Partial Fulfillment of the Requirements for the Degree of

Analyzes of inverse problems become essential in the evaluation of medium properties or boundary conditions. The flow chart is given to highlight the working principle of the GA in the inverse problem.

Contents

List of Tables

D1Q2 one-dimensional and two-way model D2Q9 two-dimensional and nine-way model D3Q15 three-dimensional and fifteen-way model.

Introduction

Overview of Inverse Problems

In most cases, the initial and boundary conditions and properties of the medium are known, and the objective is to determine the temperature and/or heat flow distributions. However, there are many situations in which the initial condition, boundary conditions, and/or properties of the medium are unknown, but the temperature and/or heat flux distributions are known.

Literature Review on Inverse Problems – Applications and its Solution

In addition, inverse problems are studied in composite materials design [10, 11], robotics [12] and many manufacturing processes [13-19]. From the measured output radiation intensities, he used an iterative procedure along with CGM in inverse analysis.

Suitability and Selection of Efficient Method for Direct and Inverse Heat Transfer Problems

Selection of Direct Method Required in the Inverse Problem
Selection of Optimization Method Required in the Inverse Problem

Therefore, in the present work, FVM has been used in most cases involving radiant heat transfer. In the next section, we discuss some of the optimization methods used for solving inverse problems.

Selection of the Genetic Algorithm (GA) as an Optimization Tool for the Present Work

Based on the above discussion, in the present work, on the following pages, we further justify the General Assembly's preference. For this, an objective function needs to be written and the same can be called by the GA function.

Objectives

Furthermore, in the field of inverse problems, the suitability of a combination involving an efficient set of methods such as the LBM and the FVM/DTM in conjunction with the GA has not been investigated. Therefore, below we state the objectives for the present investigation involving inverse analysis of inverse transient conduction-radiation problems. radiation and we also investigate the implications of different GA parameters. The use of the LBM in conjunction with the GA has been investigated for inverse problems.

Organization of the Thesis

The formulation of the objective function using the LBM-FVM and its minimization using the GA are studied in detail. In Chapter 4, the use of the LBM-FVM-GA was extended to a 2-D Cartesian geometry.

Summary

Introduction

The inherent advantages of the GA as an optimization tool have already been mentioned in the previous chapter. On the following pages, we first briefly describe the mathematical formulations of the LBM and the FVM to solve the immediate problem.

Energy Equation

We then give a methodology to solve the inverse problem using GA as an optimization tool.

Lattice Boltzmann Method (LBM)

Derivation of Lattice Boltzmann Equation

In heat transfer problems, the calculation of the equilibrium particle distribution function and the methodology for obtaining the temperature (by adding f over all directions), the following relationship holds [115-125]. On the following pages we describe the methodology of the FVM to calculate the radiation information .∇qR.

Figure 2.1: D1Q2 lattice of the LBM and control volume of the FVM used in a 1-D planar geometry

Finite Volume Method (FVM)

In the case of a 1-D planar geometry, the radiation is azimuthally symmetric, so in this case, the incident radiation G and the net heat flux q are given by and calculated from the following expressions R [120],. while marching from any of the angles, the evaluation of the draw. 2.41) requires knowledge of the boundary intensity. It is calculated from the following[120] b. 2.51), the first and second terms represent the emitted and reflected components of the boundary intensity, respectively.

Solution Methodology for the Direct Problem

In the direct method, temperature distributions θ are obtained from given values of environmental properties together with known initial and boundary conditions. In the inverse method, we minimize the difference between known temperature distributions and some temperature distributions initially guessed using GA.

Principle of Genetic Algorithm

Generation of Initial Population
Evaluation of Fitness
Reproduction, Crossover and Mutation
Termination Condition

The mutation operator maintains diversity by randomly changing the genes in the set of generated offspring. In the present work, the theoretically determined global minimum value for the objective function is zero.

Figure 2.4: Flowchart of the genetic algorithm.

Formulation of Inverse Problem

The other, but most unlikely, mode of termination is to reach a global maximum or global minimum of the value of the objective function. Therefore, if at any point the value of the objective function becomes zero, the algorithm will automatically terminate and therefore it will extract the best individual or solution.

Solution Methodology to Solve Inverse Problem

Define the desired value of the objective function. Check if the objective function value changes in successive iterations. If the objective function value remains nearly constant for a sufficiently limited number of iterations, terminate the algorithm. If the objective function value keeps changing, discard the old values and assume other values of parameters and go to step 2.

Figure 2.5: Solution methodology to solve the inverse problem.

Summary

Parameter Retrieval in a 1-D Transient Conduction-Radiation Problem

Introduction

To establish the correctness of the estimated parameters in the inverse method using LBM-FVM-GA, temperature fields are evaluated based on the estimated parameters and compared with those obtained from the direct method. Additionally, a comparison of CPU times involved in the direct method and the inverse method is also made.

Formulation

Detailed formulations for the analysis of a transient conduction radiation problem in the direct and inverse methods are presented in Chapter 2. In the following pages we briefly present the relevant formulations used for the inverse analysis of a 1-D transient conduction radiation problem with heat transfer.

Results and Discussion

Effect of the GA Parameters

In this work, we have studied three different values of crossover probabilities, ie and 0.60. In Table 3.2, we also compare the results for three different combinations of the conductivity-emissivity parameter N and the western boundary emissivity εW, ie, (N, εW).

Table 3.1: Combination of genetic parameters and measurement errors for different runs used in the inverse method

Parameter Retrieval in a 2-D Transient Conduction-Radiation Problem

Introduction

Formulation

In the inverse method, a simultaneous estimation of three parameters has been carried out, namely the scattering albedoω, the southern limit emissivityεS and the conduction radiation parameter N. With the inclusion of measurement errors ( )E, the measured temperature profile (θmeasured = +θɶ E) and the exact temperature are minimized at the same way as discussed in Chapter 3.

Results and Discussion

Retrieval of Parameters without Measurement Error

It can be seen that the results of the inverse method are in close agreement with the results of the direct method. To illustrate the variation of the objective function with increasing number of generations, the comparison was shown in fig.

Table 4.1: Comparison of exact and estimated values of parameters for ( ω , N , ε S ) =

Inverse Analysis of Conduction-Radiation Problems with Varying Complexities

Introduction

In non-Fourier transient line radiation problem, we have simultaneously estimated two unknown parameters, namely the extinction coefficient and the line radiation parameter. However, in the inverse analysis of transient conduction radiation with mixed boundary condition, we have simultaneously estimated two unknown boundary conditions, namely the boundary heat flux and the boundary emissivity.

Formulation of a Non-Fourier Conduction-Radiation Problem

In the transient conductivity-irradiance problem involving variable conductivity, we simultaneously estimated different combinations of two unknown parameters, namely, the extinction coefficient, the scattering albedo and the conductivity-irradiance parameter. On the following pages we provide the results of the non-Fourier conduction-radiation problem.

Results and Discussion on Inverse Non-Fourier Transient Conduction Radiation Problem

Effects of GA Parameters for Inverse Non-Fourier Conduction-Radiation Problem
Comparison of Reconstructed and Exact Temperature Fields
Comparison of Computational Time
Variation of Estimated Parameters

To demonstrate the accuracy of the estimated parameters obtained in the inverse method, in Fig. 5.5, we present an equation for the simultaneous estimation of the conduction-radiation parameter, N and the extinction coefficient, β.

Table 5.1: The effect of the number of lattices in the LBM and control volumes in the FVM and number of discrete directions on temperature distribution at three different locations; ξ = 0.60, β = 0.5, ω = 0.8, N= 0.01 and

Formulation of an Inverse Transient Conduction-Radiation Problem with Variable Thermal Conductivity

In the simultaneous evaluation of the conductivity-radiation parameter and the scattering albedo, the emissivity and the extinction coefficient are fixed. While for estimation of extinction coefficient and albeo scattering, the conduction-radiation parameter and emissivity are determined.

Results and Discussion on Inverse Problem involving Variable Thermal Conductivity

Effects of GA Parameters
Comparison of Computational Time

5.8, we present the variation of the best fit with the number of generations for simultaneously obtaining the conductivity-emissivity parameter N and the scattering albedo ω. To study the effect of CPU time involved in the direct method and the inverse method, we present a comparison in Table 5.8.

Figure 5.6: Comparison of the temperature θ distributions at different locations for different variable thermal conductivity parameter γ with Talukdar and Mishra [69]

Formulation of an Inverse Transient Conduction-Radiation Problem with Mixed Boundary Condition

Below we briefly discuss the procedure for treating the mixed boundary condition type problem. In implementing the boundary condition in the LBM, conservation of energy is applied to the half-size boundary grid.

Figure 5.13: 1-D planar medium with D1Q2 lattice subjected to heat flux at its west boundary

Results and Discussions on the Inverse Transient Conduction- Radiation Problem with Mixed Boundary Condition

Effect of the GA Parameters
Comparison of CPU Time

In Table 5.11 we present the comparison of the elapsed CPU times in the direct method (LBM-FVM) and the inverse method (LBM-FVM-GA). This comparison is made for three different values of the measurement errors, namely E= 0.0, 1.0 and 2.0 are taken.

Table 5.9: Comparison of temperature θ distributions for 1.0, 0.5, E W 0.5 and N 0.05.

5. 8 Summary

A comparison of the CPU times involved in the direct and the inverse method was also done. The accuracy of the estimated parameters was checked by comparing the temperature distributions obtained using the direct and the inverse method.

Simultaneous Reconstruction of Thermal Field and Retrieval of Parameters in a Cylindrical Enclosure

Introduction

Since in a concentric cylindrical case, the DTM formulation is simpler for the radiation heat transfer problem, the same was done using DTM and GA. For the conduction problem, an unknown radius ratio was estimated that would give the desired temperature distribution.

Formulation of an Inverse Problem in a Cylindrical Geometry

LBM Formulation for Conduction Problem in a Cylindrical Geometry

While in the radiative transfer problem, two parameters viz. simultaneously estimated the extinction coefficient and the radius ratio that would provide a desired heat flux distribution in the medium. The CPU times paid out in the direct method and the inverse method were also compared.

Results and Discussions

Effects of the GA Parameters
Variation of Estimated Parameters
Comparison of CPU Time

Table-6.2 Comparison of the estimated and the exact values of the radius ratio along with the measurement error. In Table 6.2 we compare the exact and approximate values of the radius ratio for a conduction problem in a concentric cylindrical medium.

Figure 6.1: Physical geometry of the problem; initial condition and boundary conditions are θ = 0.1 ; ξ > 0 : θ 1 = 0

Summary

The accuracy of the estimated parameters was checked by comparing the heat flux fields obtained using the direct and the inverse methods. In this case, the ratio between the CPU times involved in the direct method and the inverse method ranged approximately between.

Conclusions and Scope for Future Work

Conclusions

In the estimation of the conduction-radiation parameter and the scattering albedo this was found to be +28% and -26% respectively. While, in the evaluation of the extinction coefficient and the distribution albedo, the accuracy was +29% and -22%, respectively.

Scope for the Future Work

Takara, Analysis of combined conduction-radiation heat transfer in a two-dimensional rectangular enclosure with gray medium, J. Lankadasu, Analysis of transient conduction and radiation heat transfer using the Lattice Boltzmann method and the discrete transfer method, Numer.