A Multi-scale Finite Element Approach for the Random Mechanical Response of Honeycomb-cored Structures

This paper studies the uncertainty in the mechanical response of foam-filled honeycomb cores by means of a computational multi-scale approach. A finite element framework is adopted to determine the response of a periodic arrangement of aluminum honeycomb core filled with PVC foam. By considering uncertainty in the geometric properties of the microstructure, a significant computational cost is added to the solution of a large set of microscopic equilibrium problems. In order to tackle this high cost, we combine two strategies. Firstly, we make use of symmetry conditions present in a representative volume element of material. Secondly, we build a statistical approximation to the output of the computer model, known as a Gaussian process emulator. Following this double approach, we are able to reduce the cost of performing uncertainty analysis of the mechanical response. In particular, we are able to estimate the 5-th, 50-th, and 95-th percentile of the mechanical response without resorting to more computationally expensive methods such as Monte Carlo simulation. Nomenclature C(·, ·) covariance structure G virtual work functional m∗∗(·) predictive mean uµ local displacement field ũµ displacement fluctuation field x arbitrary point of the macroscopic continuum y arbitrary point of the microscopic domain ε macroscopic or homogenised strain tensor εµ microscopic strain tensor field η(·) emulator σ macroscopic or homogenised stress tensor σµ microscopic stress tensor Ωµ Microscopic domain I