Séminaire : Anthony Gravouil

Non-smooth elastodynamics as an example for setting up an explicit variational time integrator

Accurate numerical simulations of dynamical systems are essential in practically all areas of physics. However, most traditional numerical methods do not take into account the underlying geometric structure of the physical system, leading to simulation results that may suggest non-physical behaviors. The field of numerical geometric integration (NGI) concerns numerical methods that respect physics of a problem by preserving the geometric properties of the differential equations associated. Research conducted over more than 40 years has produced Geometric Integrators so accurate that they are now used to de ne benchmarks for long-time simulations conservative dynamical systems. However, the general construction of Geometric Integrators for mechanical problems of great complexity (non-regular problems, dissipative phenomena) and of high dimension is always an open question. Geometric Integrators seduce physicists, mathematicians and engineers for many reasons: Numerical methods based on discrete variational principles can exhibit superior numerical stability and preservation capabilities of the underlying structure (symplectic structure, Lie group, Dirac structure, discrete differential geometry, discrete exterior calculus). These methods can advance the design and analysis based on models while preserving the fidelity of the physical system in continuous time, allowing, for example, more accurate predictions of energy transfer between subsystems. Since the emergence of numerical methods, fundamental properties such as precision, stability, convergence and computational efficiency were considered crucial in deciding the usefulness of a numerical algorithm. Today, the aspects related to the preservation of the underlying structure are considered essential, in addition to these fundamental properties. One of key ideas of approximate structure preservation is to treat the numerical method as a discrete dynamical system that approximates the flow of the continuous differential equation instead of focusing on the numerical approximation of a single trajectory. Such an approach allows a better understanding of the invariants and the qualitative properties of the numerical method. Variational geometric integrators (more recent than symplectic integrators) use the concepts of discrete mechanics, i.e. a discrete analogue of Lagrangian mechanics/ Hamiltonian continuous in time. Due to their variational nature, these methods can be extended to non- conservative mechanics by discretization of the Lagrange-d'Alembert principle (discrete action). The stationary points of this discrete action give the discrete trajectories in time of the dynamical system. Thus, fixed time step variational geometric integrators have been proposed in the literature in order to verify in the discrete sense the linear and angular momentum balance equations of the discrete Lagrangian, with a decreasing exponential error on the discrete energy. These integrators with remarkable properties over long times, however, do not verify exactly the conservation of the discrete energy. This property is known in the literature as name of Ge and Marsden's paradox. In this contribution, it is proposed to illustrate the power of discrete variational approach in the particular case of non-smooth large strain elastodynamics as an example for the implementation of a new explicit variational time integrator. In a first part, a general variational Lagrangian approach is introduced in order to illustrate the multi-symplectic character of large strain elasto-dynamics. In a second step, the same approach is applied to non-smooth large strain elasto-dynamics. All balance equations are systematically obtained from variational calculations. In a third part, a space-time discrete Lagrangian is introduced. It is based on the space finite element method and the time explicit central difference method with Lagrange multipliers for possible impacts. At first, a fixed time step is considered, and the discrete multi-symplectic structure is studied. Finally, in order to remove the Ge and Marsden's paradox, the time step is considered as an unknown of the discrete problem in order to keep the energy exactly at the discrete sense.

[1] Hamilton WR. VII. Second essay on a general method in dynamics. Philosophical Transactions of the Royal Society of London, 1835. [2] Noether E. Invariante Variations Probleme, Nachr. Ges. Wiss. Gottingen Math-Phys Kl. 1918.
[3] Onsager L. Phys Rev 1931.
[4] P.A.M. Dirac. Generalized hamiltonian dynamics. Canad. J. Math., 1950.
[5] De Vogelaere R. Methods of integration which preserve the contact transformation property of the Hamilton equations, 1956.
[6] Moreau J.J. Fluid dynamics and the calculus of horizontal variations. Int. J. Engng Sci., 1982.
[7] Ge Z, Marsden JE. Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory. Phys Lett A. 1988.
[8] Gonzalez O, Simo JC. On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems, cmame, 1996. [9] M. Grmela, H.C. Ottinger, Dynamics and thermodynamics of complex fluids I. Development of a general formalism, Phys. Rev., 1997.
[10] Marsden JE, Pekarsky S, Shkoller S,West M. Variational methods, multisymplectic geometry and continuum mechanics. J. of Geo. and Phys, 2001. [11] Orden JC, Romero I. Energy–entropy–momentum integration of discrete thermo-visco-elastic dynamics. Eur. J. of Mech. - A/Solids 2012.
[12] Betsch P, Becker C. Conservation of generalized momentum maps in mechanical optimal control problems with symmetry, ijnme, 2017.