Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 9

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 9 Sven Bossuyt Gary Schajer Alberto Carpinteri Proceedings of the 2015 Annual Conference on Experimental and Applied Mechanics River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics Bethel, CT, USA

River Publishers Sven Bossuyt • Gary Schajer • Alberto Carpinteri Editors Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 9 Proceedings of the 2015 Annual Conference on Experimental and Applied Mechanics

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-923-8 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2016 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems represents one of nine volumes of technical papers presented at the 2015 SEM Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Costa Mesa, CA, June 8–11, 2015. The complete Proceedings also include volumes on: Dynamic Behavior of Materials; Challenges in Mechanics of TimeDependent Materials; Advancement of Optical Methods in Experimental Mechanics; Experimental and Applied Mechanics; MEMS and Nanotechnology; Mechanics of Biological Systems and Materials; Mechanics of Composite & Multifunctional Materials; and Fracture, Fatigue, Failure and Damage Evolution. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics, Residual Stress, Thermomechanics and Infrared Imaging, Hybrid Techniques and Inverse Problems being three of these areas. Residual stresses have a great deal of importance in engineering systems and design. The hidden character of residual stresses often causes them to be underrated or overlooked. However, they profoundly influence structural design and substantially affect strength, fatigue life and dimensional stability. Since residual stresses are induced during almost all materials processing procedures, for example, welding/joining, casting, thermal conditioning and forming, they must be taken seriously and included in practical applications. In recent years, the applications of infrared imaging techniques to the mechanics of materials and structures have grown considerably. The expansion is marked by the increased spatial and temporal resolution of the infrared detectors, faster processing times and much greater temperature resolution. The improved sensitivity and more reliable temperature calibrations of the devices have meant that more accurate data can be obtained than were previously available. Advances in inverse identification have been coupled with optical methods that provide surface deformation measurements and volumetric measurements of materials. In particular, inverse methodology was developed to more fully use the dense spatial data provided by optical methods to identify mechanical constitutive parameters of materials. Since its beginnings during the 1980s, creativity in inverse methods has led to applications in a wide range of materials, with many different constitutive relationships, across material heterogeneous interfaces. Complex test fixtures have been implemented to produce the necessary strain fields for identification. Force reconstruction has been developed for high strain rate testing. As developments in optical methods improve for both very large and very small length scales, applications of inverse identification have expanded to include geological and atomistic events. Helsinki, Finland Sven Bossuyt Vancouver, British Columbia, Canada Gary Schajer Torino, Italy Alberto Carpinteri v

Contents 1 Reconstruction of Spatially Varying Random Material Properties by Self-Optimizing Inverse Method.................................................... 1 Joshua M. Weaver and Gunjin J. Yun 2 Performance Assessment of Integrated Digital Image Correlation Versus FEM Updating.......... 11 A.P. Ruybalid, J.P.M. Hoefnagels, O. van der Sluis, and M.G.D. Geers 3 IGMU: A Geometrically Consistent Framework for Identification from Full Field Measurement ............................................................ 17 J.-E. Dufour, J. Schneider, F. Hild, and S. Roux 4 Characterization of the Dynamic Strain Hardening Behavior from Full-field Measurements . . . . . . . . 23 J.-H. Kim, M.-G. Lee, F. Barlat, and F. Pierron 5 Bridging Kinematic Measurements and Crystal Plasticity Models in Austenitic Stainless Steels ........................................................... 29 A. Guery, F. Latourte, F. Hild, and S. Roux 6 Inverse Identification of Plastic Material Behavior Using Multi-Scale Virtual Experiments . . . . . . . . . 37 D. Debruyne, S. Coppieters, Y. Wang, P. Eyckens, T. Kuwabara, A. Van Bael, and P. Van Houtte 7 An Effective Experimental-Numerical Procedure for Damage Assessment of Ti6Al4V............. 43 L. Cortese, F. Nalli, G.B. Broggiato, and T. Coppola 8 Identification of the YLD2000-2D Model with the Virtual Fields Method....................... 51 Marco Rossi, Fre´de´ric Barlat, Fabrice Pierron, Marco Sasso, and Attilio Lattanzi 9 Identification of Post-necking Strain Hardening Behavior of Pure Titanium Sheet ................ 59 S. Coppieters, S. Sumita, D. Yanaga, K. Denys, D. Debruyne, and T. Kuwabara 10 Challenges for High-Pressure High-Temperature Applications of Rubber Materials in the Oil and Gas Industry.................................................. 65 Allan Zhong 11 Full-Field Strain Imaging of Ultrasonic Waves in Solids .................................... 81 C. Devivier, F. Pierron, P. Glynne-Jones, and M. Hill 12 Acoustic Emission Analysis in Titanium Grade 5 Samples During Fatigue Test .................. 87 C. Barile, C. Casavola, G. Pappalettera, and C. Pappalettere 13 Analysis of High-Frequency Vibrational Modes Through Laser Pulses ......................... 93 G. Lacidogna, S. Invernizzi, B. Montrucchio, O. Borla, and A. Carpinteri 14 Acquisition of Audio Information from Silent High Speed Video............................. 105 Jason Quisberth, Zhaoyang Wang, and Hieu Nguyen vii

15 Overview of the Effects of Process Parameters on the Accuracy in Residual Stress Measurements by Using HD and ESPI ................................................. 113 C. Barile, C. Casavola, G. Pappalettera, and C. Pappalettere 16 Near Weld Stress Analysis with Optical and Acoustic Methods .............................. 119 Sanichiro Yoshida, Tomohiro Sasaki, Masaru Usui, Shuuichi Sakamoto, Ik-keun Park, Hyunchul Jung, and Kyeongsuk Kim 17 Nondestructive Characterization of Thin Film System with Dual-Beam Interferometer............ 129 Hae-Sung Park, David Didie, Daniel Didie, Sanichiro Yoshida, Ik-Keun Park, Seung Bum Cho, and Tomohiro Sasaki 18 Effect of Horn Tip Geometry on Ultrasonic Cavitation Peening.............................. 139 Tomohiro Sasaki, Kento Yoshida, Masayuki Nakagawa, and Sanichiro Yoshida 19 Fatigue Damage Analysis of Aluminum Alloy by ESPI ..................................... 147 Tomohiro Sasaki, Shun Hasegawa, and Sanichiro Yoshida 20 Dynamic Failure Mechanisms in Woven Ceramic Fabric Reinforced Metal Matrix Composites During Ballistic Impact ............................................. 155 Brandon A. McWilliams, Jian H. Yu, and Mark Pankow 21 Digital Image Correlation Analysis and Numerical Simulation of the Aluminum Alloys under Quasi-static Tension after Necking Using the Bridgman’s Method.................. 161 Jian H. Yu, Brandon A. McWilliams, and Robert P. Kaste 22 Robust Intermediate Strain Rate Experimentation Using the Serpentine Transmitted Bar.......... 167 W.R. Whittington, A.L. Oppedal, D.K. Francis, and M.F. Horstemeyer 23 Testing Program for Crashworthiness Assessment of Cutaway Buses .......................... 175 Michal Gleba, Jeff Siervogel, Jerry W. Wekezer, and Sungmoon Jung 24 In-Situ DIC and Strain Gauges to Isolate the Deficiencies in a Model for Indentation Including Anisotropic Plasticity............................................. 183 Jacob S. Merson, Michael B. Prime, Manuel L. Lovato, and Cheng Liu 25 Analysis of Laser Weld Induced Stress in a Hermetic Seal .................................. 199 Ryan D. Jamison, Pierrette H. Gorman, Jeffrey Rodelas, Danny O. MacCallum, Matthew Neidigk, and J. Franklin Dempsey 26 A Summary of Failures Caused by Residual Stresses ...................................... 209 E.J. Fairfax and M. Steinzig 27 Comparative Analysis of Shot-Peened Residual Stresses Using Micro-Hole Drilling, Micro-Slot Cutting, X-ray Diffraction Methods and Finite-Element Modelling................... 215 B. Winiarski, M. Benedetti, V. Fontanari, M. Allahkarami, J.C. Hanan, G.S. Schajer, and P.J. Withers 28 Thermal Deformation Analysis of an Aluminum Alloy Utilizing 3D DIC....................... 225 Jarrod L. Smith, Jeremy D. Seidt, and Amos Gilat 29 Hybrid Full-Field Stress Analysis of Loaded Perforated Asymmetrical Plate.................... 235 S. Kurunthottikkal Philip and R.E. Rowlands 30 Automated Detection of CFRP Defects by Infrared Thermography and Image Analysis ........... 243 Terry Yuan-Fang Chen and Guan-Yu Lin 31 Modelling the Residual Stress Field Ahead of the Notch Root in Shot Peened V-Notched Samples ................................................................ 249 M. Benedetti, V. Fontanari, M. Allahkarami, J.C. Hanan, B. Winiarski, and P.J. Withers 32 Numerical Prediction of Temperature and Residual Stress Fields in LFSW..................... 263 C. Casavola, A. Cazzato, V. Moramarco, and C. Pappalettere viii Contents

33 Residual Stress Measurement on Shot Peened Samples Using FIB-DIC........................ 275 Enrico Salvati, Matteo Benedetti, Tan Sui, and Alexander M. Korsunsky 34 Residual Stress in Injection Stretch Blow Molded PET Bottles ............................... 285 Masoud Allahkarami, Sudheer Bandla, and Jay C. Hanan 35 Applying Infrared Thermography and Heat Source Reconstruction for the Analysis of the Portevin-Le Chatelier Effect in an Aluminum Alloy........................... 291 Didier Delpueyo, Xavier Balandraud, and Michel Gre´diac 36 Applying a Gad Filter to Calculate Heat Sources from Noisy Temperature Fields ................ 297 Cle´ment Beitone, Xavier Balandraud, Michel Gre´diac, Didier Delpueyo, Christophe Tilmant, and Fre´de´ric Chausse 37 Contour Method Residual Stress Measurement Uncertainty in a Quenched Aluminum Bar and a Stainless Steel Welded Plate........................................ 303 Mitchell D. Olson, Adrian T. DeWald, Michael B. Prime, and Michael R. Hill 38 On the Separation of Complete Triaxial Strain/Stress Profiles from Diffraction Experiments . . . . . . . 313 H. Wern and E. J€ackel 39 Residual Stress Mapping with Multiple Slitting Measurements ............................... 319 Mitchell D. Olson, Michael R. Hill, Jeremy S. Robinson, Adrian T. DeWald, and Victor Sloan 40 A Novel Approach for Biaxial Residual Stress Mapping Using the Contour and Slitting Methods ............................................................... 331 Mitchell D. Olson and Michael R. Hill 41 Measurement of Residual Stresses in B4C-SiC-Si Ceramics Using Raman Spectroscopy............ 341 Phillip Jannotti and Ghatu Subhash 42 Hole Drilling Determination of Residual Stresses Varying Along a Surface..................... 347 Alberto Makino and Drew Nelson 43 Sensitivity Analysis of i-DIC Approach for Residual Stress Measurement in Orthotropic Materials ............................................................ 355 Antonio Baldi 44 Stress Measurement Repeatability in ESPI Hole-Drilling................................... 363 Theo Rickert 45 Some Aspects of the Application of the Hole Drilling Method on Plastic Materials ................ 371 Arnaud Magnier, Andreas Nau, and Berthold Scholtes Contents ix

Chapter 1 Reconstruction of Spatially Varying Random Material Properties by Self-Optimizing Inverse Method Joshua M. Weaver and Gunjin J. Yun Abstract In this paper, a new methodology for reconstructing spatially varying random material properties is presented by combining stochastic finite element (SFE) models with Self-Optimizing Inverse Method (Self-OPTIM). The Self-OPTIM can identify model parameters based on partial boundary force and displacement data from experimental tests. Statistical information (i.e. spatial mean, variance, correlation length and Gaussian normal random variables) of spatially varying random fields (RFs) are parameterized by Karhunen-Loe`ve (KL) expansion method and integrated into SFE models. In addition, a new software framework is also presented that can simultaneously utilize any number of remote computers in a network domain for the Self-OPTIM simulation. This can result in a significant decrease of computational times required for the optimization task. Two important issues in the inverse reconstruction problem are addressed in this paper: (1) effects of the number of internal measurements and (2) non-uniform reaction forces along the boundary on the reconstruction accuracy. The proposed method is partially proven to offer new capabilities of reconstructing spatially inhomogeneous material properties and estimating their statistical parameters from incomplete experimental measurements. Keywords Inverse analysis • Inverse reconstruction • Self-optimizing inverse method • Parameter estimation • Parallel computations 1.1 Introduction Understanding and characterizing behavior of the materials we use is one of the fundamental aspects of engineering design. Constitutive models are often used to predict the response of such materials under various loadings. Theses constitutive models can contain various parameters that are used to approximate their mechanical response of materials. With increasing complexity of the constitutive models, it becomes very difficult to determine all material parameters with one simple experimental test. This causes the parameter identification procedure to become costly and time comprehensive. The parameter identification of material constitutive models has been studied by many researchers within the same inverse problem setting: naming a few representative researches, Mahnken, et al. [1–5], Saleeb, et al. [6–9], Geriach et al. [10, 11], Akerstrom et al. [12] and Castello, et al. [13]. Various methods for parameter estimation of the constitutive models have been researched including the finite element model updating method (FEMU) [14], the constitutive equation gap method (CEGM) [15, 16], the virtual fields method (VFM) [17, 18], the equilibrium gap method (EGM) [19] and the reciprocity gap method (RGM) [20]. All of these methods require full-field measurements of displacements. Recently, Yun et al. developed Self-Optimizing Inverse Method called Self-OPTIM [21]. The Self-OPTIM was used to identify the constitutive material parameters of an elasto-plasticity model based upon one experimental material test [22]. The Self-OPTIM is suitable for parameter estimations using a large-scale finite element model under general loading conditions since its algorithm is designed to minimize errors of full-field and inhomogeneous stresses and strains computed from two parallel finite element simulations subjected to experimentally measured boundary force and displacement data, respectively, rather than a least squares functional between experimental response (e.g. displacements) along limited boundaries and corresponding modelbased predictions. For example, a comparison with a least-square functional between experimental and FE-simulated displacements showed that Self-OPTIM can offer better convexity of the optimization problem [23]. A group of German mathematicians provided a strong mathematical basis proving the existence of a global minimum of the Self-OPTIM’s objective function for the case of linear non-homogeneous and non-isotropic elasticity in the stationary case [24]. Self-OPTIM was also successfully applied to identification of damping of highway bridge embankments from real earthquake acceleration response [23]. J.M. Weaver • G.J. Yun (*) Department of Civil Engineering, The University of Akron, 244 Sumner St. ASEC 210, Akron, OH 44325-3905, USA e-mail: gy3@uakron.edu #The Society for Experimental Mechanics, Inc. 2016 S. Bossuyt et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 9, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-21765-9_1 1

Many times a material is modeled as a homogenous material across the entirety of the model. However, in reality this is not the case. Uncertainties in raw materials, geometric heterogeneity of constituents in composite materials, manufacturing process, long-term operation under uncertain environmental and loading conditions cause a non-uniform distribution of material properties. Therefore, reconstruction of heterogeneous material properties is important for condition assessments. Several approaches used for parameter estimation have been applied for damage identification problems. The EGM method was used for identifying heterogeneous map of elastic moduli [19]. Heterogeneous elasto-plastic properties could be identified based on full-field measurements by the CEGM method [25]. Generally, distributions of damage over the structures are inhomogeneous and the number of parameters thus increases in proportional to the size of the finite element model. Therefore, compared to parameter estimation problems for constitutive models, updating a set of damage parameters defined at a large number of degrees of freedom and/or elements yields a much more challenging inverse problem. To reduce the number of updating parameters, the concept of damage function like finite element shape functions were used in dynamic modal based FE modeling [26–28]. However, in their studies, spatial correlation between identified local properties could not be taken into account. Recently, Adhikari and Friswell applied Karhunen-Loe`ve expansion (KLE) method to modalbased model updating for identifying distributed properties (i.e. mass and stiffness) of beam structures [29]. In their study, statistical properties (i.e. spatial mean, variance and correlation length) are assumed to be a priori known properties. The KLE method have also been applied to model updating for identifying distributions and statistical parameters of spatially correlated random variables in 2D and 3D complex structures [30–32] whereby stochastic finite element with spatial varying material properties was integrated with Self-OPTIM [30]. In spite of these efforts, stochastic inverse material parameter characterization and/or damage detection still requires identifying a large number of updating parameters from incomplete measurements. Global optimization methods (e.g. evolutionary algorithms) are considered suitable for the large number of unknown parameters. However, they usually entail a large number of runs of finite element analyses. This paper present a new methodology for reconstructing spatially varying random material properties by combining stochastic finite element (SFE) models with Self-Optimizing Inverse Method (Self-OPTIM) and a new software framework that can simultaneously utilize any number of remote computers in a network domain for the Self-OPTIM simulation. This software framework offers efficiency of optimization as well as the ability for Self-OPTIM to easily interact with finite element analysis program. Because the optimization of complex models can be time comprehensive, a parallelization method will be created and used to dramatically reduce the time required for optimization. Two important issues in the inverse reconstruction problem are addressed in this paper: (1) effects of the number of internal measurements and (2) nonuniform reaction forces along the boundary on the reconstruction accuracy. The proposed method is partially proven to offer new capabilities of reconstructing spatially inhomogeneous material properties and estimating their statistical parameters from incomplete experimental measurements. 1.2 Random Field Modeling and Self-OPTIM Methodology In this paper, Self-OPTIM was integrated with stochastic finite element models for the purpose of characterizing the random distribution of material properties. For this method, the Karhunen-Loe`ve expansion (KLE) method was used to discretize the spatially varying random fields. This method produces two dimensional (2D) random fields with non-zero mean values. The 2D random field is broken down into deterministic and stochastic parts as follows: E x, θ ð Þ¼ E xð ÞþX M i¼1 ffiffiffifi λi p φi xð Þξi θð Þ ð1:1Þ where Mis the truncation number of KLE terms; x is the position vector over the domain; θ is the primitive randomness; E¯ is the mean value of the random variable; λi and φi are the eigenvalue and eigenfunction of an assumed covariance kernel (e.g. an exponential covariance kernel); and ξi is the statistically uncorrelated normal variable. Assuming that the random field is second-order homogeneous, E¯(x) is constant and therefore can be simplified as the mean value E¯. The Galerkin finite element approach is used to approximate the covariance kernel by the eigensolutions by discretizing the problem domain. Details on its formulation can be referred to [33]. This paper used the exponential covariance kernel expressed as 2 J.M. Weaver and G.J. Yun

Ci x1; x2 ð Þ¼σ 2 i exp x1 x2 j j lx y1 y2 j j ly ð 1:2Þ where lx and ly are the correlation length parameters in x and y directions, respectively; and σi 2 is the variance. Stochastic finite element models with spatial variations of material parameters are used in Self-OPTIM analysis. In Self-OPTIM, x= E, lx, ly, σ2, <ξ i>i =1, ::, k will be a set of unknown statistical parameters to be identified. Self-OPTIM runs two parallel finite element simulations under experimentally measured boundary forces and displacements, respectively. The variation of the full-field stress and strain values between the two simulations exhibit errors in the material parameters and are used in the objective function to determine the correct material parameters of the constitutive model. The problem in Self-OPTIM is a nonlinear unconstrained optimization problem as follows Minimize Π xð Þ¼ X n i¼1 RMSEε i F; D ð ÞþMAE ε i F; D ð Þ Rε i 2 F; D ð Þþ1 þX n i¼1 RMSEσ i F; D ð ÞþMAE σ i F; D ð Þ Rσ i 2 F; D ð Þþ1 ð 1:3Þ where R A; B ð Þ¼ Xn An A Bn B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn An A 2 Xn Bn B 2 r ð1:4Þ RMSE A; B ð Þ¼ Xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifffii n An Bn ð Þ 2 n s ð 1:5Þ MAE A; B ð Þ¼Xn An Bn j j n ð 1:6Þ The unconstrained nonlinear optimization problem of Self-OPTIM uses the implicit objective function in (1.3), which utilizes the stress and strain values obtained from finite element simulations of the two models; force driven and displacement driven analyses. The root mean square error (RMSE), mean absolute error (MAE) and correlation (R) between the stress and strains of the two models make up the objective function and are calculated as by (1.4–1.6). The stress and strains of the two models must be normalized before the statistical measures are calculated for the objective function. This allows the strain values to become equally valuable in the determination of the objective function. Normalization is done by determining the maximum and minimum values of the force and displacement based simulations for each component of stress and strain and then applying (1.7) to each component using those maximum and minimum values. This is done for each simulation of the force and displacement based models. Equation (1.7) also allows the normalization process to identify the high stress regions; whether they are positive or negative. The low stress regions can then be removed from the objective function calculation by using a cutoff value. For example, if a cutoff value of 0.2 is used, any stress value less than twenty percent of the maximum stress is removed from the calculation of the objective function. σij ¼ σij max σij max , σij min ð 1:7Þ In addition to (1.3), Self-OPTIM has the ability to incorporate displacements at internal degrees of freedom from sources such as Digital Image Correlation (DIC) in the objective function as expressed in (1.8). The additional objective functional is the statistical measure between the displacement of the force driven finite element simulation and the internal displacements from the experimental tests. This provides Self-OPTIM with the ability to optimize the material parameters more accurately. 1 Reconstruction of Spatially Varying Random Material Properties. . . 3

Minimize ΠDIC xð Þ¼ X n i¼1 RMSEε i F; D ð ÞþMAE ε i F; D ð Þ Rε i 2 F; D ð Þþ1 þX n i¼1 RMSEσ i F; D ð ÞþMAE σ i F; D ð Þ Rσ i 2 F; D ð Þþ1 þX n i¼1 RMSEUi F; DIC ð ÞþMAE Ui F, DIC ð Þ RUi 2 F; DIC ð Þþ1 ð1:8Þ In this paper, firefly algorithm is used to identify unknown parameters [34]. 1.3 Development of Self-OPTIM Software Framework The Self-OPTIM framework was created in the C# programming language and utilizes various components of the .NET framework to expedite the optimization process. The use of multiple threads and the ability to create asynchronous TCP/IP sockets to connect to remote clients prove invaluable to the optimization speed of Self-OPTIM. Another benefit to using the C# programming language is the ability to create a graphical user interface (GUI) which allows the user to effectively create any optimization run in a time efficient manner. The development of the Self-OPTIM framework was made in such a way that additional modules could easily be created to implement the Self-OPTIM procedure. Any module that implements the abstract base class of the Self-OPTIM framework shown in Fig. 1.1 can be used in the optimization process. For example, the primary FEA tool utilized by the Self-OPTIM framework at this time is ABAQUS. However, other third party FEA tools could also be implemented if desired. For this implementation several sub-modules have been created that utilize ABAQUS. The two primary ones are the Standard module, which utilize the input file, and the UMAT module, which uses a Fortran file for user defined materials. The final module that we will use in this paper is the Stochastic module which is a derivative of the UMAT module. The diagram in Fig. 1.1 shows the modules utilized in Self-OPTIM and provides an idea of how additional modules can be created. Note that a module needs to implement the abstract methods identified in the abstract Self-OPTIM class in order for it to be used in the framework. 1.3.1 Parallelization of Self-OPTIM Simulations Depending upon the scale of the model, a single analysis in ABAQUS can be quite demanding on a computer’s resources and can be time comprehensive. Since each firefly has a unique solution, every firefly must be analyzed using ABAQUS. The total number of times of ABAQUS analyses on the model is Nr=2 NF NIter where NF and NIter indicate the number of Fig. 1.1 Self-OPTIM module diagram 4 J.M. Weaver and G.J. Yun

fireflies and iterations set by the user. As a result of the number of runs that need to be implemented, the total time required to run an optimization using Self-OPTIM can be very large. Therefore, the implementation of multiple computers to run simulations was determined as a reasonable method to reduce the time requirements for optimization. A single computer could be considered the server as illustrated in Fig. 1.1 Self-OPTIM Module Diagram Fig. 1.2. Any additional computer that has Self-OPTIM installed on it can then connect to the server, establishing a connection that can be used to send the fireflies out from the server and receive the objective function value back from the client; each firefly contains a unique solution to the optimization problem. Any number of client computers can be connected to the server. Also, the server itself can also act like a client and receive fireflies and send back results using an IP loopback. The fireflies are sent asynchronously to the clients; meaning that as soon as a client is done with an analysis it will receive a new firefly to analyze. Depending upon the performance of the individual computers, the time required to optimize the material parameters can be reduced dramatically with additional computers. 1.3.2 Optimization Algorithm The firefly algorithm is used by Self-OPTIM as the optimization algorithm to determine the constitutive material parameters. The algorithm is based upon the characteristics of fireflies in the night sky. The light intensity of a flash of a specific firefly is directly related to the objective function that is to be optimized. The fireflies in the solution domain move with some randomness in the direction of the nearest firefly with a light intensity greater then itself. The fireflies will move around the domain until a local or global minimum is acquired [34]. The number of fireflies dictates how many unique solutions will be analyzed per iteration. The more fireflies that there are, the greater the possibility of determining the global optimum, or in this case the correct material parameters. The number of iterations determines how many times each firefly will be analyzed. The Alpha parameter dictates the randomness of the firefly’s movements. The value of Alpha must be between 0 and 1. Beta is another value that determines the characteristics of the firefly’s movements. Lastly, the light absorption coefficient Gamma characterizes the variation of the attractiveness of the fireflies through the sky. Essentially, this allows the light intensity of a firefly to decrease relative to another firefly based upon the distance they are from one another. This parameter is important in determining the speed at which the convergence of the optimization will take place. However, the faster the optimization takes place, the less likely that the global optimum is the returned solution. Fig. 1.2 Diagramof server–client interaction 1 Reconstruction of Spatially Varying Random Material Properties. . . 5

1.4 Reconstruction of Random Fields by Self-OPTIM 1.4.1 Effects of the Number of DIC Measurements To determine the effect of the number of DIC (Digital Image Correlation) points on the ability of Self-OPTIM to estimate the random field of the material strength parameter, a series of tests with varying number of DIC points was conducted. An 8 4 rectangular model was used with pinned restraints on the left edge of the model and the uniaxial loading applied on the right edge. The synthetic model was created using a fixed displacement and a synthetic reference random field distribution with a normalized median of 1, log normal variance of 0.05, and a correlation length of 5. The resulting forces were then used to create the corresponding force driven model for Self-OPTIM. For this specific test, the forces were taken at the nodal points and used to calculate a uniformly distributed load on the specimen. This will show that the Self-OPTIM can determine distribution of random fields even when more realistic boundary conditions are used. Using a series of DIC points ranging from 0 to 15, the effect of the ability of Self-OPTIM to obtain the correct random distribution could be obtained. The 15 DIC points that were used in this test can be seen in Fig. 1.3a. For a specific test, the DIC points used are equivalent to the number of points desired and all numbers below. For example, if 8 nodes are used in the experiment then nodes 1–8 are used. Five KLE terms (ξi, i=1, ::, 5) were included as unknown parameters. Using a series of DIC points ranging from 0 to 15, the effect of the ability of Self-OPTIM to obtain the correct random distribution could be obtained. In order to measure the errors in each of the tests, the Self-OPTIM objective function, as shown in (1.3), was used to calculate the errors between the simulated test and the identification model at each Gaussian point. In this way, the error in the random field could be quantified. It can be seen from Fig. 1.3b that a minimum of two DIC points is sufficient in this test for a semi-accurate result from Self-OPTIM. Using five or more DIC points resulted in the least amount of error between the reference and identification simulations. An error of 0.0132 was obtained when 5 DIC points were used. This indicates that Self-OPTIM can identify distributions of random fields from incomplete measurements. 1.4.2 Effect of Non-uniform Reaction Forces on Reconstruction of Random Fields In case of reconstruction of spatially inhomogeneous random fields, it is difficult to measure inhomogeneous reaction forces along the boundary. Simplification of non-uniform distribution of reaction forces to uniform distribution in the identification model needs to be addressed for practical applications of Self-OPTIM. It is worth noting that there will be negative effects on the identification results from uniform distribution of reaction forces with varying degrees of extent. However, negative Fig. 1.3 (a) Location of DIC points and reference distribution of elastic modulus (b) changes of random field error ( = RMSE rid; rre f þMAE rid; rre f = 1þR rid; rre f vs. the number of DIC points 6 J.M. Weaver and G.J. Yun

effects of discrepancies between FE model boundary and real-life boundary can be reduced by avoiding extraction of stress and strain fields away from the boundary. In addition to the above results with a varying number of DIC points, an additional test with the non-uniform reaction force obtained from the synthetic model and 15 DIC points was conducted. A plot of the uniform and non-uniform force distributions along the right edge of the model can be seen in Fig. 1.4. The purpose of this test is to show that with the synthetic force data a more accurate random field realization can be obtained even though convergence of the random field has already been established at five DIC points per Fig. 1.3b. Also, this shows that the results in Fig. 1.3b are truly optimized for real life situations. The resulting random field distribution can be seen in Fig. 1.5c and when compared to the reference distribution in Fig. 1.5a, it can be seen that the results are very accurate. The error produced between the test and reference random fields was calculated to be 0.0000569, which is significantly lower than the error of 0.00962 obtained when a uniform force distribution was used. 1.5 Conclusions In this paper, a networked parallel software framework was presented for the purpose of characterizing and reconstructing spatially varying random fields by Self-OPTIM methodology with incomplete measurements. The proposed software framework has benefits of significantly reducing computational times by using remote client computers connected through a local network. A global optimization algorithm called firefly algorithm was used. Spatially varying random fields were modeled by Karhunen-Loe`ve (KL) expansion method and used in the identification model within Self-OPTIM analysis. Fig. 1.4 Uniform and nonuniform force distribution along right edge of specimen Fig. 1.5 Effect of non-uniform reaction forces on reconstruction of elastic modulus by Self-OPTIM, (a) reference distribution, (b) reconstruction of elastic modulus with 15 DIC points and uniform reaction forces, (c) with 15 DIC points and non-uniform reaction forces 1 Reconstruction of Spatially Varying Random Material Properties. . . 7

Using the proposed software framework, two critical aspects in the inverse reconstruction problems were addressed. According to Self-OPTIM analysis results presented in this paper, the Self-OPTIM methodology could potentially reconstruct inhomogeneous distributions of material properties with incomplete full-field measurements and partial boundary force and displacement. It was shown that simplification of non-uniform distributions of the boundary forces can induce errors in the reconstruction results. The proposed method is partially proven to offer new capabilities of reconstructing spatially inhomogeneous material properties and estimating their statistical parameters from incomplete experimental measurements. References 1. R. Mahnken, A comprehensive study of a multiplicative elastoplasticity model coupled to damage including parameter identification. Comput. Struct. 74(2), 179–200 (2000) 2. R. 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28. M.M.A. Wahab, G. De Roeck, B. Peeters, Parameterization of damage in reinforced concrete structures using model updating. J. Sound Vib. 228(4), 717–730 (1999) 29. S. Adhikari, M.I. Friswell, Distributed parameter model updating using the karhunen-loeve expansion. Mech. Syst. Signal Process 24(2), 326–339 (2010) 30. S. Shang, Stochastic material characterization of heterogeneous media with randomly distributed material properties. Department of Civil Engineering 2012, The University of Akron: Doctoral Dissertation 31. G.J. Yun, L. Zhao, E. Iarve, Probabilistic mesh-independent discrete damage analyses of laminate composites. Comp. Sci. Tech. (2015) 32. S. Shang, G.J. Yun, Stochastic material characterization for spatially varying random macroscopic material properties by stochastic self-optimizng inverse method. Probab. Eng. Mech. (2014) 33. S. Shang, G.J. Yun, Stochastic finite element with material uncertainties: implementation in a general-purpose simulation program. Finite Elem. Anal. Des. 64, 65–78 (2013) 34. X.S. Yang, Firefly algorithms for multimodal optimization. Stochastic Algorithms Foundations Appl. Proc. 5792, 169–178 (2009) 1 Reconstruction of Spatially Varying Random Material Properties. . . 9

Chapter 2 Performance Assessment of Integrated Digital Image Correlation Versus FEM Updating A.P. Ruybalid, J.P.M. Hoefnagels, O. van der Sluis, and M.G.D. Geers Abstract Full-field identification methods can adequately identify constitutive material parameters, by combining Digital Image Correlation (DIC) with Finite Element (FE) simulation. It is known that interpolation within the DIC procedure is an important error source for DIC-results. In this study, the influence of these errors on the eventual identification results is investigated. Virtual experiments are conducted from which constitutive parameters are identified by two approaches: the commonly used method of Finite Element Model Updating (FEMU) and the more recent method of Integrated Digital Image Correlation (IDIC), in which the utilized interpolation functions are varied, and the influence on the identified parameters is investigated. It was found that image-interpolation has a significant effect on the accuracy of both methods. However, the observed differences in results between the two methods of FEMU and IDIC cannot be explained by interpolation errors. Keywords Digital Image Correlation (DIC) • Inverse parameter identification • Finite Element Method (FEM) • Interpolation errors • Integrated DIC • FEMU 2.1 Introduction To properly describe the mechanical behavior of materials, constitutive parameters must be identified, which is best done by using full-field kinematic data in the form of (microscopic) images of the deformation process, and combining simulation with in-situ experimentation. This is particularly interesting for the solid sate lighting industry where, ideally, dense, complex material stacks must be characterized from one test. The most intuitive and widely used full-field identification method is that of Finite Element Model Updating (FEMU) [1]. In this technique, parameters are optimized by comparing displacement fields from finite element (FE) simulation with measured displacement fields acquired through (subset-based) Digital Image Correlation (DIC) on experimental images containing speckle-patterns. A more recently developed method [2], termed Integrated Digital Image Correlation (IDIC), intimately integrates mechanical descriptions of a material with full-field measurements to identify model parameters. The method eliminates the need for calculating displacements from images before parameter identification can be realized. Instead, digital images are directly correlated by optimizing the mechanical parameters that govern the deformation of the imaged material. Mechanical knowledge drives the correlation procedure, and can be introduced to the problem through FE-simulation. In essence, the correlation procedure and identification procedure are integrated into a one-step approach, making it distinct from FEMU, which is a two-step approach in which post-processing of experimental images precedes the identification procedure. Overviews of both methods are shown in Figs. 2.1 and 2.2. 2.2 Systematic Error The dense material stacks in microelectronics only exhibit small displacements upon material or interface failure. Such fine kinematics make full-field, DIC-based identification methods prone to systematic errors that result in biased solutions. An important systematic error source results from inevitable interpolation steps needed in (1) the DIC procedure to correlate A.P. Ruybalid • J.P.M. Hoefnagels (*) • O. van der Sluis • M.G.D. Geers Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, Eindhoven 5600, The Netherlands e-mail: j.p.m.hoefnagels@tue.nl #The Society for Experimental Mechanics, Inc. 2016 S. Bossuyt et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 9, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-21765-9_2 11

images, and (2) transposing displacement fields from the FE-discretized space to the image pixel space (in case of IDIC) or the subset space of the measured displacement field (in case of FEMU). To illustrate this issue, a virtual tensile test, described later in the subsequent section, on a linear elastic, cubic orthotropic material, described by three known parameters, Ex, νxy, and Gxy, reveals the sensitivity of IDIC and FEMU to systematic errors, when initialized with the known, reference parameters, as shown in Fig. 2.3. The goal of this study is to further explore the influence of these systematic errors on the accuracy of the identification results of FEMU and IDIC. Fig. 2.1 Overview of the two-step method of FEMU, which consists of two iteration loops for (1) DIC and (2) identification of parameters Fig. 2.2 Overview of the one-step method of IDIC, which integrates DIC and the identification of parameters, resulting in one iteration loop 12 A.P. Ruybalid et al.

2.3 Virtual Experimentation In order to quantitatively assess the parameter errors caused by interpolation in IDIC and FEMU and exclude effects of errors that cannot be quantified, virtual experiments were conducted. In such an experiment, an artificial speckle pattern in reference image f x!; t 0 , is deformed by numerically simulated displacements, to produce subsequent images g x !; t . The altering speckle pattern represents a virtual material that mechanically deforms over time. The finite element method was used in this study to simulate the displacement fields by which the speckle patterns were deformed to generate images. The acquired images were subsequently used in IDIC and FEMU, with which the mechanical parameters that govern the virtual material’s deformation, were identified. Since the reference parameters used for virtual experimentation are known, the accuracy of parameters, identified by IDIC and FEMU, can be quantitatively assessed. Uniaxial tensile tests (2D plane stress simulations) were performed on a virtual tensile bar of 30 46 0.9mm3, with two circular notches with a radius of 11 mm, of which an illustration is shown in Fig. 2.4. The artificial speckle pattern was stored in an 8-bit, gray-valued image of 2048 1536 pixels, corresponding to an imaged region of 27 20 mm2. The pattern is built from random gray values drawn from three standard normal distributions with different widths that are superposed, establishing a combination of pattern features of 2, 18, and 150 pixels in width. The test-case corresponds to a cubic orthotropic, linear elastic material, which is described by three independent model parameters [3]; Young’s modulus Ex ¼130 GPa, Poisson’s ratioνxy ¼0.28, and the shear modulus Gxy ¼79.6GPa. The virtual tensile bar is loaded in x -direction by an applied horizontal tensile force of 2500 N. This results in sub-pixel displacements throughout the specimen with an average value of 0.64 pixel, and a maximum strain of 0.25 % in the center of the specimen (where the strain is largest due to the presence of the notches). Such small deformations, which are realistic in materials, such as silicon, used in the microelectronics industry, are challenging for the identification methods, since it puts high demands on the required resolution of the DIC method, which must be capable of capturing these fine kinematics. To focus the study on the influence of interpolation errors on the relative parameter errors for IDIC and FEMU, the iterative identification procedures are initialized with perfect initial guesses for these parameters, and noiseless images are used. Different choices are thereby used for image-interpolation and displacement field interpolation. For the former, cubic spline and linear functions are investigated, and for the latter, cubic and linear functions are tested. Fig. 2.3 The relative errors of the identified parameters for the IDIC and FEMU methods, initialized with perfect initial guesses for the parameters. When larger deformations are imposed during the virtual test, the relative parameter errors decrease, showing that systematic errors become especially influential in case of small displacements 2 Performance Assessment of Integrated Digital Image Correlation Versus FEM Updating 13

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