Lecture 6: Models and measurements for thermal systems, types of inverse problems

Models involved in practical inversion of measurements are presented and classified. They belong to the white, grey or black box categories. The quantities they are based on are introduced and great care is given to the notation, in order to be able to understand the different causes of errors in the subsequent inversion output. The specific application in heat transfer is considered and the different types of inverse problems are presented. The notions of discretization of the observations and of parameterization of the functions that have to be retrieved through inversion are introduced. Reduction of a multidimensional heat transfer problem is presented by introduction of models of lower space dimensions: it allows some kind of regularization prior to any inversion. Introduction Modelling constitutes a very general activity in engineering. A system can be considered as modelled if its response to a well-known excitation can be predicted. So prediction is one of the natural outcomes of modelling. Structure is a very important feature of a model in heat transfer or in any other field. It is a theoretical representation of the physical reality in the case of a material system where fluxes of extensive quantities (matter, energy, momentum, .) are exchanged . In this lecture we do not deal with the special cas of information systems, that is systems where data are exchanged and processed. In heat transfer, the choice is quite large and the model structure should be selected according to the objectives of the model-builder. The model-builder can have in mind an optimal design problem, a parameter estimation problem based on measurements, a control problem to define the best excitation shape for a given desired output, or a model reduction or a system identification problem, just to quote a few applications. In heat transfer, the choice of the model structure depends on peculiar considerations: • State variables and observed quantities In a heat diffusion problem, temperature is the quantity that constitutes the state variable, in the thermodynamics sense. In order to calculate temperature and flux at METTI 6 Advanced School: Thermal Measurements and Inverse Techniques Biarritz, March 16, 2015 Lecture 6: Models and measurements – page 2 any time t and at any point P, one has to know the initial temperature field (at time t = 0) at the local scale, as well as the history of the different thermal disturbances from times 0 up to t. So, one has to define what is a local point P and a local scale. For instance, if heat transfer is intented to be studied at the nanoscale in a metal, Fourier’s law, relating the heat flux to the temperature gradient, may no longer be valid. In such a case, the model-builder could refer to the Boltzmann’s based equations or to a two temperatures model respectively for the electron gas and the lattice. Such a detailed state model will be necessary if observations or predictions are looked for at the nano scale or at the picosecond time scale. • State definition The continuous state equations have then to be defined for the modeling problem at stake: it can be a partial differential equation, the heat equation (state = temperature), or an integro-differential equation, the radiative transfer equation (state = radiative intensity), or both coupled equations. Their solution, that is constituted by both temperature and intensity fields in the third case, should be calculated everywhere and any time starting from the initial time. It can also be a differential equation, or a system of differential equations for lumped systems where only space averaged quantities are used (an average temperature instead of local temperatures, heat flow rates instead of local heat fluxes). Everytime state equations, based on conservation equations are used, one deals with internal representation and it is possible to talk of models of the white box type. These conservation equations are associated with constitutive laws (Fourier’s, Fick’s, Ohm’s, Darcy’s or rheological laws, .) corresponding to the considered quantity (heat, species, electrical charge, momentum in porous matrerials or in fluid mechanics). • Quantities of the direct problem We focus here on the diffusion heat equation in a medium composed of one or several homogeneous materials, with its associated initial, boundary and interface equations. Its solution, the state variable, here the continuous temperature field P) ( , t T , has first to be found and the desired observed quantities, that is the (theoretical) output of the model at a given point P, P) ( ) ( , t T t ymo = , has to be calculated next. The quantities that are required for solving the direct problem are the structural parameters of the system (conductivities, volumetric heat capacities, heat exchange coefficients, emissivities of walls,.), the thermal excitation and the initial temperature field P) 0 ( , t T = . Let us note here that it is possible to make a physical reduction of a model based on the three dimensional (3D) transient heat equation to get simpler models of lower dimensionality. The thermal fin (1D) or the bulk temperature representation (0D) constitute such reduced models. This type of reduction may also reduce the number of parameters defining the excitations. • Numerical/analytical models METTI 6 Advanced School: Thermal Measurements and Inverse Techniques Biarritz, March 16, 2015 Lecture 6: Models and measurements – page 3 There are many ways for solving the heat equation and finding a state model for the observations: analytical solutions provide the temperature field explicitely as a function of the structural parameters of the system, the excitation and the initial state. They can be constructed if the heat equation in each material and the associated conditions are all linear and the corresponding geometry simple. The other class of state models relies on numerical solution of the heat equation: one can quote the nodal, boundary element, finite elements and finite volumes methods, for example. State models rely on an internal representation of the system: the temperature field has to be found first and the observations are calculated next. • Internal/external representation The models based on state equations, which are described above, corespond to an internal representation of the system: they are of the white box type and their structural parameters (thermophysical or optical properties, geometrical quantities associated to the shape or material structure of the physical system) have an intrinsic character since their definition does not depend on the structure that is chosen for the model. External representations, that short circuit the state variable and link directly the observation(s) to the excitation(s), constitute another class of models. They can be of the grey box type if the structure of the corresponding model is imposed by the nature of the governing equations: if the relationship between observed quantities (output) and stimulations (input) is linear, for physical reasons, the structure of the model is imposed and its structural parameters are the coefficients of a matrix once the input discretized and the output parameterized (see below). If, furthermore, the physical structure of the material system does not vary neither with time nor with temperature, this linear relationship becomes a convolution product and a transfer function can be defined, see Tutorials 1 and 11. In that last case the model can be represented using the parametric approach presented in Tutorial 1. Another type of model of the grey box type is used in the case of thermal radiation in an enclosed cavity with opaque walls, where radiative fluxes depend on the diferences of the fourth powers of the wall temperatures. If no information whatsoever is available on the nature of the relationship between input and output, black box models, that link observations to inputs and ad’hoc parameters (the structure of the model) has to be found. Models of this nature can be neuronal networks (multidimensional abacus), stochastic models, . • Parameterization for inverse problem solution Parameterization of the data of the direct problem constitutes another characteristic of the structure of a model: structural parameters, thermal excitations, and the initial temperature field are, in the very general case, functions of different explanatory variables: space, time, temperature. The conversion of functions into vectors of finite dimensions does not pose much problem in the direct problem (calculation of the observations, the model output, as a function of the input). It is no more the case when the inverse problem is considered. This point will be discussed in section 2.3.2 METTI 6 Advanced School: Thermal Measurements and Inverse Techniques Biarritz, March 16, 2015 Lecture 6: Models and measurements – page 4 and in lecture 8 of this school. One of the objectives of mathematical reduction methods is to construct a reduced model that will have a reduced number of structural parameters, starting from a detailed reference model, see [1, chapter 13] for details on model reduction, while physical reduction also change the definitions of both output and excitations, see section 3.2. 2. Physical system, model, direct and inverse problems 2.1 Fourier’s law: a reminder In the following, we will assume the following two assumptions : a unique temperature, as well as a temperature gradient, can be defined, which means that the investigated medium is considered in the sense of a continuum; the Fourier law relating the local heat flux and the temperature gradient is valid. This asumption is justified for time and space scales that are not too small: typically, for times larger than 10 s and lengths larger than 10 m. This corresponds to what is called a mesoscopic description of a thermal system. However, even with such considerations, special attention has to be taken for the definition of local tem