Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

Abstract

We present a new penalty term approximating the Ciarlet-Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet-Nečas condition. Moreover, the penalization can be chosen in such a way that for all low energy deformations, self-interpenetration is completely avoided already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.

Numerical solution of a bending-torsion model for elastic rods

“…Computationally, this case is particularly challenging since strong forces related to bending effects have to by compensated by repulsive forces related to the tangentpoint functional to avoid self-intersections. Regularization approaches guaranteeing global injectivity have been successfully implemented in different fields, see Krömer and Valdman [39] for an example in the context of elasticity.…”

Numerical approximation of von Kármán viscoelastic plates

“…. This yields (27) and also (31)(ii). Since also D is continuous with respect to ρ, (31)(i) holds as well.…”

Sobolev Gradients for the Möbius Energy

“…Potential applications for self-repulsive energies are manifold as they can be employed as barriers for shape optimization problems and physical simulation with self-contact: They arise, for instance, in mechanics [21,29,47,80,81,93] and in molecular biology [22,23,34,35,52,53]. The Möbius energy can also be considered as differentiable relaxation of curve thickness.…”