A probabilistic multi-scale model for polycrystalline MEMS resonators

The size of micro-electro-mechanical systems (MEMS) is only one or two orders of magnitude higher than the size of their micro-structure, i.e. their grain size. As a result, the structural properties exhibit a scatter. As an example we study the beam resonator illustrated in Fig. 1(a), made of poly-silicon material, in which each grain has a random orientation. Solving the problem with a full direct numerical simulation combined to a Monte-Carlo method allows the
probability density function to be computed as illustrated in Fig. 1(b). However this methodology
is computationally expensive due to the number of degrees of freedom required to study one sample, motivating the development of a non-deterministic 3-scale approach [3]. In a multiscale approach, at each macro-point of the macro-structure, the resolution of a microscale
boundary value problem relates the macro-stress tensor to the macro-strain tensor. At the micro-level, the macro-point is viewed as the center of a Representative Volume Element (RVE). The resolution of the micro-scale boundary problem can be performed using finite-element simulations, as in the computational homogenization framework, e.g. [2]. However,
to be representative, the micro-volume-element should have a size much bigger than the microstructure size.
In the context of the MEMS resonator, this representativity is lost and Statistical Volume Elements (SVE) are considered. These SVEs are generated under the form of a Voronoi tessellation with a random orientation for each silicon grain. Hence, a Monte-Carlo procedure
combined with a homogenization technique allows a distribution of the material tensor at the
meso-scale to be estimated. The correlation between the meso-scale material tensors of two
SVEs separated by a given distance can also be evaluated.
A generator at the meso-scale based on the spectral method [4] is implemented. The generator
[3] accounts for a lower bound [1] of the meso-scale material tensor in order to ensure the
existence of the second-order moment of the Frobenius norm of the generated material tensor
inverse [5].
Using the random meso-scale field obtained with the meso-scale generator, which accounts
for the spatial correlation, a Monte-Carlo method can be used at the macro-scale to predict the
probabilistic behavior of the MEMS resonator.