Babuška's paradox in a nonlinear bending-folding model

TL;DR

The paper analyzes Babuška's paradox in both linear and nonlinear bending–folding models, showing that polygonal domain approximations can fail variational convergence due to boundary-curvature terms. It develops remedies by relaxing boundary conditions (Γ-convergence) and by introducing slits/fattened crease lines to permit correct convergence, even with simplicial meshes. A key theoretical contribution is an angle–curvature framework and a nonexistence result for folded $H^2$-isometries along polygonal creases, explaining why polygonal approximations under full continuity fail. The numerical study, using a saddle-point/HHJ formulation and HHJ finite elements, demonstrates the practical impact: curved or vertex-continuous crease treatments converge correctly, while polygonal creases with full continuity exhibit locking and spurious stress concentrations.

Abstract

The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff--Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite element methods to avoid incorrect convergence is devised.

Figures (9)

• Figure 1: Interpolant of the exact deflection (left), approximation imposing the boundary condition along the entire boundary (middle), and approximation obtaind imposing the boundary condition in the corner points (right); pictures taken from BarTsc24.
• Figure 2: Crease line $\gamma$ (left), polygonal approximation $\gamma_m$ (middle), and fattened polygonal crease line ${\widehat{\gamma}}_m$ (right). In each case the lines define partitions of the domain $\omega$. For the fattened crease line ${\widehat{\gamma}}_m$ we have ${\widehat{\omega}}^i_m \subset \omega^i$, $i=1,2$.
• Figure 3: Bending of an elastic plate via compressing the plate at the end-points of a crease line. Curved crease line (left), singularities occur when a polygonal approximation is used (middle), these disappear if slits are introduced along the straight segments (right).
• Figure 4: Deformations and energy densities as coloring in the simulation of folding and bending experiments using the symmetry of the problem along the long midline. A correct discrete deformation is obtained for a curved approximation of the crease line (left), while the polygonal approximation leads to flatter pieces and a singularity (middle), introducing discontinuities along the straight segments provides another correct approximation (right).
• Figure 5: Folding angle between the jumping normals along a crease which partitions the deformed sheet into parts of opposite curvatures and defines two isometries of the subdomains. The induced Darboux frames specify curvature and torsion quantities for the deformed folding arc $u(\gamma)$.

Theorems & Definitions (9)

• Proposition 2.1: Compactness
• proof
• Proposition 2.2: Gamma convergence
• proof
• Corollary 2.3: Convergence of almost-minimizers
• Proposition 3.1: Folding angle, Horn23-pre
• Proposition 3.2: Nonexistence
• proof
• Remark 3.3