Inverse Modeling: Glue-Package-Die Problem

Abstract

In mathematical modeling, the physical or geometrical parameters are often affected by uncertainties. For example, imperfections of an industrial manufacturing generate undesired variations in the produced devices. We consider an uncertainty quantification, where parameters are replaced by random variables. Consequently, the probability distributions of the parameters have to be predetermined as an input to the stochastic model. However, the variability of input parameters is often not directly accessible by measurements, whereas the output quantities are available. We investigate a problem from nanoelectronics: a piece of glue connecting a die and a package. A randomness in both the formation and the quality of the piece of glue cause uncertain geometrical parameters and material parameters. The task consists in a fitting of input probability distributions for the random parameters to measurements of the output. This problem can be seen as a form of a stochastic inverse problem. The cumulative distribution function is approximated by a piecewise linear function. We apply a minimization, which yields a nonlinear least squares problem. Numerical results are illustrated for this test problem.