Inverse Heat Conduction Ill-Posed Problems

A comprehensive reference on the field of inverse heat conduction problems (IHCPs), now including advanced topics, numerous practical examples, and downloadable MATLAB codes.

The First Edition of the classic book Inverse Heat Conduction: III-Posed Problems, published in 1985, has been used as one of the primary references for researchers and professionals working on IHCPs due to its comprehensive scope and dedication to the topic. The Second Edition of the book is a largely revised version of the First Edition with several all-new chapters and significant enhancement of the previous material. Over the past 30 years, the authors of this Second Edition have collaborated on research projects that form the basis for this book, which can serve as an effective textbook for graduate students and as a reliable reference book for professionals. Examples and problems throughout the text reinforce concepts presented.

The Second Edition continues emphasis from the First Edition on linear heat conduction problems with revised presentation of Stolz, Function Specification, and Tikhonov Regularization methods, and expands coverage to include Conjugate Gradient Methods and the Singular Value Decomposition method. The Filter Matrix concept is explained and embraced throughout the presentation and allows any of these solution techniques to be represented in a simple explicit linear form. Two direct approaches suitable for non-linear problems, the Adjoint Method and Kalman Filtering, are presented, as well as an adaptation of the Filter Matrix approach applicable to non-linear heat conduction problems.

In the Second Edition of Inverse Heat Conduction: III-Posed Problems, readers will find:

• A comprehensive literature review of IHCP applications in various fields of engineering
• Exact solutions to several fundamental problems for direct heat conduction problems, the concept of the computational analytical solution, and approximate solution methods for discrete time steps using superposition of exact solutions which form the basis for the IHCP solutions in the text
• IHCP solution methods and comparison of many of these approaches through a common suite of test problems
• Filter matrix form of IHCP solution methods and discussion of using filter-form Tikhonov regularization for solving complex IHCPs in multi-layer domain with temperature-dependent material properties
• Methods and criteria for selection of the optimal degree of regularization in solution of IHCPs
• Application of the filter concept for solving two-dimensional transient IHCP problems with multiple unknown heat fluxes
• Estimating the heat transfer coefficient, h, for lumped capacitance body and bodies with temperature gradients
• Bias in temperature measurements in the IHCP and correcting for temperature measurement bias

Inverse Heat Conduction is a must-have resource on the topic for mechanical, aerospace, chemical, biomedical, or metallurgical engineers who are active in the design and analysis of thermal systems within the fields of manufacturing, aerospace, medical, defense, and instrumentation, as well as researchers in the areas of thermal science and computational heat transfer.

Keith A. Woodbury is Professor Emeritus of Mechanical Engineering at the University of Alabama, where his research in inverse heat conduction supported investigations into quenching and metal casting. Dr. Woodbury is a life-long member of ASME and has organized numerous technical sessions on inverse problems through the Heat Transfer Division’s K-20 Committee. He is the editor of the Inverse Engineering Handbook (2003).

Hamidreza Najafi is Associate Professor of Mechanical Engineering and Director of the Heat Transfer Lab at Florida Institute of Technology. He has authored and co-authored numerous articles in the areas of inverse heat conduction problems, computational heat transfer, and designing and optimization of energy/thermal systems. Dr. Najafi is an active member of ASME and ASHRAE and has served in various leadership roles in multiple technical committees.

Filippo de Monte is Professor of Mechanical Engineering at the University L’Aquila, Italy. He served as a full-time Visiting Ph.D. student at the Department of Engineering, University of Cambridge, UK, in 1992, and a seasonal Visiting Associate Professor at the Department of Mechanical Engineering, Michigan State University, USA, in 2007 up to 2014. Also, he is a Member of American Society of Mechanical Engineers (ASME) and holds editorial positions at the Journal of Verification, Validation and Uncertainty Quantification (ASME) and Heat Transfer Engineering. He was the Chairman of the 10^th International Conference on Inverse Problems in Engineering (ICIPE 22), May 15-19, 2022, Francavilla al Mare (Chieti), Italy, and is co-editor of the Elsevier 1^st Edition book: Modeling of Mass Transport Processes in Biological Media (July 2022).

James V. Beck (1930-2022) was Professor Emeritus of Mechanical Engineering at Michigan State University (MSU), a Fellow of ASME and one of the pioneers of the fields of inverse problems and parameter estimation. Dr. Beck was honored with the MSU Distinguished Faculty Award (1987) and the ASME Heat Transfer Memorial Award (1998). He was the originator of the Inverse Problems Symposium and was the inventor, with Professor Litkouhi, of the numbering system for heat conduction solutions. Professor Beck made outstanding pioneering contributions to the field of heat transfer with numerous refereed journal articles and books.

Preface First Edition

Preface Second Edition

Nomenclature

1              Inverse Heat Conduction Problems: An Overview              1-1

1.1          Introduction       1-1

1.2          Basic Mathematical Description 1-3

1.3          Classification of Methods             1-5

1.4          Function Estimation Versus Parameter Estimation             1-7

1.5          Other Inverse Function Estimation Problems       1-7

1.6          Early Works on IHCPs     1-9

1.7          Applications of IHCPS: A Modern Look    1-10

1.7.1      Manufacturing Processes             1-10

1.7.1.1   Machining Processes      1-10

1.7.1.2   Milling and Hot Forming                1-11

1.7.1.3   Quenching and Spray Cooling      1-12

1.7.1.4   Jet Impingement              1-13

1.7.1.5   Other Manufacturing Applications            1-14

1.7.2      Aerospace Applications 1-15

1.7.3      Biomedical Applications 1-16

1.7.4      Electronics Cooling          1-17

1.7.5      Instrumentation, Measurement and Non-Destructive Testing      1-18

1.7.6      Other Applications           1-19

1.8          Measurements 1-20

1.8.1      Description of Measurement Errors         1-20

1.8.2      Statistical Description of Errors   1-21

1.9          Criteria for Evaluation of IHCP Methods 1-23

1.10        Scope of Book   1-24

1.11        Chapter Summary            1-24

1.12        References         1-25

1.13        List of Figures    1-33

1.14        List of Tables      1-34

2              Analytical Solutions of Direct Heat Conduction Problems 2.1

2.1          Introduction       2.1

2.2          Numbering System         2.2

2.3          One-Dimensional Temperature Solutions              2.3

2.3.1      Generalized One-Dimensional Heat Transfer Problem      2.3

2.3.2      Cases of Interest              2.4

2.3.3      Dimensionless variables 2.5

2.3.4      Exact analytical solution 2.6

2.3.5      The concept of computational analytical solution               2.7

2.3.5.1   Absolute and relative errors        2.8

2.3.5.2   Deviation time   2.10

2.3.5.3   Second deviation time   2.10

2.3.5.4   Quasi-steady and steady-state times       2.11

2.3.5.5   Solution for large times 2.11

2.3.5.6   Intrinsic verification        2.12

2.3.6      X12B10T0 case  2.12

2.3.6.1   Computational analytical solution             2.13

2.3.6.2   Computer code and plots             2.14

2.3.7      X12B20T0 case  2.16

2.3.7.1   Computational analytical solution             2.17

2.3.7.2   Computer code and plots             2.18

2.3.8      X22B10T0 case  2.19

2.3.8.1   Computational analytical solution             2.20

2.3.8.2   Computer code and plots             2.21

2.3.9      X22B20T0 case  2.23

2.3.9.1   Computational analytical solution             2.24

2.3.9.2   Computer code and plots             2.25

2.4          Two-Dimensional Temperature Solutions              2.27

2.4.1      Dimensionless variables 2.28

2.4.2      Exact Analytical Solution               2.29

2.4.3      Computational analytical solution             2.29

2.4.3.1   Absolute and relative errors        2.31

2.4.3.2   One- and two-dimensional deviation times           2.32

2.4.3.3   Quasi-steady time           2.35

2.4.3.4   Number of terms in the quasi-steady solution with eigenvalues in the homogeneous direction     2.37

2.4.3.5   Number of terms in the quasi-steady solution with eigenvalues in the nonhomogeneous direction             2.38

2.4.3.6   Deviation distance along x            2.40

2.4.3.7   Deviation distance along y            2.42

2.4.3.8   Number of terms in the complementary transient solution            2.43

2.4.3.9   Computer code and plots             2.46

2.5          Chapter Summary            2.48

2.6          References         2.50

2.7          Problems             2.52

2.8          List of Figures    2.55

2.9          List of Tables      2.56

3              Approximate Methods for Direct Heat Conduction Problems        3-1

3.1          Introduction       3-1

3.1.1      Various Numerical Approaches  3-1

3.1.2      Scope of Chapter             3-1

3.2          Superposition Principles                3-2

3.2.1      Green’s function solution interpretation 3-3

3.2.2      Superposition example – step pulse heating         3-4

3.3          One-Dimensional Problem with Time-Dependent Surface Temperature   3-4

3.3.1      Piecewise-constant approximation          3-5

3.3.1.1   Superposition-based numerical approximation of the solution     3-6

3.3.1.2   Sequential-in-time nature and sensitivity coefficients      3-7

3.3.1.3   Basic ‘building block’ solution     3-8

3.3.1.4   Computer code and example      3-8

3.3.1.5   Matrix form of the superposition-based numerical approximation             3-11

3.3.2      Piecewise-linear approximation 3-12

3.3.2.1   Superposition-based numerical approximation of the solution     3-13

3.3.2.2   Sequential-in-time nature and sensitivity coefficients      3-14

3.3.2.3   Basic ‘building block’ solutions   3-16

3.3.2.4   Computer code and examples    3-16

3.3.2.5   Matrix form of the superposition-based numerical approximation             3-20

3.4          One-Dimensional Problem with Time-Dependent Surface Heat Flux          3-21

3.4.1      Piecewise-constant approximation          3-22

3.4.1.1   Superposition-based numerical approximation of the solution     3-23

3.4.1.2   Heat flux-based sensitivity coefficients   3-23

3.4.1.3   Basic ‘building block’ solution     3-24

3.4.1.4   Computer code and example      3-24

3.4.1.5   Matrix form of the superposition-based numerical approximation             3-26

3.4.2      Piecewise-linear approximation 3-27

3.4.2.1   Superposition-based numerical approximation of the solution     3-28

3.4.2.2   Heat flux-based sensitivity coefficients   3-29

3.4.2.3   Basic ‘building block’ solutions   3-30

3.4.2.4   Computer code and examples    3-30

3.4.2.5   Matrix form of the superposition-based numerical approximation             3-33

3.5          Two-Dimensional Problem with Space-Dependent and Constant Surface Heat Flux             3-34

3.5.1      Piecewise-uniform approximation            3-35

3.5.1.1   Superposition-based numerical approximation of the solution     3-37

3.5.1.2   Heat flux-based sensitivity coefficients   3-37

3.5.1.3   Basic ‘building block’ solution     3-38

3.5.1.4   Computer code and examples    3-38

3.5.1.5   Matrix form of the superposition-based numerical approximation             3-40

3.6          Two-Dimensional Problem with Space- and Time-Dependent Surface Heat Flux   3-41

3.6.1      Piecewise-uniform approximation            3-42

3.6.1.1   Numerical approximation in space            3-42

3.6.2      Piecewise-constant approximation          3-43

3.6.2.1   Numerical approximation in time              3-44

3.6.3      Superposition-based numerical approximation of the solution     3-44

3.6.3.1   Sequential-in-time nature and sensitivity coefficients      3-46

3.6.3.2   Basic ‘building block’ solution     3-47

3.6.3.3   Computer code and example      3-47

3.6.3.4   Matrix form of the superposition-based numerical approximation             3-51

3.7          Chapter Summary            3-52

3.8          References         3-53

3.9          Problems             3-53

3.10        List of Figures    3-59

3.11        List of Tables      3-60

4              Inverse Heat Conduction Estimation Procedures 4.1

4.1          Introduction       4.1

4.2          Why is the IHCP Difficult?             4.2

4.2.1      Sensitivity to Errors         4.2

4.2.2      Damping and Lagging     4.3

4.3          Ill-Posed Problems          4.4

4.3.1      An Exact Solution             4.4

4.3.2      Discrete System of Equations      4.7

4.3.3      The Need for Regularization        4.8

4.4          IHCP Solution Methodology        4.8

4.5          Sensitivity Coefficients  4.9

4.5.1      Definition of Sensitivity Coefficients and Linearity              4.9

4.5.2      One-Dimensional Sensitivity Coefficient Examples            4.11

4.5.2.1   X22 Plate Insulated on One Side 4.11

4.5.2.2   X12 Plate Insulated on One Side, fixed boundary temperature     4.14

4.5.2.3   X32 Plate Insulated on One Side, fixed heat transfer coefficient   4.15

4.5.3      Two-Dimensional Sensitivity Coefficient Example              4.15

4.6          Stolz Method:  Single Future Time Step Method 4.19

4.6.1      Introduction       4.19

4.6.2      Exact Matching of Measured Temperatures         4.20

4.7          Function Specification Method  4.23

4.7.1      Introduction       4.23

4.7.2      Sequential Function Specification Method            4.23

4.7.2.1   Piecewise Constant Functional Form        4.24

4.7.2.2   Piecewise Linear Functional Form             4.31

4.7.3      General Remarks about Function Specification Method  4.33

4.8          Tikhonov Regularization Method              4.35

4.8.1      Introduction.     4.35

4.8.2      Physical Significance of Regularization Terms.      4.35

4.8.2.1   Continuous formulation 4.35

4.8.2.2   Discrete Formulation      4.36

4.8.3      Whole Domain TR Method          4.37

4.8.3.1   Matrix Formulation.        4.37

4.8.4      Sequential TR Method   4.43

4.8.5      General Comments about Tikhonov Regularization           4.43

4.9          Gradient Methods           4.44

4.9.1      Conjugate Gradient Method       4.44

4.9.1.1   Fletcher-Reeves CGM    4.46

4.9.1.2   Polak-Ribiere CGM          4.46

4.9.2      Adjoint Method (non-linear problems)   4.48

4.9.2.1   Some necessary mathematics    4.49

4.9.2.2   The Continuous Form of IHCP     4.50

4.9.2.3   The sensitivity problem 4.50

4.9.2.4   The Lagrangian and the Adjoint Problem               4.51

4.9.2.5   The Gradient Equation   4.53

4.9.2.6   Summary of IHCP solution by Adjoint Method     4.53

4.9.2.7   Comments about Adjoint Method            4.54

4.9.3      General Comments about CGM 4.55

4.10        Truncated Singular Value Decomposition Method             4.55

4.10.1    SVD Concepts    4.56

4.10.2    TSVD in the IHCP              4.56

4.10.3    General Remarks about TSVD     4.58

4.11        Kalman Filter      4.58

4.11.1    Discrete Kalman Filter Derivation              4.59

4.11.2    Two concepts for Applying Kalman Filter to IHCP 4.61

4.11.3    Scarpa and Milano Approach      4.61

4.11.3.1                Kalman Filter      4.62

4.11.3.2                Smoother            4.63

4.11.4    General Remarks about Kalman Filtering                4.64

4.12        Chapter Summary            4.66

4.13        References         4.67

4.14        Problems             4.71

4.15        List of Figures    4.74

4.16        List of Tables      4.75

5              Filter Form of IHCP Solution         5-1

5.1          Introduction       5-1

5.2          Temperature Perturbation Approach      5-1

5.3          Filter Matrix Perspective              5-3

5.3.1      Function Specification Method  5-3

5.3.2      Tikhonov Regularization 5-6

5.3.3      Singular Value Decomposition    5-9

5.3.4      Conjugate Gradient        5-12

5.4          Sequential Filter Form    5-15

5.5          Using Second Temperature Sensor as Boundary Condition             5-18

5.5.1      Exact Solution for the Direct Problem      5-19

5.5.2      Tikhonov Regularization Method as IHCP Solution             5-21

5.5.3      Filter Form of IHCP Solution         5-22

5.6          Filter Coefficients for Multi-Layer Domain             5-26

5.6.1      Solution Strategy for IHCP in Multi-Layer Domain               5-27

5.6.1.1   Inner Layer         5-27

5.6.1.2   Outer Layer        5-28

5.6.1.3   Combined Solution          5-30

5.6.2      Filter Form of the Solution           5-30

5.7          Filter Coefficients for Non-Linear IHCP: Application for Heat Flux Measurement Using Directional Flame Thermometer   5-33

5.7.1      Solution for the IHCP      5-34

5.7.1.1   Back Layer (Insulation)  5-35

5.7.1.2   Front Layer (Inconel plate)           5-35

5.7.1.3   Combined Solution          5-36

5.7.2      Filter form of the solution            5-36

5.7.3      Accounting for Temperature-Dependent Material Properties       5-37

5.7.4      Examples             5-39

5.8          Chapter Summary            5-46

5.9          Problems             5-46

5.10        References         5-47

5.11        List of Figures    5-49

5.12        List of Tables      5-51

6              Optimal Regularization  6.1

6.1          Preliminaries      6.1

6.1.1      Some Mathematics         6.1

6.1.2      Design vs. Experimental Setting 6.2

6.2          Two Conflicting Objectives          6.2

6.2.1      Minimum Deterministic Bias        6.2

6.2.2      Minimum Sensitivity to Random Errors   6.3

6.2.3      Balancing Bias and Variance        6.3

6.3          Mean Squared Error        6.4

6.4          Minimize Mean Squared Error in Heat Flux           6.5

6.4.1      Definition of       6.6

6.4.2      Expected value of             6.6

6.4.3      Optimal Regularization using E( )               6.10

6.5          Minimize Mean Squared Error in Temperature    6.13

6.5.1      Definition of       6.13

6.5.2      Expected value of             6.14

6.5.3      Morozov Discrepancy Principle   6.14

6.6          The L-curve        6.17

6.6.1      Definition of L-curve       6.17

6.6.2      Using Expected Value to Define L-curve 6.18

6.6.3      Optimal Regularization using L-curve       6.19

6.7          Generalized Cross Validation       6.20

6.7.1      The GCV Function            6.21

6.8          Chapter Summary            6.24

6.9          References         6.26

6.10        Problems             6.27

6.11        List of Figures    6.28

6.12        List of Tables      6.29

7              Evaluation of IHCP Solution Procedures 7.1

7.1          Introduction       7.1

7.2          Test Cases           7.3

7.2.1      Introduction       7.3

7.2.2      Step Change in Surface Heat Flux              7.4

7.2.3      Triangular Heat Flux        7.5

7.2.4      Quartic Heat Flux             7.7

7.2.5      Random Errors  7.11

7.2.6      Temperature Perturbation          7.11

7.2.7      Test Cases with Units      7.12

7.3          Function Specification Method  7.13

7.3.1      Step Change in Surface Heat Flux              7.13

7.3.2      Triangular Heat Flux        7.15

7.3.3      Quartic Heat Flux             7.15

7.3.4      Temperature Perturbation          7.17

7.3.5      Function Specification Test Case Summary            7.21

7.4          Tikhonov Regularization 7.22

7.4.1      Step Change in Surface Heat Flux              7.22

7.4.2      Triangular Heat Flux and Quartic Heat Flux            7.24

7.4.3      Temperature Perturbation          7.26

7.4.4      Tikhonov Regularization Test Case Summary        7.29

7.5          Conjugate Gradient Method       7.29

7.5.1      Step Change in Surface Heat Flux              7.30

7.5.2      Triangular Heat Flux and Quartic Heat Flux            7.31

7.5.3      Temperature Perturbation          7.34

7.5.4      Conjugate Gradient Test Case Summary 7.37

7.6          Truncated Singular Value Decomposition              7.37

7.6.1      Step Change in Surface Heat Flux              7.38

7.6.2      Triangular and Quartic Heat Flux               7.39

7.6.3      Temperature Perturbation          7.41

7.6.4      TSVD Test Case Summary             7.43

7.7          Kalman Filter      7.44

7.7.1      Step Change in Surface Heat Flux              7.44

7.7.2      Triangular and Quartic Heat Flux               7.45

7.7.3      Temperature Perturbation          7.47

7.7.4      Kalman Filter Test Case Summary             7.51

7.8          Chapter Summary            7.51

7.9          References         7.55

7.10        Problems             7.55

7.11        List of Figures    7.57

7.12        List of Tables      7.61

8              Multiple Heat Flux Estimation     8-1

8.1          Introduction       8-1

8.2          The forward and the inverse problems   8-1

8.2.1      Forward Problem             8-2

8.2.2      Inverse Problem               8-3

8.2.3      Filter form of the solution            8-5

8.3          Examples             8-7

Example 8.1        8-8

Example 8.2        8-11

8.4          Chapter Summary            8-15

8.5          References         8-16

8.6          Problems             8-16

8.7          List of Figures    8-17

8.8          List of Tables      8-19

9              Heat Transfer Coefficient Estimation       9-1

9.1          Introduction       9-1

9.1.1      Recent Literature             9-1

9.1.2      Basic approach  9-2

9.2          Sensitivity Coefficients  9-4

9.3          Lumped Body Analyses  9-8

9.3.1      Exact Matching of the Measured Temperatures 9-8

9.3.2      Filter Coefficient Solution             9-10

9.3.3      Estimating constant heat transfer coefficient       9-12

9.4          Bodies with Internal Temperature Gradients       9-15

9.5          Chapter Summary            9-18

9.6          References         9-18

9.7          Problems             9-20

9.8          Figures 9-21

9.9          Tables   9-22

10           Temperature Measurement        10.1

10.1        Introduction       10.1

10.1.1    Subsurface Temperature Measurement 10.1

10.1.2    Surface Temperature Measurement        10.2

10.2        Correction Kernel Concept           10.3

10.2.1    Direct Calculation of Surface Heat Flux    10.3

10.2.2    Temperature Correction Kernels               10.3

10.2.3    2-D Axisymmetric Model              10.5

10.2.4    High Fidelity Models and Thermocouple Measurement  10.13

10.2.5    Experimental Determination of Sensitivity Function          10.15

10.3        Unsteady surface element method          10.16

10.3.1    Intrinsic thermocouple  10.17

10.4        Chapter Summary            10.22

10.5        References         10.23

10.6        Problems             10.25

10.7        Figures 10.27

10.8        Tables   10.27

Appendices

A             Numbering System         A.1

A.1         Dimensionality, coordinate system, and types of boundary condition        A.1

A.2         Boundary condition information                A.2

A.2.1      Finite-in-time boundary condition            A.4

A.2.2      Partial-in-space boundary condition         A.4

A.3         Initial temperature distribution A.5

A.3.1      Partial-in-space initial condition A.6

A.4         REFERENCES       A.6

B             Exact Solution X22B(y1pt1)0Y22B00T0     B.1

B.1          Exact analytical solution. Short-time form             B.1

B.1.1      Short-time form with semi-infinite solution along y           B.1

B.1.2      Short-time form with semi-infinite solution along x           B.2

B.1.3      Two-dimensional semi-infinite solution  B.3

B.2          Exact analytical solution. Large-time form             B.4

B.2.1      Quasi-steady part with eigenvalues in the homogeneous direction            B.5

B.2.2      Quasi-steady part with eigenvalues in the nonhomogeneous direction    B.6

B.2.3      Complementary transient part   B.7

B.3          References         B.8

C             Green’s functions Solution Equation        C-1

C.1          Introduction       C-1

C.2          One-Dimensional Problem with Time-Dependent Surface Temperature   C-1

C.2.1      Piecewise-constant approximation          C-2

C.2.1.1  Numerical approximation of the GF equation.     C-4

C.2.2      Piecewise-linear approximation C-4

C.2.2.1  Superposition of solutions at the (i-1)-th time step.           C-6

C.2.2.2  Superposition of solutions at the i-th time step. C-7

C.2.2.3  Numerical approximation of the GF equation.     C-9

C.3          One-Dimensional Problem with Time-Dependent Surface Heat Flux          C-9

C.3.1      Piecewise-constant approximation          C-10

C.3.1.1  Numerical approximation of the GF equation.     C-11

C.3.2      Piecewise-linear approximation C-11

C.3.2.1  Superposition of solutions at the (i-1)-th time step.           C-13

C.3.2.2  Superposition of solutions at the i-th time step. C-13

C.3.2.3  Numerical approximation of the GF equation.     C-13

C.4          Two-Dimensional Problem With Space- And Time-Dependent Surface Heat Flux  C-14

C.5          References         C-16

C.6          List of Figures    C-16

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