A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries

TL;DR

The paper develops a general, rotation-free isogeometric thin shell formulation capable of handling arbitrary nonlinear material laws through either 2D surface formulations or through-thickness extraction from 3D models. It introduces a unified treatment of G^1 patch continuity, symmetry, and rotational boundary conditions using penalty and Lagrange multiplier methods, all formulated in curvilinear coordinates for computational efficiency. The framework encompasses Koiter, Canham, and Helfrich-type models and demonstrates robustness through a suite of linear and nonlinear benchmarks, including multi-patch and kinked geometries. A key finding is that the simple mixed Koiter model, which avoids thickness integration, achieves accuracy comparable to thickness-based 3D models, offering substantial efficiency gains. The work provides a practical, flexible, and extensible platform for rotation-free shell analysis with clear pathways to extend to liquid shells and anisotropic materials.

Abstract

This paper presents a general non-linear computational formulation for rotation-free thin shells based on isogeometric finite elements. It is a displacement-based formulation that admits general material models. The formulation allows for a wide range of constitutive laws, including both shell models that are extracted from existing 3D continua using numerical integration and those that are directly formulated in 2D manifold form, like the Koiter, Canham and Helfrich models. Further, a unified approach to enforce the $G^1$-continuity between patches, fix the angle between surface folds, enforce symmetry conditions and prescribe rotational Dirichlet boundary conditions, is presented using penalty and Lagrange multiplier methods. The formulation is fully described in the natural curvilinear coordinate system of the finite element description, which facilitates an efficient computational implementation. It contains existing isogeometric thin shell formulations as special cases. Several classical numerical benchmark examples are considered to demonstrate the robustness and accuracy of the proposed formulation. The presented constitutive models, in particular the simple mixed Koiter model that does not require any thickness integration, show excellent performance, even for large deformations.

Figures (13)

• Figure 1: Mapping between parameter domain $\mathcal{P}$, reference surface $\mathcal{S}_0$ and current surface $\mathcal{S}$ of a Kirchhoff--Love shell. The boundaries of the physical shell are shown by solid red lines. A shell layer is denoted by $\mathcal{S}^*$ and $\mathcal{S}_0^*$ in the current and reference configuration, respectively, and is shown by dashed blue lines.
• Figure 2: Edge rotation conditions: a. $G^1$-continuity constraint, b. fixed surface folds (e.g. V-shapes and L-shapes), c. symmetry (or clamping) constraints, d. symmetry constraint at a kink, e. rotational Dirichlet boundary condition. Surface edge $\mathcal{L}$, shown by a filled circle, is perpendicular to the plane and parallel to the inward pointing unit direction $\hbox{\boldmath$\tau$}$.
• Figure 3: Pinching of a hemisphere: a. Undeformed configuration with boundary conditions. Here, the blue curves denote the symmetry lines. b. Deformed configuration (scaled $50$ times), colored by radial displacement. Radial displacement at the point load vs. mesh refinement for c. the Koiter and d. the projected shell model, considering various NURBS orders.
• Figure 4: Simply supported plate under sinusoidal pressure: a. Initial configuration with boundary conditions, b. deformed configuration (scaled $10^4$ times) colored by the vertical displacement, c. displacement of the plate center normalized w.r.t. the analytical solution and d. relative error of the displacement.
• Figure 5: Pinching of the cylinder with rigid end diaphragms: a. Setup of the computation model, b. deformed shell (scaled $10^6$ times) colored by the radial displacement, c. normalized radial displacement at the point load and d. error w.r.t. the reference solution as the mesh is refined.