Optimal Reconstruction of Material Properties in Complex Multiphysics Phenomena

TL;DR

This paper presents an efficient PDE-constrained optimization method to reconstruct temperature-dependent viscosity in complex thermo-fluid systems, using level set integrals for gradient computation and validated through computational tests.

Contribution

It introduces a novel gradient computation technique involving level set integrals for PDE-constrained optimization in thermo-fluid inverse problems.

Findings

The method accurately reconstructs viscosity from noisy temperature data.

Level set integral techniques are effective for gradient evaluation.

Computational tests validate the approach's efficiency and robustness.

Abstract

We develop an optimization-based approach to the problem of reconstructing temperature-dependent material properties in complex thermo-fluid systems described by the equations for the conservation of mass, momentum and energy. Our goal is to estimate the temperature dependence of the viscosity coefficient in the momentum equation based on some noisy temperature measurements, where the temperature is governed by a separate energy equation. We show that an elegant and computationally efficient solution of this inverse problem is obtained by formulating it as a PDE-constrained optimization problem which can be solved with a gradient-based descent method. A key element of the proposed approach, the cost functional gradients are characterized by mathematical structure quite different than in typical problems of PDE-constrained optimization and are expressed in terms of integrals defined over the level sets of the temperature field. Advanced techniques of integration on manifolds are required to evaluate numerically such gradients, and we systematically compare three different methods. As a model system we consider a two-dimensional unsteady flow in a lid-driven cavity with heat transfer, and present a number of computational tests to validate our approach and illustrate its performance.