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

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.

💡 Research Summary

The paper introduces a novel fourth‑order multi‑scale (FOMS) computational framework for the analysis of composite Kirchhoff plates with highly periodic microstructures. Recognizing that traditional multi‑scale methods (typically second‑ or third‑order) are insufficient for fourth‑order governing equations, the authors develop a systematic asymptotic expansion in the small periodicity parameter ε, separating the macroscopic variable x and the microscopic variable y = x/ε. The displacement field ωε(x) is expanded as

ωε(x)=ω⁽⁰⁾(x)+ε ω⁽¹⁾(x,y)+ε² ω⁽²⁾(x,y)+ε³ ω⁽³⁾(x,y)+ε⁴ ω⁽⁴⁾(x,y)+O(ε⁵).

By inserting this expansion into the Kirchhoff plate equation

∂i∂j(D⁽ε⁾_{ijkl}(x) ∂k∂l ωε)=q,

and applying the chain‑rule operators B₀,…,B₄, a hierarchy of equations for each power of ε is obtained. The leading term ω⁽⁰⁾ is shown to be independent of y, yielding the homogenized macroscopic solution. The next two orders vanish (ω⁽¹⁾=0), while the ε⁻² and ε⁻¹ orders give rise to second‑ and third‑order cell functions N²(y) and N³(y) that satisfy periodic boundary value problems on the unit cell Y. These cell functions capture the influence of the microscopic stiffness tensor D_{ijkl}(y) on the macroscopic behavior.

At order ε⁰ the macroscopic homogenized equation is derived:

−∂i∂j( bD_{ijkl} ∂k∂l ω⁽⁰⁾ ) = q,

with effective stiffness

bD_{ijkl}=|Y|⁻¹∫_Y