The First Edition of Inverse Heat Conduction: Ill-Posed Problems has long been thought of as the “gold standard” on the topic. Now, in the long-awaited Second Edition, the leading experts in the field explain the latest inverse heat conduction problems in a way that is accessible and understandable for today’s generation of engineers and researchers. In addition to discussing established IHCP techniques like the Stoltz method, Function Specification method, Tikhonov Regularization method, and Singular Value Decomposition method, the book has been thoroughly updated to cover new and advanced IHCP topics like Optimal Regularization, Temperature Measurement, Multiple Heat Flux Estimation, Temperature Dependent Material Properties, and IHCP in Multi-Dimensional Domains. Updated figures are featured throughout the book and all plots have been re-generated using MATLAB for enhanced quality. New industrial examples connect theory to applications such as directional flame thermometers, casting, thermal protection of space vehicles during atmospheric entry, and more. Each chapter features examples and problems to reinforce concepts and the book is accompanied by downloadable MATLAB codes covering all solution methods for every example in the book.

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