A multisymplectic integrator for elastodynamic frictionless impact problems

Abstract

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational description of nonsmooth Lagrangian continuum mechanics, combined with generalized Lagrange multipliers to handle inequality constraints. We test the resulting explicit integrator for the longitudinal impact of two elastic linear bar models, and for the collision of a nonlinear geometrically exact beam model with a rigid plane. Numerical simulations for various physical parameters are presented to illustrate the behavior and performance of our approach.

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

“…Thus, we can apply Theorem 5. We give below only the equations at contact time; see Demoures et al [30] for the full discrete equations of motion.…”

“…We refer to Demoures et al [30] for further numerical tests and discussions of the results. The configuration of the beam in contact, without friction, is illustrated in Figure 7.…”

“…This rule is applied to the examples in Section 5. See also Demoures et al [30], where we give a detailed algorithm in two concrete examples. 4.2.5.…”

“…This Lagrangian differs from the one in Demoures et al [29] which is based only on triangles j a . The choice (87) greatly improves the behavior of the integrator, especially concerning the boundaries (see Demoures et al [30]). …”

“…The discrete action functional S ns d (φ 1d , φ 2d ) is of the form (76). Given the discrete impenetrability condition See Demoures et al [30] for the full discrete equations of motion outside of contact.…”