Finite element discretizations of bending plates with prestrained microstructure

TL;DR

The paper develops a rigorous fully discrete $\Gamma$-convergence framework for a prestrained, microstructured bending plate model. It combines a macro discretization using Discrete Kirchhoff Triangle elements with a micro discretization of the corrector problems on axis-aligned grids, deriving effective quantities $Q_{\text{hom}}^{\gamma}$ and $B_{\text{eff}}^{\gamma}$ via correctors on the RVE and reformulating the 2D energy in terms of the Hessian to facilitate numerics. The main result shows that the fully discrete energy converges to the continuous energy as $(h,H)\to(0,0)$ under mild regularity and quadrature assumptions, with corollaries guaranteeing convergence when the micro-scale and macro-scale discretizations are refined sequentially and that these limits commute. This extends prior results to prestrained composites and accommodates local microstructure variation, possibly discontinuous, while allowing flexibility in the microdiscretization method beyond first-order finite elements.

Abstract

We investigate a finite element discretization of an elastic bending-plate model with an effective prestrain. The model has been obtained via homogenization and dimension reduction by Bönlein at al. (2023). Its energy functional is the $Γ$-limit of a three-dimensional nonlinear microstructured elasticity functional. In the derived effective model, the microstructure is incorporated as a local corrector problem, a system of linear elliptic partial differential equations posed on a three-dimensional representative volume element. The discretization uses Discrete Kirchhoff Triangle elements for the macroscopic bending-plate problem on a mesh of scale $H$, and first-order Lagrange elements for the microscopic corrector problem on an axis-aligned mesh of scale $h$. We show that the discretized model $Γ$-converges to the continuous one as $(h,H)\to 0$,provided that there exists a microstructure mesh such that the elasticity tensor is Lipschitz continuous on each mesh element. This extends earlier results by Rumpf et al. (2024) to prestrained composites. Our argument does not require any rate of convergence for the microscopic discretization error. As a corollary, we also obtain convergence when $h \to 0$ and $H \to 0$ consecutively, and we prove that these limit processes commute.

Figures (3)

• Figure 1: The two-dimensional domain $S$ and the representative volume element (RVE) $\Box$ are used to model the microstructure of the material on three-dimensional reference domain $\Omega_\eta$.
• Figure 2: Example of an admissible hexahedral partition $\mathcal{T}_h^\Box$. The mesh resolves jumps in the coefficients. The coefficients can vary within each element, but have to remain Lip-schitz continuous.
• Figure 3: Degrees of freedom of the deformation space $\mathcal{W}_H$ and the image of $\nabla_H$ in the rotation space $\Theta_H$

Theorems & Definitions (34)

• Remark 2.3
• Lemma 2.5: Existence and uniqueness of a corrector field boehnlein2023homogenized
• Definition 2.6: Effective bending stiffness $Q_{\textup{hom}}^{\gamma}$
• Definition 2.7: Effective prestrain $B_{\textup{eff}}^{\gamma}$
• Lemma 2.8
• Lemma 2.9
• proof
• Lemma 3.1: Existence and energy estimate
• proof
• Definition 3.2: Approximate effective bending stiffness $Q_{\textup{hom}}^{\gamma,h}$