Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Institution: Technische Universität Dresden
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Awards: G1595172143, G5511866461
OpenAlex ID: W7084116773
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