Computational Inverse Techniques in Nondestructive Evaluation

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ISBN: 9780849315237 | 0849315239
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Copyright: 6/27/2003

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Ill-posedness. Regularization. Stability. Uniqueness. To many engineers, the language of inverse analysis projects a mysterious and frightening image, an image made even more intimidating by the highly mathematical nature of most texts on the subject. But the truth is that given a sound experimental strategy, most inverse engineering problems can be well-posed and not difficult to deal with.Computational Inverse Techniques in Nondestructive Evaluation sets forth in clear, easy-to-understand terms the principles, computational methods, and algorithms of inverse analyses based on elastic waves or the dynamic responses of solids and structures. After describing the features of inverse problems, the authors discuss the regularization methods useful in handling ill-posed problems. The book also presents practical optimization algorithms, including some developed and successfully tested by his research group.Inverse analyses are fast becoming one of the engineer's most powerful tools in nondestructive evaluation and testing. With straightforward examples, a wealth of specific applications, and clear exposition written by engineers for engineers, this book offers an outstanding opportunity to overcome any trepidation and begin using inverse analysis in practice.

1 Introduction

(10)

1.1 Forward and Inverse Problems Encountered in Structural Systems

(1)

1.2 General Procedures to Solve Inverse Problems

(3)

1.3 Outline of the Book

(6)

2 Fundamentals of Inverse Problems

(44)

2.1 A Simple Example: A Single Bar

(4)

2.1.1 Forward Problem

(1)

2.1.2 Inverse Problem

(1)

2.2 A Slightly Complex Problem: A Composite Bar

(12)

2.2.1 Forward Problem

(3)

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

(1)

2.2.3 Inverse Problem Case I-2: Load/Force Identification with No Unique Solution (Under-Posed System)

(2)

2.2.4 Inverse Problem Case II-1: Material Property Identification with Unique Solution (Even-Posed System)

(2)

2.2.5 Inverse Problem Case II-2: Material Property Identification with No Unique Solution (Over-Posed System)

(1)

2.2.6 Inverse Problem Case III: Geometry Identification with Unique Solution

(1)

2.2.7 Inverse Problem Case IV, Boundary Condition Identification

(1)

2.2.8 Points to Note

(1)

2.3 Type III Ill-Posedness

(5)

2.3.1 Forward Problem

(1)

2.3.2 Differential Operation: Magnification of Error

(2)

2.3.3 Definition of Type III Ill-Posedness

(1)

2.3.4 A Simple Solution for Type III Ill-Posed Inverse Problems

(1)

2.3.5 Features of Ill-Posedness

(1)

2.4 Types of Ill-Posed Inverse Problems

(1)

2.5 Explicit Matrix Systems

(2)

2.6 Inverse Solution for Systems with Matrix Form

(6)

2.6.1 General Inversion of System Matrix

(2)

2.6.2 Under-Posed Problems: Minimum Length Solution

(2)

2.6.3 Even-Posed Problems: Standard Inversion of Matrix

(1)

2.6.4 Over-Posed Problems: Least-Squares Solution

(2)

2.7 General Inversion by Singular Value Decomposition (SVD)

(6)

2.7.1 Property of Transformation and Type II Ill-Posedness

(2)

2.7.2 SVD Procedure

(2)

2.7.3 Ill Conditioning

(1)

2.7.4 SVD Inverse Solution

(2)

2.8 Systems in Functional Forms: Solution by Optimization

(2)

2.9 Choice of the Outputs or Effects

(1)

2.10 Simulated Measurement

(1)

2.11 Examination of Ill-Posedness

(1)

2.12 Remarks

(3)

3 Regularization for Ill-Posed Problems

(22)

3.1 Tikhonov Regularization

(5)

3.1.1 Regularizing the Norm of the Solution

(2)

3.1.2 Regularization Using Regularization Matrix

(1)

3.1.3 Determination of the Regularization Matrix

(2)

3.1.4 Tikhonov Regularization for Complex Systems

(1)

3.2 Regularization by SVD

(1)

3.3 Iterative Regularization Method

(1)

3.4 Regularization by Discretization (Projection)

(9)

3.4.1 Exact Solution of the Problem

(1)

3.4.2 Revealing the Ill-Posedness

(2)

3.4.3 Numerical Method of Discretization for Inverse Problem

(2)

3.4.3.1 Finite Element Solution

(1)

3.4.3.2 Inverse Force Estimation

(1)

3.4.4 Definition of the Errors

(2)

3.4.5 Property of Projection Regularization

(1)

3.4.6 Selecting the Best Mesh Density

(1)

3.5 Regularization by Filtering

(3)

3.5.1 Example I: High-Frequency Sine Noise

(1)

3.5.2 Example II: Gaussian Noise

(2)

3.6 Remarks

(2)

4 Conventional Optimization Techniques

(30)

4.1 The Role of Optimization in Inverse Problems

(2)

4.2 Optimization Formulations

(2)

4.3 Direct Search Methods

(9)

4.3.1 Golden Section Search Method

(1)

4.3.2 Hooke and Jeeves' Method

(4)

4.3.2.1 Exploratory Moves

(1)

4.3.2.2 Pattern Moves

(1)

4.3.2.3 Algorithm

(1)

4.3.2.4 Example

(2)

4.3.3 Powell's Conjugate Direction Method

(3)

4.3.3.1 Conjugate Directions

(1)

4.3.3.2 Example

(1)

4.4 Gradient-Based Methods

(7)

4.4.1 Cauchy's (Steepest Descent) Method

(2)

4.4.2 Newton's Method

(3)

4.4.3 Conjugate Gradient Method

(2)

4.5 Nonlinear Least Squares Method

(5)

4.5.1 Derivations of Objective Functions

(1)

4.5.2 Newton's Method

(1)

4.5.3 The Gauss-Newton Method

(2)

4.5.4 The Levenberg-Marquardt Method

101

(1)

4.5.5 Software Packages

101

(1)

4.6 Root Finding Methods

102

(3)

4.6.1 Newton's Root Finding Method

102

(1)

4.6.2 Levenberg-Marquardt Method

103

(2)

4.7 Remarks

105

(1)

4.8 Some References for Optimization

106

(1)

5 Genetic Algorithms

107

(68)

5.1 Introduction

107

(1)

5.2 Basic Concept of GAs

108

(13)

5.2.1 Coding

108

(1)

5.2.2 Genetic Operators

109

(4)

5.2.2.1 Selection

109

(1)

5.2.2.2 Crossover

110

(2)

5.2.2.3 Mutation

112

(1)

5.2.3 A Simple Example

113

(5)

5.2.3.1 Solution

113

(1)

5.2.3.2 Representation (Encoding)

113

(1)

5.2.3.3 Initial Generation and Evaluation Function

114

(1)

5.2.3.4 Genetic Operations

114

(3)

5.2.3.5 Results

117

(1)

5.2.4 Features of GAs

118

(1)

5.2.5 Brief Reviews on Improvements of GAs

119

(2)

5.3 Micro GAs

121

(16)

5.3.1 Uniform µGA

123

(1)

5.3.2 Real Parameter Coded µGA

123

(14)

5.3.2.1 Four Crossover Operators

125

(2)

5.3.2.2 Test Functions

127

(1)

5.3.2.3 Performance of the Test Functions

127

(10)

5.4 Intergeneration Projection Genetic Algorithm (IP-GA)

137

(18)

5.4.1 Modified µGA

137

(2)

5.4.2 Intergeneration Projection (IP) Operator

139

(1)

5.4.3 Hybridization of Modified µGA with IP Operator

140

(3)

5.4.4 Performance Tests and Discussions

143

(12)

5.4.4.1 Convergence Performance of the IP-GA

143

(3)

5.4.4.2 Effect of Control Parameters a and 3

146

(2)

5.4.4.3 Effect of the IP Operator

148

(2)

5.4.4.4 Comparison with Hybrid GAs Incorporated with Hill-Climbing Method

150

(5)

5.5 Improved IP-GA

155

(7)

5.5.1 Improved IP Operator

155

(1)

5.5.2 Implementation of the Improved IP Operator

156

(1)

5.5.3 Performance Test

157

(5)

5.5.3.1 Performance of the Improved IP-GA

158

(1)

5.5.3.2 Effect of the Mutation Operation

159

(1)

5.5.3.3 Effect of the Coefficients a and ß

160

(1)

5.5.3.4 Effect of the Random Number Seed

161

(1)

5.6 IP-GA with Three Parameters (IP3-GA)

162

(2)

5.6.1 Three-Parameter IP Operator

163

(1)

5.6.2 Performance Comparison

163

(1)

5.7 GAs with Search Space Reduction (SR-GA)

164

(3)

5.8 GA Combined with the Gradient-Based Method

167

(4)

5.8.1 Combined Algorithm

167

(1)

5.8.2 Numerical Example

168

(3)

5.9 Other Minor Tricks in Implementation of GAs

171

(2)

5.10 Remarks

173

(1)

5.11 Some References for Genetic Algorithms

173

(2)

6 Neural Networks

175

(20)

6.1 General Concepts of Neural Networks

175

(2)

6.2 Role of Neural Networks in Solving Inverse Problems

177

(1)

6.3 Multilayer Perceptrons

178

(4)

6.3.1 Topology

178

(1)

6.3.2 Back-Propagation Training Algorithm

179

(2)

6.3.3 Modified BP Training Algorithm

181

(1)

6.4 Performance of MLP

182

(5)

6.4.1 Number of Neurons in Hidden Layers

182

(1)

6.4.2 Training Samples

183

(1)

6.4.3 Normalization of Training Data Set

184

(1)

6.4.4 Regularization

184

(3)

6.5 A Progressive Learning Neural Network

187

(2)

6.6 A Simple Application of NN

189

(3)

6.6.1 Inputs and Outputs of the NN Model

190

(1)

6.6.2 Architecture of the NN Model

191

(1)

6.6.3 Training and Performance of the NN Model

191

(1)

6.7 Remarks

192

(1)

6.8 References on Neural Networks

193

(2)

7 Inverse Identification of Impact Loads

195

(28)

7.1 Introduction

195

(2)

7.2 Displacement as System Effects

197

(1)

7.3 Identification of Impact Loads on the Surface of Beams

198

(6)

7.3.1 Finite Element Model

198

(1)

7.3.2 Validation of FE Model with Experiment

199

(3)

7.3.3 Estimation of Loading Time History

202

(2)

7.3.4 Boundary Effects

204

(1)

7.4 Line Loads on the Surface of Composite Laminates

204

(13)

7.4.1 Hybrid Numerical Method

206

(1)

7.4.2 Why HNM?

207

(1)

7.4.3 TransWave©

207

(1)

7.4.4 Comparison between HNM and FEM

208

(2)

7.4.5 Kernel Functions

210

(2)

7.4.6 Identification of Time History of Load Using Green's Functions

212

(1)

7.4.7 Identification of Line Loads

213

(3)

7.4.7.1 Identification of Time Function

214

(1)

7.4.7.2 Identification of the Spatial Function

214

(1)

7.4.7.3 Identification of the Time and Spatial Functions

215

(1)

7.4.8 Numerical Verification

216

(1)

7.5 Point Loads on the Surface of Composite Laminates

217

(4)

7.5.1 Inversion Operation

217

(2)

7.5.2 Concentrated Point Load

219

(2)

7.6 Ill-Posedness Analysis

221

(1)

7.7 Remarks

222

(1)

8 Inverse Identification of Material Constants of Composites

223

(44)

8.1 Introduction

223

(2)

8.2 Statement of the Problem

225

(1)

8.3 Using the Uniform µGA

226

(19)

8.3.1 Solving Strategy

226

(1)

8.3.2 Parameter Coding

227

(1)

8.3.3 Parameter Settings in µGA

227

(1)

8.3.4 Example I: Engineering Elastic Constants in Laminates

227

(6)

8.3.4.1 Laminate [G0/+45/-45]s

228

(2)

8.3.4.2 Laminate [C0/+45/-45/90/-45/+45]s

230

(1)

8.3.4.3 Regularization by Projection

231

(1)

8.3.4.4 Regularization by Filtering

231

(1)

8.3.4.5 Discussion

232

(1)

8.3.5 Example II: Fiber Orientation in Laminates

233

(7)

8.3.5.1 Eight-Ply Symmetrical Composite Laminates

234

(2)

8.3.5.2 Ten-Ply Symmetrical Composite Laminates

236

(1)

8.3.5.3 Further Investigations

237

(3)

8.3.6 Example III: Engineering Constants of Laminated Cylindrical Shells

240

(8)

8.3.6.1 [GO/+45/-45/90/-45/+45]s Cylindrical Shell

241

(3)

8.3.6.2 [GO/-30/30/90/-60/+60]s Cylindrical Shell

244

(1)

8.3.6.3 [CO/30/-30/90/-60/60]s Cylindrical Shell

244

(1)

8.4 Using the Real µGA

245

(1)

8.5 Using the Combined Optimization Method

246

(2)

8.6 Using the Progressive NN for Identifying Elastic Constants

248

(18)

8.6.1 Solving Strategy and Statement of the Problem

248

(1)

8.6.2 Inputs of the NN Model

249

(11)

8.6.3 Training Samples

260

(1)

8.6.4 Results and Discussion

261

(3)

8.6.5 A More Complicated Case Study

264

(2)

8.7 Remarks

266

(1)

9 Inverse Identification of Material Property of Functionally Graded Materials

267

(40)

9.1 Introduction

267

(2)

9.2 Statement of the Problem

269

(1)

9.3 Rule of Mixture

269

(2)

9.4 Use of Gradient-Based Optimization Methods

271

(5)

9.4.1 Example 1: Transversely Isotropic FGM Plate

271

(3)

9.4.1.1 Approach I: Identification of Material Property at Discrete Locations

272

(1)

9.4.1.2 Approach II: Identification of Parameterized Values

273

(1)

9.4.2 Example 2: SiC-C FGM Plate

274

(2)

9.4.2.1 Identification of Parameterized Values

274

(2)

9.4.2.2 Approach III: Identification of Volume Fractions

276

(1)

9.5 Use of Uniform µGA

276

(10)

9.5.1 Material Characterization of FGM Plate

278

(5)

9.5.1.1 Parameters Used in the Uniform µGA

280

(2)

9.5.1.2 Test of GAs' Performance

282

(1)

9.5.1.3 Search Range

282

(1)

9.5.2 Material Characterization of FGM Cylinders

283

(3)

9.6 Use of Combined Optimization Method

286

(4)

9.7 Use of Progressive NN Model

290

(14)

9.7.1 Material Characterization of SiC-C FGM Plate

290

(8)

9.7.1.1 Inputs of the NN Model

290

(1)

9.7.1.2 Training Samples

291

(3)

9.7.1.3 Results and Discussion

294

(4)

9.7.2 Material Characterization of SS-SN FGM Cylinders

298

(13)

9.7.2.1 Inputs of the NN Model

298

(3)

9.7.2.2 Training Samples

301

(1)

9.7.2.3 Results and Discussions

302

(2)

9.8 Remarks

304

(3)

10 Inverse Detection of Cracks in Beams Using Flexural Waves

307

(84)

10.1 Introduction

308

(3)

10.2 Beams with Horizontal Delamination

311

(10)

10.2.1 The SEM

311

(1)

10.2.2 Why SEM?

311

(1)

10.2.3 Brief on SEM Formulation

312

(3)

10.2.4 Experimental Study

315

(3)

10.2.5 Sensitivity Study and Rough Estimation of Crack in Isotropic Beams

318

(3)

10.2.5.1 Crack Length

318

(1)

10.2.5.2 Crack Depth

319

(2)

10.3 Beam Model of Flexural Wave

321

(17)

10.3.1 Basic Assumptions

323

(1)

10.3.2 Homogeneous Solution

324

(2)

10.3.3 Particular Solution

326

(1)

10.3.4 Continuity Conditions

327

(2)

10.3.5 Comparison between SEM and Beam Model

329

(4)

10.3.6 Experimental Verification

333

(5)

10.4 Beam Model for Transient Response to an Impact Load

338

(6)

10.4.1 Beam Model Solution

338

(1)

10.4.2 Experimental Study on Impact Response

339

(3)

10.4.3 Comparison Study

342

(2)

10.5 Extensive Experimental Study

344

(13)

10.5.1 Test Specimens

345

(2)

10.5.2 Test Setup

347

(1)

10.5.3 Effect of Crack Depth

348

(3)

10.5.4 Effect of Crack Length

351

(3)

10.5.5 Effect of Excitation Frequency

354

(1)

10.5.6 Effect of Location of the Excitation Point

355

(1)

10.5.7 Study on Beams of Anisotropic Material

356

(1)

10.6 Inverse Crack Detection Using Uniform µGAs

357

(7)

10.6.1 Use of Simulated Data from SEM

358

(1)

10.6.2 Use of Experimental Data

359

(5)

10.7 Inverse Crack Detection Using Progressive NN

364

(23)

10.7.1 Procedure Outline

364

(3)

10.7.2 Composite Specimen

367

(1)

10.7.3 Delamination Detection

368

(4)

10.7.4 Effect of Different Training Data

372

(2)

10.7.5 Use of Beam Model and Harmonic Excitation

374

(6)

10.7.6 Use of Beam Model and Impact Excitation

380

(2)

10.7.7 FEM as Forward Solver

382

(5)

10.8 Discussion on Ill-Posedness

387

(2)

10.9 Remarks

389

(2)

11 Inverse Detection of Delaminations in Composite Laminates

391

(36)

11.1 Introduction

391

(2)

11.2 Statement of the Problem

393

(1)

11.3 Delamination Detection Using Uniform µGA

394

(4)

11.3.1 Horizontal Delamination

394

(3)

11.3.2 Vertical Crack

397

(1)

11.4 Delamination Detection Using the IP-GA

398

(3)

11.5 Delamination Detection Using the Improved IP-GA

401

(2)

11.6 Delamination Detection Using the Combined Optimization Method

403

(13)

11.6.1 Implementation of the Combined Technique

404

(7)

11.6.1.1 Formulations of Objective Functions

404

(3)

11.6.1.2 Switch from µGA to BCLSF

407

(2)

11.6.1.3 Effect of Noise

409

(1)

11.6.1.4 Ill-Posedness Analysis

409

(2)

11.6.1.5 Regularization by Filtering

411

(1)

11.6.2 Horizontal Delamination in [C90/G45/G-45]s Laminate

411

(5)

11.6.2.1 Noise-Free Cases

412

(1)

11.6.2.2 Noisy Cases

413

(1)

11.6.2.3 Discussion

414

(2)

11.7 Delamination Detection Using the Progressive NN

416

(9)

11.7.1 Implementation

416

(3)

11.7.2 Noise-Free Case

419

(3)

11.7.3 Noise-Contaminated Case

422

(2)

11.7.4 Discussion

424

(1)

11.8 Remarks

425

(2)

12 Inverse Detection of Flaws in Structures

427

(24)

12.1 Introduction

427

(1)

12.2 Inverse Identification Formulation

428

(4)

12.2.1 Damaged Element Identification

429

(1)

12.2.2 Stiffness Factor Identification

430

(2)

12.2.2.1 Objective Function with Weight

431

(1)

12.2.2.2 Direct Formulation

431

(1)

12.3 Use of Uniform µGA

432

(4)

12.3.1 Example I: Sandwich Beam

432

(1)

12.3.2 Example II: Sandwich Plate

433

(3)

12.4 Use of Newton's Root Finding Method

436

(9)

12.4.1 Calculation of Jacobian Matrix

437

(1)

12.4.2 Iteration Procedure

438

(1)

12.4.3 Example I: Cantilever Beam

438

(5)

12.4.3.1 Stiffness of Cantilever Beam

438

(3)

12.4.3.2 Performance Comparison with µGA

441

(1)

12.4.3.3 Noise-Contaminated Case

442

(1)

12.4.4 Example II: Plate

443

(2)

12.5 Use of Levenberg-Marquardt Method

445

(3)

12.6 Remarks

448

(3)

13 Other Applications

451

(64)

13.1 Coefficient Identification for Electronic Cooling Systems

451

(22)

13.1.1 Using the Golden Section Search Method

452

(4)

13.1.1.1 Natural Convection Problem

452

(1)

13.1.1.2 Numerical Results

453

(2)

13.1.1.3 Summary

455

(1)

13.1.2 Using GAs

456

(9)

13.1.2.1 Forward Modeling

457

(2)

13.1.2.2 Inverse Analysis of a PCB Board

459

(1)

13.1.2.3 A Complex Example

460

(3)

13.1.2.4 Summary

463

(2)

13.1.3 Using NNs

465

(8)

13.1.3.1 Coefficient Identification of a Telephone Switch Model

465

(2)

13.1.3.2 Coefficient Identification for IC Chips

467

(5)

13.1.3.3 Summary

472

(1)

13.2 Identification of the Material Parameters of a PCB

473

(7)

13.2.1 Introduction

473

(1)

13.2.2 Problem Definition

473

(1)

13.2.3 Objective Functions

474

(1)

13.2.4 Finite Element Representation

475

(1)

13.2.5 Numerical Results and Discussion

475

(5)

13.2.5.1 Sensitivity Analysis

477

(1)

13.2.5.2 Identification Using Natural Frequencies

478

(1)

13.2.5.3 Identification Using Frequency Response

479

(1)

13.2.6 Summary

480

(1)

13.3 Identification of Material Property of Thin Films

480

(6)

13.3.1 Noise-Free Cases

482

(2)

13.3.2 Noisy Cases

484

(1)

13.3.3 Discussion

485

(1)

13.4 Crack Detection Using Integral Strain Measured by Optic Fibers

486

(9)

13.4.1 Introduction

486

(1)

13.4.2 Numerical Calculation of Integral Strain

487

(1)

13.4.3 Inverse Procedure

488

(3)

13.4.3.1 Crack Expression

489

(1)

13.4.3.2 Remesh Technique

489

(1)

13.4.3.3 Definition of Objective Functions

489

(2)

13.4.4 Numerical Results

491

(3)

13.4.4.1 Different Dimensions of Cracks

491

(2)

13.4.4.2 Different Locations of Cracks (Case C)

493

(1)

13.4.4.3 Different Materials (Case D)

493

(1)

13.4.4.4 Different Applied Loads (Case E)

493

(1)

13.4.4.5 Different Boundary Conditions (Case F)

493

(1)

13.4.5 Summary

494

(1)

13.5 Flaw Detection in Truss Structure

495

(1)

13.6 Protein Structure Prediction

496

(7)

13.6.1 Protein Structural Prediction

498

(1)

13.6.2 Parameters for Protein Structures

499

(1)

13.6.3 Confirmation Energy

499

(1)

13.6.4 Lattice Model

499

(3)

13.6.4.1 Cubic Lattice Model

500

(1)

13.6.4.2 Random Energy Model

500

(1)

13.6.4.3 Lattice Structure Prediction by 1P3-GA

501

(1)

13.6.5 Results and Discussion

502

(1)

13.6.6 Summary

502

(1)

13.7 Fitting of Interatomic Potentials

503

(5)

13.7.1 Introduction

503

(1)

13.7.2 Fitting Model

504

(2)

13.7.3 Numerical Result

506

(1)

13.7.4 Summary

507

(1)

13.8 Parameter Identification in Valveless Micropumps

508

(4)

13.8.1 Introduction

508

(1)

13.8.2 Valveless Micropump

509

(1)

13.8.3 Flow-Pressure Coefficient Identification

510

(1)

13.8.4 Numerical Examples

511

(1)

13.8.5 Summary

512

(1)

13.9 Remarks

512

(3)

14 Total Solution for Engineering Systems: A New Concept

515

(28)

14.1 Introduction

515

(2)

14.2 Approaching a Total Solution

517

(5)

14.2.1 Procedure for a Total Solution

519

(1)

14.2.2 Forward Solver

519

(2)

14.2.3 System Parameters

521

(1)

14.2.4 Mathematical Representation

521

(1)

14.3 Inverse Algorithms

522

(7)

14.3.1 Sensitivity Matrix-Based Method (SMM)

522

(6)

14.3.1.1 Sensitivity-Based Equations

522

(2)

14.3.1.2 Algorithms

524

(1)

14.3.1.3 Solution Procedure

525

(1)

14.3.1.4 Comments on SMM

526

(2)

14.3.2 Neural Network

528

(1)

14.4 Numerical Examples

529

(12)

14.4.1 Vibration Analysis of a Circular Plate

530

(6)

14.4.1.1 SMM Solution

531

(1)

14.4.1.2 Progressive NN Solution

532

(4)

14.4.2 Identification of Material Properties of a Beam

536

(5)

14.4.2.1 SMM Solution

537

(2)

14.4.2.2 Progressive NN Solution

539

(2)

14.5 Remarks

541

(2)

References

543

(18)

Index

561

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Computational Inverse Techniques in Nondestructive Evaluation

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Computational Inverse Techniques in Nondestructive Evaluation

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