Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1996 Finite element modeling of ultrasonic wave propagation with application to acoustic microscopy Tianji Xue Iowa State University Follow this and additional works at:https://lib.dr.iastate.edu/rtd Part of theApplied Mechanics Commons,Electrical and Computer Engineering Commons, and theMaterials Science and Engineering Commons Recommended Citation Xue, Tianji, "Finite element modeling of ultrasonic wave propagation with application to acoustic microscopy " (1996).Retrospective Theses and Dissertations. 11348. https://lib.dr.iastate.edu/rtd/11348 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. 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UMI A Bell & Howell Infonnation Company 300 North Zed) Road, Ann Aibcr MI 48106-1346 USA 313/761-4700 800/521-0600 Finite element modeling of ultrasonic wave propagation with application to acoustic microscopy by Tianji Xue A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Electrical and Computer Engineering Major: Electrical Engineering (Electromagnetics) Af^proyed: Mernbers 9f the Committee: Signature was redacted for privacy. Signature was redacted for privacy. In enlarge of Major W k Signature was redacted for privacy. Signature was redacted for privacy. Signature was redacted for privacy. For the Major De Signature was redacted for privacy. Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1996 Copyright © Tianji Xue, 1996. All rights reserved. UMI Number: 9626077 UMI Microfonn 9626Vn Copyright 1996, by UMI Company. All rights reserved. This microfonn edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii TABLE OF CONTENTS CHAPTER 1. INTRODUCTION 1 Ultrasonic NDE and Acoustic Microscopy 1 Review of Solution Methods 5 Analytical methods 6 Numerical methods 8 Objectives of Dissertation 11 CHAPTER 2. ACOUSTIC WAVES IN ELASTIC MEDIA 13 Deformation and Strain 13 Traction Force and Stress 17 Dynamic Equation of Motion 19 Stress-Strain Relationship 21 Major Wave Types in Isotropic Media 23 Bulk waves 23 Surface and plate waves 25 CHAPTERS. FINITE ELEMENT MODELING 28 Semi-discretized Finite Element Formulation 28 Governing equations 28 Variational formulation 30 iii Finite element approximation 31 Two-Dimensional and Axisymmetric Approximations 34 Two-dimensional approximation 34 Axisymmetric approximation 38 Modal Superposition Method 40 Direct Integration Methods 44 The Newmark method 44 The central difference method 45 Stability analysis 49 Absorbing Boundary Conditions 52 Viscous boundaries 52 Non-reflecting boundaries 54 CHAPTER 4. TRANSIENT FIELDS OF PULSED TRANSDUC­ ERS IN SOLIDS 56 Numerical Modeling 59 Model assumptions 59 Form of the forcing function 61 Artificial boundary conditions 63 Farfield of a Point-like Source 64 Fields of a Finite Aperture Transducer 67 Transducer Field Interaction with a Flaw 74 Additional Numerical Examples 85 Finite aperture transducer fields scattered from a penny shaped crack 85 Plane waves scattering from crack in 2D geometry 89 iv CHAPTER 5. TREATMENT FOR ACOUSTIC MEDIA AND FLUID/SOLID INTERFACES 94 Governing Equations from Linear Acoustics 95 Acoustic Finite Element Formulation 98 Direct Galerkin weighted residual approach 98 Pressure analog approach 100 Mock fluid elements 105 Fluid/Solid Coupling 106 Numerical Examples of Transient Wave Phenomena at Solid/Fluid Inter­ faces with Axisymmetric Geometries 109 Simulation of Leaky Rayleigh Waves with 2D Geometries 116 Case 1 117 Case 2 120 Case 3 125 CHAPTER 6. FIELDS OF ULTRASONIC TRANSDUCERS AND TIME-DELAY ARRAYS 128 Numerical Approaches 129 The impulse response approach 129 The finite element approach 134 Nearfields of the Planar Transducer 135 Field of the Time-delay Array 142 Effects of Array Structural Parameters 151 CHAPTER 7. WAVE ANALYSIS IN ACOUSTIC MICROSCOPY 160 Fields of the Spherically Curved Transducer 162 V Modeling of the Spherical Lens 168 Focused Fields Probing a Fluid/Solid Interface 168 Focused Fields Probing a Solid/Solid Interface 171 CHAPTERS. CONCLUSIONS 180 BIBLIOGRAPHY 183 ACKNOWLEDGEMENTS 195 vi LIST OF FIGURES Figure 1.1: System diagram for the scanning acoustic microscope 3 Figure 1.2: The ultrasonic probe configuration 4 Figure 3.1: Wave profiles ai t — 7jus based on (a) bilinear elements (b) quadratic elements 48 Figure 4.1: Axisymmetric geometry for modeling the pulsed transducer with or without defects in the solid 60 Figure 4.2: Excitation pulses of different bandwidth 62 Figure 4.3: Simulated longitudinal pulses showing variation with different number of nodes per shear wavelength 64 Figure 4.4: Wave profiles for the point-like source at t=3 //s, (a) Z-displacement, (b) R-displacement 65 Figure 4.5: Farfield directivity pattern for the point-like source, (a) lon­ gitudinal wave, (b) shear wave 66 Figure 4.6: Z-displacement profiles for the pulsed transducer at time in­ stants (a) 0.5, (b) 1.0, (c) 1.5, (d) 3.0 and (e) 6.0 /zs 68 Figure 4.7: R-displacement profiles for the pulsed transducer at time in­ stants (a) 0.5, (b) 1.0, (c) 1.5, (d) 3.0 and (e) 6.0 /ws 69