Spectral Finite Element Method: Wave Propagation...:Spectral Finite Element Method: Wave Propagation, Diagnostics and Control in Anisotropic and Inhomogeneous Structures (Computational Fluid and Solid Mechanics)
By Srinivasan Gopalakrishnan, Debiprosad Roy Mahapatra, Abir Chakraborty
Publisher: Springer
Number Of Pages: 440
Publication Date: 2008-01
ISBN-10 / ASIN: 1846283558
ISBN-13 / EAN: 9781846283550
Binding: Hardcover
In recent times, the use of composites and functionally graded materials (FGMs) in structural applications has increased. FGMs allow the user to design materials for a specified functionality and therefore have numerous uses in structural engineering. However, the behaviour of these structures under high-impact loading is not well understood. Spectral Finite Element Method: Wave Propagation,
Health Monitoring and Control in Composite and Functionally Graded Structures focuses on some of the wave propagation and transient dynamics problems with this complex media which had previously been thought unmanageable.
By using state-off-the-art computational power, the Spectral Finite Element Method (SFEM) can solve many practical engineering problems. This book is the first to apply SFEM to inhomogeneous and anisotropic structures in a unified and systematic manner. The authors discuss the different types of SFEM for regular and damaged 1-D and 2-D waveguides, various solution techniques, different methods of detecting the presence of damages and their locations, and different methods available to actively control the wave propagation responses. The theory is supported by tables, figures and graphs; all the numerical examples are so designed to bring out the essential wave behaviour in these complex structures. Some case studies based on real-world problems are also presented.
This book is intended for senior undergraduate students and graduate students studying wave propagation in structures, smart structures, spectral finite element method and structural health monitoring. Readers will gain a complete understanding of how to formulate a spectral finite element; learn about wave behaviour in inhomogeneous and anisotropic media; and, discover how to design some diagnostic tools for monitoring the health or integrity of a structure. This important contribution to the engineering mechanics research community will also be of value to researchers and practicing engineers in structural integrity.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Solution Methods for Wave Propagation Problems . . . . . . . . . . . 1
1.2 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Continuous Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 What is the Spectral Element Method?. . . . . . . . . . . . . . . . . . . . . 19
1.5 Outline and Scope of Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Introduction to the Theory of Anisotropic and
Inhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Introduction to Composite Materials . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Theory of Laminated Composites . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Micromechanical Analysis of a Lamina . . . . . . . . . . . . . . . 25
2.2.2 Strength of Materials Approach to Determination of
Elastic Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Stress–Strain Relations for a Lamina . . . . . . . . . . . . . . . . . 29
2.2.4 Stress–Strain Relation for a Lamina with Arbitrary
Orientation of Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Introduction to Smart Composites . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Modeling Inhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Idealization of Wave Propagation and Solution Techniques. 41
3.1 General Form of the Wave Equations . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Characteristics of Waves in Anisotropic Media . . . . . . . . . . . . . . 42
3.3 General Form of Inhomogeneous Wave Equations . . . . . . . . . . . . 43
3.4 Basic Properties and Solution Techniques . . . . . . . . . . . . . . . . . . . 43
3.5 Spectral Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . 44
3.6 Efficient Computation of the Wavenumber and Wave Amplitude 48
x Contents
3.6.1 Method 1: The Companion Matrix and the SVD
Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6.2 Method 2: Linearization of PEP . . . . . . . . . . . . . . . . . . . . . 50
3.7 Spectral Element Formulation for Isotropic Material . . . . . . . . . 51
3.7.1 Spectral Element for Rods . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.7.2 Spectral Element for Beams . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Wave Propagation in One-dimensional Anisotropic
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Wave Propagation in Laminated Composite Thin Rods and
Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Governing Equations and PEP . . . . . . . . . . . . . . . . . . . . . . 56
4.1.2 Spectrum and Dispersion Relations . . . . . . . . . . . . . . . . . . 58
4.2 Spectral Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Finite Length Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Throw-off Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Numerical Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Impact on a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Effect of the Axial–Flexural Coupling . . . . . . . . . . . . . . . . 63
4.3.3 Wave Transmission and Scattering Through an
Angle-joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Wave Propagation in Laminated Composite Thick Beams:
Poisson’s Contraction and Shear Deformation Models . . . . . . . . 69
4.4.1 Wave Motion in a Thick Composite Beam . . . . . . . . . . . . 70
4.4.2 Coupled Axial–Flexural Shear and Thickness
Contractional Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.3 Correction Factors at High Frequency Limit . . . . . . . . . . 74
4.4.4 Coupled Axial–Flexural Shear Without the Thickness
Contractional Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.5 Modeling Spatially Distributed Dynamic Loads . . . . . . . 79
4.5 Modeling Damping Using Spectral Element . . . . . . . . . . . . . . . . . 81
4.5.1 Proportional Damping Through a Discretized Finite
Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.2 Proportional Damping Through the Wave Equation . . . 83
4.6 Numerical Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6.1 Comparison of Response with Standard FEM . . . . . . . . . 91
4.6.2 Presence of Axial–Flexural Shear Coupling . . . . . . . . . . . 93
4.6.3 Parametric Studies on a Cantilever Beam. . . . . . . . . . . . . 96
4.6.4 Response of a Beam with Ply-drops . . . . . . . . . . . . . . . . . . 96
4.7 Layered Composite Thin-walled Tubes . . . . . . . . . . . . . . . . . . . . . 99
4.7.1 Linear Wave Motion in Composite Tube . . . . . . . . . . . . . . 102
4.8 Spectral Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.8.1 Short and Long Wavelength Limits for Thin Shell and
Limitations of the Proposed Model . . . . . . . . . . . . . . . . . . 107
4.8.2 Comparison with Analytical Solution . . . . . . . . . . . . . . . . 114
Contents xi
4.9 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.9.1 Time Response Under Short Impulse Load and the
Effect of Fiber Orientations . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 Wave Propagation in One-dimensional Inhomogeneous
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1 Length-wise Functionally Graded Rod. . . . . . . . . . . . . . . . . . . . . . 124
5.1.1 Development of Spectral Finite Elements . . . . . . . . . . . . . 126
5.1.2 Smoothing of Reflected Pulse . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Depth-wise Functionally Graded Beam . . . . . . . . . . . . . . . . . . . . . 135
5.2.1 Spectral Finite Element Formulation . . . . . . . . . . . . . . . . . 137
5.2.2 The Spectrum and Dispersion Relation . . . . . . . . . . . . . . . 137
5.2.3 Effect of Gradation on the Cut-off Frequencies . . . . . . . 139
5.2.4 Computation of the Temperature Field . . . . . . . . . . . . . . . 142
5.3 Wave Propagation Analysis: Depth-wise Graded Beam (HMT) 142
5.3.1 Validation of the Formulated SFE . . . . . . . . . . . . . . . . . . . 143
5.3.2 Lamb Wave Propagation in FSDT and HMT Beams . . 148
5.3.3 Effect of Gradation on Stress Waves . . . . . . . . . . . . . . . . . 151
5.3.4 Coupled Thermoelastic Wave Propagation . . . . . . . . . . . . 153
5.4 Length-wise Graded Beam: FSDT . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4.1 Spectral Finite Element Formulation . . . . . . . . . . . . . . . . . 158
5.4.2 Effect of Gradation on the Spectrum and Dispersion
Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.4.3 Effect of Gradation on the Cut-off Frequencies . . . . . . . . 160
5.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.5.1 Effect of the Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.5.2 Elimination of the Reflection from Material Boundary. . 165
6 Wave Propagation in Two-dimensional Anisotropic
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1 Two-dimensional Initial Boundary Value Problem . . . . . . . . . . . 172
6.2 Spectral Element for Doubly Bounded Media . . . . . . . . . . . . . . . 176
6.2.1 Finite Layer Element (FLE) . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.2 Infinite Layer Element (ILE) . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.3 Expressions for Stresses and Strains . . . . . . . . . . . . . . . . . 178
6.2.4 Prescription of Boundary Conditions . . . . . . . . . . . . . . . . . 179
6.2.5 Determination of Lamb Wave Modes . . . . . . . . . . . . . . . . . 179
6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.3.1 Propagation of Surface and Interface Waves . . . . . . . . . . . 181
6.3.2 Propagation of Lamb Wave . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Wave Propagation in Two-dimensional Inhomogeneous
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.1 SLE Formulation: Inhomogeneous Media . . . . . . . . . . . . . . . . . . . 195
7.1.1 Exact Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
xii Contents
7.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2.1 Propagation of Stress Waves . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2.2 Propagation of Lamb Waves . . . . . . . . . . . . . . . . . . . . . . . . 204
7.3 SLE Formulation: Thermoelastic Analysis . . . . . . . . . . . . . . . . . . 208
7.3.1 Inhomogeneous Anisotropic Material . . . . . . . . . . . . . . . . . 209
7.3.2 Discussion on the Properties of Wavenumbers . . . . . . . . . 212
7.3.3 Finite Layer Element (FLE) . . . . . . . . . . . . . . . . . . . . . . . . 215
7.3.4 Infinite Layer Element (ILE) . . . . . . . . . . . . . . . . . . . . . . . . 216
7.3.5 Homogeneous Anisotropic Material . . . . . . . . . . . . . . . . . . 217
7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.4.1 Effect of the Relaxation Parameters - Symmetric
Ply-layup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.4.2 Interfacial Waves: Thermal and Mechanical Loading . . . 220
7.4.3 Propagation of Stress Waves . . . . . . . . . . . . . . . . . . . . . . . . 221
7.4.4 Propagation of Thermal Waves . . . . . . . . . . . . . . . . . . . . . . 226
7.4.5 Effect of Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.5 Wave Motion in Anisotropic and Inhomogeneous Plate . . . . . . . 229
7.5.1 SPE Formulation: CLPT . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.5.2 Computation of Wavenumber: Anisotropic Plate . . . . . . . 234
7.5.3 Computation of Wavenumber: Inhomogeneous Plate . . . 237
7.5.4 The Finite Plate Element . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.5.5 Semi-infinite or Throw-off Plate Element . . . . . . . . . . . . 242
7.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.6.1 Wave Propagation in Plate with Ply-drop . . . . . . . . . . . . 243
7.6.2 Propagation of Lamb waves . . . . . . . . . . . . . . . . . . . . . . . . . 246
8 Solution of Inverse Problems: Source and System
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.1 Force Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.1.1 Force Reconstruction from Truncated Response . . . . . . . 250
8.2 Material Property Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.2.1 Estimation of Material Properties: Inhomogeneous Layer254
9 Application of SFEM to SHM: Simplified Damage Models . 259
9.1 Various Damage Identification Techniques . . . . . . . . . . . . . . . . . . 259
9.1.1 Techniques for Modeling Delamination . . . . . . . . . . . . . . . 260
9.1.2 Modeling Issues in Structural Health Monitoring . . . . . . 261
9.2 Modeling Wave Scattering due to Multiple Delaminations
and Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
9.3 Spectral Element with Embedded Delamination . . . . . . . . . . . . . 265
9.3.1 Modeling Distributed Contact Between Delaminated
Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.4 Numerical Studies on Wave Scattering due to Single
Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4.1 Comparison with 2-D FEM . . . . . . . . . . . . . . . . . . . . . . . . . 271
Contents xiii
9.4.2 Identification of Delamination Location from Scattered
Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9.4.3 Effect of Delamination at Ply-drops . . . . . . . . . . . . . . . . . . 274
9.4.4 Sensitivity of the Delaminated Configuration . . . . . . . . . . 276
9.5 A Sublaminate-wise Constant Shear Kinematics Model . . . . . . . 279
9.6 Spectral Elements with Embedded Transverse Crack . . . . . . . . . 284
9.6.1 Element-internal Discretization and Kinematic
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.6.2 Modeling Dynamic Contact Between Crack Surfaces . . . 288
9.6.3 Modeling Surface-breaking Cracks . . . . . . . . . . . . . . . . . . . 290
9.6.4 Distributed Constraints at the Interfaces Between
Sublaminates and Hanging Laminates . . . . . . . . . . . . . . . . 291
9.7 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.7.1 Comparison with 2-D FEM . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.7.2 Identification of Crack Location from Scattered Wave . . 294
9.7.3 Sensitivity of the Crack Configuration . . . . . . . . . . . . . . . . 296
9.8 Spectral Finite Element Model for Damage Estimation . . . . . . . 297
9.8.1 Spectral Element with Embedded Degraded Zone. . . . . . 300
9.9 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
10 Application of SFEM to SHM: Efficient Damage
Detection Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.1 Strategies for Identification of Damage in Composites . . . . . . . . 307
10.2 Spectral Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
10.2.1 Properties of Spectral Power . . . . . . . . . . . . . . . . . . . . . . . . 312
10.2.2 Measurement of Wave Scattering due to Delaminations
and Inclusions Using Spectral Power . . . . . . . . . . . . . . . . . 314
10.3 Power Flow Studies on Wave Scattering . . . . . . . . . . . . . . . . . . . . 314
10.3.1 Wave Scattering due to Single Delamination . . . . . . . . . . 314
10.3.2 Wave Scattering due to Length-wise Multiple
Delaminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
10.3.3 Wave Scattering due to Depth-wise Multiple
Delaminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
10.4 Wave Scattering due to Strip Inclusion . . . . . . . . . . . . . . . . . . . . . 319
10.4.1 Power Flow in a Semi-infinite Strip Inclusion with
Bounded Media: Effect of Change in the Material
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
10.4.2 Effect of Change in the Material Properties of a Strip
Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
10.5 Damage Force Indicator for SFEM. . . . . . . . . . . . . . . . . . . . . . . . . 323
10.6 Numerical Simulation of Global Identification Process . . . . . . . . 327
10.6.1 Effect of Single Delamination . . . . . . . . . . . . . . . . . . . . . . . 327
10.6.2 Effect of Multiple Delaminations . . . . . . . . . . . . . . . . . . . . 329
10.6.3 Sensitivity of Damage Force Indicator due to Variation
in Delamination Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
xiv Contents
10.6.4 Sensitivity of Damage Force Indicator due to Variation
in Delamination Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
10.7 Genetic Algorithm (GA) for Delamination Identification . . . . . . 337
10.7.1 Objective Functions in GA for Delamination
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
10.7.2 Displacement-based Objective Functions . . . . . . . . . . . . . 338
10.7.3 Power-based Objective Functions . . . . . . . . . . . . . . . . . . . . 343
10.8 Case Studies with a Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . 346
10.8.1 Identification of Delamination Location . . . . . . . . . . . . . . 346
10.8.2 Identification of Delamination Size . . . . . . . . . . . . . . . . . . . 348
10.8.3 Identification of Delamination Location and Size . . . . . . 349
10.8.4 Identification of Delamination Location, Size and Depth 349
10.8.5 Effect of Delamination Near the Boundary . . . . . . . . . . . . 350
10.9 Neural Network Integrated with SFEM . . . . . . . . . . . . . . . . . . . . . 352
10.10Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 357
11 Spectral Finite Element Method for Active Wave Control . 365
11.1 Challenges in Designing Active Broadband Control Systems . . 365
11.1.1 Strategies for Vibration and Wave Control . . . . . . . . . . . . 366
11.1.2 Active LAC of Structural Waves . . . . . . . . . . . . . . . . . . . . 371
11.2 Externally Mounted Passive/Active Devices . . . . . . . . . . . . . . . . . 372
11.3 Modeling Distributed Transducer Devices . . . . . . . . . . . . . . . . . . . 377
11.3.1 Plane Stress Constitutive Model of Stacked and
Layered Piezoelectric Composite . . . . . . . . . . . . . . . . . . . . 378
11.3.2 Constitutive Model for Piezoelectric Fiber Composite
(PFC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
11.3.3 Design Steps for Broadband Control . . . . . . . . . . . . . . . . . 391
11.4 Active Spectral Finite Element Model . . . . . . . . . . . . . . . . . . . . . . 394
11.4.1 Spectral Element for Finite Beams. . . . . . . . . . . . . . . . . . . 394
11.4.2 Sensor Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
11.4.3 Actuator Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
11.4.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 397
11.5 Effect of Broadband Distributed Actuator Dynamics . . . . . . . . . 398
11.6 Active Control of Multiple Waves in Helicopter Gearbox
Support Struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
11.6.1 Active Strut System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
11.6.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
11.7 Optimal Control Based on ASFEM and Power Flow . . . . . . . . . 415
11.7.1 Linear Quadratic Optimal Control Using Spectral Power416
11.7.2 Broadband Control of a Three-member Composite
Beam Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Spectral Finite Element Method Wave Propagation.part1
ص
[ 本帖最后由 drjiachen 于 2008-12-24 11:47 编辑 ]
Spectral Finite Element Method Wave Propagation.part2-4
[ 本帖最后由 drjiachen 于 2008-12-24 12:03 编辑 ]
see see ：23de
好书。。。
谢谢楼主了
:30bb :30bb
看看~~~~:30bb :30bb
:53bb :53bb :53bb
Thank you for the job,just wait for downloading
nice book, thanks LZ for your sharing
很好的书籍，大家一起欣赏吧 ..........................................
很好的书籍，一起学习学习吧 ..........................................
感谢楼主分享
:11bb :11bb :11bb :11bb
下来看看！！！谢谢分享！！！:31bb :31bb
马上下来珍藏起来,以后慢慢看,要学习的东西还好多
good book, 3ks LZ
楼主共享的书每天都一箩筐啊， 多谢了
好書
當然值得看看
感謝分享
新书一定要下:11bb :11bb
谢谢分享！！！！！！！！！！！！！！！！！！！！！
:27bb :27bb :27bb
thanks very much!!!!!!!!!!!!!!!!!!!!!
:27bb :27bb :27bb
看看~~~~:30bb :30bb
谢谢
:11bb :31bb :31bb :27bb
正在学校有限元法中 ！！！！！！！！！！
:14bb :14bb :14bb !!!!!!!!!!
一本好书,感射楼主提供!!!!!!!!
感谢楼主的辛勤工作，都是好书!:30bb
内容看上去还是不错的，download 学一下，谢谢！
think you very much!
thanks.........................
价格实惠量又足，好数哈，呵呵呵呵呵
Thank you very much!!
:11bb :11bb :11bb
楼主的精神值得我们学习啊!
楼主的精神值得我们学习啊!
感谢楼主分享!!!!!!!!!!!!!!!!!
谢谢分享！
kankankankan
dfds dfasdf fdafsd
thank you !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
本帖最后由 huangfeihong88 于 2009-10-25 14:50 编辑
nice book, thanks LZ
本帖最后由 huangfeihong88 于 2009-10-25 14:50 编辑
希望大家一起学习讨论。
：18de:27bb:27bb
真么多好书，楼主调料不起了:16bb
下来看一看，用到电磁场
谢谢分享！下来看看
goood thanks
:53bb
:27bb:27bb
好像是结构方面的东西呀
走过路过千万别错过。谢谢分享！
感谢楼主，正需要呢！！！
辛辛苦苦待了24小时才能回复
谢谢楼主分享~~~
收下了，谢谢版主
感謝了，我先收下好好学习一下！
thank you very much for this book. you are a great guy
很好的书籍，一起学习学习吧 ..........................................
下来看看！！！谢谢分享！
这是一本好书，谢谢
感谢微网的推荐
看看哈
谢谢啊，先看看啊。哈哈哈
多谢楼主，太好的资料了
谢谢您的分享了。
没听过，下来学习学习
{:7_1234:}
很好的书籍，一起学习学习吧
看看,谢谢!!!
回复 drjiachen 的帖子
good good good good good
找了好久了，谢谢楼主分享
Spectral Finite Element Method Wave Propagation
好书啊，谢谢楼主的分享
{:7_1235:}
{:7_1234:}
谢谢~~~~~~~~~~~~~~~~~~~~~~~~~~~
好东西，看看
十分感谢楼主！:16bb
行动纲领
谢谢分享！！！！！
各向异性看看此书。。。。。。
thanks for sharing.
Spectral Finite Element Method Wave Propagation
thanks
谢谢楼主
thanks LZ for your sharing
大感謝阿~好書!!
very good and thanks
楼主辛苦了哈哈哈
very good and thanks
34提高44太阳光4其他
Spectral Finite Element Method: Wave Propagation...: Spectral Finite Element Method Wave Propagation.part1.rar
