A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities

Abstract

We present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is the stationarity condition of a augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. The residual and the (symmetrised) tangent follow from this functional by successive variations. Automatic differentiation (AD) generates the first Piola--Kirchhoff stress tensor and the elasticity tensor directly from the scalar energy density, avoiding manual re-derivation when exchanging hyperelastic models. To our knowledge, this is the first unfitted finite-strain scheme combining an energy-only, model-independent construction with AD and an accuracy analysis at unfitted boundaries. Analysis of the linearised problem solved at each Newton step establishes cut-independent coercivity, continuity, and an $O(h^{-2})$ condition number bound, yielding a quasi-optimal convergence theorem for regular solutions through the Brezzi--Rappaz--Raviart framework. Numerically, the method attains optimal $h$-convergence for linear, quadratic, and cubic elements on a smooth test case. Furthermore, we quantify the method's accuracy limit at mixed Dirichlet--Neumann junctions using the Kolosov--Muskhelishvili characteristic equation. The exact solution's corner singularity caps the convergence rate identically for fitted and unfitted methods. We demonstrate that local mesh refinement removes this bound, with the unfitted discretisation inheriting the recovered optimal rates and cut-independent constants.