A fourth-order multi-scale computational method and its convergence analysis for composite Kirchhoff plates with microscopic periodic configurations

TL;DR

The paper introduces a fourth-order multi-scale (FOMS) framework for simulating composite Kirchhoff plates with highly periodic microstructures. It derives a detailed multiscale expansion, solves a hierarchy of cell problems, and proves an explicit $H^2$-norm convergence rate for the fourth-order solution. An efficient FE-based algorithm using Morley and HCT spaces is developed to compute the macroscopic solution and fourth-order corrections, with extensive numerical experiments confirming high accuracy and significant computational savings over direct DNS. The approach enables robust, high-fidelity analysis of complex plate composites and shows promise for large-scale and future dynamic or nonlinear extensions.

Abstract

The Kirchhoff plate model plays a vital role in modeling, computing and analyzing the mechanical behaviors of thin plate structures. This study propose a novel fourth-order multi-scale (FOMS) computational method for high-accuracy and efficient simulation of composite Kirchhoff plates with highly periodic heterogeneities. At first, two-scale asymptotic expansion theory is employed to establish the high-accuracy fourth-order multi-scale computation model with novel fourth-order correctors for composite Kirchhoff plates, which are governed by fourth-order partial differential equation (PDE) with periodically oscillatory and highly discontinuous coefficients. Then, the locally point-wise error analysis is derived to theoretically illustrate the local balance preserving of fourth-order multi-scale model enabling high-accuracy multi-scale computation. Furthermore, a global error estimation with an explicit order for fourth-order multi-scale solutions is first demonstrated under appropriate assumptions. In contrast to the second- and third-order multi-scale solutions, only the fourth-order one is capable of providing an explicit error order estimate. Additionally, an efficient numerical algorithm is developed to conduct high-accuracy simulation for heterogeneous plate structures. Extensive numerical examples are provided to confirm the theoretical results for the computational convergence and accuracy of the proposed method. This work offers a higher-order (fourth-order) multi-scale computational framework that enables robust simulation and high-accuracy analysis to composite Kirchhoff plates.

Figures (15)

• Figure 1: The symmetry and boundaries of microscopic PUC $\mathbf{Y}$.
• Figure 2: (a) The multi-scale structure of composite plate $\Omega$; (b) the microscopic unit cell $\mathbf{Y}$; (c) the macroscopic homogenization structure.
• Figure 3: The transverse displacement of composite Kirchhoff plate computed by Morley finite element: (a) $\omega^{(0)}$; (b) $\omega^{(2\epsilon)}$; (c) $\omega^{(3\epsilon)}$; (d) $\omega^{(4\epsilon)}$; (e) $\omega^\epsilon_{\text{D}}$.
• Figure 4: The transverse displacement of composite Kirchhoff plate computed by HCT finite element: (a) $\omega^{(0)}$; (b) $\omega^{(2\epsilon)}$; (c) $\omega^{(3\epsilon)}$; (d) $\omega^{(4\epsilon)}$; (e) $\omega^\epsilon_{\text{D}}$.
• Figure 5: The $x_1$-direction gradient of transverse displacement of composite Kirchhoff plate computed by Morley finite element: (a) $\frac{\partial\omega^{(0)}}{\partial x_1}$; (b) $\frac{\partial\omega^{(2\epsilon)}}{\partial x_1}$; (c) $\frac{\partial\omega^{(3\epsilon)}}{\partial x_1}$; (d) $\frac{\partial\omega^{(4\epsilon)}}{\partial x_1}$; (e) $\frac{\partial\omega^\epsilon_{\text{D}}}{\partial x_1}$.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2