Bayesian identification of the random elasticity field of a heterogeneous microstructure
Résumé
This paper presents a statistical inverse method for identifying the mechanical properties of a heterogeneous microstructure, which is modeled by a random elastic media. To this end, several experimental tests are performed on a series of specimens made of the same material. For each experiment, the applied force field is supposed to be imposed, and the induced displacement field is measured on the contours of the specimens only. In parallel, for given properties of the random media, it is possible to simulate independent realizations of the elasticity random field, and to approximate (using the Finite Element Method) the displacements that are induced by the experimental force field. Based on the comparison of the statistical properties of the displacement fields on the contours of the specimens in the experimental and the simulated cases, a method is thus proposed to identify the most likely properties of the random media characterizing the heterogeneous microstructure (such as the mean elasticity field, the dispersion and the correlation lengths). It should be noted that the elasticity field is not a real-valued random field, but a tensor-valued random field, and that the different components of this random field cannot be identified separately due to algebraic constraints. Additionally, the quantity of interest on which the identification procedure is based is not a scalar, but a high-dimension vector. This requires the introduction of dedicated reduction techniques. Last but not least, the number of code evaluations that is generally required for the identification procedure can quickly become burdensome. To circumvent this problem, statistical extrapolation techniques and iterative procedures will be presented to maximize the precision of the identification at a reduced computational cost. Validations of the procedure are eventually presented using simulated data in two and three dimensions.