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

TL;DR

The paper develops a coordinate-free, mixed-hybrid finite element approach for Kirchhoff-Love shells that treats the bending moment tensor $oldsymbol m_ ext{Γ}$ as a primary variable alongside displacement, enabling equal-order $C^0$ interpolation and avoiding $C^1$ continuity. Boundary conditions are fully incorporated via Lagrange multipliers on edges, and element-wise static condensation via the Schur complement yields a smaller, positive-definite global system in $(oldsymbol u,oldsymbol ω_t)$. The method delivers higher-order convergence, with $ ext{O}(p+1)$ in the primary fields and $ ext{O}(p)$ for derived stress resultants, demonstrated across classical benchmarks and new smooth test cases. The coordinate-free formulation and reliance on standard FE spaces offer a practical path to high-fidelity shell simulations, with promising extensions to trace- and isogeometric-based geometries.

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.

Figures (24)

• Figure 1: Some vectors on an arbitrary surface $\Gamma$ with boundary $\partial\Gamma$. The normal vector $\boldsymbol n$ on the inner point $\boldsymbol P_1$ is shown in blue. The normal vector $\boldsymbol n$, the tangential vector $\boldsymbol t$ and the conormal vector $\boldsymbol q$ on the boundary point $\boldsymbol P_2$ are shown in blue, red, and green, respectively.
• Figure 2: Sketch of the kinematic situation between the undeformed middle surface $\Gamma$ and the deformed middle surface $\bar{\Gamma}$ of a shell. The displacement $\boldsymbol u_{\Omega}$ of a point $\boldsymbol P$ in the continuum $\Omega$ and other related quantities are depicted.
• Figure 3: Boundary quantities at an arbitrary point on the boundary $\partial\Gamma$ with displacement and force components in terms of the local triad $\boldsymbol t$, $\boldsymbol n$, and $\boldsymbol q$, and rotation and moment around the tangential direction. Normal displacements and Kirchhoff forces are depicted on the corners $C_1$ and $C_2$. Quantities in tangential, normal, and conormal directions are shown in red, blue, and green, respectively.
• Figure 4: Moments $m_{\boldsymbol q}(C_i^-)$ and $m_{\boldsymbol q}(C_i^+)$ around their conormal directions at the two sides of some arbitrary corner $C_i$, respectively, and the resulting Kirchhoff force.
• Figure 5: For some surface $\Gamma$, (a) an example surface mesh composed by $4 \times 4$ cubic elements representing $\Gamma^h$ is shown, (b) the discretized boundary $\partial\Gamma^h$, and (c) the skeleton $\Psi^h$ composed by the element edges.