Stochastic multiscale method applied to thermo-elasticity of polycrystalline micro-structures

A stochastic 3-scale approach is developed in order to predict the probabilistic behavior of micro-resonators made of polycrystalline materials. In this method, stochastic volume elements (SVEs) [1] are defined from Voronoï tessellations using experimental measurements of the grain size, orientation, and surface roughness [2]. For each SVE realization, the mesoscopic apparent thermo-elastic properties such as elasticity tensor, thermal conductivity tensor, and thermal dilatation tensor are extracted using a coupled homogenization theory [3, 4]. A stochastic model is then built from the homogenized properties extracted from Voronoï tessellations using a moving window technique in order to generate spatially correlated meso-scale random fields. These random fields are then used as input for stochastic finite element simulations. As a result, the probabilistic distribution of micro-resonator properties can be extracted. The applications are two-fold: either a stochastic thermo-elastic homogenization is coupled to thermo-elastic 3D models of the micro-resonator in order to extract the probabilistic distribution of the Quality factor of the micro-resonators [5], or a stochastic second-order mechanical homogenization is coupled to a plate model of the micro-resonator in order to extract the effect of the uncertainties related to the surface roughness of the polycrystalline structures [1].

References

[1] Lucas, V., Golinval, J.-C., Voicu, R., Danila, M., Gravila, R., Muller, R., Dinescu, A., Noels, L., & Wu, L. (in press). Propagation of material and surface profile uncertainties on MEMS micro-resonators using a stochastic second-order computational multi-scale approach. International Journal for Numerical Methods in Engineering.
[2] Ostoja-Starzewski, M., Wang, X. (1999) Stochastic finite elements as a bridge between random material microstructure and global response, Computer Methods in Applied Mechanics and Engineering, 168, 35-49, 1999
[3] Temizer, I., Wriggers, P. (2011) Homogenization in finite thermoelasticity, Journal of the Mechanics and Physics of Solids 59 (2), 344-372
[4] Nguyen, V. D., Wu, L., Noels, L. (in press). Unified treatment of boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method. Computational Mechanics.
[5] Wu, L., Lucas, V., Nguyen, V. D., Golinval, J.-C., Paquay, S., & Noels, L. (2016) A Stochastic Multi-Scale Approach for the Modeling of Thermo-Elastic Damping in Micro-Resonators. Computer Methods in Applied Mechanics & Engineering, 310, 802-839.