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I NVERSE H EAT C ONDUCTION I II-posed P roblems J AMES V . B ECK D"pClfl",etrl o f Medlatrkal M it'IIiga" Slclle Utrivc·r,o;iIJ' £"yi"c'('r;tr~1 BEN B LACKWELL Sanaia NUlicmal Laboratories Alhuqu('rqllt'. N"It' M exim C HARLES R. ST. CLAIR. J R. or Depurlme", Medlatrit'al £trg;""('r;"y Michigan Slale Ullivl'rsil), A W ilev-Interscience P ublication N ew Y ork • C hichester • B risbane • T oronto • S ingapore To my wife, Barbara; children, Sharon and Douglas; and father and mother, Peter and Louise Beck J.Y.B To my wife, Betty; and children, Jeffrey and Gregory Blackwell B.E Copyril:hl © 1985 by John Wiley & Sons. Inc. All righls reserved. Published simullaneously in Canada. Reproduclion or lranslalion o f any pari o f lhis work beyond lhal pcrmilled by Seclion 107 o r 108 o f lhe . . 1976 Uniled Slales Copyrighl ACI w ilhoullhe pcrmls."on o f lhe copyrighl owner is unlawful. Requesls for permission o r furlher informalion sh~uld be addressed 10 lhe Permissions Deparl~enl, J ohn Wiley & Sons, Inc. Librllry o f C ongrus CIIIIIIOf/ing in Publicli/ion DIIIII: Beck, J . V. (James Vere), 1930Inverse heal conduclion . " A Wiley·lnlerscience publicalion." Includes bibliographies and index. \. Heal .-Conduclion . 2. Numerical analysis -· Improperly posed problems. I. Blackwell, Ben. II . Sl . Clair, Charles R . I ll. Tille. QC320.B4 1985 536' ,23 85-5391 ISBN 0-471-08319-4 Prinled in lhe Uniled Slales o f America 10 9 8 7 6 5 4 3 2 I T o the greatest engineers in the family : Charles R. St. Clair, Sr. and Deborah S. Short (my daughter). And to those for whom the engineers live and labor: my mother Erla, my wife Jeanette, and our children and grandchildren from oldest to youngest with the greatest in doubt: Charles III, Scott, Judy, Timothy B. Short, Gregory, and Kevin. And to those who are as family: John and Ann Polomsky C.R.S P REFACE This book presents a study o f the Inverse Heat Conduction Problem ( IHCP) which is the estimation o f t he surface heat flux history o f a heat conductinr body. Transient temperature measurements inside t he body are utilized in the calculational procedure. The presence o f e rrors in the measurements as well as the ill-posed nature o f the problem lead to " estimates" r ather than the "true" surface heat flux a nd/or temperature. This book was written because o f the importance and practical nature o f the I HCP; furthermore, at the time o f writing there is no available book on the subject written in English. T he specific problem treated is only one o f man} ill-posed problems but the techniques discussed herein c an be applied to many others. The basic objective is to estimate a function given measurements that are " remote" in s ome sense. O ther applications include remote sensing, oil exploration, nondestructive evaluation o f materials, and determination o f the Earth's interior structure. The authors became interested in the I HCP over two decades ago while employed in the aerospace industry. O ne o f the applications was the determination o f the surface heat flux histories o f reentering heat shields. This book is written as a textbook in engineering with numerical examples and exercises for students. These examples will be useful to practicing engineers who use the book to become acquainted with the problem and methods o f solution. A companion book, Parameter Estimation in Engineering (lnd Science by J . V. Beck and K . J . A rnold (Wiley, 1977), discusses estimation o f certain constants o r p arameters rather than functions as in t he I HCP. T hough many o f the ideas relating to least squarcs and sensitivity coefficients are present in both books, the present book docs not re,!uire a mastery o f p aramcter estimation. The book is written at the advanced B.S. o r the M.S. level. A course in heat conduction a t the M .S. level o r courses in partial differential equations and numerical methods are recommended as prerequisite materials. vii .'!" v iii PREFACE O ur philosophy in writing this book was to emphasize general techniques rather than specialized procedures unique to the IHCP. F or example, basic techniques developed in Chapter 4 can be applied either to integral equation representations o f the heat diffusion phenomena o r t o finite difference (or element) approximations o f the heat conduction equation. The basic procedures in C hapter 4 can treat nonlinear cases, mUltiple sensors, nonhomogeneous media, multidimensional bodies, and many equations, in addition to the transient heat conduction equation . The two general procedures that are used are called (a) function specification and (b) regularization. A method o f combining these (the trial function method) is also suggested. One o f the important contributions o f this book is the demonstration t hat all o f these methods can be implemented in a sequential manner. The sequential method in some case gives nearly the same result as whole domain estimation and yet is much more computationally efficient. One o f o ur goals was to provide the reader with an insight into the basic procedures that provide analytical tools to compare various procedures. We do this by using the concepts o f sensitivity coefficients, basic test cases, and the mean squared error. The reader is also shown that optimal estimation involves the compromise between minimum sensitivity to random measurement errors and the minimum bias. Preliminary notes have been used for an ASME short course and for a graduate course a t Michigan State University. There are many people who have helped in the preparation o f this text and to whom we express o ur appreciation. These include D. Murio, M. Raynaud, and other colleagues and students who have read and commented on the notes. Thanks are also due to Judy Duncan, Phyllis Murph, Terese Stuckman, Alice Montoya, a nd J eana Pineau, who have aided in typing the manuscript. James V. Beck wishes to express appreciation for the contributions to his education made by Kenneth Astill o f T ufts University, Warren Rohsenow o f Massachusetts Institute o f Technology, and A. M. Dhanak o f Michigan State University. Ben Blackwell would like to acknowledge the contributions that several people made to his heat transfer education : H. Wolf o f the University o f Arkansas, M. W. Wildin o f the University o f New Mexico, and W. M. Kays o f S tanford University. A special a nd deep appreciation is extended to George A. Hawkins for the education a nd philosophy that he imparted to Charles R. St. Clair, Jr. as his ' graduate student. JAMES V. B ECK, BEN BLACKWELL CHARLES R. S T. C LAIR, J R. Eos/ Lallsing. Michigan Alhuquerque. Nitl\' M exico East Lal/sillg. Michigan August 1985 C ONTENTS N omenclature 1. xv D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM 1 1 .1 1 .2 Introduction. 1 Examples o f Inverse P roblems. 3 1.2.1 Inverse H eat C onduction P roblem Examples. 3 1 .2.2 . O ther Inverse F unction Estimation Problems 7 1 .3 Function Estimation Versus Parameter Estimation 9' 1 .4 Measurements. 9 . 1.4.1 Description of M easurement Errors 9 1 .4 .2. S tatistical Description of Errors. 1 0' 1.5 Why IS t he IHCP Difficult? 1 3 1 .5.1 Sensitivity to Errors. 1 3 1 .5.2 Exam?'es of Damping a nd L agging; Exact Solutions. 1 3 1.6 S ensitivity Coefficients. 1 9 1 .6.1 Definition of Sensitivity C oefficients a nd Linearity. 19 1.6.2 O ne- Dimensional Sensitivity C oefficient Examples. 22 Lumped Body Case. 22 1.6.2 .2 Semi-Infinite Body. 25 1.6.2 .3 Plate I nsulated o n O ne S ide 3 0 1 .6.3 T wo-Dimensional S ensitivity Coeffici~nt Example. 31 1 .7 Classification of Methods. 3 6 1 .8 Criteria for Evaluation of IHCP M ethods 3 8 1 .9 S cope o f B ook. 39 . References. 4 0 P roblems. 4 3 ix CONTENTS CONTENTS x 2. EXACT SOLUTIONS OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM 4. 51 I ntroduction. 51 Steady-State Solution. 52 Transient Analysis of Bodies w ith Small Internal Thermal Resistance. 54 2.3.1 Exact Solution. 54 2.3.2 Approximate Solutions. 54 2.3.3 Temperature Errors and Approximate Solutions. 55 2.4 Heat Flux From Measured Surface Temperature History. 59 2.4.1 Exact Results for C ontinuous Surface Temperature History. 59 2.4.2 Approximate Results f or S emi-Infinite B ody w ith Surface Temperature Measured at Discrete Times. 61 2.4 .3 Temperature Error Propagation in Eq . (2.4 .8). 63 2.5 Exact Solutions o f Inverse Heat C onduction Problems. 67 2.5.1 Literature Review. 67 2.5.2 Derivation of Exact Solution f or Planar Geometry. 67 2.5.3 Expressions for Cylinders and Spheres. 71 2 .5.4 Example Results for Planar Geometry. 7 2 References. 75 Problems. 76 3. A PPROXIMATE M ETHODS FOR DIRECT H EAT C ONDUCTION P ROBLEMS I NVERSE H EAT C ONDUCTION E STIMATION PROCEDURES 4 .1 4.2 2.1 2 .2 2.3 4.3 4.4 4.5 78 Introduction. 78 3.1.1 Various Numerical Approaches. 78 3.1.2 Scope o f Chapter. 79 3.2 Duhamel's Theorem. 80 3 .2.1 D erivation of D uhamel's Theorem. 8 0 . 3.2.2 Numerical Approximation o f D uhamel's Theorem. 82 3.2.3 M atrix Form o f D uhamel's Theorem . 8 3 3.3 Difference Methods. 87 3.3 .1 Finite C ontrol V olume Procedure for C onstant Property Planar Geometries. 87 3.3.2 Other B oundary C onditions and Material Interfaces. 93 3.3 .3 Numerical Techniques for Solving Systems of First -Order Ordinary Differential Equations. 9 4 3.3.4 General Form of Difference Equations for Heat C onduction i n Planar Body. 96 3.3.5 Standard Form for Temperature Equation for I HCP. 99 References. 1 02 Problems. 1 03 3 .1 4.6 4.7 4.8 i xi 108 Introduction. 1 08 111- Posed Problems. 1 10 4.2.1 Partial Differential Equation Perspective. 1 10 4.2.2 I ntegral Equation Perspective. 112 4.2.3 Difference Equati on Perspective. 1 13 S ingle Future Time Step M ethod. 115 4.3.1 I ntroduction. 115 4.3.2 Exact M atching o f Measured Temperatures (Single Sensor) . 1 15 4 .3.3 M ultiple Temperature Sensors. 1 18 F unction Specification M ethod . 119 4.4 .1 Introduction. 1 19 4.4.2 Whole Domain Estimation. 1 19 Smoothly C hanging H eat Flux. 1 20 4.4.2 .2 A bruptly C hanging Heat Flux Histories. 122 4.4 .3 Sequential Estimation. 125 4.4 .3.1 C onstant Heat Flux Functional Form. 125 4.4 .3.2 Linear Heat Flux Functional Form. 131 4.4 .3.3 Alternative Interpretation. 1 33 Regularization M ethod . 1 34 4 .5.1 Introduction. 1 34 4 .5.2 Physical Significance of Regularization Terms. 135 4.5.3 Whole Domain Regularization Method. 1 37 4 .5.3.1 Algebra ic Formulation. 1 37 4 .5.3.2 M atrix F ormulation . 1 38 4.5.3 .3 Selection o f Regularization Parameter. 1 40 4 .5.4 Sequential Regularization Method. 141 Trial Function M ethod . 1 45 4 .6.1 I ntroduction. 1 45 4.6 .2 Matrix Analysis. 1 45 4 .6.3 Zeroth-Order Regularization M ethod . 1 47 4 ,6.4 Generalized Sequential Function Specification M ethod . 1 47 Filter Form o f Linear I H CPo 1 48 4 .7.1 I ntroduction. 1 48 4 .7.2 Sequential Filter Algorithm. 1 48 4 .7.3 Prefiltering Temperature Measurements. 1 53 T wo C onflicting Objectives. 1 53 4 .8.1 M in imum Deterministic Bias. 1 53 4 .8.2 M inimum S ensitivity to Random Errors. 1 54 4.8.3 Mean Squared Error. 1 54 4 .8.4 Variance o f Estimated Heat Flux Component. 1 56 4 .8.5 Estimate o f Determin istic Error in Surface Heat Flux. 1 57 C ONTENTS xii C ONTENTS 5.5 Digital Filter Algorithm. 196 5.5.1 Introduction. 1 96 5.5.2 Function Specification- Based Filter. 197 Finite Plate Case. 1 97 S emi-Infinite Body. 200 5.5.3 W hole Domain Regularization Filter. 201 5.6 Optimal Considerations. 203 5.6.1 Optimal Function Specificatio'n Algorithm. 204 5.6.2 Optimal Whole Domain Regularization Method. 210 References. 212 Problems. 213 References. 159 Problems. 161 5. I NVERSE C ONVOLUTION P ROCEDURES FOR A S INGLE S URFACE H EAT F LUX 5.1 5.2 5.3 5.4 xiii 1 65 I ntroduction. 165 Test Cases. 1 67 5.2.1 I ntroduction. 1 67 5.2.2 Step Change in Surface Heat Flux. 1 68 5.2.3 Triangular Heat Flux. 169 5.2.4 Random Errors. 1 70 5.2.5 Heat Flux Impulse Test Case (C5qM/Dqf)' 1 73 5.2.6 Temperature Impulse Test Case (i>qM/CW f ). 1 74 5.2.7 Test Cases w ith Units. 1 74 Function Specification Algorithms. 176 5.3.1 I ntroduction. 1 76 5.3.2 Single Future Temperature Algorithm (Stolz Method). 176 Step Heat Flux Test Case. 1 77 Triangular Heat Flux Test Case. 1 77 Heat Flux Impulse Test Case (OqM/Oqf)' 1 78 Temperature Impulse Test Case (OC,M/cW f ).179 5.3.3 M ultiple Future Temperatures Algorithm. 181 Step Heat Flux Test Case. 181 Triangular Heat Flux Test Case. 182 Heat Flux Impulse Test Case (oq M/Oq f )' 184 Temperature Impulse Test Case (fJqM/fJY f ). 1 84 Regularization Algorithms. 186 5.4.1 I ntroduction. 1 86 5.4.2 W hole Domain Regularization Method. 187 Triangular Heat Flux Test Case. 189 Heat Flux Impulse Test Case. 1 90 Temperature Impulse Test Case ( oqM/oY f ).191 5.4.3 Sequential Regularization Method. 191 Triangular Heat Flux Test Case. 1 93 Heat Flux Impulse Test Case. 194 Temperature Impulse Test Case. 1 94 Comparison of Whole Domain and Sequential Regularization Methods. 196 Comparison of Sequential Regularization and Function Specification Methods. 196 6. D IFFERENCE M ETHODS FOR T HE S OLUTION O F T HE O NE-DIMENSIONAL I NVERSE H EAT C ONDUCTION P ROBLEM 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 I ntroduction. 218 Sensitivity Coefficients and Their Calculation by Difference Methods. 219 Single Temperature Sensor. Function Specification ( q=C). Single Future Time Step (Exact Matching o f Data). 222 6.3.1 M odification o f Difference Equations o f the Direct Heat C onduction Problem for the Solution o f the IHCP. 222 6.3.2 Sensitivity Coefficient Approach for Exactly Matching Data from a Single Sensor. 223 M ultiple Temperature Sensors. Function Specification ( q=C). Single Future Time Step. 230 W hole Domain Estimation With Difference Methods. 233 Single Temperature Sensor. Function Specification ( q=C). r Future Time Steps. 237 M ultiple Temperature Sensors; Function Specification ( q=C). Arbitrary Future Time Steps. 241 Single Temperature Sensor. Function Specification. Linear Heat Flux (Connected Segments). 242 Second Order Sequential Regularization Methods. 243 Space Marching Techniques for One-Dimensional Problems. 247 6.10.1 Analytical Solution. 248 6.10.2 M ethod o f D·Souza. 249 6.10.3 M ethod o f Weber. 252 6.10.4 M ethod of Raynaud and Bransier. 253 6.10.5 M ethod o f Hills and Hensel. 254 6.10.6 Comparison w ith Prior Methods. 256 Numerical Calculations. 256 Computer Programs. 262 218 C ONTENTS x iv / 'J ~ References. 264 Problems. 265 7. M ULTIPLE H EAT F LUX E STIMATION 267 7.1 7.2 I ntroduction. 267 T wo Independent Heat Fluxes Case. 268 7.2.1 Sequential Function Specification M ethod . 271 7.2.2 Sequential Regularization Method. 273 7.3 Multiple Heat Flux Case . 275 7.3.1 Sequential Function Specification Method for Multiple Heat Flux Components. 276 7.3.2 Sequential Regularization Method for Multiple Heat Fluxes. 277 References. 279 Problems. 279 8. N OMENCLATURE H EAT T RANSFER C OEFFICIENT E STIMATION 281 8.1 I ntroduction. 281 8 .2 Sensitivity Coefficients. 283 8.2.1 Lumped Body Case . 283 8.2.2 Semi - Infinite Body. 287 8.3 Lumped Body Analyses . 290 8.3.1 Exact Matching of the Measured Temperatures. 290 8.3.2 Regression Method. 292 8.3.3 Function. Specification Procedure w ith q = Constant. 293 8.3.4 Function Specification Procedure w ith h =Constant. 294 8.4 Bodies With Internal Temperature Gradients. 297 8.4.1 Analysis for r Future Temperatures -Using q =C Function Specification Method. 297 8.4.2 Examples. 299 8.5 Estimation of Contact Conductance. 301 References. 301 Problems. 301 A uthor I ndex 3 05 " c cov (., .) ej e rf erfe E E (· ) jj If G h he H o• H b ierfc 3 03 S ubject I ndex a, b, c, d, e,1 Coefficients in tridiagonal matrix algorithm; see (6.3.9-10) a J ... Radius o f cylinder o r sphere Heat capacity Covariance o perator; see (104.8) Residual temperature e rror; see (104.7) E rror function Complementary e rror function Sensor depth below heated surface Expected value operator Filter coefficient; see (4.7.2) Dimensionless filter coefficient Green's function Convective heat transfer coefficient Contact conductance Regularization matrices; see (4.5.16) Integral error function A vector o f ones Identity matrix Number o f t emperature sensors Bessel functions o f the first kind, order 0, I, . . _ Thermal conductivity Gain coefficient at time I j ; see (4.4.25) = (q,L/k)K j , q ,= I in consistent units, dimensionless gain coefficien t xv NOMENCLATURE x vi G ain coefficient for sensor j a t time I j Slab thickness General time index a il tl!:,.x2, grid scale Fourier number Heat flux C onstant value o f heat flux H eat flux a t time Ij Estimated value o f qj H eat flux a t time t j t hat exactly matches the temperature d ata Y j ; see ( Estimated heat flux for interval 1 M-I t o 1M H eat flux vector; see (4.6.4) Heat flux vector; see (4.6.4) Trial value o f q N umber o f future time steps Radial coordinate qM q ql q* r r r; rIa S Least square function Time Dimensionless time, atl L 2 atla 2 T emperature Initial temperature Ambient temperature a t which convection o r r adiation is taking place ( T - To)/(qLlk) Estimated value o f T Vector o f estimated temperatures for q = 0 T emperature corresponding t o q = q*; see (6.3.8) T emperature response function for a body at zero initial temperature and subjected to a unit step in surface temperature; see (2.4.1) Variance operator Heat flux weighting factor; see (4.4.40c) Regularization constants; see (4 .5.1) Spatial coordinate x lL Sensitivity coefficient for heat flux pulse = (klx)oTloq< T To T.., T+ T t\q=O t u(x, I) v (·) Wj W o, W h X x+ X X+ ·•• NOMENCLATURE X Yj Z Matrix o f sensitivity coefficients Measured value o f t emperature at time I j Sensitivity coefficient for heat flux step GREEK S YMBOLS PI,P2,· " y fiq, c5Y, c5qM f iYj I lt illM !:,.x i l¢(r, In) ilT1 ; o }. p a ¢ (x, I ) w T hermal diffusivity, also regularization parameter See (5.6.14) Parameters; see (4.4.1-5) Thermal wave speed Heat flux impulse Temperature impulse o r e rror Heat flux e rror for a uni"t e rror in t emperature Time step = IM+I-I M Spatial grid size for difference methods = ¢(r, I n + , )-¢(r, t n ) a illlE 2 T emperature e rror; see (1.4.1) Time difference weighting parameter Dummy time variable Density Standard deviation Temperature response to a unit step in heat flux Tridiagonal matrix coefficient; see (6.3.10) x vii