Two-Scale Finite Element Approximation of a Homogenized Plate Model

TL;DR

The work addresses numerically approximating a two-scale, homogenized plate model obtained as a $\Gamma$-limit of 3D nonlinear elasticity, with a bending energy that is quadratic in the second fundamental form. It combines a heterogeneous multiscale method (HMM) for the microscopic cell problems with a nonconforming Discrete Kirchhoff Triangle (DKT) discretization for the macroscopic isometry-constrained problem, and proves convergence of the fully discrete scheme to the continuous two-scale limit via $\Gamma$-convergence arguments. The key contributions include a rigorous discretization framework for the unit-cell problem, explicit error estimates showing first-order (and, under a consistency condition, second-order) convergence of the microscopic corrector and the effective tensor $Q^{2,\gamma}$, and a robust macroscopic solver using IPOPT to handle the isometry constraint. The numerical experiments validate the theoretical results, demonstrate convergence rates, and qualitatively compare the simulations with experiments on microstructured sheets of paper, highlighting the method’s effectiveness in predicting anisotropic bending behavior due to microstructure.

Abstract

This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić in 2014. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proven for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.

Figures (5)

• Figure 1: Thin plates with thickness $\delta$ and periodic in-plane microstructure with size $\varepsilon$, left: homogeneous microstructure, right: Macroscopically varying microstructure.
• Figure 1: Plots of the different components of the homogenized tensor $\mathcal{C}^{2,\gamma}$ for varying $\gamma$.
• Figure 2: Top row: numerically computed deformed configurations of a plate, clamped on the left side, under a uniform vertical load, with homogeneous material, with Lamé constants scaled with $1.0$ (left), $0.286$ (middle) and $0.02$ (right). Bottom row: photos of physical deformations of paper. Left: thick paper ($300g/m^2$) glued on thinner paper ($120g/m^2$). Middle: only thin paper ($120 g/m^2$).
• Figure 3: Top row: deformed configurations under a uniform vertical load for stripe-type microstructures (left and middle) and a microstructure with radial rays (right). Bottom row: photos of physical experiments with stripes of thicker paper ($300 g/m^2$, left and middle: red, right: yellow) glued on thinner paper ($120g/m^2$, left and middle: green, right: purple).
• Figure 4: Top left: deformed configuration under compression enforced by the boundary conditions \ref{['eq:bdexpA']} with soft material in blue and hard material in orange, bottom left: corresponding experiment with stripes of thicker paper ($300g/m^2$, orange) glued on thin paper ($120g/m^2$blue); top middle: deformed configuration for a truss type microstructure under the same boundary conditions, with soft material in red and hard material in white, top right material distribution on the rescaled microscopic cell, bottom middle: a correspondingly deformed sheet of paper (red) with glued layer structure of the thicker paper (white), bottom right: undeformed experimental plate configuration viewed from above.

Theorems & Definitions (8)

• Proposition 1
• Proof 1
• Proposition 1
• Proof 2
• Theorem 1
• Proof 3
• Remark 1
• Remark 2