Multidimensional inverse heat conduction problems solution via lagrange theory and model size reduction techniques

Abstract

This article is concerned with inverse heat conduction problems (IHCP) involving large-scale linear systems (i.e. multidimensional systems) with unknown sources or boundary conditions. The inverse problem is stated as an optimisation problem with a regularised quadratic objective functional, and it is solved using Lagrange theory. We demonstrate that the IHCP solution is obtained by simple time integration of a state-variable model: the inverse model, which is analytically derived from the so called direct and adjoint problems. In addition, we show that the number of differential equations in the inverse model can be strongly reduced without a significant loss of precision. The inversion is then carried out in three main steps: calculation of the full-order inverse model, size reduction, and time integration of the resulting reduced-order inverse model. A numerical example of heat sources identification in a 2D diffusion problem shows the performances and the accuracy of the proposed approach.

A non-parametric method for estimating enthalpy-temperature functions of shape-stabilized phase change materials

“…[26]) that the minimum of J is achieved when the following three problems are satisfied simultaneously: ðaÞ Direct problem : dTðtÞ=dt ¼ ATðtÞ þ BWðtÞ; ð12Þ ðbÞ Adjoint problem : ÀdkðtÞ=dt ¼ A 0 kðtÞ þ C 0 CTðtÞ À C 0 y m ðtÞ; ð13Þ ðcÞ Stationarity condition : WðtÞ ¼ ÀR À1 B 0 kðtÞ ð 14Þ with T(0) = T o and k(t f ) = 0. T(t) (n  1) and k(t) (n  1) are respectively the PCM thermal state vector and the so called co-state vector (as previously, n is the number of nodes in the discretisation mesh).…”

“…This problem can be easily solved using the ''sweep method'' [26,27], which assumes that kðtÞ and T(t) satisfy an affine relation like k(t) = S 1 T(t) + v(t) for some as yet unknown matrix function S 1 and vector function v(t). The solution of the inverse problem is then given by (cf.…”

Genetic Algorithm-Based Method for Determination of Temperature-Dependent Thermophysical Properties

“…Zmywaczyk [4] applied the modified Newton-Raphson method to determine temperature-dependent thermophysical properties (k r , k z , ρc p ) of an orthotropic material. Del Barrio [5] identified heat sources in a 2D diffusion problem by the Lagrange theory. Pohanka et al [6] applied the simplex method to determine the temperature-dependent thermal properties (k, ρc p ) of a fused silica shell.…”

Thermal characterization of materials using Karhunen–Loève decomposition techniques – Part I. Orthotropic materials

“…Powerful model reduction techniques based on PCA/KLD/SVD have been also proposed for lowdimensional description of problems described by partial differential equations, mostly in the field of turbulent flows [12,13]. In thermal analysis, SVD-based methods have been developed for efficient reduction of linear and non-linear heat transfer problems [14][15][16][17][18], as well as for solving heat transfer inverse problems dealing with unknown heat sources [19][20][21]. For a fairly comprehensive introduction to PCA/KLD/SVD, we recommend the books by Jolliffe [22] and Deprettere [8].…”