liu g.r., han x. - computational inverse techniques in nondestructive evaluation(2003)(592)

8/13/2019 Liu G.R., Han X. - Computational Inverse Techniques in Nondestructive Evaluation(2003)(592)

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COMPUTATIONALINVERSE TECHNIQUES

in

NONDESTRUCTIVEEVALUATION

2 3 by CRC Press LLC


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CRC PR ESS

Boca Raton London New York Washington, D.C.

COMPUTATIONALINVERSE TECHNIQUES

in

NONDESTRUCTIVEEVALUATION

G.R. Liu

X. Han



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This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

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Library of Congress Cataloging-in-Publication Data

Liu, G.R.

Computational inverse techniques in nondestructive evaluation / G.R., Liu, X. Han,

p. cm.

Includes bibliographical references and index.

ISBN 0-8493-1523-9 (alk. paper)

1. Non-destructive testingMathematics. I. Han, X. II. Title.

TA417.2.L58 2003

620.1127dc21 2003043554

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Dedication

To Zuona

Yun, Kun, Run,

and my family

for the time they gave to me

G. R. Liu

To Zhenglin, Weiqi

and my family for their support

To my mentor, Dr. Liu

for his guidance

X. Han



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as using the simplest examples to reveal true meanings of these abstruseterminologies and the mechanisms of some important phenomena. Manyexample problems and practical engineering problems are presented,

together with many numerical tests as well as some experimental verification.The authors hope that this book can help readers face inverse problems

comfortably and tackle them with ease, without being frightened off. Thetruth is that many engineering inverse problems are not that difficult becausethey can be well posed if they are properly formulated with a sound exper-imental strategy.

Properly formulating and solving an inverse problem demands that theanalyst have (1) very good understanding of the physical problem, (2) goodexperimental strategy and quality measurement data, and (3) most impor-

tantly, effective computational techniques. Without a good understanding ofthe physics of the problem, basically nothing can be done. This book willnot help much in this context, except to emphasize the importance of thisunderstanding. Quality measurement data are essential because they willdecide the quality of the solution of the inverse problem. This includes notonly the accuracy of the experimental (or test or observational) data, but alsothe precise knowledge of the characteristics of the measurement data in termsof the noise content (noise level, frequency, etc.). Apart from modern high-tech experimental equipment, acquiring such quality experimental data

depends highly on understanding the physics involved in the problem andthe process of measurement. Although, this book will cover some of theissues in measuring wave and vibration responses of structures, they are notits focus.

This book emphasizes the key to solution of any practical and complexinverse problems: computational techniques. These techniques concern howto obtain what is needed from given experimental data efficiently and accu -rately. Without the computer and effective computational techniques, it isnot possible to perform a decent inverse analysis of a complex engineering

problem. Forward solver is also very important, but this book generallyassumes that a reliable forward solver to the physical problem is available.Thus, only sources of and a brief introduction to forward solvers are pro-vided here. It is the task of the analyst to use these forward solvers properlyand produce reliable results for the inverse analysis by no means an easytask, but not the focus of this volume. Readers may refer to earlier books byDr. Liu or other related literature.

The authors work in the area of inverse analysis has been profoundlyinfluenced and guided by many existing works reported in the open litera-ture, which are partially listed in the references. Without those significantcontributions to this area, this book would not exist. The authors would liketo thank all the authors of the excellent papers and books published in areasrelated to this books topic.

Many colleagues and students have supported and contributed to thewriting of this book. Dr. Liu expresses sincere thanks to all of them, withspecial appreciation to Y.G. Xu, Z.L. Yang, Z.P. Wu, S.I. Ishak, H.M. Shang,



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S.P. Lim, W.B. Ma, Irwan Bin Karim, S.C. Chen, and H.J. Ma. Many of themhave contributed examples to this book in addition to their hard work incarrying out a number of projects related to inverse problems.

Finally, the authors would also like to thank A*STAR, Singapore, for itspartial financial sponsorship for research projects related to the topic of thisbook that were undertaken by the authors and their teams.

G.R. Liu and X. Han



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Authors

Dr. G.R. Liureceived his Ph.D. from Tohoku Univer-sity, Japan, in 1991. He was a postdoctoral fellow atNorthwestern University, Evanston, Illinois. He iscurrently the director of the center for advanced com-putations in engineering science (ACES), National

University of Singapore and an associate professorin the Department of Mechanical Engineering,National University of Singapore. He is also currentlythe president of the Association for ComputationMechanics (Singapore). Dr. Liu has provided consul-tation services to many national and internationalorganizations and authored more than 300 technical publications, includingmore than 180 international journal papers. He has written five books,including the popular book Mesh-Free Method: Moving beyond the Finite Ele-ment Method. He serves as an editor and a member of editorial boards of fivescientific journals. Dr. Liu is the recipient of the Outstanding UniversityResearchers Award, the Defense Technology Prize, and the Silver Award atCrayQuest (nationwide competition). His research interests include compu-tational mechanics, mesh-free methods, nanoscale computation, microbio-system computation, vibration and wave propagation in composites,mechanics of composites and smart materials, inverse problems, and numer-ical analysis.

Dr. X. Han obtained his bachelors and masters

degrees in engineering mechanics from Harbin Insti-tute of Technology, China, in 1990 and 1997, respec-tively, and his doctorate in mechanical engineeringfrom National University of Singapore in 2001. Hewas a research fellow at the School of Mechanical andProduction Engineering, Nanyang Technology Uni-versity, Singapore. Dr. Han has been working on thedevelopment of numerical analysis techniques forwave propagation problems and computational

inverse techniques. He is currently the manager of thecenter for advanced computations in engineering science (ACES), Depart-ment of Mechanical Engineering, National University of Singapore. Dr.Hans research interests include structural dynamics of advanced compositeand smart materials, inverse problems and numerical analysis. He is theauthor or co-author of approximately 30 referenced journal papers.



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2.7 General Inversion by Singular Value Decomposition (SVD)2.7.1 Property of Transformation and Type II Ill-Posedness2.7.2 SVD Procedure

2.7.3 Ill Conditioning2.7.4 SVD Inverse Solution

2.8 Systems in Functional Forms: Solution by Optimization2.9 Choice of the Outputs or Effects2.10 Simulated Measurement2.11 Examination of Ill-Posedness2.12 Remarks

3 Regularization for Ill-Posed Problems3.1 Tikhonov Regularization

3.1.1 Regularizing the Norm of the Solution3.1.2 Regularization Using Regularization Matrix3.1.3 Determination of the Regularization Matrix3.1.4 Tikhonov Regularization for Complex Systems

3.2 Regularization by SVD3.3 Iterative Regularization Method3.4 Regularization by Discretization (Projection)

3.4.1 Exact Solution of the Problem

3.4.2 Revealing the Ill-Posedness3.4.3 Numerical Method of Discretization for Inverse Problem

3.4.3.1 Finite Element Solution3.4.3.2 Inverse Force Estimation

3.4.4 Definition of the Errors3.4.5 Property of Projection Regularization3.4.6 Selecting the Best Mesh Density

3.5 Regularization by Filtering3.5.1 Example I: High-Frequency Sine Noise

3.5.2 Example II: Gaussian Noise3.6 Remarks

4 Conventional Optimization Techniques4.1 The Role of Optimization in Inverse Problems4.2 Optimization Formulations4.3 Direct Search Methods

4.3.1 Golden Section Search Method4.3.2 Hooke and Jeeves Method

4.3.2.1 Exploratory Moves4.3.2.2 Pattern Moves4.3.2.3 Algorithm4.3.2.4 Example

4.3.3 Powells Conjugate Direction Method4.3.3.1 Conjugate Directions

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4.3.3.2 Example4.4 Gradient-Based Methods

4.4.1 Cauchys (Steepest Descent) Method

4.4.2 Newtons Method4.4.3 Conjugate Gradient Method

4.5 Nonlinear Least Squares Method4.5.1 Derivations of Objective Functions4.5.2 Newtons Method4.5.3 The GaussNewton Method4.5.4 The LevenbergMarquardt Method4.5.5 Software Packages

4.6 Root Finding Methods

4.6.1 Newtons Root Finding Method4.6.2 LevenbergMarquardt Method

4.7 Remarks4.8 Some References for Optimization

5 Genetic Algorithms5.1 Introduction5.2 Basic Concept of GAs

5.2.1 Coding

5.2.2 Genetic Operators5.2.2.1 Selection5.2.2.2 Crossover5.2.2.3 Mutation

5.2.3 A Simple Example5.2.3.1 Solution5.2.3.2 Representation (Encoding)5.2.3.3 Initial Generation and Evaluation Function5.2.3.4 Genetic Operations

5.2.3.5 Results5.2.4 Features of GAs5.2.5 Brief Reviews on Improvements of GAs

5.3 Micro GAs5.3.1 Uniform GA5.3.2 Real Parameter Coded GA

5.3.2.1 Four Crossover Operators5.3.2.2 Test Functions5.3.2.3 Performance of the Test Functions

5.4 Intergeneration Projection Genetic Algorithm (IP-GA)5.4.1 Modified GA5.4.2 Intergeneration Projection (IP) Operator5.4.3 Hybridization of Modified GA with IP Operator5.4.4 Performance Tests and Discussions

5.4.4.1 Convergence Performance of the IP-GA

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5.4.4.2 Effect of Control Parameters and 5.4.4.3 Effect of the IP Operator5.4.4.4 Comparison with Hybrid GAs Incorporated

with Hill-Climbing Method5.5 Improved IP-GA

5.5.1 Improved IP Operator5.5.2 Implementation of the Improved IP Operator5.5.3 Performance Test

5.5.3.1 Performance of the Improved IP-GA5.5.3.2 Effect of the Mutation Operation5.5.3.3 Effect of the Coefficients and 5.5.3.4 Effect of the Random Number Seed

5.6 IP-GA with Three Parameters (IP3-GA)5.6.1 Three-Parameter IP Operator5.6.2 Performance Comparison

5.7 GAs with Search Space Reduction (SR-GA)5.8 GA Combined with the Gradient-Based Method

5.8.1 Combined Algorithm5.8.2 Numerical Example

5.9 Other Minor Tricks in Implementation of GAs5.10 Remarks

5.11 Some References for Genetic Algorithms

6 Neural Networks6.1 General Concepts of Neural Networks6.2 Role of Neural Networks in Solving Inverse Problems6.3 Multilayer Perceptrons

6.3.1 Topology6.3.2 Back-Propagation Training Algorithm6.3.3 Modified BP Training Algorithm

6.4 Performance of MLP6.4.1 Number of Neurons in Hidden Layers6.4.2 Training Samples6.4.3 Normalization of Training Data Set6.4.4 Regularization

6.5 A Progressive Learning Neural Network6.6 A Simple Application of NN

6.6.1 Inputs and Outputs of the NN Model6.6.2 Architecture of the NN Model

6.6.3 Training and Performance of the NN Model6.7 Remarks6.8 References on Neural Networks

7 Inverse Identification of Impact Loads7.1 Introduction

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7.2 Displacement as System Effects7.3 Identification of Impact Loads on the Surface of Beams

7.3.1 Finite Element Model

7.3.2 Validation of FE Model with Experiment7.3.3 Estimation of Loading Time History7.3.4 Boundary Effects

7.4 Line Loads on the Surface of Composite Laminates7.4.1 Hybrid Numerical Method7.4.2 Why HNM?7.4.3 TransWave

7.4.4 Comparison between HNM and FEM7.4.5 Kernel Functions

7.4.6 Identification of Time History of Load UsingGreens Functions

7.4.7 Identification of Line Loads7.4.7.1 Identification of Time Function7.4.7.2 Identification of the Spatial Function7.4.7.3 Identification of the Time and Spatial Functions

7.4.8 Numerical Verification7.5 Point Loads on the Surface of Composite Laminates

7.5.1 Inversion Operation

7.5.2 Concentrated Point Load7.6 Ill-Posedness Analysis7.7 Remarks

8 Inverse Identification of Material Constants of Composites

8.1 Introduction8.2 Statement of the Problem8.3 Using the Uniform GA

8.3.1 Solving Strategy8.3.2 Parameter Coding8.3.3 Parameter Settings in GA8.3.4 Example I: Engineering Elastic Constants in Laminates

8.3.4.1 Laminate [G0/+45/45]s8.3.4.2 Laminate [C0/+45/45/90/45/+45]s8.3.4.3 Regularization by Projection8.3.4.4 Regularization by Filtering8.3.4.5 Discussion

8.3.5 Example II: Fiber Orientation in Laminates8.3.5.1 Eight-Ply Symmetrical Composite Laminates8.3.5.2 Ten-Ply Symmetrical Composite Laminates8.3.5.3 Further Investigations

8.3.6 Example III: Engineering Constants of LaminatedCylindrical Shells

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8.3.6.1 [G0/+45/45/90/45/+45]sCylindrical Shell8.3.6.2 [G0/30/30/90/60/+60]sCylindrical Shell8.3.6.3 [C0/30/30/90/60/60]sCylindrical Shell

8.4 Using the Real GA8.5 Using the Combined Optimization Method8.6 Using the Progressive NN for Identifying Elastic Constants

8.6.1 Solving Strategy and Statement of the Problem8.6.2 Inputs of the NN Model8.6.3 Training Samples8.6.4 Results and Discussion8.6.5 A More Complicated Case Study

8.7 Remarks

9 Inverse Identification of Material Property of FunctionallyGraded Materials

9.1 Introduction9.2 Statement of the Problem9.3 Rule of Mixture9.4 Use of Gradient-Based Optimization Methods

9.4.1 Example 1: Transversely Isotropic FGM Plate9.4.1.1 Approach I: Identification of Material Property at

Discrete Locations9.4.1.2 Approach II: Identification of Parameterized

Values9.4.2 Example 2: SiC-C FGM Plate

9.4.2.1 Identification of Parameterized Values9.4.2.2 Approach III: Identification of Volume Fractions

9.5 Use of Uniform GA9.5.1 Material Characterization of FGM Plate

9.5.1.1 Parameters Used in the Uniform GA

9.5.1.2 Test of GAs Performance9.5.1.3 Search Range

9.5.2 Material Characterization of FGM Cylinders9.6 Use of Combined Optimization Method9.7 Use of Progressive NN Model

9.7.1 Material Characterization of SiC-C FGM Plate9.7.1.1 Inputs of the NN Model9.7.1.2 Training Samples9.7.1.3 Results and Discussion

9.7.2 Material Characterization of SS-SN FGM Cylinders9.7.2.1 Inputs of the NN Model9.7.2.2 Training Samples9.7.2.3 Results and Discussions

9.8 Remarks

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10 Inverse Detection of Cracks in Beams Using Flexural Waves

10.1 Introduction

10.2 Beams with Horizontal Delamination10.2.1 The SEM10.2.2 Why SEM?10.2.3 Brief on SEM Formulation10.2.4 Experimental Study10.2.5 Sensitivity Study and Rough Estimation of Crack in

Isotropic Beams10.2.5.1 Crack Length10.2.5.2 Crack Depth

10.3 Beam Model of Flexural Wave10.3.1 Basic Assumptions10.3.2 Homogeneous Solution10.3.3 Particular Solution10.3.4 Continuity Conditions10.3.5 Comparison between SEM and Beam Model10.3.6 Experimental Verification

10.4 Beam Model for Transient Response to an Impact Load10.4.1 Beam Model Solution

10.4.2 Experimental Study on Impact Response10.4.3 Comparison Study

10.5 Extensive Experimental Study10.5.1 Test Specimens10.5.2 Test Setup10.5.3 Effect of Crack Depth10.5.4 Effect of Crack Length10.5.5 Effect of Excitation Frequency10.5.6 Effect of Location of the Excitation Point

10.5.7 Study on Beams of Anisotropic Material10.6 Inverse Crack Detection Using Uniform GAs10.6.1 Use of Simulated Data from SEM10.6.2 Use of Experimental Data

10.7 Inverse Crack Detection Using Progressive NN10.7.1 Procedure Outline10.7.2 Composite Specimen10.7.3 Delamination Detection10.7.4 Effect of Different Training Data10.7.5 Use of Beam Model and Harmonic Excitation10.7.6 Use of Beam Model and Impact Excitation10.7.7 FEM as Forward Solver

10.8 Discussion on Ill-Posedness10.9 Remarks

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11 Inverse Detection of Delaminations in Composite Laminates

11.1 Introduction

11.2 Statement of the Problem11.3 Delamination Detection Using Uniform GA

11.3.1 Horizontal Delamination11.3.2 Vertical Crack

11.4 Delamination Detection Using the IP-GA11.5 Delamination Detection Using the Improved IP-GA11.6 Delamination Detection Using the Combined Optimization

Method11.6.1 Implementation of the Combined Technique

11.6.1.1 Formulations of Objective Functions11.6.1.2 Switch from GA to BCLSF11.6.1.3 Effect of Noise11.6.1.4 Ill-Posedness Analysis11.6.1.5 Regularization by Filtering

11.6.2 Horizontal Delamination in [C90/G45/G45]sLaminate11.6.2.1 Noise-Free Cases11.6.2.2 Noisy Cases11.6.2.3 Discussion

11.7 Delamination Detection Using the Progressive NN11.7.1 Implementation11.7.2 Noise-Free Case11.7.3 Noise-Contaminated Case11.7.4 Discussion

11.8 Remarks

12 Inverse Detection of Flaws in Structures12.1 Introduction

12.2 Inverse Identification Formulation12.2.1 Damaged Element Identification12.2.2 Stiffness Factor Identification

12.2.2.1 Objective Function with Weight12.2.2.2 Direct Formulation

12.3 Use of Uniform GA12.3.1 Example I: Sandwich Beam12.3.2 Example II: Sandwich Plate

12.4 Use of Newtons Root Finding Method

12.4.1 Calculation of Jacobian Matrix12.4.2 Iteration Procedure12.4.3 Example I: Cantilever Beam

12.4.3.1 Stiffness of Cantilever Beam12.4.3.2 Performance Comparison with GA12.4.3.3 Noise-Contaminated Case

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12.4.4 Example II: Plate12.5 Use of LevenbergMarquardt Method12.6 Remarks

13 Other Applications13.1 Coefficient Identification for Electronic Cooling Systems

13.1.1 Using the Golden Section Search Method13.1.1.1 Natural Convection Problem13.1.1.2 Numerical Results13.1.1.3 Summary

13.1.2 Using GAs13.1.2.1 Forward Modeling

13.1.2.2 Inverse Analysis of a PCB Board13.1.2.3 A Complex Example13.1.2.4 Summary

13.1.3 Using NNs13.1.3.1 Coefficient Identification of a Telephone

Switch Model13.1.3.2 Coefficient Identification for IC Chips13.1.3.3 Summary

13.2 Identification of the Material Parameters of a PCB

13.2.1 Introduction13.2.2 Problem Definition13.2.3 Objective Functions13.2.4 Finite Element Representation13.2.5 Numerical Results and Discussion

13.2.5.1 Sensitivity Analysis13.2.5.2 Identification Using Natural Frequencies13.2.5.3 Identification Using Frequency Response

13.2.6 Summary

13.3 Identification of Material Property of Thin Films13.3.1 Noise-Free Cases13.3.2 Noisy Cases13.3.3 Discussion

13.4 Crack Detection Using Integral Strain Measured by OpticFibers13.4.1 Introduction13.4.2 Numerical Calculation of Integral Strain13.4.3 Inverse Procedure

13.4.3.1 Crack Expression13.4.3.2 Remesh Technique13.4.3.3 Definition of Objective Functions

13.4.4 Numerical Results13.4.4.1 Different Dimensions of Cracks13.4.4.2 Different Locations of Cracks (Case C)

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13.4.4.3 Different Materials (Case D)13.4.4.4 Different Applied Loads (Case E)13.4.4.5 Different Boundary Conditions (Case F)

13.4.5 Summary13.5 Flaw Detection in Truss Structure13.6 Protein Structure Prediction

13.6.1 Protein Structural Prediction13.6.2 Parameters for Protein Structures13.6.3 Confirmation Energy13.6.4 Lattice Model

13.6.4.1 Cubic Lattice Model13.6.4.2 Random Energy Model

13.6.4.3 Lattice Structure Prediction by IP3-GA13.6.5 Results and Discussion13.6.6 Summary

13.7 Fitting of Interatomic Potentials13.7.1 Introduction13.7.2 Fitting Model13.7.3 Numerical Result13.7.4 Summary

13.8 Parameter Identification in Valveless Micropumps

13.8.1 Introduction13.8.2 Valveless Micropump13.8.3 Flow-Pressure Coefficient Identification13.8.4 Numerical Examples13.8.5 Summary

13.9 Remarks

14 Total Solution for Engineering Systems: A New Concept

14.1 Introduction14.2 Approaching a Total Solution

14.2.1 Procedure for a Total Solution14.2.2 Forward Solver14.2.3 System Parameters14.2.4 Mathematical Representation

14.3 Inverse Algorithms14.3.1 Sensitivity Matrix-Based Method (SMM)

14.3.1.1 Sensitivity-Based Equations

14.3.1.2 Algorithms14.3.1.3 Solution Procedure14.3.1.4 Comments on SMM

14.3.2 Neural Network14.4 Numerical Examples

14.4.1 Vibration Analysis of a Circular Plate

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14.4.1.1 SMM Solution14.4.1.2 Progressive NN Solution

14.4.2 Identification of Material Properties of a Beam

14.4.2.1 SMM Solution14.4.2.2 Progressive NN Solution

14.5 RemarksReferences

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1Introduction

1.1 Forward and Inverse Problems Encountered in Structural

SystemsIn engineering, computer-aided design (CAD) tools are used to designadvanced structural systems. Computational simulation techniques are oftenused in such tools to calculate the displacement, deflection, strains, stresses,natural frequencies, and vibration modes, etc. in the structural system forgiven loading, initial and boundary conditions, geometrical configuration,material properties, etc. of the structure. These types of problems are calledforward problems and are often governed by ordinary or partial differentialequations(ODE or PDE) with unknownfield variables. For structure mechanics

problems, the field variable is basically the displacements; the constants inthe ODE or PDE and problem domain are known a priori. The source or thecause of the problem or phenomenon governed by the ODE or PDE and therelevant initial and boundary conditions are also known. To solve a forwardproblem is, in fact, to solve the ODE or PDE subjected to these initial and

boundary conditions.Many solution procedures, especially the computational procedures, have

been developed, such as:

Finite deference method (FDM; see e.g., Hirsch, 1988; Anderson, 1995) Finite element method (FEM; see e.g., Zienkiewicz and Taylor, 2000;

Liu and Quek, 2003)

Strip element method (Section 10.3)

Boundary element method (BEM; see e.g., Brebbia et al., 1984)

FEM/BEM (see e.g., Liu, Achenbach et al., 1992)

Mesh-free methods (see e.g., Liu, 2002a; Liu and Liu, 2003)

Wave propagation solvers (see e.g., Liu and Xi, 2001)

These methods for solving forward problems have been well established,although the mesh-free methods are still in a stage of rapid development.Using these methods, the displacements in the structure and then the strains



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and stresses (outputs) can be obtained as long as the material property,the geometric configuration of the structure, and the loading, initial, and

boundary conditions (inputs) are given.

Another class of often encountered practical problems is called inverseproblems. In an inverse problem, the effects or outputs(displacement, velocity,acceleration, natural frequency, etc.) of the system may be known (by exper-iments, for example), but the parameters of the loading profile (inputs),material property, geometric feature of the structure, boundary conditions,or a combination of these may need to be determined. Solving this class ofproblem is obviously extremely useful for many engineering applications.

One of the earliest inverse problems in mechanical engineering is theinverse problem in wave propagation. These problems are formulated based

on the fact that mechanical (elastic) waves (Achenbach, 1973; Liu and Xi,2001) traveling in materials are scattered from the boundaries and interfacesof materials, and propagate over distance to encode the information ontheir path such as the domain boundaries, martial properties, and the wavesource (loading excitation, etc.). It must be possible to decode some of theinformation encoded in the waves that are recorded as wave responses. Asystematic method to decode the information is to formulate and solveinverse problems. Problems of this nature arise from nondestructive evalu-ation (NDE) using waves and ultrasounds, ocean acoustics, earth and space

exploration, biomedical examination, radar guidance and detection, solarastrophysics, and many other areas of science, technology, and engineering.The nature of inverse problems requires proper formulations and solution

techniques in order to perform the decoding successfully. In this book,approaches to formulating inverse problems, inverse analysis procedures, andcomputational techniques are discussed. Many engineering inverse problemsare formulated and investigated using these techniques and many importantissues related to inverse problems are examined and revealed by using simpleexamples. Methods for dealing with these issues are also presented.

Note that many types of inverse problems exist in engineering. Some ofthem can only be formulated in an under-posed form (see Chapter 2), dueto the difficulty or cost of obtaining more experimental data or observations.Solving this class of under-posed inverse problems will be discussed but isnot the major focus of this book. This book focuses on inverse problems ofeven- and over-posed problems because, for many engineering systems,sufficient experimental readings can be produced, at least in numbers, toformulate the problem in even- or over-posed forms.

1.2 General Procedures to Solve Inverse Problems

Thegeneral procedure of solving an inverse problem is illustrated inFigure1.1. The details are as follows:



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Define the problem define the purpose and objectives of theproject with an analysis on the available budget, resources, and

timeframe. An overall strategy and feasible schedule should bedetermined for later effective execution. Efforts must be made at alltimes to (1) reduce the number of unknowns to be inversely identi-fied and (2) confine all the parameters in the smallest possible region.Made at the very first step, these two efforts can often lead to aneffective reduction on possibilities of ill-posed inverse problems,and, thus, drastically increase the chance of success and improve theefficiency and accuracy of the inversion operation.

Create the forward model a physical model should be established

to capture the physics of the defined problem. The outputs or effectsof the system should be as sensitive as possible to the system param-eters to be inversely identified. The parameters should be indepen-dently influential to the outputs or the effects of the system.Enforcing more conditions can help to well-pose the inverse prob-lems. Mathematical and computational models should be developed

FIGURE 1.1

General procedure to solve inverse problems.

Define the problem

Create the forward model

Sensitivity analysis between the inputs and outputs

Experiment design

Minimize measurement error (e.g., filtering)

Solution verificationNo

Yes

END

Inverse analysisGeneral inversion or optimization or NN(Chapters 4-6); regularization techniques(Chapter 3) may be used



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for the underlined forward problem. Possible standard computa-tional methods are FEM, FDM, FVM (finite volume method), mesh-free method, wave solvers, etc.

Analyze sensitivity between the effects or outputs and the parame-ters make sure that the outputs of the problem and parameters(including the inputs) to be inversely identified are well correlated.Ensuring high sensitivity of the outputs to the parameters is one ofthe most effective approaches to reducing ill-posedness in the laterstage of inverse analysis. The analysis should be done using theforward model created without the need of experiments that may

be expensive. Based on the sensitive analysis, modification to theforward model and to the choices of parameters may be made.

Design the experiment decide on proper measurement methods,type of equipment for testing and recording, and data analysis. Thenumber of the measurements or readings should be at least morethan the number of unknowns to be inversely identified, which canlead to at least an even-posed problem. An over-posed system (usingmore outputs) is usually preferred so as to improve the property ofthe system equation and reduce the ill-posedness of the problem.An over-posed formulation can usually accommodate higher levelsof noise contamination in the experimental data. However, too

heavily over-posed systems may result in a poor output reproduc-ibility that can be checked later by computing the output reproduc-ibility after obtaining the inverse solution.

Minimize measurement noise (e.g., through filtering) errors in themeasurement data should be eliminated as much as possible becausethey can trigger the ill-posedness of the problem and can be mag-nified in the inverse solution, or even result in an unstable solution.Properly designed filters can be used to filter out the errors beforethe measurement data are used for the inverse analysis. The princi-

ple is to use a low pass filter to filter out all the noise with frequencyhigher than the frequency or wavelength shorter than the wave-length of the effects of the problem. The frequency or the wavelengthof the effects of the problem can often be estimated by the forwardsolver. Details will be covered in Chapter 3.

Apply the inverse solver if the system can be formulated in anexplicit matrix form, general inversion of the system (or transforma-tion) matrix can be performed to obtain the inverse solution. Forcomplex systems that cannot be formulated in an explicit matrix

form, a functional of error can always be established using a propernorm, and optimization/minimization techniques should be used tosearch for the solution that minimizes the error norm. These optimi-zation techniques will be discussed in detail in Chapter 4 and Chap-ter 5. Proper regularization techniques may be used for ill-posedinverse problems. The regularization techniques are very important



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for obtaining stable solutions for the ill-posed inverse problems. Notealso that the use of some of the regularization techniques should bethe last resort to remedy the ill-posedness of the problem. Some side

effects will occur in using many of the regularization methods andmisuse of regularization techniques can also lead to erroneousresults. Regularization methods will be detailed in Chapter 3.

Verify the solution this is important to ensure that the inversesolution obtained is physically meaningful. All possible methodswith proper engineering judgments should be employed to makesure that the solution obtained is reliable. Checking on the outputand input reproducibility matrices can give some indications on thequality of the solution. Modifications of the inverse and experimen-

tal strategy may be needed, and the preceding steps may be repeateduntil the inverse solution is satisfactory. Note that many of the ver-ifications can be done computationally, and experimental verifica-tions need to be done at the final stage.

1.3 Outline of the BookThis book details the theory, principles, computational methods and algo-rithms, and practical techniques for inverse analyses using elastic wavespropagating in solids and structures or the dynamic responses of solids andstructures. These computational inverse methods and procedures will beexamined and tested numerically via a large number of examples of force/source reconstructions, crack detection, flaw characterization, material char-acterization, heat transfer coefficients identification, protein structure pre-diction, interatomic potential construction, and many other applications.

Some of these techniques have been confirmed with experiments conductedby the authors and co-workers in the past years. Discussions of regulariza-tion methods for the treatment of ill-posed inverse problems will be easy tounderstand.

The book also discusses many robust and practical optimization algo-rithms that are very efficient for inverse analysis and optimization, espe-cially algorithms developed through the combination of different types ofthe existing optimization methods such as gradient based methods withgenetic algorithms, intergeneration projection genetic algorithms, real

number coded microgenetic algorithms, and progressive neural networks.The efficiency and features of all these optimization algorithms will bedemonstrated using benchmark objective functions as well as actualinverse problems. Table 1.1 gives a concise summary of the applicationsof those computational inverse techniques for actual inverse problemsstudied in this book.



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TABLE 1.1

Summary of Applications of Computational Inverse Techniques for Actual Inverse Prob

Computational Inverse Techniques

Conventionaloptimizationtechniques (Chapter 4)

Golden section search method (Section 4.3.1) Section 13.1: Coefficien

Conjugate gradient method (Section 4.4.3) Section 7.4.7: Identifica

Nonlinear least square method (Section 4.5)

1. Section 7.4.6: Ident2. Section 9.4: Identifi

materials2.1 Section 9.4.1: T2.2 Section 9.4.2: S

3. Used frequently in

Newtons root finding method (Section 4.6.1) 1. Section 12.4.3: Iden2. Section 12.4.4: Iden

Levenberg-Marquardt root finding method(Section 4.6.2)

1. Section 12.5: Flaw d2. Section 13.5: Flaw d

Genetic algorithms (GA)(Chapter 5)

Binary micro-GA (GA) (Section 5.3.1)

1. Section 8.3.4: Ident1.1 Glass/epoxy [01.2 Carbon/epoxy

2. Section 8.3.5: Ident2.1 Eight-ply symm2.2 Ten-ply symme2.3 Complex case

3. Section 8.3.6: Ident

3.1 Glass/epoxy [03.2 Glass/epoxy [03.3 Carbon/epoxy

4. Section 9.5: Materia4.1 Section 9.5.1: 4.2 Section 9.5.2: F

5. Section 10.6: Crack5.1 Section 10.6.1: 5.2 Section 10.6.2:

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6. Section 11.3: Delamination de6.1 Section 11.3.1: Horizonta6.2 Section 11.3.2: Vertical cr

7. Section 12.3: Flaw detection i7.1 Sandwich beams7.2 Sandwich plates

8. Section 13.4: Crack detection u

Real GA (Section 5.3.2) 1. Section 8.4: Identification of m

Intergeneration projection GA (IP-GA)(Section 5.4)

1. Section 11.4: Delamination de2. Section 13.3: Identification of3. Section 13.8: Parameter ident

Improved IP-GA (Section 5.5) 1. Section 11.5: Delamination de2. Section 13.7: Fitting of interat

IP-GA with three parameters (IP3-GA)(Section 5.6)

1. Section 13.2: Identification of2. Section 13.6: Protein structure

GA with search space reduction (SR-GA)

(Section 5.7)

1. Section 13.1.2.2: Thermal coef

system2. Section 13.1.2.3: Thermal coe

Genetic algorithm combined with the gradient-based methods (Section5.8)

1. Section 8.5: Identification of m2. Section 9.6: Identification of m

materials3. Section 11.6: Delamination de

Neural network(Chapter 6)

Plain neural network 1. Section 13.1.3: Coefficient ide

Progressive neural network

1. Section 8.6: Identification of m2. Section 9.7: Identification of m

materials

2.1 Section 9.7.1: FGM plate2.2 Section 9.7.2: FGM cylin

3. Section 10.7: Crack detection 3.1 Sections 10.7.2-10.7.4: Us3.2 Section 10.7.5: Using bea3.3 Section 10.7.6: Using bea3.4 Section 10.7.7: Using FEM

4. Section 11.7: Delamination de

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The book is organized as follows:

Chapter 1 provides a general description and procedure of inverse

analysis and backgrounds and motivations that led to the develop-ment of these methods for nondestructive evaluation, as well asdevelopment of this book.

In Chapter 2 the general definition of forward problem as well asinverse problem will be presented. Ill-posed inverse problems areclassified into three types. Issues related to these three types of ill-posedness are revealed and discussed using very simple examples.The formulation of inverse problems will be presented, and thegeneral procedure to solve inverse problems that can be formulated

in explicit matrix forms will be provided. Chapter 3 offers a brief introduction of five regularization methods

for ill-posed inverse problems. These regularization methodsinclude the Tikhonov regularization, regularization by singularvalue decomposition, iterative regularization methods, regulariza-tion by projection, and regularization by filtering.

In Chapter 4 some conventional optimization techniques, includingdirect search algorithms as well as gradient-based algorithms, areintroduced because engineering inverse problems are usually for-mulated and solved as optimization problems. These techniques areprovided in a concise and insightful manner with the help of simpleexamples.

Chapter 5 describes the basic concept of genetic algorithms (GAs)and some modified GAs, with an emphasis on the intergenerationproject GA (IP-GA) as well as the method that combines GAs withgradient-based methods.

In Chapter 6, the basic terminology, concepts, and procedures of the

neural network (NN) will be briefly introduced. A typical NN modeland multilayer perceptrons (MLP), along with the back-propagationlearning algorithm, will be detailed. Some practical computationalissues on NNs as well as the progressive NN model are also discussed.

Chapter 7 through Chapter 12 present a number of computationalinverse techniques using elastic waves propagating in compositestructures or dynamic responses of structures. Practical complexnondestructive evaluation problems of force function reconstruction,material property identification, and crack (delamination, flaw)

detection have been examined in detail in the following order: Chapter 7 presents inverse procedures for identification of im-

pact loads in composite laminates. Traditional optimizationmethods are employed for the inverse analysis and numericalexamples of identification of impact loads applied on beam andplate types of structures are presented. Experimental studies



28/573

have also been presented for the verification of the inversesolution.

In Chapter 8 material constants include the elastic constants orthe engineering constants required in the constitutive law forcomposites, and fiber orientation of composite laminates will beinversely identified from the dynamic displacement responsesrecorded at only one receiving point on the surface of compositelaminated structures.

Chapter 9 discusses the computational inverse techniques formaterial property characterization of functionally graded mate-rials (FGMs) from the dynamic displacement response recordedon the surface of the FGM structures.

In Chapter 10, numerical analysis and experimental studies onthe use of flexural waves for nondestructive detection of cracksand delaminations in beams of isotropic and anisotropic materi-als are introduced. Computational inverse procedures employingthe GAs and NNs are detailed for determining the geometricalparameters of the crack and delaminations.

Computational inverse techniques using elastic wave responsesof displacement for delamination detection in composite lami-

nates are introduced in Chapter 11. Horizontal delaminations aswell as vertical cracks will be considered. GAs and NNs areemployed for the inverse analysis; the strip element method isused as the forward solver to compute the wave response. Ex-amples of practical applications are presented to demonstrate theefficiency of computational inverse techniques for delaminationdetection in composite laminates.

Chapter 12 considers the detection of flaws in beams or plates;special considerations and treatment for the detection of flaws

in sandwich structures are also provided. The finite elementmodel is used for forward analysis. GAs andNewtons root find-ing method, as well as the LevenbergMarquardt method, areused for the inverse analysis. A number of numerical examplesare provided to demonstrate the application of these computa-tional inverse techniques.

Several other application examples of the computational inversetechniques are presented in Chapter 13. These topics range fromthe electronic system (heat transfer coefficient identifications), use

of integral optical fibers, MEMS, and interatomic potential to theprotein structure. These applications provide a landscape view onthe broadness of the applications of the inverse techniques.

Chapter 14 introduces a concept of total solution for engineeringmechanics problems as an extension of the inverse analysis. Theapproach for obtaining a total solution is to formulate practical engi-



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neering problems as a parameter identification problem. All theparameterized unknown information is determined through an iter-ative procedure of conducting alternately forward and inverse (or

mixed) analyses. This chapter suggests a new approach to formulateand deal with practical engineering problems.

The background of and many terminologies used in this book are definedin Chapter 1 through Chapter 3. These chapters will be useful in understand-ing Chapter 7 through Chapter 14, and therefore should be read first beforeproceeding to other chapters. Chapter 4 through Chapter 6 can be readseparately. In fact, these materials are useful not only for inverse problems

but also for general optimization problems. Readers who are familiar with

these optimization techniques may skip these chapters. Chapter 7 throughChapter 14 can be read in any order, based on the interest of the reader,

because proper cross references for commonly used materials are provided.The book is written primarily for senior university students, postgraduate

students and engineers in civil, mechanical, geographical and aeronauticalengineering, and engineering mechanics. Students in mathematics and com-putational science may also find the book useful. Anyone with an elemen-tary knowledge of matrix algebra and basics of mechanics should be ableto understand its contents fairly easily.



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2Fundamentals of Inverse Problems

Using simple examples that can largely be treated manually, this chapterreveals some important and fundamental issues in formulating and solvinginverse problems. This will prepare readers for dealing with complex inverseproblems presented in later chapters. The general definition of the often usedtechnologies for forward problems, as well as inverse problems, are pre-sented in this chapter. Issues related to the ill-posedness of problems arediscussed and ill-posed inverse problems are classified into three types.Features of these three types are examined, and general approaches andsteps to deal with them are then discussed.

Detailed methods are introduced for dealing with a class of inverse prob-lems whose input, output, and system can be expressed explicitly in matrixforms. The properties of this class of inverse problems as well as the general

procedure to solve them are then provided. Formulations of other complexinverse problems are also introduced. This chapter is written referencingworks by Santamarina and Fratta (1998), Tosaka et al. (1999), Engl et al.(2000).

2.1 A Simple Example: A Single Bar

Consider now a simple mechanical system of a straight bar with uniformcross-sectional areaAand length l, as shown inFigure 2.1.The bar is madeof elastic material with Youngs modulus of E. It is subjected to force f1atnode 1 and f2at node 2. The axial displacement of the bar is denoted by u1at node 1 and u2at node 2. The governing equation for the bar member can

be written as

(2.1)

EA

l

EA

lEAl

EAl

uu

ff

=

1

2

1

2



31/573

or

(2.2)

where

(2.3)

is the tensional stiffness of the bar.Equation 2.2 can be written in a standard matrix form of

(2.4)

where

(2.5)

is called the stiffness matrix, U is the nodal displacementvector that collectsthe displacements at these two nodes of the bar:

(2.6)

and Fis the nodal forcevector that collects the forces acting at the two nodesof the bar:

FIGURE 2.1A straight bar of uniform cross-sectional areaAand length l. The bar is made of elastic materialwith Youngs modulus of E. The bar is subjected to forces f1 at node 1 and f2 at node 2. Theaxial displacement of the bar is denoted by u1at node 1 and u2at node 2.

E, A, l

f2f1u1 u

2

Initial status

Stressed status

1 2

1 2

k k

k k

u

u

f

f

=

1

2

1

2

k

EA

l=

K U F2 2 2 1 2 1 =

K=

k k

k k

U=

uu

1

2



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(2.7)

For complex engineering systems, a set of discrete system equations likeEquation 2.4 can always be created using the standard and well-establishedfinite element method (see, for example, Zienkiewicz and Taylor, 2000; Liuand Quek, 2003), as well as finite difference methods, element free methods(Liu, 2002a), or any other type of numerical methods. If the total degrees offreedom (DOF) are N, the standard discrete system equation can be givenin the form of

(2.8)

2.1.1 Forward Problem

In forward problems, it is assumed that the following parameters are known:Geometrical parameters:

(2.9)

Material property parameter:

(2.10)

External force:

(2.11)

where ^ stands for the parameters whose values are specified. This isintentionally utilized, particularly in this chapter, to help us to distinguishexplicitly the knowns and unknowns in the process of establishing the con-cept of forward and inverse problems.

For the forward problem, the unknown are the displacements u1and u2,and it is only necessary to solve the linear algebraic Equation 2.4 for theunknown. However, because the stiffness matrix Kis singular, the solutionwill not be unique. To obtain a unique solution, the bar must be properlysupported or constrained, which provides additional conditions calledboundary conditions.

Consider the problem shown inFigure 2.2. The bar is now fixed at oneend, so that a boundary condition exists:

(2.12)

F=

f

f1

2

K U FN N N N =1 1

A A l l= = ,

E E=

f f2 2=

u u1 1 0= =



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The u2is now the only unknown. Using the second equation in Equation2.4, u2can then be obtained easily:

(2.13)

where

(2.14)

Equation 2.13 gives

(2.15)

After u2is obtained,f1, termed the reaction force, can be obtained using thefirst equation in Equation 2.2:

(2.16)

This simple example has demonstrated that a forward problem can besolved for the unique solution provided the boundary conditions are givensufficiently for the problem to be well-defined or well-posed. Otherwise, theforward problem can be nonunique or ill-posed.

2.1.2 Inverse Problem

Consider now that, somehow (e.g., via experiment), the value of isknown, and the geometrical information of the bar (Equation 2.9), boundarycondition (Equation 2.12), and external force at nodes 2 (Equation 2.11) arestill known. However, the material property parameter Youngs modulus

FIGURE 2.2A straight bar of uniform cross-sectional area A and length l clamped at node 1. The bar ismade of elastic material with Youngs modulus of Eand is subjected to force f2at node 2.

E, A, lf2

u2

1 2

x

+ = k u k u f 10

2 2

k

EA

l=

u f

k2

2=

f k u k u k fk

f1 10

22

2= = =

u u2 2= ,



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E(Equation 2.10) is not known. Using the second equation in Equation2.2, it can then be obtained easily:

(2.17)

which gives

(2.18)

which is simply an inverse expression of Equation 2.15. Using Equation 2.14,

(2.19)

The problem of solving for the unknown of material property using themeasured displacement is an often encountered inverse problem. This exam-ple, in fact, is the standard procedure used in practice for determining theYoungs modulus of materials. Because the problem is very simple, no special

techniques are usually needed to resolve it. The fact that this is an inverseproblem may not even be obvious. Other inverse problems related to thisexample could be those of finding force applied on the bar, area, or lengthof the bar. They are all equally trivial and can all be solved very easily forthis simple example without any difficulty.

The terms of forward problems and inverse problems are naturally usedfollowing the physics of the problem or the convention of looking at theproblem.

2.2 A Slightly Complex Problem: A Composite Bar

Consider now the slightly more complex problem shown inFigure 2.3. Thegoverning equation of this system can be easily obtained by assembling thesetwo bar members using Equation 2.2:

(2.20)

where

+ = k u k u f 10

2 2

k f

u=

2

2

E kl

A

l f

Au= =

2

2

k kk k k k

k k

uu

u

ff

f

1 1

1 1 2 2

2 2

1

2

3

1

2

3

0

0

+

=



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(2.21)

and

(2.22)

Details about the assembly of the matrices of the members can be found inany FEM textbook (e.g., Liu and Quek, 2003).

Using the boundary condition, u1= 0, Equation 2.20 becomes

(2.23)

2.2.1 Forward Problem

First examine the conventional forward problem with the conditions giveninTable 2.1.Solving Equation 2.23 for the displacements gives

(2.24)

Using the first equation in Equation 2.20, the reaction force at node 1 isfound to be

(2.25)

FIGURE 2.3A straight bar made of two uniform cross-sectional bar members clamped at node 1. The baris subjected to forces f2at node 2 and f3at node 3.

E1, A1, l1f3

u2

1 2

u1

u3

f2

E2, A2, l2 3

k E A

l11 1

1

=

k E A

l22 2

2

=

k k kk k

u

u ff1 2 2

2 2

2

3

2

3+

=

uu

k k k

k k

f

f

k k

k

k k

k k

f

f

k

2

3

1 2 2

2 2

12

3

1 1

1

1 2

1 2

2

3

1

1 1

1

1

= +

=

+

=

ff f

k

f k k

k

f

2 3

1

21 2

2

3

1

+( )

+ +

f k u f f1 1 2 2 3= = +( )



36/573

It is shown again that, for given conditions Equation 2.9 through Equation2.12, the forward problem can be solved, and the displacements can beuniquely determined. The solution given in Equation 2.24 can be written in

the following general form:

(2.26)

From the mechanics point of view, the vector X in this case is the force vectorgiven by

TABLE 2.1

Cases of Problems for the Composite Bar

Cases

Boundary

Conditions

Geometry

Parameters

Material

Property

Parameter

External

Causes

(Forces)

Effects(Displacements,

Natural

Frequency/

Modes)

Forward problem

u1= 0

Inverse problem

Case I-1 even-

posed u1= 0

Inverse problem

Case I-2 under-

posedu1= 0

Inverse problem

Case II-1 even-

posedu1= 0

Inverse problemCase II-2 over-

posedu1= 0

Inverse problem

Case III-1 even-posed

u1= 0

Inverse problem

Case III-2 even-posed

u1= 0

Inverse problem

Case IV even-posed

u1= ?f1= ?

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E E

E E

1 1

2 2

=

=

f f

f f

2 2

3 3

=

=

u

u

2

3

=

=

?

?

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E E

E E

1 1

2 2

=

=

f

f

2

3

=

=

?

?

u u

u u

2 2

3 3

=

=

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E E

E E

1 1

2 2

=

=

f

f

2

3

=

=

?

?u u3 3=

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E

E

1

2

=

=

?

?

f f

f f

2 2

3 3

=

=

u u

u u

2 2

3 3

=

=

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E E

E

1 1

2

=

=

?

f f

f f

2 2

3 3

=

=

u u

u u

2 2

3 3

=

=

A l l

A l l

1 1 1

2 2 2

= =

= =

?,

?,

E E

E E

1 1

2 2

=

=

f f

f f

2 2

3 3

=

=

u u

u u

2 2

3 3

=

=

A A l

A A l

1 1 1

2 2 2

= =

= =

, ?

, ?

E E

E E

1 1

2 2

=

=

f f

f f

2 2

3 3

=

=

u u

u u

2 2

3 3

=

=

A A l l

A A l l

1 1 1 1

2 2 2 2

= =

= =

,

,

E E

E E

1 1

2 2

=

=

f f

f f

2 2

3 3

=

=

u u

u u

2 2

3 3

=

=

Y S X2 1 2 2 2 1 =



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(2.27)

and Sis the system matrix (known as the flexibility matrix) obtained as

(2.28)

which depends on the material property and the geometrical parameters ofthe system. The vector Yin this case is the displacement vector (effect of thesystem) given by

(2.29)

Mathematically, the vector X is viewed as an input vector, S is termed a

transformation matrix, and Yis an output vector, as illustrated inFigure 2.4.Note that, if Equation 2.20 is to be solved without using the boundary

condition, this forward problem is also ill-posed and cannot be solved for aunique solution. Also, if k1or k2are zero, Equation 2.24 still cannot providea unique solution. It seems unlikely to happen in this example but, mathe-matically, it can always be argued that k1or k2could be zero, and the solutioncould be nonunique. In fact, there are such problems in practice. The so-called locking problem in mechanics, e.g., shear locking (Zienkiewicz

FIGURE 2.4A simple schematic illustration on forward and inverse problems.

X=

f

f2

3

S=+

1 1

11 1

1

1 2

1 2

k k

k

k k

k k

Y=

u

u2

3

System formulated as a

transformation matrix S(smoothing operator)Input X Output Y

Inverse Problem: Y= S-gX(Harshening operator on X)

Forward Problem: Y= SX(Smoothing operator on X)



38/573

and Taylor, 2000), is exactly of this nature. The point here is that the forwardproblem can also be ill-posed, which is usually said to be not well-defined.

2.2.2 Inverse Problem Case I-1: Load/Force Identification withUnique Solution (Even-Posed System)

Consider now the first case of theinverse problem of load/force identifica-tion. The conditions are given inTable 2.1. In this case, the outputs or theeffects (displacement) of the system as well as other conditions, such as the

boundary condition, geometrical parameters, and material properties, aresomehow known, but not the input (load/force). Using the boundary con-

dition, Equation 2.20 becomes

(2.30)

These are the two nodal forces input to the system to produce the outputsof nodal displacements The case I inverse problem is thereforesuccessfully solved, and the solution is unique. Equation 2.30 can be written

in the general form of

(2.31)

where is the system matrix of the forward problem model given inEquation 2.26. Therefore, when the model of the forward problem is given,the output of the system is somehow obtained (via measurement, for exam-ple), and the forward transformation matrix is given and invertible, the

solution of the inverse problem is obtained by simple matrix inversion.Because the number of unknowns and knowns is the same, this problem

is said to be even-posed.

2.2.3 Inverse Problem Case I-2: Load/Force Identification with NoUnique Solution (Under-Posed System)

Consider again the first case of inverse problem of load/force identification.The conditions, knowns, and unknowns are also listed inTable 2.1.In thiscase, the boundary condition and geometrical and material properties areknown, but only partial output of the system, that is, is known. The input(load/force) must be identified based on Equation 2.30. Because is notknown, it must be removed from these equations. To do this, Equation 2.30is first changed to

,u1 0=

f

fk k k

k k

u

uk k u k u

k u k u2

3

1 2 2

2 2

2

3

1 2 2 2 3

2 2 2 3

= +

= +( )

+

.u u2 3and

X S Y2 1 2 21

2 1

=

S2 2

u3u2



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(2.32)

Eliminating in this equation by adding the preceding two equationstogether yields

(2.33)

In this case the corresponding forward model becomes

(2.34)

where the force (input) vector Fhas the form of

(2.35)

The system transformation matrix Sis given by

(2.36)

and the displacement (output) vector becomes

(2.37)

In this inverse problem, the input Xfrom the given output Ymust be foundbased on the forward model given by Equation 2.34. From Equation 2.33, itis clear that multiple solutions for the two inputs of f2andf3 exist, because

two unknowns must be determined with only one equation. It is necessaryto obtain the inverse of the system transformation matrix Sthat is fat withdimension of 1 2, so that the solution can be given by

(2.38)

1

1

1 2

2

2

3

22

1 2

3

2 3

k kf

k f

u k

k ku

u u

+

=

+ +

u2

uk

k k

k k

f

f31

1 2

1 2

2

3

1

Y

SX

= +

Y S X1 1 1 2 2 1 =

X=

f

f2

3

S=

+

1

1

1 2

1 2k

k k

k k

Y=u3

X S Y2 1 2 1 1 1

= g



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41/573

experimentally. Rewrite Equation 2.23 in the following form with k1and k2the unknowns:

(2.41)

The matrix form of the forward model is then obtained as

(2.42)

In this type of problem, the system transformation matrix depends on themeasured displacements. The output vector is the external forces, and theinput vector refers to the material property and the geometrical parametersof the system. This fact reveals an important feature of inverse problems:the system matrix is not limited to representing the characteristics of the

structure system, and it can be formed using the field variables of themechanics problem. In addition, the input vector X for this model is thestiffness of the structure system related to the material property and thegeometrical parameters of the system.

Solving the above equation for input Xgives

(2.43)

where

(2.44)

Therefore,

(2.45)

u k u u k f

u u k f

2 1 2 3 2 2

2 3 2 3

+ ( ) =

+( ) =

f

f

u u u

u u

k

k2

3

2 2 3

2 3

1

20

=

+

Y S

X

X S Y2 1 2 21

2 1

=

S

=

+

=

+

1 2 2 3

2 3

1

2 2

2 3

0

1 1

01

u u u

u uu u

u u

X S Y=

= =

=

+( )

k

ku u

u u

f

f

u f f

u u f

1

22 2

12 1

2 2

3 2

2

3

22 3

3 23

1 1

01

1

1



42/573

Using the preceding equation, the Youngs modulus of these two barmembers can be determined:

(2.46)

If E2 cannot be determined, and if E1 cannot be deter-mined. This reveals another very important feature of inverse problems:

situations can exist in which the solution process fails. In addition, when is very small and erroneous, it can be easily seen that the

error in estimated E1 (or E2) can be magnified and even unstable (a smallchange in could result in a big change in E1). This reveals another veryimportant feature of inverse problems: the error in the solution can be mag-nified or the solution can be unstable. This instability is responsible for theill-posedness of inverse problems. This book classifies this type of problemas Type II ill-posed inverse problem.

Note that this instability or ill-posedness is caused mathematically by the

rank of the system transformation matrix S defined in Equation 2.42. It isclearly seen that when Shas only a rank of 1. The physicalcause of this ill-posedness is that E2is not sensitive to any measurement thatproduces because such a measurement will not cause any defor-mation in the bar number 2. Therefore, there is no way to determine E2fromsuch a measurement. Similarly E1 is not sensitive to any measurement thatproduces because such a measurement will not cause any deforma-tion in the bar number 1.

Because the unknowns and knowns are equal in number, this problem is

said to be even-posed. Note that an even-posed system does not necessarilyguarantee a stable solution for the inverse problem due to the possible TypeII ill-posedness of the problem mentioned previously. Even-posed problemscan also be ill-posed.

Note that the Type II ill-posedness in the forward problem has beenobserved with a solution of Equation 2.24 when k1 or k2 is zero (see thediscussion in the last paragraph of Section 2.2.1).

2.2.5 Inverse Problem Case II-2: Material Property Identification withNo Unique Solution (Over-Posed System)

Consider again case II-1, but assume that as shown inTable 2.1.In this case, Youngs modulus E2 can be posed using the following twoequations:

E

E

l

A k

l

A k

l

A u f f

l

A u u f

1

2

1

1

1

2

2

2

1

1 2

2 3

2

2 3 2

3

=

=

+( )

( )

,u u2 3= ,u2 0=

)u u u2 2 3(or

u2

,u u u= =0 2 3or

,u u2 3=

u2 0=

E E1 1= ,



43/573

(2.47)

or in the matrix form of

(2.48)

It is seen that this system is over-posed because, for one unknown, thereare two equations. Two different contradicting solutions for E2could exist.Therefore, strictly speaking, no solutions satisfy both equations in Equation2.48. To obtain the input X, it is necessary to perform the inversion of thesystem transformation matrix Sthat is slim with dimension of 2 1, andthe solution can be given by

(2.49)

wh

liu g.r., han x. - computational inverse techniques in nondestructive evaluation(2003)(592)

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