Shape optimisation of nonlinear Naghdi shells on discrete geometries

2026-08-10

AI Summary

This work addresses the inability of traditional linear shell models to accurately capture stiffness variations under large deformations, which often leads to distorted shape optimization results. For the first time, a fully geometrically nonlinear Naghdi shell formulation is implemented directly on discrete triangular meshes for shape optimization, eliminating the need for mid-surface parametrization and circumventing limitations inherent in isogeometric analysis. The approach integrates a five-parameter nonlinear shell model—stabilized via selectively reduced integration—with automated residual and tangent operator generation in Firedrake, adjoint-based sensitivity analysis enabled by automatic differentiation, and the Fireshape/ROL trust-region optimizer. The framework successfully reproduces benchmark cases from Sze and Abaqus, accurately capturing stiffness stiffening effects, and achieves an 87% reduction in elastic strain energy for a sheet-metal bracket and a 78% decrease in average deflection for a hemicylindrical shell.

Abstract

A thin shell carries load through its shape and, at finite deflections, its stiffness changes with the load itself, so a shape optimum found with a linear model can be far from optimal. We present an automated framework that embeds the fully geometrically nonlinear shell response in the shape-optimisation loop. The forward model, a five-parameter nonlinear Naghdi shell stabilised against locking by partial selective reduced integration, operates directly on a discrete (faceted) triangulation with a numerically recovered director field -- dispensing with the exact mid-surface parameterisation of isogeometric approaches, a chart that ceases to exist once the geometry itself is the design variable. Implemented in Firedrake, the model generates its residual, consistent tangent and adjoint automatically; shape derivatives, computed by algorithmic differentiation through the full load-continuation solve, drive the Fireshape/ROL trust-region optimiser. The forward solver reproduces the Sze/Abaqus benchmark for a clamped semi-cylindrical shell under a point load, capturing the progressive stiffening that a geometrically linear model cannot reproduce. The optimisation is validated against the COMSOL benchmark, a sheet-metal bracket under bending: the framework develops the same off-mid-plane corrugation mechanism and attains an 87% reduction of elastic strain energy within the prescribed displacement budget, matching the benchmark's magnitude and area change. Applied to the curved semi-cylinder, it forms a smooth stiffening crease that reduces the shell's average deflection under load by 78%.

Problem

• shape optimisation
• nonlinear shells
• geometric nonlinearity
• discrete geometry
• Naghdi shell

Innovation

• nonlinear Naghdi shell
• shape optimization
• discrete geometry
• algorithmic differentiation
• locking-free formulation