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 derived from a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. At each nonlinear iterate, the residual is the exact first variation of this functional with the adaptive Nitsche weight frozen, while the correction uses a symmetric, coercive approximation of its Hessian. 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 quasi-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 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. Local mesh refinement removes this bound: we verify the recovery of optimal rates numerically for first-order elements, and prove that the unfitted discretisation inherits the rate the underlying refinement attains, with cut-independent constants.
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BibTeX
@misc{wichrowski2026unified,
title = {{A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities}},
author = {Michał Tomasz Wichrowski and Ella Godiva Noomen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.02334},
eprint = {2607.02334},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.02334},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities
AU - Michał Tomasz Wichrowski
AU - Ella Godiva Noomen
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2607.02334
UR - https://arxiv.org/abs/2607.02334
AB - We present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is derived from a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. At each nonlinear iterate, the residual is the exact first variation of this functional with the adaptive Nitsche weight frozen, while the correction uses a symmetric, coercive approximation of its Hessian. 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 quasi-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 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. Local mesh refinement removes this bound: we verify the recovery of optimal rates numerically for first-order elements, and prove that the unfitted discretisation inherits the rate the underlying refinement attains, with cut-independent constants.
ER -