Authors

Harsh Sharma, Mayuresh Patil, Craig Woolsey

Abstract

Accurate numerical simulation of dynamical systems is essential in applications ranging from particle physics to geophysical fluid flow to space hazard analysis. However, most traditional numerical methods do not account for the underlying geometric structure of the physical system, leading to simulation results that may suggest nonphysical behavior. The field of geometric numerical integration (GNI) is concerned with numerical methods that respect the fundamental physics of a problem by preserving the geometric properties of the governing differential equations. Research over the past two decades has produced GNI methods that are so accurate that they are now used for benchmarking purposes for long-time simulation of conservative dynamical systems. However, their utility for large-scale engineering problems is still an open question. This paper presents a review of structure-preserving numerical methods with focus on their engineering applications. The purpose of this paper is to provide an overview of different classes of GNI methods for mechanical systems while providing a survey of practical examples from numerical simulation of realistic engineering problems.

Keywords

Structure-preserving numerical methods; Engineering applications; Geometric numerical integration; Variational integrators; Energy–momentum integrators; Lie group methods

Citation

  • Journal: Computer Methods in Applied Mechanics and Engineering
  • Year: 2020
  • Volume: 366
  • Issue:
  • Pages: 113067
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.cma.2020.113067

BibTeX

@article{Sharma_2020,
  title={{A review of structure-preserving numerical methods for engineering applications}},
  volume={366},
  ISSN={0045-7825},
  DOI={10.1016/j.cma.2020.113067},
  journal={Computer Methods in Applied Mechanics and Engineering},
  publisher={Elsevier BV},
  author={Sharma, Harsh and Patil, Mayuresh and Woolsey, Craig},
  year={2020},
  pages={113067}
}

Download the bib file

References

  • Marsden, (2013)
  • Holm, (2009)
  • Lee, (2017)
  • Doolin, (2013)
  • Boothby, (1986)
  • Newton, (1833)
  • Lagrange, Application de la méthode exposée dans le mémoire précédent à la solution des problèmes de dynamique différents. (1762)
  • Lagrange, (1853)
  • Hamilton, (1834)
  • Hamilton, VII. Second essay on a general method in dynamics. Philos. Trans. R. Soc. Lond. (1835)
  • Marsden, J. E., Pekarsky, S., Shkoller, S. & West, M. Variational methods, multisymplectic geometry and continuum mechanics. Journal of Geometry and Physics vol. 38 253–284 (2001) – 10.1016/s0393-0440(00)00066-8
  • Goldstein, (1980)
  • Noether, Invariante variations probleme, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. (1918)
  • Newton, (2013)
  • Abraham, (1978)
  • Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. & Murray, R. M. Nonholonomic mechanical systems with symmetry. Archive for Rational Mechanics and Analysis vol. 136 21–99 (1996) – 10.1007/bf02199365
  • Marsden, (1990)
  • Fetecau, R. C., Marsden, J. E., Ortiz, M. & West, M. Nonsmooth Lagrangian Mechanics and Variational Collision Integrators. SIAM Journal on Applied Dynamical Systems vol. 2 381–416 (2003) – 10.1137/s1111111102406038
  • Lázaro-Camí, (2007)
  • Bismut, Mécanique aléatoire. (1982)
  • Bou-Rabee, N. & Owhadi, H. Stochastic variational integrators. IMA Journal of Numerical Analysis vol. 29 421–443 (2008) – 10.1093/imanum/drn018
  • Gotay, (2004)
  • Marsden, J. E., Patrick, G. W. & Shkoller, S. Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs. Communications in Mathematical Physics vol. 199 351–395 (1998) – 10.1007/s002200050505
  • Feng, (2010)
  • Atherton, R. W. & Homsy, G. M. On the Existence and Formulation of Variational Principles for Nonlinear Differential Equations. Studies in Applied Mathematics vol. 54 31–60 (1975) – 10.1002/sapm197554131
  • Ibragimov, N. H. Integrating factors, adjoint equations and Lagrangians. Journal of Mathematical Analysis and Applications vol. 318 742–757 (2006) – 10.1016/j.jmaa.2005.11.012
  • Ibragimov, N. H. A new conservation theorem. Journal of Mathematical Analysis and Applications vol. 333 311–328 (2007) – 10.1016/j.jmaa.2006.10.078
  • Kraus, M. & Maj, O. Variational integrators for nonvariational partial differential equations. Physica D: Nonlinear Phenomena vol. 310 37–71 (2015) – 10.1016/j.physd.2015.08.002
  • De Vogelaere, (1956)
  • Ruth, R. D. A Can0nical Integrati0n Technique. IEEE Transactions on Nuclear Science vol. 30 2669–2671 (1983) – 10.1109/tns.1983.4332919
  • Lasagni, F. M. Canonical Runge-Kutta methods. ZAMP Zeitschrift f�r angewandte Mathematik und Physik vol. 39 952–953 (1988) – 10.1007/bf00945133
  • Sanz-Serna, J. M. Runge-kutta schemes for Hamiltonian systems. BIT vol. 28 877–883 (1988) – 10.1007/bf01954907
  • Suris, On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems. (1988)
  • Reich, S. Momentum conserving symplectic integrators. Physica D: Nonlinear Phenomena vol. 76 375–383 (1994) – 10.1016/0167-2789(94)90046-9
  • Reich, S. Backward Error Analysis for Numerical Integrators. SIAM Journal on Numerical Analysis vol. 36 1549–1570 (1999) – 10.1137/s0036142997329797
  • Zhong, G. & Marsden, J. E. Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Physics Letters A vol. 133 134–139 (1988) – 10.1016/0375-9601(88)90773-6
  • Maeda, Lagrangian formulation of discrete systems and concept of difference space. Math. Japon. (1982)
  • Maeda, Extension of discrete noether theorem. Math. Japon. (1981)
  • Veselov, A. P. Integrable discrete-time systems and difference operators. Functional Analysis and Its Applications vol. 22 83–93 (1988) – 10.1007/bf01077598
  • Veselov, A. P. Integrable Lagrangian correspondences and the factorization of matrix polynomials. Functional Analysis and Its Applications vol. 25 112–122 (1991) – 10.1007/bf01079590
  • Moser, J. & Veselov, A. P. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Communications in Mathematical Physics vol. 139 217–243 (1991) – 10.1007/bf02352494
  • Marsden, J. E. & West, M. Discrete mechanics and variational integrators. Acta Numerica vol. 10 357–514 (2001) – 10.1017/s096249290100006x
  • Wendlandt, J. M. & Marsden, J. E. Mechanical integrators derived from a discrete variational principle. Physica D: Nonlinear Phenomena vol. 106 223–246 (1997) – 10.1016/s0167-2789(97)00051-1
  • Hairer, (2006)
  • Lew, A., Marsden, J. E., Ortiz, M. & West, M. Variational time integrators. International Journal for Numerical Methods in Engineering vol. 60 153–212 (2004) – 10.1002/nme.958
  • Hairer, E. & Lubich, C. The life-span of backward error analysis for numerical integrators. Numerische Mathematik vol. 76 441–462 (1997) – 10.1007/s002110050271
  • Leimkuhler, (2004)
  • Kane, C., Marsden, J. E., Ortiz, M. & West, M. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering vol. 49 1295–1325 (2000) – 10.1002/1097-0207(20001210)49:10<1295::aid-nme993>3.0.co;2-w
  • Kane, C., Marsden, J. E. & Ortiz, M. Symplectic-energy-momentum preserving variational integrators. Journal of Mathematical Physics vol. 40 3353–3371 (1999) – 10.1063/1.532892
  • Lee, T. D. Can time be a discrete dynamical variable? Physics Letters B vol. 122 217–220 (1983) – 10.1016/0370-2693(83)90687-1
  • Shibberu, (2006)
  • Shibberu, (2005)
  • Sharma, H., Patil, M. & Woolsey, C. Energy-preserving variational integrators for forced Lagrangian systems. Communications in Nonlinear Science and Numerical Simulation vol. 64 159–177 (2018) – 10.1016/j.cnsns.2018.04.015
  • Bloch, (2003)
  • Cortés, (2002)
  • Kobilarov, Geometric discretization of nonholonomic systems with symmetries. Discrete Contin. Dyn. Syst. Ser. S (2010)
  • Milstein, G. N., Repin, Yu. M. & Tretyakov, M. V. Numerical Methods for Stochastic Systems Preserving Symplectic Structure. SIAM Journal on Numerical Analysis vol. 40 1583–1604 (2002) – 10.1137/s0036142901395588
  • Milstein, G. N., Repin, Yu. M. & Tretyakov, M. V. Symplectic Integration of Hamiltonian Systems with Additive Noise. SIAM Journal on Numerical Analysis vol. 39 2066–2088 (2002) – 10.1137/s0036142901387440
  • Bou-Rabee, (2007)
  • Holm, D. D. & Tyranowski, T. M. Stochastic discrete Hamiltonian variational integrators. BIT Numerical Mathematics vol. 58 1009–1048 (2018) – 10.1007/s10543-018-0720-2
  • Fetecau, Variational multisymplectic formulations of nonsmooth continuum mechanics. (2003)
  • Johnson, G., Leyendecker, S. & Ortiz, M. Discontinuous variational time integrators for complex multibody collisions. International Journal for Numerical Methods in Engineering vol. 100 871–913 (2014) – 10.1002/nme.4764
  • Lall, S. & West, M. Discrete variational Hamiltonian mechanics. Journal of Physics A: Mathematical and General vol. 39 5509–5519 (2006) – 10.1088/0305-4470/39/19/s11
  • Leok, M. & Zhang, J. Discrete Hamiltonian variational integrators. IMA Journal of Numerical Analysis vol. 31 1497–1532 (2011) – 10.1093/imanum/drq027
  • Schmitt, J. M. & Leok, M. Properties of Hamiltonian variational integrators. IMA Journal of Numerical Analysis vol. 38 377–398 (2017) – 10.1093/imanum/drx010
  • Gotay, (1998)
  • Lew, A., Marsden, J. E., Ortiz, M. & West, M. Asynchronous Variational Integrators. Archive for Rational Mechanics and Analysis vol. 167 85–146 (2003) – 10.1007/s00205-002-0212-y
  • Belytschko, T. & Schoeberle, D. F. On the Unconditional Stability of an Implicit Algorithm for Nonlinear Structural Dynamics. Journal of Applied Mechanics vol. 42 865–869 (1975) – 10.1115/1.3423721
  • Gonzalez, O. & Simo, J. C. On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Computer Methods in Applied Mechanics and Engineering vol. 134 197–222 (1996) – 10.1016/0045-7825(96)01009-2
  • LaBudde, R. A. & Greenspan, D. Discrete mechanics—A general treatment. Journal of Computational Physics vol. 15 134–167 (1974) – 10.1016/0021-9991(74)90081-3
  • LaBudde, R. A. & Greenspan, D. Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numerische Mathematik vol. 25 323–346 (1975) – 10.1007/bf01396331
  • LaBudde, R. A. & Greenspan, D. Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numerische Mathematik vol. 26 1–16 (1976) – 10.1007/bf01396562
  • Simo, J. C. & Tarnow, N. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP Zeitschrift f�r angewandte Mathematik und Physik vol. 43 757–792 (1992) – 10.1007/bf00913408
  • Simo, J. C., Tarnow, N. & Wong, K. K. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering vol. 100 63–116 (1992) – 10.1016/0045-7825(92)90115-z
  • Betsch, P. & Steinmann, P. Conservation properties of a time FE method. Part I: time-stepping schemes forN-body problems. International Journal for Numerical Methods in Engineering vol. 49 599–638 (2000) – 10.1002/1097-0207(20001020)49:5<599::aid-nme960>3.0.co;2-9
  • Betsch, P. & Steinmann, P. Conservation properties of a time FE method—part II: Time‐stepping schemes for non‐linear elastodynamics. International Journal for Numerical Methods in Engineering vol. 50 1931–1955 (2001) – 10.1002/nme.103
  • Betsch, P. & Steinmann, P. Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints. International Journal for Numerical Methods in Engineering vol. 53 2271–2304 (2002) – 10.1002/nme.347
  • Groß, M., Betsch, P. & Steinmann, P. Conservation properties of a time FE method. Part IV: Higher order energy and momentum conserving schemes. International Journal for Numerical Methods in Engineering vol. 63 1849–1897 (2005) – 10.1002/nme.1339
  • Armero, F. & Petőcz, E. A new dissipative time-stepping algorithm for frictional contact problems: formulation and analysis. Computer Methods in Applied Mechanics and Engineering vol. 179 151–178 (1999) – 10.1016/s0045-7825(99)00036-5
  • Hughes, Analysis of transient algorithms with particular reference to stability behavior. (1983)
  • Kuhl, D. & Ramm, E. Generalized Energy–Momentum Method for non-linear adaptive shell dynamics. Computer Methods in Applied Mechanics and Engineering vol. 178 343–366 (1999) – 10.1016/s0045-7825(99)00024-9
  • Gonzalez, O. Time integration and discrete Hamiltonian systems. Journal of Nonlinear Science vol. 6 449–467 (1996) – 10.1007/bf02440162
  • McLachlan, R. I., Quispel, G. R. W. & Robidoux, N. Geometric integration using discrete gradients. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences vol. 357 1021–1045 (1999) – 10.1098/rsta.1999.0363
  • Gonzalez, O. Mechanical systems subject to holonomic constraints: Differential–algebraic formulations and conservative integration. Physica D: Nonlinear Phenomena vol. 132 165–174 (1999) – 10.1016/s0167-2789(99)00054-8
  • Celledoni, Energy-preserving integrators applied to nonholonomic systems. J. Nonlinear Sci. (2016)
  • McLachlan, (2013)
  • Celledoni, E. et al. Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. Journal of Computational Physics vol. 231 6770–6789 (2012) – 10.1016/j.jcp.2012.06.022
  • Iserles, (2015)
  • Crouch, P. E. & Grossman, R. Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science vol. 3 1–33 (1993) – 10.1007/bf02429858
  • Munthe-Kaas, H. Runge-Kutta methods on Lie groups. BIT Numerical Mathematics vol. 38 92–111 (1998) – 10.1007/bf02510919
  • Munthe-Kaas, H. High order Runge-Kutta methods on manifolds. Applied Numerical Mathematics vol. 29 115–127 (1999) – 10.1016/s0168-9274(98)00030-0
  • Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. & Zanna, A. Lie-group methods. Acta Numerica vol. 9 215–365 (2000) – 10.1017/s0962492900002154
  • Celledoni, E., Marthinsen, H. & Owren, B. An introduction to Lie group integrators – basics, new developments and applications. Journal of Computational Physics vol. 257 1040–1061 (2014) – 10.1016/j.jcp.2012.12.031
  • Lewis, D. & Simo, J. C. Conserving algorithms for the dynamics of Hamiltonian systems on lie groups. Journal of Nonlinear Science vol. 4 253–299 (1994) – 10.1007/bf02430634
  • Bobenko, A. I. & Suris, Yu. B. Discrete Time Lagrangian Mechanics on Lie Groups,¶with an Application to the Lagrange Top. Communications in Mathematical Physics vol. 204 147–188 (1999) – 10.1007/s002200050642
  • Marsden, J. E., Pekarsky, S. & Shkoller, S. Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity vol. 12 1647–1662 (1999) – 10.1088/0951-7715/12/6/314
  • Lee, A lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum. (2005)
  • Lee, T., Leok, M. & McClamroch, N. H. Lie group variational integrators for the full body problem. Computer Methods in Applied Mechanics and Engineering vol. 196 2907–2924 (2007) – 10.1016/j.cma.2007.01.017
  • Lee, T., Leok, M. & McClamroch, N. H. Lagrangian mechanics and variational integrators on two‐spheres. International Journal for Numerical Methods in Engineering vol. 79 1147–1174 (2009) – 10.1002/nme.2603
  • Demoures, (2012)
  • Demoures, F., Gay-Balmaz, F., Kobilarov, M. & Ratiu, T. S. Multisymplectic Lie group variational integrator for a geometrically exact beam in. Communications in Nonlinear Science and Numerical Simulation vol. 19 3492–3512 (2014) – 10.1016/j.cnsns.2014.02.032
  • Demoures, F. et al. Discrete variational Lie group formulation of geometrically exact beam dynamics. Numerische Mathematik vol. 130 73–123 (2014) – 10.1007/s00211-014-0659-4
  • Sharma, Energy-preserving, adaptive time-step Lie group variational integrators for the attitude dynamics of a rigid body. (2019)
  • Sharma, Energy-preserving, adaptive time-step Lie group variational integrators for rigid body motion in SE(3). (2019)
  • Feng, The symplectic methods for the computation of Hamiltonian equations. (1987)
  • Channell, P. J. & Scovel, C. Symplectic integration of Hamiltonian systems. Nonlinearity vol. 3 231–259 (1990) – 10.1088/0951-7715/3/2/001
  • Forest, E. & Ruth, R. D. Fourth-order symplectic integration. Physica D: Nonlinear Phenomena vol. 43 105–117 (1990) – 10.1016/0167-2789(90)90019-l
  • Wisdom, J. & Holman, M. Symplectic maps for the n-body problem. The Astronomical Journal vol. 102 1528 (1991) – 10.1086/115978
  • Wisdom, J. The origin of the Kirkwood gaps - A mapping for asteroidal motion near the 3/1 commensurability. The Astronomical Journal vol. 87 577 (1982) – 10.1086/113132
  • Wisdom, J. Chaotic behavior and the origin of the 31 Kirkwood gap. Icarus vol. 56 51–74 (1983) – 10.1016/0019-1035(83)90127-6
  • Yoshida, Recent progress in the theory and application of symplectic integrators. (1993)
  • Gladman, B., Duncan, M. & Candy, J. Symplectic integrators for long-term integrations in celestial mechanics. Celestial Mechanics and Dynamical Astronomy vol. 52 221–240 (1991) – 10.1007/bf00048485
  • Kinoshita, H., Yoshida, H. & Nakai, H. Symplectic integrators and their application to dynamical astronomy. CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY vol. 50 59–71 (1991) – 10.1007/bf00048986
  • Imre, E. & Palmer, P. L. High-Precision, Symplectic Numerical, Relative Orbit Propagation. Journal of Guidance, Control, and Dynamics vol. 30 965–973 (2007) – 10.2514/1.26846
  • Chambers, J. E. A hybrid symplectic integrator that permits close encounters between massive bodies. Monthly Notices of the Royal Astronomical Society vol. 304 793–799 (1999) – 10.1046/j.1365-8711.1999.02379.x
  • Varadi, F., De La Barre, C. M., Kaula, W. M. & Ghil, M. Singularly weighted symplectic forms and applications to asteroid motion. Celestial Mechanics and Dynamical Astronomy vol. 62 23–41 (1995) – 10.1007/bf00692067
  • Liu, A numerical study of the orbits of near earth asteroids with symplectic algorithm. Chin. J. Astron. Astrophys. (1999)
  • Levison, H. F. & Duncan, M. J. The Long-Term Dynamical Behavior of Short-Period Comets. Icarus vol. 108 18–36 (1994) – 10.1006/icar.1994.1039
  • Emel’yanenko, V. Celestial Mechanics and Dynamical Astronomy vol. 84 331–341 (2002) – 10.1023/a:1021125829276
  • Tsuda, Y. & Scheeres, D. J. Computation and Applications of an Orbital Dynamics Symplectic State Transition Matrix. Journal of Guidance, Control, and Dynamics vol. 32 1111–1123 (2009) – 10.2514/1.42358
  • Wisdom, Symplectic correctors. (1996)
  • Farr, W. M. & Bertschinger, E. Variational Integrators for the GravitationalN‐Body Problem. The Astrophysical Journal vol. 663 1420–1433 (2007) – 10.1086/518641
  • Lee, T., Leok, M. & Harris McClamroch, N. High-fidelity numerical simulation of complex dynamics of tethered spacecraft. Acta Astronautica vol. 99 215–230 (2014) – 10.1016/j.actaastro.2014.02.021
  • Lee, Lagrangian mechanics and Lie group variational integrators for spacecraft with imbalanced reaction wheels. (2014)
  • Hall, J. & Leok, M. Spectral variational integrators. Numerische Mathematik vol. 130 681–740 (2014) – 10.1007/s00211-014-0679-0
  • Palacios, L. & Gurfil, P. Variational and symplectic integrators for satellite relative orbit propagation including drag. Celestial Mechanics and Dynamical Astronomy vol. 130 (2018) – 10.1007/s10569-018-9826-8
  • Simo, Recent results on the numerical integration of infinite-dimensional hamiltonian systems. (1994)
  • Gonzalez, O. Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering vol. 190 1763–1783 (2000) – 10.1016/s0045-7825(00)00189-4
  • Leyendecker, S., Betsch, P. & Steinmann, P. Objective energy–momentum conserving integration for the constrained dynamics of geometrically exact beams. Computer Methods in Applied Mechanics and Engineering vol. 195 2313–2333 (2006) – 10.1016/j.cma.2005.05.002
  • Sansour, C., Wriggers, P. & Sansour, J. Nonlinear Dynamics vol. 13 279–305 (1997) – 10.1023/a:1008251113479
  • Lew, (2003)
  • Kale, K. G. & Lew, A. J. Parallel asynchronous variational integrators. International Journal for Numerical Methods in Engineering vol. 70 291–321 (2006) – 10.1002/nme.1880
  • GarcÍa Orden, J. C. & Goicolea, J. M. Multibody System Dynamics vol. 4 225–244 (2000) – 10.1023/a:1009871728414
  • Betsch, P. & Leyendecker, S. The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics. International Journal for Numerical Methods in Engineering vol. 67 499–552 (2006) – 10.1002/nme.1639
  • Leyendecker, S., Betsch, P. & Steinmann, P. The discrete null space method for the energy-consistent integration of constrained mechanical systems. Part III: Flexible multibody dynamics. Multibody System Dynamics vol. 19 45–72 (2007) – 10.1007/s11044-007-9056-4
  • Betsch, P. & Steinmann, P. Constrained integration of rigid body dynamics. Computer Methods in Applied Mechanics and Engineering vol. 191 467–488 (2001) – 10.1016/s0045-7825(01)00283-3
  • Betsch, P. & Uhlar, S. Energy-momentum conserving integration of multibody dynamics. Multibody System Dynamics vol. 17 243–289 (2007) – 10.1007/s11044-007-9043-9
  • Uhlar, S. & Betsch, P. On the derivation of energy consistent time stepping schemes for friction afflicted multibody systems. Computers & Structures vol. 88 737–754 (2010) – 10.1016/j.compstruc.2010.03.003
  • Leyendecker, S., Marsden, J. E. & Ortiz, M. Variational integrators for constrained dynamical systems. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik vol. 88 677–708 (2008) – 10.1002/zamm.200700173
  • Leyendecker, A variational approach to multirate integration for constrained systems. (2013)
  • Brüls, O. & Cardona, A. On the Use of Lie Group Time Integrators in Multibody Dynamics. Journal of Computational and Nonlinear Dynamics vol. 5 (2010) – 10.1115/1.4001370
  • Brüls, O., Cardona, A. & Arnold, M. Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory vol. 48 121–137 (2012) – 10.1016/j.mechmachtheory.2011.07.017
  • Jonghoon Park & Wan-Kyun Chung. Geometric integration on Euclidean group with application to articulated multibody systems. IEEE Transactions on Robotics vol. 21 850–863 (2005) – 10.1109/tro.2005.852253
  • Terze, Z., Müller, A. & Zlatar, D. Lie-group integration method for constrained multibody systems in state space. Multibody System Dynamics vol. 34 275–305 (2014) – 10.1007/s11044-014-9439-2
  • Perot, B. Conservation Properties of Unstructured Staggered Mesh Schemes. Journal of Computational Physics vol. 159 58–89 (2000) – 10.1006/jcph.2000.6424
  • Elcott, S., Tong, Y., Kanso, E., Schröder, P. & Desbrun, M. Stable, circulation-preserving, simplicial fluids. ACM Transactions on Graphics vol. 26 4 (2007) – 10.1145/1189762.1189766
  • Arnold, V. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Annales de l’institut Fourier vol. 16 319–361 (1966) – 10.5802/aif.233
  • Cotter, C. J., Holm, D. D. & Hydon, P. E. Multisymplectic formulation of fluid dynamics using the inverse map. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 463 2671–2687 (2007) – 10.1098/rspa.2007.1892
  • Mullen, P., Crane, K., Pavlov, D., Tong, Y. & Desbrun, M. Energy-preserving integrators for fluid animation. ACM Transactions on Graphics vol. 28 1–8 (2009) – 10.1145/1531326.1531344
  • Pavlov, D. et al. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena vol. 240 443–458 (2011) – 10.1016/j.physd.2010.10.012
  • Gawlik, E. S., Mullen, P., Pavlov, D., Marsden, J. E. & Desbrun, M. Geometric, variational discretization of continuum theories. Physica D: Nonlinear Phenomena vol. 240 1724–1760 (2011) – 10.1016/j.physd.2011.07.011
  • Zeitlin, V., Gay-Balmaz, F., Gawlik, E. S. & Desbrun, M. Variational discretization for rotating stratified fluids. Discrete and Continuous Dynamical Systems vol. 34 477–509 (2013) – 10.3934/dcds.2014.34.477
  • Bauer, (2017)
  • Junge, O., Marsden, J. E. & Ober-Blöbaum, S. DISCRETE MECHANICS AND OPTIMAL CONTROL. IFAC Proceedings Volumes vol. 38 538–543 (2005) – 10.3182/20050703-6-cz-1902.00745
  • Ober-Blöbaum, S., Junge, O. & Marsden, J. E. Discrete mechanics and optimal control: An analysis. ESAIM: Control, Optimisation and Calculus of Variations vol. 17 322–352 (2010) – 10.1051/cocv/2010012
  • Leyendecker, S., Ober-Blöbaum, S., Marsden, J. E. & Ortiz, M. Discrete mechanics and optimal control for constrained systems. Optimal Control Applications and Methods vol. 31 505–528 (2010) – 10.1002/oca.912
  • Kobilarov, Optimal control using nonholonomic integrators. (2007)
  • Betsch, P. & Becker, C. Conservation of generalized momentum maps in mechanical optimal control problems with symmetry. International Journal for Numerical Methods in Engineering vol. 111 144–175 (2016) – 10.1002/nme.5459
  • Manns, P. & Mombaur, K. Towards Discrete Mechanics and Optimal Control for Complex Models. IFAC-PapersOnLine vol. 50 4812–4818 (2017) – 10.1016/j.ifacol.2017.08.966
  • Junge, Optimal reconfiguration of formation flying spacecraft—a decentralized approach. (2006)
  • Lee, Attitude maneuvers of a rigid spacecraft in a circular orbit. (2006)
  • Shareef, Z. & Trächtler, A. Simultaneous path planning and trajectory optimization for robotic manipulators using discrete mechanics and optimal control. Robotica vol. 34 1322–1334 (2014) – 10.1017/s0263574714002318
  • Pekarek, Discrete mechanics and optimal control applied to the compass gait biped. (2007)
  • Kai, T., Yamaki, K. & Koike, S. Development of Discrete Mechanics for Distributed Parameter Mechanical Systems and Its Application to Vibration Suppression Control of a String. Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics 492–498 (2016) doi:10.5220/0005978204920498 – 10.5220/0005978204920498
  • Shareef, Optimal trajectory planning for robotic manipulators using discrete mechanics and optimal control. (2014)
  • Manchester, Z., Doshi, N., Wood, R. J. & Kuindersma, S. Contact-implicit trajectory optimization using variational integrators. The International Journal of Robotics Research vol. 38 1463–1476 (2019) – 10.1177/0278364919849235
  • Kern, D. & Groß, M. Variational Integrators and Optimal Control for a Hybrid Pendulum‐on‐Cart‐System. PAMM vol. 18 (2018) – 10.1002/pamm.201800088
  • McLachlan, R. & Marsland, S. Discrete Mechanics and Optimal Control for Image Registration. ANZIAM Journal vol. 48 1 (2007) – 10.21914/anziamj.v48i0.82
  • Johnson, E. R. & Murphey, T. D. Scalable Variational Integrators for Constrained Mechanical Systems in Generalized Coordinates. IEEE Transactions on Robotics vol. 25 1249–1261 (2009) – 10.1109/tro.2009.2032955
  • Lee, (2016)
  • Fan, (2019)
  • Kobilarov, Solvability of geometric integrators for multi-body systems. (2014)
  • Leyendecker, S., Betsch, P. & Steinmann, P. Energy-conserving integration of constrained Hamiltonian systems ? a comparison of approaches. Computational Mechanics vol. 33 174–185 (2004) – 10.1007/s00466-003-0516-2
  • Betsch, P., Hesch, C., Sänger, N. & Uhlar, S. Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics. Journal of Computational and Nonlinear Dynamics vol. 5 (2010) – 10.1115/1.4001388