Higher-order, mixed-hybrid finite elements for Kirchhoff-Love shells

TL;DR

This paper introduces a mixed-hybrid finite element method for Kirchhoff-Love shells that allows the use of classical higher-order elements with reduced continuity requirements, enabling efficient and accurate numerical analysis.

Contribution

The novel mixed-hybrid formulation reduces continuity constraints and enables equal-order interpolation of displacements and moments in Kirchhoff-Love shell analysis.

Findings

Achieves optimal higher-order convergence rates with smooth solutions

Enables element-wise static condensation of moments

Proposes new benchmark test cases for shell analysis

Abstract

A novel mixed-hybrid method for Kirchhoff-Love shells is proposed that enables the use of classical, possibly higher-order Lagrange elements in numerical analyses. In contrast to purely displacement-based formulations that require higher continuity of shape functions as in IGA, the mixed formulation features displacements and moments as primary unknowns. Thereby the continuity requirements are reduced, allowing equal-order interpolations of the displacements and moments. Hybridization enables an element-wise static condensation of the degrees of freedom related to the moments, at the price of introducing (significantly less) rotational degrees of freedom acting as Lagrange multipliers to weakly enforce the continuity of tangential moments along element edges. The mixed model is formulated coordinate-free based on the Tangential Differential Calculus, making it applicable for explicitly and implicitly defined shell geometries. All mechanically relevant boundary conditions are considered. Numerical results confirm optimal higher-order convergence rates whenever the mechanical setup allows for sufficiently smooth solutions; new benchmark test cases of this type are proposed.