A Shape Newton Scheme for Deforming Shells with Application to Capillary Bridges

TL;DR

This paper introduces a second-order numerical scheme based on a shape Newton method to accurately compute capillary bridges between complex solids by directly minimizing interface energy using shape Hessians.

Contribution

It develops a novel shape Newton scheme with explicit shape Hessian representation for shells, avoiding volume reformulation and curvature approximation.

Findings

Explicit shape Hessian for shell functionals derived

Effective combination of triangulated interface and level set for complex geometries

Finite element implementation in FEniCS successfully computes capillary bridges

Abstract

We present a second order numerical scheme to compute capillary bridges between arbitrary solids by minimizing the total energy of all interfaces. From a theoretical point of view, this approach can be interpreted as the computation of generalized minimal surfaces using a Newton-scheme utilizing the shape Hessian. In particular, we give an explicit representation of the shape Hessian for functionals on shells involving the normal vector without reverting back to a volume formulation or approximating curvature. From an algorithmic perspective, we combine a resolved interface via a triangulated surface for the liquid with a level set description for the constraints stemming from the arbitrary geometry. The actual shape of the capillary bridge is then computed via finite elements provided by the FEniCS environment, minimizing the shape derivative of the total interface energy.