Continuous and discrete inf-sup conditions for surface incompressibility of a deformable continuum

TL;DR

This paper establishes continuous and discrete inf-sup conditions for surface inextensibility constraints in deformable materials, ensuring well-posedness and aiding finite element method convergence.

Contribution

It provides an elementary proof of the inf-sup condition for surface inextensibility and extends it to include volume preservation, with implications for numerical methods.

Findings

Proves well-posedness of surface inextensibility problems

Derives a modified discrete inf-sup condition for finite element methods

Ensures convergence of stabilized finite element schemes

Abstract

Surface incompressibility, also called inextensibility, imposes a zero-surface-divergence constraint on the velocity of a closed deformable material surface. The well-posedness of the mechanical problem under such constraint depends on an inf-sup or stability condition for which an elementary proof is provided. The result is also shown to hold in combination with the additional constraint of preserving the enclosed volume, or isochoricity. These continuous results are then applied to prove a modified discrete inf-sup condition that is crucial for the convergence of stabilized finite element methods.