On Nonlinear Inverse Problems of Heat Transfer with Radiation Boundary Conditions: Application to Dehydration of Gypsum Plasterboards Exposed to Fire

A Belmiloudi,F Mahé

doi:10.1155/2014/634712

Abstract

The paper investigates boundary optimal controls and parameter estimates to the well-posedness nonlinear model of dehydration of thermic problems. We summarize the general formulations for the boundary control for initial-boundary value problem for nonlinear partial differential equations modeling the heat transfer and derive necessary optimality conditions, including the adjoint equation, for the optimal set of parameters minimizing objective functions J. Numerical simulations illustrate several numerical optimization methods, examples, and realistic cases, in which several interesting phenomena are observed. A large amount of computational effort is required to solve the coupled state equation and the adjoint equation (which is backwards in time), and the algebraic gradient equation (which implements the coupling between the adjoint and control variables). The state and adjoint equations are solved using the finite element method.

Highlights

We summarize the general formulations for the boundary control for initial-boundary value problem for nonlinear partial differential equations modeling the heat transfer and derive necessary optimality conditions, including the adjoint equation, for the optimal set of parameters minimizing objective functions J
A large amount of computational effort is required to solve the coupled state equation and the adjoint equation, and the algebraic gradient equation
One of the building materials presenting the best fire resistance is gypsum plasterboard, which in turn is due to the dehydration phenomenon

Summary

Introduction and Mathematical Setting of the Problem

One of the building materials presenting the best fire resistance is gypsum plasterboard, which in turn is due to the dehydration phenomenon This material presents the particularity to undergoing two chemical reactions of dehydration during its heating. A priori, most researchers who have worked on the modeling of the behavior of gypsum board (literature in the public domain in this field is sparse; see, e.g., [3,4,5, 7, 11, 15]), have assumed the convective heat transfer coefficients h0 and h1 and the relative emissivity ε0 as constants or/and neglected the relative emissivity ε1 on cold surface. The resultant emissivities and convective heat transfer coefficients depend on temperature and large uncertainties exist in regard to the quality of the data reported. It is important to have a consistent set of values for these data

The Inverse Problem Formulation

Numerical Analysis and Simulations

Method Constant step Decreasing step

Findings

Commentary and Conclusion