Continuous and discrete inf-sup conditions for surface incompressibility of a deformable continuum
This work provides foundational theoretical guarantees for numerical methods in mechanics of thin structures and biological membranes, but is incremental as it extends known inf-sup theory to a specific constraint.
The paper provides an elementary proof of the inf-sup condition for surface incompressibility (inextensibility) in deformable continua, and extends it to include volume preservation. These results are used to prove a modified discrete inf-sup condition essential for the convergence of stabilized finite element methods.
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.