Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Abstract

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations

“…For instance, in classical mechanics, a time discretization of the Lagrangian variational formulation allows for the derivation of numerical schemes, called variational integrators, that are symplectic, exhibit good energy behavior, and inherit a discrete version of Noether's theorem which guarantees the exact preservation of momenta arising from symmetries, see [25]. An extension of this approach to the context of certain partial differential equations may be made through an appeal to their spacetime variational formulation resulting in multisymplectic schemes, [26], [21], see, e.g., [12], [13], [18] for recent developments in variational multisymplectic integrators.…”

A multisymplectic integrator for elastodynamic frictionless impact problems

“…These conditions are obtained via the generalized Lagrange multiplier approach to enforce inequality constraints (see, e.g., Clarke [1983], Moreau [1988], Rockafellar [1993], Rockafellar and Wets [1997]). Following Demoures, Gay-Balmaz, and Ratiu [2016] (Theorem 3), we find:…”

“…The modern treatment of describing the motion of a body in the presence of unilateral contact (see Clarke [1990] for a concise exposition) has been largely based on the work of Moreau and Rockafellar in nonsmooth convex analysis (Moreau [1962], Moreau [1964], Moreau [1966], Rockafellar [1963], Rockafellar [1966]). The resulting computational contributions to contact mechanics include Curnier [1993], Laursen and Chawla [1997], Armero and Petocz [1998], Kane, Repetto, Ortiz, and Marsden [1999], Pandolfi, Kane, Marsden, Ortiz [2002], Wriggers [2002], Laursen [2002], Laursen and Love [2003], Cirak and West [2005], Wriggers and Laursen [2007], Johnson, Leyendecker, and Ortiz [2014]; see Demoures, Gay-Balmaz, and Ratiu [2016] for an account of the differences between these various methods.…”

“…Collisions are numerically handled via a generalized Lagrange multiplier approach for constrained optimization problems as formulated in Rockafellar and Wets [1997]; we do not consider penalty methods or the theory of augmented Lagrangians in this work. Based on the recent theoretical results in Demoures, Gay-Balmaz, and Ratiu [2016], we develop a fully space-time variational formulation of both the continuous and discrete settings, thus advancing further the vision originally outlined in :…”

“…The notation N µ refers to the normal vector field to the singularity submanifold D X , and the jump • is defined as the difference between the values on each side of the singularity submanifold D X 2 . We refer to and Demoures, Gay-Balmaz, and Ratiu [2016] for the derivation of these jump terms in the unconstrained and constrained cases.…”