Modeling the flash-heat experiment on porous domains

Abstract

We discuss a mathematical model for the flash-heat experiment in homogeneous isotropic media. We then use this model to investigate the use of homogenization techniques in approximating models for interrogation via flash-heating in porous materials. We represent porous materials as both randomly perforated domains and periodically perforated domains.

Parameter estimation for the heat equation on perforated domains

“…Moreover, in our earlier work we did not consider random error which is often associated with experimental data. In this paper we present initial results suggesting that indeed the homogenization approximate mathematical models developed in [2] and [1] will perform well in an inverse problem setting. We do this with noisy simulated data in the context of mathematical and statistical parameter estimation procedures discussed and developed in [6], [13] and also in [4].…”

“…. n r , which are generated using methods described in [1], [2] and [18]. We assume that these pores do not intersect with each other nor the boundaries ofΩ.…”

“…In [2], we discussed the approximation of the heat equation on the random domain Ω with results derived from homogenization theory (see [2], [8], [11], [9], [12] and [10] and the references therein for details). Using the results of homogenization theory, we obtain the limit system…”

“…As noted, this model was too computationally intensive for direct use in parameter estimation or inverse problem techniques. We thus considered an approximation for a heat equation on a perforated domain which was derived through limiting processes in homogenization theories for the forward problem in [2]. Though the results of [2] were encouraging, good behavior of an approximation in the forward problem is not necessarily indicative of the behavior of the approximation in inverse problems.…”

“…We thus considered an approximation for a heat equation on a perforated domain which was derived through limiting processes in homogenization theories for the forward problem in [2]. Though the results of [2] were encouraging, good behavior of an approximation in the forward problem is not necessarily indicative of the behavior of the approximation in inverse problems. Moreover, in our earlier work we did not consider random error which is often associated with experimental data.…”