J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63--116, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(14.32) a n # = a n 1 (14.33) The displacement and velocity quantities at t n 1 are updated using the Newmark formulas given above. This solution option is selected using the command TRANsient ALPHa beta gamma alpha # = 0.5, # = 1, and # = 0.5. 3. An energy conserving form of the alpha method [31, 32, 33] (i.e. similar to the HHT method) with the acceleration given as: a n # = 1 #t [ v n 1 v n ] 14.34) This solution option is selected using the command TRANsient CONServe beta gamma alpha # = 0.5, # = 1, and # = 0.5. Note that the conserving form does not involve the accelerations in the ....

J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63--116, 1992.

.... or relate to a particular sub class of the present framework of timediscretiz ed operators as described subsequently) Both energy conserving and momentum conserving schemes for rigid dynamics and nonlinear dynamic problem appear in various e#orts proposed by Simo and Wong [7] and Simo et al. [8], in which the implicit Newmark method or thetrapez idal rule is shown not to be angular momentum conserving although it inherits total energy conservation, whereas the explicit central di#erence timediscretiz ed operator (within the stability limit) pertaining to the Newmark family is both ....

J.C. Simo, N. Tarnow, K.K. Wang, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Computer Methods in Applied Mechanics and Engineering 100 (1992) 63.

.... that preserve the Hamiltonian or Poisson structure, 10, 12, 21, 35, 46, 9, 45, 2, 3, 29, 6] variational integrators that utilize the variational character of Lagrangian and canonical Hamiltonian systems, 4, 5, 31, 29] conservative integrators that preserve rst integrals or conservation laws, [19, 20, 27, 42, 38, 39, 40, 25, 1], and symmetric integrators that preserve symmetries of the system, 7, 11, 17, 36] A geometric integrator will track solutions over short time intervals as well as a standard scheme of the same order, e.g. a Runge Kutta algorithm, while the extra expense required to construct and implement it ....

.... The rotation group plays a crucial role in many formulations of elasticity and plasticity and the advantages of exact rotations in numerical simulations of such materials, implemented via either the Rodriguez formula for the true exponential or the Cayley transform, have been amply demonstrated [41, 42, 40, 37]. Discretizations of speci c dynamical systems on Lie groups that preserve not only the group structure, but additional geometric structures, have been used in the study of integrable systems; see, e.g. Moser and Veselov [31] Lewis and Simo [25] and McLaughlin and Scovel [28] for schemes ....

Simo, J.C., Tarnow, N., and Wong, K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech Eng. 100 (1992), 63-116.

....are updated using the Newmark formulas given above. This solution option is selected using the command TRANsient,ALPHa,beta,gamma,alpha The alpha parameter should be specified between zero and 1. Default values are # = 0.5, # = 1, and # = 0.5. 3. An energy conserving form of the alpha method [9, 10, 3] (i.e. similar to the HHT method) with the acceleration given as: a n # = 1 #t [ v n 1 v n ] 13.30) This solution option is selected using the command CHAPTER 13. COMMAND LANGUAGE PROGRAMS 98 TRANsient,CONServe,beta,gamma,alpha The alpha parameter should be specified between zero and ....

J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63--116, 1992. 125 BIBLIOGRAPHY 126

....over (10) Finally, in Section 4, we discuss various numerical aspects of our new method and demonstrate its properties by means of two simple numerical examples. Another approach to the long time integration of highly oscillatory Hamiltonian system has been taken by Simo and his collaborators [20]. They advocate the direct discretization of (1) by an implicit energy momentum method and the usage of a large step size. However, we are not aware of rigorous stability and convergence results for these methods when applied to the system (1) with a step size Deltat AE ffl. 2 Mathematical ....

....conserving. The method (34) is computational expensive. An effective implementation of (34) and the discretization of (32) 31) respectively, by less expensive methods can be found in [2] Note that one could also discretize (32) by a proper modification of the energy momentum methods proposed in [20]. Example 2. In this example we consider a four bead three bond structure [13] where the structure is restricted to move in a finite volume by the potential V r (q) X i K r r i oe 6 Here r i denotes the distance of each of the four beads to the origin, oe = 2, and K r = 0:1. We set ....

Simo, J., Tarnow, N., and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech. Eng., 1, 63--116, 1992.

....Q(t) of H ffl . Finally, in Section 8, we discuss numerical aspects of our new method and demonstrate its properties by means of a simple numerical example. Another approach to the long time integration of highly oscillatory Hamiltonian system has been taken by Simo and his collaborators [25]. They advocate the direct discretization of (1) by an implicit energy momentum method and the usage of a large step size. However, there do not exist rigorous stability and order of convergence results for these methods when applied to general systems of type (1) with a step size Deltat AE ffl. ....

....momentum conserving. The method (39) is computational expensive. An effective implementation of (39) and the discretization of (32) by less expensive methods can be found in [3] and [11] Note that one could also discretize (32) by a proper modification of the energy momentum methods proposed in [25]. Example 2. In this example we consider a three bead two bond structure where the structure is restricted to move in a finite volume by the potential V r (q) X i K r r i oe 12 : Here r i denotes the distance of each of the three beads to the origin, oe = 3:0, and K r = 50:0. We ....

Simo, J., Tarnow, N., and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech. Eng., 1, 63--116, 1992.

....LIN [47] ii) Methods which inaccurately resolve the highest frequencies: e.g. the implicit midpoint method with large timestep Deltat. Although to date apparently not used in MD, this method has recently been employed for other problems in nonlinear dynamics, e.g. for rigid and elastic bodies [38]. iii) Methods based on removal of the fast components through the introduction of constraints, including SHAKE [34] and RATTLE[2] iv) Methods which attempt to suppress the fast components, e.g. the method LI [29] which introduces dissipation via the implicit Euler scheme to enable larger ....

Simo, J., Rarnow, N., and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlineaer dynamics, Comp. Meth. Appl. Mech. Eng. 1, 1528--44, 1992.

....LIN [47] ii) Methods which inaccurately resolve the highest frequencies: e.g. the implicit midpoint method with large timestep Deltat. Although to date apparently not used in MD, this method has recently been employed for other problems in nonlinear dynamics, e.g. for rigid and elastic bodies [38]. iii) Methods based on removal of the fast components through the introduction of constraints, including SHAKE [34] and RATTLE[2] iv) Methods which attempt to suppress the fast components, e.g. the method LI [29] which introduces dissipation via the implicit Euler scheme to enable larger ....

Simo, J., Rarnow, N., and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlineaer dynamics, Comp. Meth. Appl. Mech. Eng. 1, 1528--44, 1992.

....Depending on boundary conditions and external loads, the nonlinear system (1) may possess various integrals related to the total linear momentum L(#, V) angular momentum J(#, V) and energy H(#,V) of the material body. These integrals are typically lost under discretization in time. In [42,43] it was shown that all members of the Newmark family preserve L, but none preserve H for arbitrary constitutive laws. Moreover, it was shown that the explicit central di#erence scheme is the only member of the Newmark family that preserves J. The main result in [42,43] was the construction of ....

....discretization in time. In [42,43] it was shown that all members of the Newmark family preserve L, but none preserve H for arbitrary constitutive laws. Moreover, it was shown that the explicit central di#erence scheme is the only member of the Newmark family that preserves J. The main result in [42,43] was the construction of implicit schemes for (1) that preserve L, J and H, but at the cost of introducing a nonlinear scalar equation at each time step. In this article we consider generalizations of the time reversible, integralpreserving time discretization schemes of [19,20] for the treatment ....

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J.C. Simo, N. Tarnow & K.K. Wong (1992) "Exact Energy-Momentum Conserving Algorithms and Symplectic Schemes for Nonlinear Dynamics," Computer Methods in Applied Mechanics and Engineering, 1, 63--116.

....discretization of the Jacobian [24] this is a difficult and essentially unsolved problem. Probably the right generalization of Hamiltonian has not yet been found. We are thus reluctantly led to consider only energy conserving discretizations. Or perhaps we should not be reluctant: Simo et al. [21] have argued and presented detailed evidence from elastodynamics that conserving energy leads to excellent nonlinear stability properties that preserving symplectic structure does not. Essentially because symplectic schemes can only brake the fast modes, whereas energy conserving schemes can also ....

J.C. Simo, N. Tarnow, and K.K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech. Eng. 100 (1992), 63--116.

....behavior suddenly worsens. Related phenomena are seen in Arnold diffusion. The energy can even blow up in finite time [130] To avoid this, it seems that the step size must be small compared to the shortest period of the system, a severe restriction if the system is a discretization of a PDE [130, 131]. ffl Quantify this observed energy breakdown, and find ways to avoid it. An alternative route is to abandon symplecticity, and require instead strict energy conservation. This approach has been explored by (among others) Simo and coworkers [79, 129, 130, 131] It is possible to simultaneously ....

....if the system is a discretization of a PDE [130, 131] ffl Quantify this observed energy breakdown, and find ways to avoid it. An alternative route is to abandon symplecticity, and require instead strict energy conservation. This approach has been explored by (among others) Simo and coworkers [79, 129, 130, 131]. It is possible to simultaneously conserve momentum, and such energy momentum integrators are very promising. At first sight, however, the resulting map on the energy momentum level set is not constrained in any way. It might not preserve a volume form, for example, and it might possess ....

Simo, J. C., N. Tarnow, and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Comp. Meth. Appl. Mech. Eng. 100 (1992), 63--116.

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