numerical Methods in Engineering 1994; 37: 2525–2550.
- 35 Simo JC, Tarnow N, Doblaré M. Nonlinear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. International Journal for Numerical Methods in Engineering 1995; 38: 1431–1474.
- 36 Gonzalez O. Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering 2000; 190: 1763–1783.
- 37 Lew A, Marsden JE, Ortiz M, West M. Asynchronous variational integrators. Archive for Rational Mechanics and Analysis 2003; 167: 85–146.
- 38 Floy PJ. Thermodynamic relations for high elastic materials. Transactions of the Faraday Society 1961; 57: 829–838.
- 39 LeTallec P, Oden JT. Existence and characterization of hydrostatic pressure in finite deformations of incompressible elastic bodies. Journal of Elasticity 1981; 11: 341–357.
- 40 Hughes TJR. The Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1987.
- 41 Lancaster P, Salkauskas K. Surface generated by moving least square methods. Mathematics of Computation 1981; 37: 141–158.
- 42 Dilts GA. Moving-least-squares-particle hydrodynamics—I. Consistency and stability. International Journal for Numerical Methods in Engineering 1999; 44: 1115–1155.
- 43 Belytschko T, Krongauz Y, Dolbow J, Gerlach C. On the completeness of meshfree particle methods. International Journal for Numerical Methods in Engineering 1998; 43: 785–819.
- 44 Chen JK, Beraun JE, Carney TC. A corrective smoothed particle method for boundary value problems in heat conduction. International Journal for Numerical Methods in Engineering 1999; 46: 231–252.
- 45 Arnold VI. Mathematical Methods of Classical Mechanics ( 2nd edn). Springer: New York, 1989.
10.1007/978-1-4757-2063-1 Google Scholar
- 46 Marsden JE, Ratiu TS. Introduction to Mechanics and Symmetry ( 2nd edn). Springer: New York, 1999.
10.1007/978-0-387-21792-5 Google Scholar
- 47 Yoshida H. Construction of higher order symplectic integrators. Physics Letters A 1990; 150: 262–268.
- 48 Reich S. Symplectic integration of constrained Hamiltonian systems by composition methods. SIAM Journal on Numerical Analysis 1996; 33: 475–491.
- 49 Reich S. On higher-order semi-explicit symplectic partitioned Runge–Kutta methods for constrained Hamiltonian systems. Numerische Mathematik 1997; 76: 249–263.
- 50 Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer: New York, 1991.
10.1007/978-1-4612-3172-1 Google Scholar
- 51 Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1990; 29: 1595–1638.
- 52 Simo JC, Armero F. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1992; 33: 1413–1449.
- 53 Dolbow J, Belytschko T. Volumetric locking in the element free Galerkin method. International Journal for Numerical Methods in Engineering 1999; 46: 925–942.
- 54 Chen J-S, Yoon S, Wang H-P, Liu WK. An improved reproducing kernel particle method for nearly incompressible finite elasticity. Computer Methods in Applied Mechanics and Engineering 2000; 181: 117–145.
- 55 Belytschko T, Guo Y, Liu WK, Xiao SP. A unified stability analysis of meshless particle methods. International Journal for Numerical Methods in Engineering 2000; 48: 1369–1400.
- 56 Mooney M. A theory of large elastic deformation. Journal of Applied Physics 1940; 11: 582–592.
- 57 Rivlin RS. Some applications of elasticity theory to rubber engineering. In Proceedings of the Rubber Technology Conference, London, TR Dawson (ed.). Heffer: Cambridge, 1948; 1–8.
- 58 Chadwick P. Continuum Mechanics: Concise Theory and Problems. George Allen and Unwin Ltd.: London, 1976.
