Finite element simulation of nonlinear bending models for thin elastic rods and plates

TL;DR

The paper develops and analyzes finite element methods for nonlinear bending models of thin elastic rods and plates, deriving dimensionally reduced theories from 3D hyperelasticity via Γ-convergence under cubic energy scaling. It introduces constrained finite element discretizations that enforce inextensibility and isometry, and couples these with energy-stable gradient-flow iterations to compute low-energy configurations. Convergence of the discrete schemes to the continuous models is established, and the linear systems arising in the iterative schemes are addressed through kernel-based reductions and preconditioning strategies. The work extends to bilayer plates, self-avoiding curves, knots, and the Föppl–von Kármán framework, and demonstrates numerical experiments on knots relaxation, Möbius strip singularities, and actuated bilayer plates, highlighting the practical relevance for complex bending phenomena.

Abstract

Nonlinear bending phenomena of thin elastic structures arise in various modern and classical applications. Characterizing low energy states of elastic rods has been investigated by Bernoulli in 1738 and related models are used to determine configurations of DNA strands. The bending of a piece of paper has been described mathematically by Kirchhoff in 1850 and extensions of his model arise in nanotechnological applications such as the development of externally operated microtools. A rigorous mathematical framework that identifies these models as dimensionally reduced limits from three-dimensional hyperelasticity has only recently been established. It provides a solid basis for developing and analyzing numerical approximation schemes. The fourth order character of bending problems and a pointwise isometry constraint for large deformations require appropriate discretization techniques which are discussed in this article. Methods developed for the approximation of harmonic maps are adapted to discretize the isometry constraint and gradient flows are used to decrease the bending energy. For the case of elastic rods, torsion effects and a self-avoidance potential that guarantees injectivity of deformations are incorporated. The devised and rigorously analyzed numerical methods are illustrated by means of experiments related to the relaxation of elastic knots, the formation of singularities in a Möbius strip, and the simulation of actuated bilayer plates.

Figures (14)

• Figure 1: The mathematical description of large bending deformations of thin objects requires the use of appropriate geometric quantities: deformed rod with circular cross-section together with an orthonormal frame that allows to measure bending and torsion effets (left); deformation of a flat plate that preserves angle and length relations (right).
• Figure 2: Continuously differentiable, piecewise cubic curves are defined by positions and tangent vectors at nodes $z_0<z_1<\dots<z_N$.
• Figure 3: Discrete deformations defined by discrete Kirchhoff triangles are defined by positions of nodes and tangent vectors at the displaced nodes.
• Figure 4: Degrees of freedom of piecewise linear, continuous and piecewise cubic, continuously differentiable finite element functions. Filled dots indicate function values and circles evaluations of derivatives.
• Figure 5: Degrees of freedom of finite element spaces using reduced cubic polynomials and quadratic vector fields. Filled dots indicate function values, circles evaluations of derivatives, and squares vectorial function values. One degree of freedom is eliminated from the set of cubic polynomials.

Theorems & Definitions (15)

• Remark 2.1
• Proposition 3.1: Convergent approximation
• proof : Proof (sketched)
• Remark 3.2
• Theorem 3.3: Convergent approximation
• proof : Proof (sketched)
• Theorem 4.2: Convergent iteration
• proof
• Proposition 4.4: Convergent iteration
• proof