Excerpt

## Contents

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SUMMARY

I. INTRODUCTION

1. Background - Inverse Problems

2. Related Work

3. Kalman Filtering and Present Problem

II. Forward Problem

1. Problem Description and Finite Difference Solution

2. Sensitivity Analysis

III. INVERSE ESTIMATION

1. Kalman Filtering

2. Adaptive State Estimation - ASE

IV. RESULTS

V. CONCLUSION

REFERENCES

List of Figures

Fig. 1 A representation of the general inverse problem solution

Fig. 2 Nodalized representative diagram of the 3D cube

Fig. 3 Temperature distribution representation with 11 x11 x 11 at three pints in the cube

Fig. 4 Temperature distribution representation for non-liner portion of time at (x, У, z) = (1,1,1)

Fig. 5 Comparison of different mesh size results with analytical solution at (x, y, z) = (1,1,1)

Fig. 6 The discrete Kalman filter cycle. The time update projects the current state estimate ahead in time. The measurement update adjusts the projected estimate by an actual measurement at that time

Fig. 7 Block diagram of Adaptive input estimator

Fig. 8 (a) True flux distribution along the z-axis at (x, y) = (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) = (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 5 sensors

Fig. 9 (a) True flux distribution along the z-axis at (x,y) = (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) = (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 1 sensor

Fig. 10 (a) True flux distribution along the z-axis at (x,y) = (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) = (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 3 sensors

Fig. 11 (a) True flux distribution along the z-axis at (x,y) = (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) = (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 5 sensors

Fig. 12 (a) True flux distribution along the z-axis at (x,y) = (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) = (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 1 sensor

Fig. 13 (a) True flux distribution along the z-axis at (x, y) - (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) - (0.5,0.5) w.r.t time (d) estimated temperature distribution, for 3 sensors

Fig. 14 (a) True flux distribution along the z-axis at (x,y) - (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) - (0.5,0.5) w.r.t time (d) estimatedtemperature distribution, for 5 sensors

Fig. 15 (a) True flux distribution along the z-axis at (x,y) - (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) - (0.5,0.5) w.r.t time (d) estimatedtemperature distribution, for 1 sensor

Fig. 16 (a) True flux distribution along the z-axis at (x,y) - (1,0.5) w.r.t time (b) estimated input flux (c) temperature distribution along x-axis at (y, z) - (0.5,0.5) w.r.t time (d) estimatedtemperature distribution, for 3 sensors

List of Tables

Table 1 Percent relative error for different mesh sizes and different simulation times

Table 2 Error table for different sensor arrangements for first scenario

Table 3 Error table for different sensor arrangements for second scenario

Table 4 Error table for different sensor arrangements for second scenario

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## SUMMARY

Inverse heat conduction problems occur in many theoretical and practical applications where it is difficult or practically impossible to measure the heat flux generated and the temperature of the layer conducting the heat flux to the body. Thus it becomes imperative to devise some means to cater for such a problem and estimate the heat flux inversely. Adaptive state estimator is one such technique which works by incorporating the semi-Markovian concept into a Bayesian estimation technique thereby developing an inverse input and state estimator consisting of a bank of parallel adaptively weighted Kalman filters. The problem presented in this study deals with a three dimensional system of a cube with one end conducting heat flux and all the other sides are insulated while the temperatures are measured on the accessible faces of the cube. The measurements taken on these accessible faces are fed into the estimation algorithm and the input heat flux and the temperature distribution at each point in the system is calculated.

A variety of input heat flux scenarios have been examined to underwrite the robustness of the estimation algorithm and hence insure its usability in practical applications. These include sinusoidal input flux, a combination of rectangular, linearly changing and sinusoidal input flux and finally a step changing input flux. The estimator’s performance limitations have been examined in these input set-ups and error associated with each set-up is compared to conclude the realistic application of the estimation algorithm in such scenarios. Different sensor arrangements, that is different sensor numbers and their locations are also examined to impress upon the importance of number of measurements and their location i.e. close or farther from the input area. Since practically it is both economically and physically tedious to install more number of measurement sensors, hence optimized number and location is very important to determine for making the study more application oriented.

Before the inverse estimation, a comprehensive mesh sensitivity analysis is given for the system’s governing equation finite difference calculations to get an optimized mesh size for the forward and inverse analysis to be correct and within the tolerable error limits and computationally undemanding at the same time.

## INTRODUCTION

### 1. Background - Inverse Problems

Inverse problems in engineering disciplines have always been an alluring area of interest for the researchers as it considerably simplifies the system identification and on the other hand is sufficiently accurate in estimating the parameters which are very difficult or impossible to measure due to system complexity, intricate geometry or many other technological or financial constraints and hence obtained data is the only source of obtaining the model parameters. A very simple representation of the inverse problem can be given as in Fig. 1 below.

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Fig. 1 A representation of the general inverse problem solution

The transformation from data to model parameters is a result of the interaction of a physical system, e.g., the Earth, the atmosphere, gravity etc. Inverse problems arise for example in heat conduction problems, geophysics, medical imaging (such as electrical impedance/resistance/capacitance tomography), remote sensing, ocean acoustic tomography, nondestructive testing, and astronomy.

Inverse problems are typically ill posed, as opposed to the well-posed problems more typical when modeling physical situations where the model parameters or material properties are known. Of the three conditions for a well-posed problem, i.e. existence, uniqueness and stability of the solution or solutions, the condition of stability is most often violated. In the sense of functional analysis, the inverse problem is represented by a mapping between metric spaces. While inverse problems are often formulated in infinite dimensional spaces, limitations to a finite number of measurements, and the practical consideration of recovering only a finite number of unknown parameters, may lead to the problems being recast in discrete form. In this case the inverse problem will typically be ill-conditioned. In these cases, regularization may be used to introduce mild assumptions on the solution and prevent overfitting.

Out of the various inverse problems involved in engineering disciplines, one of the most important is that of the inverse heat conduction problem, IHCP. It has been widely used in practical engineering problems involving the estimation of surface conditions or initial conditions as well as thermal properties of a body from known information like temperatures measured at the prescribed positions. It is sometimes necessary to calculate the transient surface heat flux and the surface temperature from a temperature measured at some location inside or outside the body. For example the case of a gun-barrel or nuclear reactor where heat flux is generated at a point which is virtually un-accessible and only the surface temperature can be measured. Keeping in view of the geometry of the body this IHCP problem can be solved using the inverse estimation techniques; however these problems are known to be severely ill-posed. Similarly IHC problems can arise in a variety of situations and there are many applications where such techniques are working wonders and are playing their role in the successful and reliable instrumentation and designing. Some of the common areas where IHC problems occur are as follows:

- Heat production and dissipation in micro-electronics where micro- devices produce heat and are covered with fins and other devices in contact.

- Heat production in nuclear reactors where the fuel is clothed in many layers of cladding and it is impossible to reach to the fuel surface.

- Designing of heat resistant materials and hence canopies of jets and rockets where it is practically impossible to install sensors on the inside and only outer surface temperatures can be measured.

- Determination of the heat transfer coefficient and outer surface conditions in the re-entry of a space vehicle.

- Designing of heat resistant gun-barrels.

- Designing of refractories and furnaces.

- Air-conditioning and refrigeration.

Related Work

Although a lot of work has been carried by different researchers in the field of IHCP, this work is usually confined to 2D problems only and we do not see much of the work carried out for a 3D domain involving IHCP. Many techniques and inverse algorithms have been proposed, applied and inspected by various investigators in the field of inverse problems in engineering. There exist many methods to solve the IHCP and the majority of researchers use the approaches where the unknowns are determined to minimize the sum of squares of the differences between the measured and the computed temperatures at the selected spatial and/or temporal points. In general, the approaches adopt the iterative scheme and the regularizations are implemented to mitigate the ill-posedness of IHCP. The Kalman filter estimation method is successfully used in predicting multidimensional and nonlinear IHCP. Zheng and Murio [1] proposed a stable algorithm for 3D IHCP problem on a slab and then they [2] developed a numerical solution for the three-dimensional inverse heat conduction problem on a finite cube by applying a mollification procedure. A fully explicit space-marching finite-difference scheme was developed and numerical simulations were provided with excellent accuracy between estimated and exact solutions. Scarpa and Milano [3] estimated the time-dependent surface heat flux at one boundary of a one-dimensional system by using the Kalman smoothing technique, given the initial temperature distribution and the time-temperature history at an interior location. The numerical results show appreciable performance of the proposed technique, which provides a comprehensive way for using future temperature measurements. Kaipio and Somersalo [4] used a regularized version of Kalman filter and inverse boundary value problem was considered for a nonstationary object, i.e. those properties of the object that one seeks to estimate change as a function of time during the measurement sequence. In recent years, many applications appeared in which Kalman filter has been used in conjunction with recursive-least square algorithm (RLSA), for example the work of Tuan et al. [5-12], deals with onedimensional and two-dimensional problems. Extending that work, recently, Jang et al. [13] has attempted to use a RLSA based on the Kalman filter to estimate the boundary heat flux varying impulsively with time by employing the finite-element scheme to discretize the problem in space, allowing multidimensional problems of various geometries to be treated.

Kalman [14] filter has been the governing filter of most of the techniques proposed in these studies, however for the higher dimension problems, the straight forward implementation of the Kalman filter becomes difficult as the size of covariance equation increases. Therefore, one of the most important prerequisites for the successful implementation of a Kalman filter for the purpose of real-time estimation is the development of a reliable low dimensional model, hence, dimension reduction techniques like Karhunen-Love Galerkin procedure is used with Kalman filter by Park and Jung [15] for solving multidimensional heat conduction problems. For the case of nonlinear IHCP, extended version of discrete Kalman filter have been used by Daouas and Radhouani [16], [17] to nonlinear IHCP to estimate surface heat flux density. Huang and Tsai [18] solved the IHCP using the conjugate gradient method. Boundary Element Method (BEM)-based inverse algorithm utilizing the iterative regularization method was successfully used to solve the Inverse Heat Conduction Problem (IHCP) for estimating the unknown transient boundary heat flux in a 2D domain with different arbitrary geometries.

An important parameter in solving the inverse problems is that of stability because of the prohibitive ill-posedness of the problem. As the IHCP finds wide applications in many thermal-related industries, it is of great practical importance to study the various effects on the stability of the inverse solutions. Surprisingly, despite so many existing inverse techniques, a systematic study of the stability of the inverse solutions has not been pursued by many researchers. Most of the techniques do not give a quantitative method for determining the computed errors due to noise in temperature measurements. The reason for a lack of studies on the stability of the solution of the inverse problem is simple. The IHCP is already very difficult and its solution instability analysis is even more. One such study has been carried out by Ling and Atluri [19]. Two matrix algebraic tools are provided for studying the solution-stabilities of inverse heat conduction problems. The propagations of the computed temperature errors, as caused by noise in temperature measurements are given and the spectral norm analysis reflects the effect of the computational time steps, the sensor locations and the number of future temperatures on the computed error levels. Hsu [20] presented an account of the inverse estimation of the boundary conditions in a 3D inverse hyperbolic heat conduction problem. Finite-difference methods are employed to discretize the problem domain, and then a linear inverse model is constructed to identify the unknown boundary condition.

It is noteworthy that Tuan et al. [5] developed the RLSA based on the Kalman filter for two-dimensional IHCP to estimate the boundary heat flux varying impulsively with time. Their approach gives good estimates for estimating unknown heat sources or heat flux inputs on the boundaries. They have proposed improvement in Kalman filter with RLSA approach by having RLSA weighted by forgetting factor to robustly extract the unknowns. The maximum likelihood type estimator combined with Huber psi-function is used to construct the weighting forgetting factor. In the context of forgetting factor, Wang et al. [21] proposed extended Kalman filter with RLSA weighted by forgetting factor to estimate nonlinear heat conduction problems.

## 3. Kalman Filtering and Present Problem

Right from its inception in 1959 by R. E. Kalman [14], there have been many modifications, additions and developments proposed in the Kalman filter and it has been proved as a building block for many revolutionary estimation techniques. The modifications to Kalman filter are in hundreds of thousands and the break-through which it has provided in the field of numerical analysis is priceless and far-reaching as well. The main contribution of Kalman filter is in the field of inverse problems and state estimation in particular. Thereby deploying the Kalman filter, Moose et al. [22] proposed an Adaptive State Estimator (ASE) for passive underwater tracking of maneuvering targets. The state estimator is designed specifically for a system containing independent unknown or randomly switching input and measurement biases. In modeling the stochastic system, it is assumed that the bias sequence dynamics for both input and measurement can be modeled by a semi-Markov process. By incorporating the semi-Markovian concept into a Bayesian estimation technique, an estimator consisting of a bank of parallel adaptively weighted Kalman filters was developed. Despite the large and randomly varying biases, the proposed estimator provides a reasonable estimate of the system states. The Bayesian computational technique has many advantages as it is able to quantify system uncertainty and random data error, to derive a probabilistic description of the inverse solution, to provide extensive spatial/temporal regularization to the ill-posedness of the inverse problem, and to allow adaptive sequential estimation. Wang and Zabaras in [23-26] developed a computational framework that integrates computational mathematics, Bayesian statistics, statistical computation, and reduced-order modeling to address data-driven inverse heat and mass transfer problems.

In the context of above mentioned filtering and estimation technique the present work deals with the input heat flux and the temperature distribution in a 3D heat conduction domain using the ASE based on Kalman filter. It is worth mentioning here that Kim et al. [27] deployed ASE to one-dimensional IHCP for estimation of input heat flux and then carrying their work forward and considering measurement bias into account, Ijaz et al. [28] have focused their research on a typical 2D heat conduction problem. Their study shows that ASE consisting of Kalman filters connected in parallel gives good performance in the presence of measurement bias also. Each filter has its operating bound limiting the range of the unknown input heat flux and measurement bias.

Traditionally, inverse problems are divided into two sequential stages: analysis and optimization. In the analysis stage, the values of unknown states are initially assumed, and then a numerical method (e.g. the finite-difference method or the finite- element method) is used to obtain the exact solution. In the optimization stage, the measurements data are compared with the results predicted in the previous stage, and are then compounded to form a non-linear problem. An optimization and filtering process is then employed to derive the optimal estimated solution. We have used the finite difference method for the discretization of the domain and a 4th-order Runge- Kutta method is deployed to get the numerical solution of the problem. ASE is deployed then for the estimation of input flux used in the previous stage to get the numerical solution and then state estimation i.e. temperature distribution estimation is obtained finally. Different types of boundary condition are used for the verification of results and different number of sensors and sensor arrangements are also analyzed to get the knowledge of estimator’s dependence on measurement numbers and measurement locations. A comprehensive mesh sensitivity analysis is provided to cater for the inaccuracy and error involved in the estimation.

## II. Forward Problem

### 1. Problem Description and Finite Difference Solution

Let us consider a three-dimensional cube, initially at temperature [Abbildung in dieser Leseprobe nicht enthalten] .For times t > 0, all the faces of the cube are kept insulated except the face [Abbildung in dieser Leseprobe nicht enthalten] which is conducting a heat flux q(t) to the cube. Fig. 1 illustrates the heat conduction problem considered along with boundary and initial conditions. The governing equation of this problem in dimensional form is given as

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where x, y, z and t are space and time coordinates respectively, a is the thermal diffusivity and L, W and H are the length, width and height of the cube respectively. If some non-dimensional parameters are defined as

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and substituted in Eq. (1), then the dimensionless form of Eq. (1) is given as:

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In the above equations, T is the dimensionless temperature and x, y, z and t are the dimensionless space and time coordinates, respectively while q is the input heat flux.

Now applying the Central Finite Difference method on Eq. (2) implies that

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M,N,0> 1 , where M, N and О are the total number of spatial nodes for x, y, z directions, respectively, and [Abbildung in dieser Leseprobe nicht enthalten] are the space intervals. Application of the Central Finite Difference method on the boundary conditions, Eqs. (3) through (8) gives

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Therefore Eq. (10) can be re-written as

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From Eqs. (12) through (17) and associated with fictitious process noise inputs [29], the continuous-time state equation can be written as

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here

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and all T jk are continuous w.r.t. time t.

The coefficient matrix T e R(MN0хш°) is given by

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**[...]**

- Quote paper
- Nauman Malik Muhamad (Author), 2009, Inverse Estimation of Heat Flux and Temperature in 3D Finite Domain, Munich, GRIN Verlag, https://www.grin.com/document/176816

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