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Fabrication-aware machine learning for accuracy control in additive manufacturing

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Fabrication-aware machine learning for accuracy control in additive manufacturing

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Content FABRICATION-AWARE MACHINE LEARNING FOR ACCURACY CONTROL IN ADDITIVE MANUFACTURING by Yuanxiang Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (INDUSTRIAL AND SYSTEMS ENGINEERING) August 2022 Copyright 2022 Yuanxiang Wang To my wife Yezhou Teng. To my beloved parents. ii Acknowledgments This work could not have happened without the help, guidance, and support of many people. First and foremost, I would like to express my deepest appreciation to my advisor Prof. Qiang Huang. Thank you for your invaluable advice, continuous support, and endless patience during my Ph.D. study. Thank you for your strict requirements and challenging questions. None of the work would have been achieved without your willingness to spend countless time on the discussions and meetings. I would also like to extend my gratitude to my dissertation and qualifying committee mem- bers, Prof. Yong Chen, Prof. John Carlsson, and Prof. Meisam Razaviyayn. Thank you for all the enthusiasm, suggestions, and constructive criticisms. I would like to express my sincere thanks to all the professors that have guided and supported me. Thank you, Prof. Xicai Guo and Prof. Junzhan Yang, for showing me my passion for research, encouraging me, and offering wisdom all the time. Thank you, Prof. Rahul Jain, Prof. Cesar Acosta, and Prof. Sima Parisay, for supporting me and giving me great advice when it was most needed. Thank you, Prof. Sid Mohasseb, for encouraging me and being a sounding board to provide insights. Special thanks to Dr. Cesar Ruiz for spending a tremendous amount of time in conversations and revisions. Thank you for your willingness to help me and provide excellent advice. I also had the great pleasure of working with Nathan Decker, Mingdong Lyu, Chris Henson, Weizhi Lin, and Minghao Gu. Thank you for your accompany, support, and inspiring discussions. Thank you, Shelly, Grace, and Roxanna, for your help and hard work. iii Finally, I would like to give my deepest thanks to my parents and wife. Thank you for your constant support, encouragement, patience, and love. iv Table of Contents Dedication ii Acknowledgments iii List of Tables viii List of Figures ix Relevant Resources xiii Abstract xv 1 Introduction 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Quality Control in AM . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Research Challenges on Quality Control in AM . . . . . . . . . . . . . 5 1.2 State of the Art on Quality Control in AM . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Physics-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Statistical-Modeling Approaches . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Engineering-Informed ML Approaches . . . . . . . . . . . . . . . . . 11 1.3 Research Task and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 A Convolution Formulation for Learning and Predicting 3D Printing Shape Accu- racy 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 A Convolution Formulation of Layer-by-Layer Shape Generation . . . . . . . . 18 2.3 Description of Shape Deviationy(x) . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Identification off(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Identification of Transfer Functiong(x) – A Deconvolution Problem . . . . . . 25 2.5.1 Identifying Transfer Functiong(x) for Vertically Printed Half Disks . . 26 2.5.2 Identifying Transfer Functiong(x) for Domes . . . . . . . . . . . . . . 41 2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 v 3 Learning and Predicting Shape Deviations of Smooth and Non-Smooth 3D Geome- tries through Mathematical Decomposition of AM 52 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 A Unified Shape Deviation Modeling Approach for Smooth and Non-Smooth Geometries in AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 Mathematical Decomposition of AM through An Additive Model for Shape Deviation Modeling . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Convolution Framework as a Baseline for Smooth Shape Deviation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.3 Association Between Smooth and Non-smooth Geometries: 3D Cookie- Cutter Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.4 Spatial Correlation Modeling with a Novel Distance Metric for Hetero- geneous Shape Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Sequential Model Estimation Procedure for the United Modeling Framework . 65 3.4 Case Study: Shape Deviation Modeling and Estimation for Domes and Thin Walls 68 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 A Shape Registration Methodology for Geometric Deviation Correction in AM 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.1 Shape Deviation Decomposition . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 Overview of the Proposed Registration Method . . . . . . . . . . . . . 82 4.2.3 Ground Points Segmentation and Alignment . . . . . . . . . . . . . . 83 4.2.4 Control Chart for Layer Selection . . . . . . . . . . . . . . . . . . . . 86 4.2.5 Final Alignment by Constrained ICP . . . . . . . . . . . . . . . . . . . 88 4.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4.1 FDM Fabricated Tilted Thin Wall . . . . . . . . . . . . . . . . . . . . 91 4.4.2 Registration to the Straight Thin Wall . . . . . . . . . . . . . . . . . . 92 4.4.3 Registration to the Tilted Thin Wall . . . . . . . . . . . . . . . . . . . 93 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5 Small-Sample Learning of 3D Printed Thin-Wall Structures Using Printing Prim- itives 96 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 A Small-Sample Learning Strategy for Thin-Wall Structures Using Printing Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2.1 Geometric Primitives and Printing Primitives . . . . . . . . . . . . . . 98 5.2.2 Deviation Modeling for Thin-Wall Structures Using Printing Primitives in AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2.3 Tensor Basis Expansion for the Layer Interaction of Thin Walls . . . . 100 5.3 Case Study for Methodology Validation . . . . . . . . . . . . . . . . . . . . . 101 5.3.1 AM Experiments and Observations . . . . . . . . . . . . . . . . . . . 101 vi 5.3.2 Initial Shape Deviation Modeling through Equally-Spaced Geometric Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Shape Deviation Model Refinement and Extension to Printing Primi- tives of Thin Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6 Discussion and Future Work 108 Bibliography 112 vii List of Tables 1.1 Additive manufacturing processes categories [3] . . . . . . . . . . . . . . . . . 4 2.1 Specifications of SLA process and design parameters . . . . . . . . . . . . . . 28 2.2 Initial model estimation through MLE . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Sequential model refinement with conjectures onn(r 0 ) and (r 0 ) . . . . . . . 32 2.4 Sequential model refinement with conjectures onn(r 0 ), (r 0 ), and(r 0 ) . . . 33 2.5 Sequential model refinement with conjectures onn(r 0 ), (r 0 ),(r 0 ), and(r 0 ) 36 2.6 Initial MLE estimation without Gaussian process model term . . . . . . . . . . 46 3.1 Parameter estimates and standard error (SE) for the deviation of dome shapes . 72 3.2 Parameter estimates and standard error (SE) for cookie-cutter and high-order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Thin wall model performance applying different cookie-cutter functions . . . . 73 3.4 Model performance comparison . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.1 Initial parameter estimates for curved beams deviations . . . . . . . . . . . . . 105 viii List of Figures 1.1 Manufacturing paradigm shifts towards personalized production [2] . . . . . . 2 1.2 Basic principles of additive manufacturing [5] . . . . . . . . . . . . . . . . . . 3 1.3 Diagram of methods for accuracy control in AM . . . . . . . . . . . . . . . . . 6 1.4 Workflow of 2D and 3D shape deviation modeling . . . . . . . . . . . . . . . . 12 2.1 The layer-by-layer fabrication process and math integral . . . . . . . . . . . . 18 2.2 System inputs, transfer function, and convolution formulation . . . . . . . . . 19 2.3 2D shape deviation represented under the PCS . . . . . . . . . . . . . . . . . . 21 2.4 Point cloud data and representation of 3D shape deviation under the spherical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Identification off(x): horizontal 2D disk . . . . . . . . . . . . . . . . . . . . 24 2.6 Half disks built vertically and identification ofg(x) . . . . . . . . . . . . . . . 25 2.7 Four half disks and corresponding shape deviation profiles in the PCS . . . . . 29 2.8 Deviation profiles and model prediction for half disks built vertically: initial model fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.9 Area ratio between a half disk and the theoretical window . . . . . . . . . . . . 31 2.10 Sequential model refinement with conjectures onn(r 0 ) and (r 0 ) . . . . . . . 33 2.11 Estimates of(r 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Sequential model refinement with conjectures onn(r 0 ), (r 0 ) and(r 0 ) . . . . 35 2.13 Estimates of(r 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ix 2.14 Sequential model refinement with conjectures onn(r 0 ), (r 0 ),(r 0 ) and(r 0 ) 36 2.15 h 1 (2:0;'),h 2 (2:0;'), andh 3 (2:0;') . . . . . . . . . . . . . . . . . . . . . . . 37 2.16 h 1 (r 0 ;'),h 2 (r 0 ;'), andh 3 (r 0 ;') . . . . . . . . . . . . . . . . . . . . . . . . 38 2.17 SDG model prediction with Gaussian process regression of residuals . . . . . . 40 2.18 Dome shape and four domes printed in a SLA process . . . . . . . . . . . . . . 41 2.19 Shape deviation measurement of four domes presented in the SCS . . . . . . . 44 2.20 Measured shape deviation (black dots) and the SDG model prediction (blue and red dots) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.21 Measured shape deviation (black dots) and the updated SDG model prediction with GPR (blue and red dots) . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.22 Final model prediction of shape deviations against the measured shape deviations 49 2.23 h 1 (),h 2 (), andh 3 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.24 h 0 (r 0 ;');r 0 2 [0:5; 1:8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.25 h 0 h 1 (r 0 ;;'),h 0 h 2 (r 0 ; 0 ;'), andh 0 h 3 (r 0 ; 0 ;') . . . . . . . . . . . . . . . . 50 3.1 (a) Domes with 0.5, 0.8, 1.5 and 1.8 inches radii and (b) Thin walls with 0.8, 1.5 and 2.0 inches radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Shape deviation measurements of two dome and two thin walls with presented in SCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Mathematical decomposition of AM to additively build the smooth base shape (dome shape outside) and subtractively carve out non-smooth shapes with sharp corners such as (a) thin wall shape and (b) cuboid shape . . . . . . . . . . . . . 56 3.4 (a) A rectangle cut from its circumcircle and corresponding (b) 2D square-wave cookie-cutter function and (c) 2D sawtooth-wave cookie-cutter function . . . . 61 3.5 A thin wall fabricated by stacking rectangles . . . . . . . . . . . . . . . . . . . 62 3.6 (a) 3D square-wave function and (b) 3D sawtooth-wave function for the 0.8- inch thin wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 x 3.7 Proposed distance metric betweenx i on shapes i (in blue) andx j on shapes j (in black) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.8 Measured shape deviation (in gray) and model prediction (in blue) for domes . 70 3.9 Measured shape deviation (in gray), training set prediction (in blue) and valida- tion set prediction (in red) for thin walls applying (a and b) square wave and (c and d) sawtooth wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.10 Measured shape deviation (in gray), training set prediction (in blue) and valida- tion set prediction (in red) after GPR with Euclidean distance . . . . . . . . . . 73 3.11 Measured shape deviation (in gray), training set prediction (in blue) and valida- tion set prediction (in red) after GPR with the proposed distance . . . . . . . . 74 3.12 (a) Measured shape deviation, (b) predicted shape deviation, and (c) residuals for the 1.5-inch thin wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Quality assessment for two tilted thin walls with measurement in black and design in blue. Bottom portions within the red boxes have no distortion. . . . . 78 4.2 Proposed shape registration methodology . . . . . . . . . . . . . . . . . . . . 84 4.3 Points on the surface (in black) are segmented from ground points with the fitted plane (in green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Simulated thin walls with different types of global distortion . . . . . . . . . . 90 4.5 The tilted thin wall fabricated through FDM . . . . . . . . . . . . . . . . . . . 92 4.6 MEWMA control chart for the thin wall, where the red line is the control limit, and indices show detected out-of-control signals. . . . . . . . . . . . . . . . . 93 4.7 Shape deviations comparing to the straight thin wall . . . . . . . . . . . . . . . 93 4.8 Shape deviations comparing to the tilted thin wall . . . . . . . . . . . . . . . . 94 4.9 Shape deviations of the compensated thin wall . . . . . . . . . . . . . . . . . . 95 5.1 Fabrication of a curved wall by stacking curved beams . . . . . . . . . . . . . 97 5.2 Two curved beams and thin walls with different radii fabricated in a FDM process102 xi 5.3 Shape deviation profiles of two curved beams and two thin walls . . . . . . . . 103 5.4 (a) Initial model prediction of curved beams, where points are the measured deviation and curved line segments are the model predictions. (b) Actual and predicted shape deviation of two curved beams. Blocks of shape primitives are represented in different colors. . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.5 (a) Smoothed model prediction of curved beams, where points are the measured deviation and curved lines are the model predictions. (b) Actual and predicted shape deviations of two curved beams with 60-degree beam deviations in black and 120-degree beam deviations in blue. . . . . . . . . . . . . . . . . . . . . . 106 5.6 (a) and (b) Primitive-based convolution model prediction of two thin walls. (b) Actual and predicted shape deviation of two curved walls with 60-degree wall deviations in black and 120-degree wall deviations in blue. . . . . . . . . . . . 107 xii Relevant Resources This dissertation is adapted from the following conference and journal articles: • Wang, Y ., and Huang, Q., “Small-Sample Learning of 3D Printed Thin-Wall Structures Using Printing Primitives”, IEEE 18th International Conference on Automation Science and Engineering (CASE), August 20 - 24, 2022, Mexico City, Mexico. • Wang, Y . and Huang, Q., “A Primitive-Based Shape Deviation Modeling Approach for 3D Freeform Geometries in Additive Manufacturing,” (Drafting). • Wang, Y ., Ruiz, C., Park, S., Shin, K., Kim, J., and Huang, Q., “A Shape Registration Methodology for Geometric Deviation Correction in Additive Manufacturing”, ASME Manufacturing Science and Engineering Conference (MSEC), June 27 - July 1, 2022, West Lafayette, Indiana. • Wang, Y ., Ruiz, C., Park, S., Shin, K., Kim, J., and Huang, Q., “Consistent Shape Regis- tration for Geometric Accuracy Assessment in Additive Manufacturing,” (Drafting). • Wang, Y ., Ruiz, C., and Huang, Q., “A Mathematical Decomposition Approach for Learn- ing and Predicting Surface Deviations of Smooth and Non-Smooth 3D Shapes in Addi- tive Manufacturing”, IEEE Transactions on Automation Science and Engineering, DOI: 10.1109/TASE.2022.3174228, in press, 2022. • Wang, Y ., Ruiz, C., and Huang, Q., “Extended Fabrication-Aware Convolution Learning Framework for Predicting 3-D Shape Deformation in Additive Manufacturing”, IEEE xiii 17th International Conference on Automation Science and Engineering (CASE), August 23-27, 2021, Lyon, France. • Huang, Q., Wang, Y ., Lyu, M., and Lin, W., “Shape Deviation Generator – A Convolution Framework for Learning and Predicting 3-D Printing Shape Accuracy”, IEEE Transac- tions on Automation Science and Engineering, vol. 17, no. 3, pp. 1486-1500, 2020. xiv Abstract As a revolutionary technology, additive manufacturing (AM) or three-dimensional (3D) printing enables producing of personalized products with highly complex geometries through layer-by-layer fabrication using various materials, including metals, ceramics, polymers, and their composites, hybrid, or functionally graded materials. Unlike traditional subtracting man- ufacturing methods, such as milling, machining, carving, and shaping, AM has the potential to build extremely complex geometries with high efficiency and low material waste. However, one major barrier to the broader adoption of AM techniques is geometric shape inaccuracy. However, due to the vast spectrum of processes, high variety of geometries, and low vol- ume of samples, accuracy control has been a daunting task for researchers and practitioners. In this dissertation research, an engineering-informed machine learning (ML) methodology is pro- posed to learn and predict the shape deviation of 3D shapes from a limited number of fabricated products. The ultimate goal is to overcome the bottleneck of inadequate shape accuracy and to enable the compensation or calibration for general 3D printing systems with a few testing artifacts. To achieve this objective, the first task is to establish a statistical modeling framework to integrate the knowledge of in-plane and out-of-plane shape deviation modeling and to learn 3D shape deviation patterns. By characterizing the layer-by-layer manufacturing process through a mathematical integration and capturing the interlayer interactions through a transfer function, a convolution learning framework is established to describe the error accumulation mechanism in xv AM. Experimental validations are conducted with the out-of-plane shape deviation of vertical- printed half-disks and 3D shape deviation of domes printed in the stereolithography process. Process insights are derived to relate the model components to the input shape effect, layer interaction effect, and gravity effect. However, due to the smoothing effect of convolution operation, the proposed convolution learning framework cannot be applied to non-smooth 3D geometries since their deviation pat- terns contain sharp transitions at the edges and corners. To further extend the model to a wider category of 3D geometries, the production of a non-smooth geometry is decomposed into two steps. First, a smooth 3D shape circumscribing the target geometry is produced. Then, each layer of the non-smooth geometry is "carved" out from the corresponding layer of the smooth shape. 3D cookie-cutter function is proposed to capture the association of smooth and non- smooth shape deviations. A unified model for heterogeneous 3D geometries is established and validated through a sequential model estimation of the domes, as the stack of circular shapes, and thin walls of half-cylindrical shape, as the stack of rectangles. Before extending the methodology to 3D freeform geometries, a new shape registration strategy is proposed for effective product qualification and error correction of AM fabricated parts, because the efficacy of learning from shape deviation data relies on proper alignment between the printed product and its intended design. Robust to distortion of the products, a novel shape deviation decomposition is proposed, and statistical control chart is applied to filter the reliable portion of the data to sequentially constrain rigid transformation parameters to reveal the true deformation of the product. The last task is to model and predict the shape deviation of 3D freeform shapes from a limited number of training products. To address the small-sample learning problem, a set of printing primitives is employed to approximate and represent freeform 3D geometries. Follow- ing a similar idea to dimension reduction, the complex target geometry can be segmented or represented by a few types of printing primitives. Thus, it is sufficient to learn the deviation patterns of these printing primitives to model and predict any untried freeform shapes. One xvi choice of the primitives would include flat surfaces, spherical patches, and corners, which fall into the category of smooth or non-smooth shapes that have been studied. These four tasks build upon each other to establish a unified learning and prediction method- ology for shape deviations of arbitrary untried geometries from a limited number of 3D printed products. This would greatly help to relieve the burden of geometric shape inaccuracy in AM. Moreover, this strategy can be applied to any AM process with all kinds of materials and print- ing mechanisms, though transfer learning methods would be desired to enable efficient learning across multiple printing processes. xvii Chapter 1 Introduction This dissertation focuses on machine learning for quality control in additive manufacturing (AM) or three-dimensional (3D) printing, particularly geometric shape accuracy. The goal is to propose a fabrication-aware machine learning (ML) approach to learn and predict the shape deviation of untried geometries from a limited number of AM fabricated products. By characterizing the physical layer-by-layer 3D printing process as the realization of math- ematical integration, a convolution formulation is proposed to describe the deviation accumula- tion from the bottom up in AM. To jointly learn the shape deviation of heterogeneous geome- tries, a cookie-cutter modeling framework is developed to associate smooth and non-smooth shapes through a mathematical decomposition of the manufacturing process. As a challenging category of 3D freeform geometries, thin-wall structures possess a large amount of deviation and warpage due to thermal stresses. A shape registration strategy is established to identify the process-induced shape deviation in the presence of global distortion of the products. Then a small-sample learning strategy is proposed to decompose or segment each thin wall into a set of printing primitives that can be modeled with the training shapes. In the remainder of this chapter, we will first introduce the background of AM and the motivations and research challenges for quality control in additive manufacturing. Then the related literature is reviewed and summarized. Our tasks and objectives are illustrated in the end. 1 1.1 Background and Motivation 1.1.1 Quality Control in AM Driven by the increasing demands of product variability, Industry 4.0 or smart manufac- turing promises the efficient manufacturing of highly personalized products at dynamic batch sizes [1]. As shown in Fig. 1.1, the manufacturing paradigm shifted from mass production to mass personalization in the past century to meet diversified needs. Figure 1.1: Manufacturing paradigm shifts towards personalized production [2] As one of the critical components of Industry 4.0, additive manufacturing (AM), or three- dimensional (3D) printing, refers to the process of joining materials layer upon layer to build parts from digital 3D models [3]. As opposed to conventional manufacturing skewed toward mass production, AM is primarily targeted as small-scale customization and personalization by enabling the direct fabrication of extremely complicated products with high efficiency and low material waste [4]. 2 Figure 1.2 demonstrated a typical AM process with three steps: (1) Generate a computer- aided design (CAD) or 3D scan describing the surface geometry of an object as a digital model like the tessellation of triangles; (2) Slice the digital model into horizontal cross-sections and provide path planning and support structures; and (3) Fabricate the physical part with necessary post-processing. The final product could be measured later for quality assurance by comparing it to the CAD model. Figure 1.2: Basic principles of additive manufacturing [5] Evolving from a rapid prototyping technology to a vital industrial-production methodology, various AM techniques have been developed with different names, principles, materials, and printing mechanisms during the past few decades. American Society for Testing and Materials (ASTM) categorizes the typical methods into seven classes in Table 1.1. Although each process obsesses distinct features and printing mechanisms, they share a large number of benefits, including direct translation from design to product, zero additional 3 Table 1.1: Additive manufacturing processes categories [3] Process Categories Principle Material Distributor Binder jetting Multi-jet material printing Polymer Print head Directed energy deposition Deposition of material Metallic Deposition nozzle Material extrusion Extrusion of melted material Polymer Deposition nozzle Material jetting Multi-jet material printing Polymer Print head Powder bed fusion Fusion of particles in a powder bed Metallic, polymer and ceramic Powder bed Sheet lamination Fusion of stacked sheets Metallic and polymer Sheet stack Vat photopolymer- ization Light reactive photopolymer curing Polymer Vat cost for geometric complexity, great reduction in overall product development, manufacturing, and assembly time, low material waste, and excellent scalability [6, 7]. Despite the high potential and great progress made in recent years, AM faces several crucial challenges. First, materials development and evaluation are needed as the basis. The devel- opment of future AM materials requires the transition from the usage of existing materials, including polymers, metals, and ceramics, to materials with physically unusual properties or even unavailable properties in nature [8]. Second, the new designing methodology and stan- dards are required to fully explore the potential of AM. One successful example is topology optimization in AM, which obtains the best geometry to satisfy certain requirements on the manufactured parts [9]. Lastly, the widespread adoption of AM needs sophisticated tools for quality inspection, monitoring, control, and optimization [10]. Various quality characteristics have been studied, including dimensional accuracy, surface roughness, grain structure, porosity, residual stress, and cracks [11, 12]. This dissertation research focuses on the geometric shape accuracy with a more detailed description in Sec. 1.2.3. 4 1.1.2 Research Challenges on Quality Control in AM To control the quality of AM fabricated products, the challenges are three folds (i) Vast spectrum of processes. As shown in Table 1.1, each type of AM process applies a unique technique to bind and stack distinct materials and incurs diverse error kinetics like material phase change, thermal gradients, and interlayer interactions. Thus, realizing quality control for all categories of the processes requires substantial efforts in under- standing the complex physical and chemical mechanisms. (ii) High variety of geometries. To meet the needs of personalized products and to fully achieve the potential of AM, extremely complicated geometries are often fabricated to reach superior functional properties and shorten the assembly time. The infinite dimen- sion of shape space, together with various covariates, produces heterogeneous patterns and complex correlation structures. (iii) Low volume of samples. A key advantage of AM is the easy customization of the printed parts. This one-of-a-kind nature limits the applicability of classical statistical approaches and large-sample ML techniques for modeling product quality. With a limited number of training samples, it is challenging to establish a reasonable model, estimate the parame- ters, and validate the assumptions. To address these challenges, many efforts have been devoted to developing quality con- trol methods in AM, including statistical modeling techniques, first-principle-based simula- tion studies, process planning and optimization strategies, and in-situ monitoring and control approaches. In this dissertation research, a fabrication-aware ML approach is proposed and illustrated to learn heterogeneous shape deviation data and predict the geometric accuracy of untried shapes from a limited number of training products. 5 1.2 State of the Art on Quality Control in AM As summarized in Fig. 1.3, there are three main categories of accuracy control methods reported in the literature: physics-based approaches like simulation studies, statistical-modeling approaches applying statistical learning methods including ML, and engineering-informed ML approaches involving process insights into the statistical models [13–17]. Figure 1.3: Diagram of methods for accuracy control in AM 1.2.1 Physics-Based Approaches To understand the process-structure-property relationship in AM, simulation strategies based on the first principles have long been studied in the literature. One widely adopted method is finite element analysis (FEA) [18], which discretizes continuous domain into a finite num- ber of elements with simplified structures. Based on scales of the target, there are three main categories of simulation studies: micro-scale, macro-scale, and multi-scale models. For the micro-scale models focus on the powder evolution process and predict grains, voids, and powder surface roughness. For example, Gong and Chou [19] investigated the dendrite structure growth and solute concentration processes in the powder bed fusion process to uncover 6 the effect of scanning speed on the microstructures. Raghavan et al. [20] built a FEA-based ther- mal model to achieve the columnar to the equiaxed transition of the microstructure during the solidification process in the direct energy deposition process. By considering complex geo- metric shapes of the particles and their distributions and interactions, Parteli and Poschel [21] studied the load behavior to reveal the relationship between coating speed and surface roughness of the power bed used in AM. Yan et al. [22] employed the Monte Carlo method to simulate the collisions among the high-speed electrons and material atoms and reveal the effect of incidence angles in a directed energy deposition process. With the development of thermal sensors, a growing body of literature seeks to model the thermal behavior at the product level, including temperature profile and molten pool character- istics. By studying the spatial and temporal distribution of the temperature field, it was reported in a few works [23–25] that the melt pool dimension is increased with lower scanning speed and higher beam or laser power in the powder bed fusion process. Huang et al. [26] related the molten pool dimension to the types of defects, including lack of fusion, balling, and keyholing, so that a defect-free region in the design space was revealed. Other common subjects in macro- scale simulation studies are residual stress, and surface roughness, and dimensional distortions. D’Amico and Peterson [27] established a simulation model for heat transfer in material extru- sion AM and found that the cooling rate first increases and then decreases with the increasing printing speed, which would further impact the residual stresses and mechanical properties. To accurately simulate the dimensional displacement of bridge parts in a fused deposition model- ing (FDM) process, Cattenone et al. [28] adopted thermal and mechanical analysis sequentially to adjust the mesh size, material model, and time step. Garg and Bhattacharya [29] found that the tensile strength first decreases with layer thickness and then increases by studying the stan- dard tensile test specimens in an FDM process. A quadratic relationship between power layer roughness and deposition speed has been found. 7 To reveal the relationship among process, structure, and property in AM, King et al. [30,31] proposed a multi-scale modeling approach based on the comprehensive physical understand- ing of both the powder scale and the part scale to predict shape accuracy, residual stress, and material properties of the powder bed fusion process. Yan et al. [32] proposed a micro-scale model for electron-material interaction, a mesoscale model for powder particle evolution, and a macro-scale model using FEA. These models can achieve voxel-level accuracy for describ- ing 3D products along the printing process but at a high computational cost. By modeling the inherent strains in the micro-scale and activating meta-layers sequentially with a constant coef- ficient of thermal expansion, Chen et al. [33] simulated the part-scale dimensional distortion and residual stress in the powder bed fusion process. Though understanding the physical kinetics is critical to achieving reliable quality control in AM, developing such models requires extensive expert knowledge and efforts, and the models are often restricted to a specific type of AM processes and materials. Moreover, there is always a balance between computational complexity and modeling accuracy, which restricted its usage in practice. Currently, simulation modeling of the end-product size is still intractable. 1.2.2 Statistical-Modeling Approaches Other than building a complete model over the entire printing process and relying heavily on the physical knowledge, many works focus on exploring the connections between process parameters and part quality characteristics through statistical models. Based on the data source and application objective, there are two categories: (1) offline process optimization and (2) online process monitoring and control. Offline Process Optimization Regression models were mostly applied in the early stage of the statistical-modeling approach. Assuming proportional relationships between process parameters, including laser 8 power, layer pitch, scan pitch, scanning speed, laser stability, and material absorption rate, Wang et al. [34] used the least-squares method to predict the part shrinkage in the powder bed fusion process. Wang [35] built a linear regression model to calculate the optimal beam offset applied to each direction so that the part shrinkage is calibrated. Using an additive model with polynomial bases to approximate the directional deviation, Tong et al. [36,37] proposed to com- pensate the CAD design in each direction independently to control the dimensional accuracy. Design of experiments techniques, including the Taguchi method, factorial experiments, response surface modeling, and Analysis of Variance (ANOV A), were popular choices among many researchers. Zhou et al. [38] adopted the Taguchi method and then response surface modeling and ANOV A table to decide the optimal settings of process parameters, including layer thickness, hatch spacing, over-cure depth, etc. to control the quality and accuracy in the stereolithography process. Grey Taguchi method, main effect and factor effect plots, and ANOV A table were applied by Sood et al. [39] to investigate the influence of layer thickness, part orientation, and raster angle on the dimensional accuracy of the fused deposition process. With carefully defined objective functions and constraints, optimization techniques can be employed to achieve better printing quality. For example, Zhou et al. [40] formulated pixel blending as a two-stage linear programming problem and found the optimal light intensity for mask image projection in the stereolithography process. Simulation and experimental studies showed better dimensional accuracy and surface quality than traditional binary mask images. With the development of machine learning strategies and increasing computational capa- bility, various thermal and optical data were employed to learn and predict the quality in AM. Francis and Bian [41] applied deep learning techniques to predict the top surface distortion of disks through a combination of convolution neural network and artificial neural network using thermal history data in the direct energy deposition process. To enable learning from different shapes, Decker et al. [42] employed a random forest method to predict the shape deviation of an untried shape by extracting position, geometry, and material expansion features from the triangular meshes of other shapes in an FDM process. 9 A growing body of efforts has been devoted to enabling model transfer across materials, fabrication processes, and shapes to improve prediction accuracy with a limited number of printed parts. Francis et al. [43] accomplished the model transfer from Ti-6Al-4V to 316L stainless steel through the effect equivalence framework proposed by Sabbaghi and Huang [44], in which the same deviation pattern can be observed in different printing processes by imposing a specific compensation plan over the product designs. Chen et al. [45] achieved the knowledge transfer across different shapes by decomposing the geometric error into shape-independent and shape-specific components and then fixing the global shape-independent parameters and shape features, respectively. Ferreira et al. [46] adopted a Bayesian extreme learning machine to auto- matically learn and predict the shape deviation pattern of 2D freeform shapes under different printing processes by using knowledge of the shape deviation patterns of simpler shapes printed under the same and different processes. By establishing a connection between process parameters and quality characteristics, offline process optimization can effectively model, predict, and compensate for the shape deviation. However, the proposed approach is mostly limited to a particular printing process and mecha- nism. Great efforts are needed to transfer the knowledge to untried shapes, materials, or fabri- cation techniques. Online Process Monitoring and Control To fully exploit the potential of AM with reduced material waste and the ability to produce parts directly without the need for extra tooling or post-processing, online process monitoring and control have attracted more attention recently [11]. Three main types of sensors adopted for in situ monitoring are optics, thermography, and acoustics. To improve the geometric shape accuracy, Hu et al. [47] employed an optoelectronic sensor to monitor the powder delivery rate and an infrared camera for the molten pool so that both powder and heat input can be controlled in a closed-loop in the powder bed fusion process. By using a laser scanner to obtain height profiles after each deposited layer, Heralic et al. [48] adopted an iterative procedure to 10 monitor the height deviation for each layer and control the wire feed rate to ensure consistent quality and sufficient meting of the wire. Wang et al. [49] proposed to apply a convolutional neural network on the multi-scale feature maps acquired from optical cameras to detect and predict voids and textures of the powder bed fusion process. By extracting molten pool features from the pyrometer and feeding it into a support vector machine, Smoqi et al. [50] achieved monitoring and prediction of product porosity levels in the powder bed fusion process. Wu et al. [51] monitored the acoustic emission signals of an FDM process to detect failure printing in real time. Multiple sensors could be employed simultaneously to acquire more information on the printing process. For instance, to integrate heterogeneous sensor data from multiple sources in an FDM process, Rao et al. [52] proposed to model each sensor with a non-parametric Bayesian Dirichlet process and then combined them using the Dempster-Shafer evidence theo- retic approach to detect the process shifts, which successfully classified the process into normal, abnormal, and failure categories. 1.2.3 Engineering-Informed ML Approaches By embedding a physical understanding of the process and products, engineering-informed machine learning approaches aim to model and predict shape deviation of untried shapes using a limited number of training samples and construct compensation plans to improve the geometric accuracy in AM [53]. This strategy has been successfully implemented for thin (in height) shapes that can be approximated as 2D shapes for purposes of shape deviation modeling. Huang et al. [13] developed a prescriptive statistical modeling approach to predict and com- pensate for the in-plane shape deviation of circular shapes considering the product size effect and machine over-exposure, which serves as a basis for a series of the following work. A key challenge to model polygonal shapes is that the deviation pattern changes dramatically on 11 the sharp corners. To transfer the knowledge from circular shape deviations to that of regu- lar polygons, Huang et al. [14] proposed the use of a cookie-cutter function to consider the polygon shape as being cut off from its circumscribed disks. With a unified model on circular and polygonal shapes, these two types of patches can be adopted to approximate or represent 2D freeform geometries. By connecting the deviation pattern of circular patches, Luan and Huang [15] established an in-plane shape deviation model for 2D freeform products and applied the compensation plan that modifies CAD design to control geometric inaccuracy. The modeling and control issues faced in printing 2D shapes are exacerbated for 3D cases since the deformation pattern can change from layer to layer due to complex process physics such as inter-layer interactions [54–57]. This dissertation research aims to extend the work from 2D to 3D, particularly for spherical and planar patches. The main reason to study these partic- ular shapes is that 3D freeform geometries can be approximated by these components. Thus, understanding of these two basic categories of 3D shapes makes it possible for 3D freeform shape prediction, following a similar workflow for 2D case [13–15]. Fig. 1.4 summarizes our previous works and the research tasks for this dissertation research. Figure 1.4: Workflow of 2D and 3D shape deviation modeling 12 1.3 Research Task and Objectives The objective of this dissertation research is to propose a fabrication-aware ML approach to model and predict the geometric shape deviation of untried AM fabricated products from limited samples of heterogeneous shapes. The four proposed tasks demonstrate the idea of incorporating engineering insights into the statistical models to achieve product qualification and deviation prediction in AM. 2D shape deviation modeling has been developed by Huang and co-authors as in-plane deviations [13–15] and out-of-plane deviation [54–57]. The first task aims to propose a unified prescriptive modeling framework for both 2D and 3D geometries, which is extended by the second task to a broader category of 3D shapes. Facing the challenge of accurate and efficient identification of shape deviation for freeform shapes, particularly for thin-wall structures, the third task focuses on consistent quality assessment in AM. The last task addresses the small- sample learning problem so that shape deviations of freeform shapes can be learned from a small batch of training samples. (i) Predicting smooth 3D shape deviations through a convolution formulation. This task aims to propose a novel ML strategy to model and learn the shape deviation of 3D shapes. Under the convolution learning framework, the previous work on horizontally and vertically built products can be fully extended to 3D cases. By characterizing the layer-by-layer manufacturing process through a mathematical integration, a convolution formulation was proposed to describe the error accumulation mechanism in AM. Specif- ically, shape deviations of smooth 3D geometries can be regarded as the stack-up of in- plane deviations of 2D horizontal layers, which provides a consistent description of 2D and 3D shape formation processes. Three components of the convolution formulation are identified: (1) description of 3D shape deviation in the spherical coordinate system, (2) input function that describes the in-plane shape deviation, and (3) transfer function that captures the interlayer interactions. Moreover, additive components of the convolution 13 model can be associated with input shape, layer interaction, and gravity effects to derive insights into the fabrication process. (ii) Predicting smooth and non-smooth 3D shape deviations through a mathematical decom- position of the manufacturing process. The goal of this task is to extend the convolution learning framework to a broader category of geometries by constructively incorporating smooth and non-smooth shapes into a unified model. Unlike smooth shapes, non-smooth 3D geometries contain edges and corners that are often deformed or distorted during the 3D printing process. Since convolution is a weighted moving average, sharp increments in the deviations of non-smooth geometries cannot be captured through the convolution formulation. To establish a unified model for heterogeneous shapes, the production of a non-smooth geometry is decomposed into two steps. First, a smooth 3D shape circum- scribing the target geometry is produced. Then, each layer of the non-smooth geometry is "carved" out from the corresponding layer of the smooth shape. 3D cookie-cutter function is proposed to capture the association of smooth and non-smooth shape deviations. (iii) Effective shape registration for error correction in AM The task put forth a new shape registration method for effective product qualification and error correction of AM fabri- cated parts. The efficacy of learning from shape deviation data relies on proper alignment between the printed product and its intended design. However, the commonly adopted registration procedures are sensitive to local shape deviation and global distortion of the final product. This leads to incorrect assessments of the shape deviation patterns and makes modeling and improvement efforts futile. To establish the shape registration method that is robust to distortions, a novel shape deviation decomposition is proposed, and statistical control chart is applied to filter the reliable portion of the data to sequen- tially constrain rigid transformation parameters to reveal the true deformation of the prod- uct. 14 (iv) Predicting freeform 3D shape deviations through printing primitives. The goal of this task is to model and predict the shape deviation of 3D freeform products with a prescriptive learning strategy independent of geometric and process complexities. Due to the one-of- a-kind nature of AM and the trend of personalized manufacturing, a very small volume of parts are often fabricated with a large variety of 3D geometries. It leads to a challenging small-sample learning problem. To efficiently learn the deviation pattern from a limited number of samples, a set of printing primitives is employed to approximate freeform 3D geometries. Following a similar idea to dimension reduction, the complex target geometry can be segmented or represented by a few types of printing primitives. Thus, it is sufficient to learn the deviation patterns of these printing primitives to model and predict any untried freeform shapes. One choice of the primitives would include flat surfaces, spherical patches, and corners, which fall into the category of smooth or non- smooth shapes that have been studied. These four tasks would work together to achieve the learning and prediction of arbitrary untried 3D shapes from a limited number of samples. This would greatly help to relieve the burden of geometric shape inaccuracy in AM. Moreover, this strategy can be applied to any AM process with all kinds of materials and printing mechanisms, though transfer learning methods would be desired to enable efficient learning across multiple printing processes. Once carefully designed shapes are printed and measured in one build plate, my research will provide knowledge and insights on shape deviation patterns. Applying the optimal compen- sation theory [53] on the predictive model would generate a compensation plan that minimizes the shape deviation of any shape and achieve the auto-calibration of the machine without chang- ing process parameters. The developed methodologies will highly contribute to the knowledge base of AI and ML for advanced manufacturing. 15 Chapter 2 A Convolution Formulation for Learning and Predicting 3D Printing Shape Accuracy 2.1 Introduction A series of works [13–15,53,56,58–61] establish a prescriptive approach to model, predict, and compensate 2D shape deviations based on geometric measurement of AM built products. In these studies, models are learned from a limited number of tested shapes, and optimal com- pensation plans for new and untried products are derived and validated. Although geometric accuracy control in AM involves 3D shapes [62–64], there is a critical lack of major progress on learning 3D shape data for improving 3D printing accuracy. Describ- ing the 3D shape formation through the layer-by-layer fabrication process has been a daunting task. Physics-based modeling and simulation approaches present voxel-level description of the 3D object formation from points to lines, lines to surfaces, and surfaces to 3D shapes [31, 65]. However, this computationally intensive modeling framework does not provide a clear structure or framework for machine learning of AM simulation or measurement data. The predominant strategy is to obtain physical understandings of critical process variables, such as tempera- ture field and melt pool geometry, as proxies of product quality through their correlation with product geometries [30, 66–68]. Since the understanding of 3D shape formation in AM is the foundation for the subsequent accuracy control activities such as compensation, it is imperative 16 to establish a data-analytical framework that not only provides an insightful description of 3D shape formation in AM, but also enables machine learning of simulation or measurement data. It is worthy of noting the recent ML advances in deep representation learning that aims to synthesize and reconstruct realistic and novel 3D shapes [69–71]. Various shape representa- tion methods such as meshes, skeletons, pre-training shape templates, voxel grids, multi-view images, point clouds or surface patches have been developed to generate digital 3D objects with less issues of low-resolution outputs, overly smoothed or discontinuous surfaces, and topologi- cal irregularities [71]. By contrast, digital 3D models in this study are given from engineering design. Our objective is to learn and predict how the physical 3D shapes are generated and built in AM processes and more importantly, how the physical 3D objects are different from their digital counterparts. As can be found out in this study, complication and variations in physical AM processes will generate different deformation patterns even for the same shapes. This has been outside the scope of consideration for the general ML community. For the purpose of predicting, learning, and compensating 3D shape deviations based on data, we propose Shape Deviation Generator (SDG), a data-analytical framework to facilitate the learning and prediction of 3D printing shape accuracy. Following the Introduction, Sec. 2.2 introduces a new convolution formulation of the 3D shape deviation generation process and defines a set of research problems. Section 2.3 addresses the issue of shape deviation repre- sentation. Section 2.4 derives the individual layer input function for the convolution integral. Section 2.5 formulates a deconvolution problem to identify transfer function which captures the inter-layer interaction and error accumulation effects in the layer-by-layer fabrication pro- cesses. A physics-informed sequential model estimation is developed in Sec. 2.5 to fully estab- lish the SDG models. Gaussian process regression is adopted to capture spatial correlation in the residuals. The printed 2D and 3D shapes via a stereolithography (SLA) process are used to demonstrate the proposed modeling framework. Summary and conclusions are given in Sec. 2.6. 17 2.2 A Convolution Formulation of Layer-by-Layer Shape Generation This section will first introduce the rationale of deriving a convolution formulation to describe the 3D shape deviation generation process. This formulation leads to the definition of a set of sub problems suitable for statistical or machine learning methods. Applicable to a wide variety of AM processes, the 3D SDG formulation is illustrated with examples from SLA processes. The layer-by-layer fabrication process and mathematical integration: Since a typical AM process builds 3D objects layer by layer (Fig. 2.1), a mathematical formulation of the shape generation in AM can naturally consider the integration along the direction of layer build-up. Equivalently, the layer-by-layer fabrication process can be viewed as a physical realization of a mathematical integration over a closed interval. Then the key question is what is the proper form of the integrand. 1 2 3 k k+1 N Build bed platform Build plate Built layers Build materials Layers to be built o x x 1 x 2 f(x) Figure 2.1: The layer-by-layer fabrication process and math integral Individual layers and transfer function: In the layer-by-layer shape formation process, individual layers can be viewed as inputs f(k) 0 s to a system with transfer function g(k) 0 s, k = 1; 2;:::;N layers. The transfer functiong(k) not only processes the input layer at the cur- rent layerk, but also modify previous layers due to the heat exchange and thermal stress between layers. Therefore, there is a “time shift" involved in the transfer function and an “accumulation" 18 of interactions leading to the final shape formation, as illustrated in Fig. 2.2. A natural integral formulation that reflects the concepts of transfer function, time shift, and accumulation effects is the convolution integral. Convolution integral for SDG in AM: We therefore propose the following convolution formu- lation for the AM SDG: y(x) = (fg)(x) + (2.1) where y(x) represents the shape deviation (1D curves or 2D surfaces for 2D or 3D shapes, respectively),x are parameters that describe the shape deviation, and is the model error term that may contain spatial correlations. g(k), g(k-1), …, g(1) 1 2 3 k-1 k … f(k) f(1) Convolution formulation Transfer function Figure 2.2: System inputs, transfer function, and convolution formulation Boundary conditions of SDG: Two boundary conditions are required to satisfy engineering constraints and limit the scope of parameter optimization and selection. B1. Ifx =r(;')! 0,y(x)! 0. If the size of a built part tends to zero, the shape deviation approaches zero as well. The system defined in Eq. (2.1) is therefore a linear causal system. B2. Ifx =r(;')!1,y(x) is bounded. Even if the size of the built part tends to infinity or becomes very large, the shape deviation of the part has to be bounded because of physical constraints on the materials and process. 19 Relation with State Space Representation of AM: Another intuitive formulation of the AM process is the state space equation with the time indexk being the layer index: x(k) = Ax(k 1) +Bu(k) y(k) = Cx(k) +Du(k) wherex(k) represents the state vector of thekth layer, andy(k) is the system response. The solution to the state space equation through the Laplace transformation is Y(s) = [D +C(sIA) 1 B]U(s). By defining the transfer function G(s) = D +C(sIA) 1 B and conducting the inverse Laplace transformation ofY(s) =G(s)U(s), we also end up with a convolution formulation consistent with Eq. (2.1): y(k) = (gu)(k) Compared to the state space model, the convolution model (2.1) has less unknown terms to be identified and learned. The compromise, however, is that the convolution model (2.1) does not enable the dynamic system representation and realtime feedback control. Note that under this convolution formulation, f(x) and g(x) represent the features of 3D shape deviation and the characteristics of the layer-by-layer fabrication process. This provides opportunities for developing change detection and correction procedures. Given the SDG formulation in model (2.1), the research problems involved in realizing the convolution formulation are: • Description of shape deviationy(x) • Identification and learning off(x) • Identification and learning of transfer functiong(x) which are to be discussed in the subsequent sections. 20 2.3 Description of Shape Deviationy(x) The system responsey represents the 3D shape (boundary) deviation of an AM built object. The measurement data of a 3D object is usually in the form of point cloud data defined in the Cartesian Coordinate System (CCS). Shape description or representation has been exten- sively investigated in the field of computer vision. It is the first stage of shape analysis such as shape matching, shape deformation, shape correspondence and shape registration for 3D objects [72–75]. Depending on applications, shape representation generally looks for effec- tive and perceptually important shape features such as point sets, curves, surfaces, level sets or deformable templates [76, 77]. For accuracy control of AM built products, the contour-based feature representations such as continuous and discrete approaches are more relevant. A feature vector derived from the integral boundary, e.g., Fourier descriptors [78], is typically applied among continuous approaches. Discrete approaches tend to break the shape boundary into seg- ments for approximation [79]. Representing a shape as diffemorphism of the unit circle to itself through conformal mapping has been a popular method for shape classification [80]. Invariant to translation and scaling, this representation approach allows one to move back and forth compu- tationally between shapes and their diffeomorphisms. Popular 3D shape representations include medial and skeletal representation, deformable templates, and geometric descriptors [81]. 0 1 2 3 4 5 6 −0.05 −0.04 −0.03 −0.02 −0.01 0.00 Observed Deformation θ Deformation (in.) 0.5'' 1'' 2'' 3'' −0.005 0.000 0.005 0.010 0.015 0.020 0.025 Regular Pentagon Deviation Profiles: R=1in & R=3in θ Shape Deviation (unit:inch) 0 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π R = 1in R = 3in 3 inches 1 inch −0.04 −0.03 −0.02 −0.01 0.00 0.01 Regular Square Shape Shrinkage θ Shape Deviation (unit:inch) 0 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π Side=1in Side=2in Side=3in Figure 2.3: 2D shape deviation represented under the PCS 21 In AM literatures the shape deviation representation methods can be generally classified in two categories: global summary and feature extraction of shape deviation, and detailed char- acterization of whole shape deviations. The first category includes volumetric shrinkage to capture overall size distortion [35, 82] and dimensional and form errors [83–85] to character- ize feature distortion of specific shapes. To enable detailed description of shape deviations for arbitrary shapes in AM, the second category provides three main methods: point-wise deviation representation [36,37,86,87], transformation of point cloud [53,56], and mesh-based deviation representation [88, 89]. Since the shape representation in AM should facilitate the development of a generic model for quality prediction of arbitrary shapes, one key consideration of describing the system response or the shape deviationy(x) is to decouple the geometric shape complexity from the modeling ofy(x). We transform the 2D shape deviations under the CCS into deviation profiles in the Polar Coordinate System (PCS) [13], and 3D shape deviations under the CCS into devia- tion surfaces in the Spherical Coordinate System (SCS) [53, 56]. This representation approach is closely related to those in computer vision, e.g., spherical harmonic description [81,90]. The main difference here is that we focus on shape deviations, as opposed to shape itself because specific shapes are known by design. For 2D shape deviations, the system responsey(x) is defined as y(x) =r()r 0 () (2.2) wherer() andr 0 () is the measured shape and design shape represented in the PCS, respec- tively. Herex = r 0 (); . Examples of deviation profiles for various 2D shapes can be found in Fig. 2.3. For 3D shapes, point cloud data under the CCS is transformed into the observed radial distancer(;') for a point with polar angle and azimuth angle'. The system responsey is 22 Figure 2.4: Point cloud data and representation of 3D shape deviation under the spherical coor- dinate system defined as the difference between measurementr(;') and the nominal design radial distance r 0 (;'), i.e., y(x) =r(;')r 0 (;') (2.3) wherex = r 0 (;');;' . Figure 2.4 shows a dome shape built by a SLA machine, its point cloud data generated by a laser scanner, and its shape deviation y under the SCS. The shape deviation pattern is clearly visible after the transformation, which facilitates the modeling. 2.4 Identification off(x) To identify the proper form of individual layer inputsf(x), we first consider a simplest case of building a single-layer horizontal disk as shown in Fig. 2.5. If we further assume that the transfer function g(x) only varies along the z direction, i.e., g(x) = g(z) or g('), then g(') only has definition atz 0 or' 0 because of the layer thickness along thez direction. 23 z x o o g() () Figure 2.5: Identification off(x): horizontal 2D disk One general form of this type of functions is Dirac’s delta function, or(x). Under the SDG formulation in Eq.(2.1) and the property of convolving with a delta function, we have (f)(x) =f(x) (2.4) Together with Eq.(2.1), Eq.(2.4) implies that individual layer inputf(x) essentially repre- sents the deviation of the 2D horizontal plate (inxy plane) with a shape defined by the input design. Note thatf(x) is not affected by inter-layer interactions. This also suggests an experi- mental design strategy for establishing the SDG defined in Eq.(2.1), i.e., horizontal plates with selected 2D shapes should be built first to understandf(x). One example off(r 0 ();) for cylindrical disks built by a SLA process is f r 0 (); =c 1 (r 0 ) +c 2 (r 0 ) cos(2) where coefficientsc 1 (r 0 ) andc 2 (r 0 ) can be obtained through estimation based on data shown in Fig. 2.3 (left panel for cylindrical disks with various sizes) [13]. Learning f(r 0 ();) for 2D freeform shapes from a small set of training shapes has been reported in [13–15,58,60,91]. Note that different statistical and machine learning methods can be introduced to obtainf(x) in general. 24 2.5 Identification of Transfer Functiong(x) – A Deconvolu- tion Problem The layer buildup along the z direction incurs the inter-layer interaction problem in AM. Complete inter-layer bonding implies full density and continuous interface between succes- sively fabricated layers, while poor bonding may cause delamination and defects like balling or beading across a layer [92]. Li and Gu [25] find that the heat accumulation effect and the remelt- ing phenomenon due to laser energy penetration will lead to the increase of the temperature and size of melt pools when the laser beam moves from bottom to top layers. Denlinger et al. [93] decomposes the overall distortion along thez direction to individual layers through in situ dis- tortion measurement and experimentally investigates the relationship between the inter-layer dwell time and accumulation of distortions for different materials. o o g() Figure 2.6: Half disks built vertically and identification ofg(x) The transfer function g(x) in the convolution formulation (2.1) intends to provide a data- analytical description of the effect of inter-layer interactions and error accumulation on build accuracy. For simplicity of methodology illustration and development, we first consider a simple case of building half disks along thez direction (Fig. 2.6), where theg(x) ends up as a univariate 25 function assigning weights for individual layer inputs. Then we will extend to a 3D case of building domes with various sizes to model the multivariate transfer functiong(x). 2.5.1 Identifying Transfer Functiong(x) for Vertically Printed Half Disks Since the half disk can be approximated as a 2D shape, Eq. (2.1) can be simply written as y(') = (fg)(') +. Here we introduce the size of the disk, i.e., nominal radiusr 0 , as an additional covariate because the size is directly proportional to the number of layers. The SDG model is rewritten as: y(r 0 ;') =(r 0 )(fg)(r 0 ;') +(r 0 ) + =(r 0 ) Z ' 0 f(r 0 ;)g(r 0 ;')d +(r 0 ) + (2.5) where(r 0 ) and(r 0 ) are scaling and location parameters depending onr 0 . Function f(r 0 ;') has been identified to be the deviation function of the same shape built horizontally (Eq. 2.4). Study has to be done first to obtainf(r 0 ;'). Givenf(r 0 ;') and mea- surement datay(r 0 ;'), identifyingg(x) is a classical deconvolution problem [94]. Signal pro- cessing, statistical model estimation, and machine learning (including neural networks) can be applied to address the deconvolution problem in this context. Here we present a model-informed estimation approach. Normalizingf(x) andg(x) for convolution integral in SDG First, by following the results in [13] for SLA processes, we can takef(r 0 ;') = c 1 (r 0 ) + c 2 (r 0 ) cos(2'). Since coefficients c 1 (r 0 ) andc 2 (r 0 ) relating to sizer 0 can be absorbed by(r 0 ) and(r 0 ) in Eq. (2.5), it is equivalent for us to takef(r 0 ;') = cos(2'), i.e., the “normalized" 26 functional basis. Since the normalization idea applies tog(x) as well, a natural choice of basis function forg(x), givenf(r 0 ;') = cos(2'), is a Fourier base: g(r 0 ;') = sin n(r 0 )' + (r 0 ) (2.6) where (r 0 ) is a phase variable andn(r 0 ) is a real number. Note that both andn are potentially related to the size covariater 0 . The SDG model (2.5) can thus be rewritten as y(r 0 ;') =(r 0 ) Z ' 0 cos(2) sin n(r 0 )(') + (r 0 ) d +(r 0 ) + (2.7) Let us defineh(r 0 ;') = R ' 0 cos(2) sin n(r 0 )(') + (r 0 ) d. The SDG model (2.5) can be conveniently expressed as y(r 0 ;') =(r 0 )h(r 0 ;') +(r 0 ) + (2.8) It is interesting to compare this SDG model (2.8) for disks built vertically with the one for horizontal disks: c 1 (r 0 ) +c 2 (r 0 ) cos(2) established in [13]. By Eq. (2.4), the model for horizontal disk is a special case of model (2.8) withg(x) =(x) in the convolution integral of h(r 0 ;'). After performing integration,h(r 0 ;') becomes h(r 0 ;') = Z ' 0 cos(2) sin n(r 0 )(') + (r 0 ) d = n(r 0 ) n(r 0 ) 2 4 cos n(r 0 )' + (r 0 ) + 1 2n(r 0 ) + 4 cos 2' (r 0 ) + 1 2n(r 0 ) 4 cos 2' + (r 0 ) (2.9) 27 Experimentation and data collection A commercial mask-image-projection-based stereolithography apparatus is used to build four half cylindrical disks vertically. This SLA process uses a digital micromirror device to project a set of mask images onto the resin surface to cure layers. After solidification of each layer, the building platform moves down at a predefined amount for the next layer. Process parameters and the design of four parts are shown in Table 2.1. Table 2.1: Specifications of SLA process and design parameters Resolution of the mask 19201200 Dimension of each pixel 0.005 00 Thickness of each layer 0.00197 00 Illuminating time of each layer 10-15s Average waiting time between layers 15s Type of the resin Perfactory SI 500 Radii of half cylinders 0.5 00 , 0.8 00 , 1.5 00 , 2.0 00 Figure 2.7 shows the four half disks and their deviation profiles under the PCS. Comparing with the repeatable deviation patterns of horizontally built disks illustrated in Fig. 2.3 (left panel for cylindrical disks with various sizes), the effect of the layer buildup and inter-layer interactions along thez direction is clearly visible in the sense that deviation patterns are not consistent when the disk size increases. SDG model estimation and physics-informed sequential model refinement Initial estimation of model (2.7) is accomplished through maximum likelihood estimation (MLE). Four separate models of (2.7) are obtained for half disks built vertically withr 0 = 0.5 00 , 0.8 00 , 1.5 00 and 2.0 00 shown in Fig. 2.7. The results of estimated parameters are presented in Table 2.2, and predicted shape deviations are shown in Fig. 2.8 as dashed lines. We can find that the predicted deviation profiles fit the data (solid lines) well, and modeling error is small and consistent. This initial model estimation in Table 2.2 also suggests that(r 0 );(r 0 );n(r 0 ), and (r 0 ) potentially vary with covariate r 0 . To build one consistent model for all disks and discover 28 ᵠ Shape deviation (inches) Figure 2.7: Four half disks and corresponding shape deviation profiles in the PCS Table 2.2: Initial model estimation through MLE r 0 (r 0 ) n(r 0 ) (r 0 ) (r 0 ) 0:5 00 0.0220 0.4493 -0.0928 0.0002 0.0007 0:8 00 0.0069 0.9158 -0.7619 0.0013 0.0008 1:5 00 0.0056 1.1916 -2.0898 0.0054 0.0008 2:0 00 0.0055 1.9238 -3.2883 0.0087 0.0008 process insights, we will first focus onn(r 0 ) and (r 0 ) which are inside the convolution inte- gralh(r 0 ;'), and conduct model estimation through a sequential model refinement procedure informed by physical knowledge. The following two conjectures are proposed forn(r 0 ) and (r 0 ). • Periodicity of transfer functiong(x) and length of interaction window: Parametern(r 0 ) determines the period of transfer functiong(x) defined in Eq. (2.6). On one hand, as the number of layers increases, the interaction of adjacent layers will be more complicated due to, for example, energy penetration and remelting. n(r 0 ) therefore is expected to increase with the number of layers orr 0 because a long period or smalln(r 0 ) will assign similar weights to adjacent layers. On the other hand, inter-layer interactions only play within a window with certain length defined by a certain number of consecutive layers. 29 Figure 2.8: Deviation profiles and model prediction for half disks built vertically: initial model fitting For instance, the depth of energy penetration or remelting is limited by the thermal prop- erties of materials and energy input. With this process understanding and the initial model fitting results in Table 2.2,n(r 0 ) is hypothesized to be n(r 0 ) = r 0 r ; (2.10) withr 0 being the nominal radius of the disk andr being the theoretical length of the inter- action window. Notice that introducing a proportional coefficient such asar 0 =r could cause over-parameterization problem in model estimation becausea can be absorbed by r . The same consideration applies to the second conjecture. • Phase shift of transfer functiong(x) and change of layer areas: Other than the effect of the number of layers, the change of layer area between adjacent layers will also affect transfer functiong(x). If all adjacent layers within an interaction window are identical, 30 Figure 2.9: Area ratio between a half disk and the theoretical window g(x) defined in Eq. (2.6) is expected to have (r 0 ) = 0 because only the number of layers varies. We hypothesize that (r 0 ) is related to ratio of the area of the half disk to the area of the theoretical “window" shown in Fig. 2.9. Since the estimated values of (r 0 ) in Table 2.2 are all negative, we finalize the conjecture as (r 0 ) = r 2 0 r 2 (2.11) With the above two conjectures,h(r 0 ;') in Eq. (2.9) is updated as: h(r 0 ;') = Z ' 0 cos(2) sin r 0 r (') r 2 0 r 2 d = r 0 r r 2 0 4r 2 cos h r 0 r ' r 2 0 r 2 i + r 2r 0 + 4r cos h 2' + r 2 0 r 2 i + r 2r 0 4r cos h 2' r 2 0 r 2 i (2.12) With updatedh(r 0 ;') in Eq. (2.12), we pool data of four half disks together to estimate one single SDG model. The model estimation through MLE is shown in Table 2.3 and Fig. 2.10. All the coefficients are significant at the 0.0001 level. Note that ^ r = 1.116 and the upper bound 31 ofr 0 for experimentation is 2.0 00 , which avoid the issue of 2r 0 4r = 0 in Eq. (2.12) because model extrapolation itself is problematic. Table 2.3: Sequential model refinement with conjectures onn(r 0 ) and (r 0 ) Parameters Estimate Standard Error r 1.11589962 0.00401990 0:5 00 0.01771531 0.00055778 0:8 00 0.00957536 0.00027614 1:5 00 0.00534351 0.00010980 2:0 00 0.00555896 0.00010031 0:5 00 0.00050797 0.00007831 0:8 00 0.00103725 0.00006886 1:5 00 0.00551954 0.00006387 2:0 00 0.00883869 0.00006383 0.00103552 0.00002059 Next we aim to determine the functional forms of(r 0 ) and(r 0 ), which are dictated by both the model fitting results in Table 2.3 and the two boundary conditions listed at the begin- ning of this section for SDG:y(r 0 ;') =(r 0 )h(r 0 ;') +(r 0 ) +. When r 0 ! 0, the first term of h(r 0 ;') in Eq. (2.12) tends to 0, while the second and the third term cancel each other. One constraint for(r 0 ) is that(r 0 )r 0 ! 0 whenr 0 ! 0. The remaining term of the expected value of y(r 0 ;') is (r 0 ). Therefore, (r 0 )! 0 when r 0 ! 0, because(r 0 ) determines the mean shape deviation. It must be close to 0 if a product with negligible size is built. So the boundary condition one will be satisfied forr 0 ! 0. When r 0 !1, the first three terms ofh(r 0 ;') in Eq. (2.12) all tends to 0. (r 0 )=r 0 ! constant whenr 0 !1. The expected value ofy(r 0 ;') will tend to(r 0 ), which has to be bounded. To summarize, the constraints for(r 0 ) and(r 0 ) are r 0 (r 0 )! 0; (r 0 )! 0; if r 0 ! 0 (r 0 ) r 0 !constant; (r 0 )!constant; if r 0 !1 (2.13) 32 Figure 2.10: Sequential model refinement with conjectures onn(r 0 ) and (r 0 ) To obtain proper forms of(r 0 ) and(r 0 ) satisfying constraints listed in Eq. (2.13), we still adopt the physics-informed sequential model refinement strategy. Since the model estimates of (r 0 ) in Table 2.3 shows strong and clear pattern illustrated in Fig. 2.11, our conjecture starts with(r 0 ) as (r 0 ) =a 2 r a 1 0 (2.14) Table 2.4: Sequential model refinement with conjectures onn(r 0 ), (r 0 ), and(r 0 ) Parameters Estimate Standard Error r 1.11636406 0.00460214 a 1 -0.85252348 0.02892385 a 2 0.00880909 0.00012932 0 :5 00 0.00042821 0.00007754 0 :8 00 0.00053241 0.00006941 1 :5 00 0.00515338 0.00006512 2 :0 00 0.00870620 0.00006698 0.00110081 0.00002261 33 0.5 1.0 1.5 2.0 0.000 0.005 0.010 0.015 0.020 r 0 α(r 0 ) 0.0177 0.0096 0.0053 0.0056 Figure 2.11: Estimates of(r 0 ) Then the refined model (2.8) takes the form ofy(r 0 ;') = a 2 r a 1 0 h(r 0 ;') +(r 0 ) + with h(r 0 ;') defined in Eq. (2.12). The MLE estimation of the updated model is shown in Table 2.4 and Fig. 2.12. All the coefficients are significant at the 0.001 level. Notice thata 1 =0:853, the boundary constraints for(r 0 ) listed in Eq.(2.13) are satisfied as well. The model estimates of(r 0 ) in Table 2.4 is illustrated in Fig. 2.13. By observing the subtle pattern and considering the boundary conditions of(r 0 ) in Eq.(2.13), we propose a sigmoid function with a logistic function baseS(x) = 1=(1 +e x ). In addition, we introduce a shift, an intercept, and a scaling term, i.e.,b 3 S(x +b 1 ) +b 2 . By satisfying(0) = 0 and grouping unknown coefficients, the final form of(r 0 ) is (r 0 ) = c 2 c 1 +e r 0 c 2 c 1 + 1 (2.15) By substituting Eq. (2.15) into Eq. (2.8), the finalized SDG model for half disks built vertically is y(r 0 ;') =a 2 r a 1 0 h(r 0 ;') + c 2 c 1 +e r 0 c 2 c 1 + 1 + (2.16) 34 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −0.005 0.000 0.005 0.010 0.015 0.020 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.5 in r 0 = 0.8 in r 0 = 0.5 in Figure 2.12: Sequential model refinement with conjectures onn(r 0 ), (r 0 ) and(r 0 ) 0.5 1.0 1.5 2.0 0.000 0.002 0.004 0.006 0.008 0.010 r 0 β(r 0 ) 0.000428 0.000532 0.005153 0.008706 Figure 2.13: Estimates of(r 0 ) The final model estimation through MLE is shown in Table 2.5 and Fig. 2.14. All the coefficients are significant at the 0.001 level. 35 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −0.005 0.000 0.005 0.010 0.015 0.020 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.5 in r 0 = 0.8 in r 0 = 0.5 in Figure 2.14: Sequential model refinement with conjectures onn(r 0 ), (r 0 ),(r 0 ) and(r 0 ) Table 2.5: Sequential model refinement with conjectures onn(r 0 ), (r 0 ),(r 0 ), and(r 0 ) Parameters Estimate Standard Error r 1.12310491 0.00528744 a 1 -0.83753308 0.02648314 a 2 0.00852358 0.00012209 c 1 -0.01077476 0.00385099 c 2 0.00130300 0.00004232 0.00118232 0.00002412 Process insights derived from SDG model for vertically built disks The proposed convolution formulation (2.1) not only provides a framework to learn models of predicting shape deviations, it also facilitates the understanding of process insights. Using the model in Eq. (2.8) for vertically built half disks as an example, the essential deviation patterns 36 are defined by the convolution integralh(r 0 ;') in Eq. (2.9). We can further define and analyze the three terms inh(r 0 ;'): h 1 (r 0 ;') = r 0 r r 2 0 4r 2 cos h r 0 r ' r 2 0 r 2 i (2.17) h 2 (r 0 ;') = r 2r 0 + 4r cos h 2' + r 2 0 r 2 i (2.18) h 3 (r 0 ;') = r 2r 0 4r cos h 2' r 2 0 r 2 i (2.19) Givenr 0 = 2:0,h 1 (r 0 ;'),h 2 (r 0 ;'), andh 3 (r 0 ;') are superimposed in Fig. 2.15. Clearly h 2 (r 0 ;') has least influence among the three terms because of its smaller weights assigned to base functionf(r 0 ;') in the convolution. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 0 1 2 3 φ Shape Deviation (inch) h 1 (2.0, φ) h 2 (2.0, φ) h 3 (2.0, φ) Figure 2.15:h 1 (2:0;'),h 2 (2:0;'), andh 3 (2:0;') Figure 2.16 further shows h 1 (r 0 ;'), h 2 (r 0 ;'), and h 3 (r 0 ;')by (i) varying r 0 and ' (left panel) and (ii) varyingr 0 and' with (r 0 ) = 0 (right panel). By observing the Fig. 2.15 and Fig. 2.16, we postulate the following interpretation which can guide further investigations. • h 1 (r 0 ;') is deemed to be the main descriptor of inter-layer interaction effects. First, h 1 (r 0 ;') change sharply with radiusr 0 , e.g., much smaller weights for disks withr 0 =0.5 00 37 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 0 1 2 3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.9 in r 0 = 1.8 in r 0 = 0.5 in 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 0 1 2 3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.9 in r 0 = 1.8 in r 0 = 0.5 in (a) h 1 (r 0 ;') (left) andh 1 (r 0 ;') with (r 0 ) = 0 (right) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.9 in r 0 = 1.8 in r 0 = 0.5 in 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 0.5 in (b) h 2 (r 0 ;') (left) andh 2 (r 0 ;') with (r 0 ) = 0 (right) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 0 1 2 3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.9 in r 0 = 1.8 in r 0 = 0.5 in 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3 −2 −1 0 1 2 3 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.9 in r 0 = 1.8 in r 0 = 0.5 in (c) h 3 (r 0 ;') (left) andh 3 (r 0 ;') with (r 0 ) = 0 (right) Figure 2.16:h 1 (r 0 ;'),h 2 (r 0 ;'), andh 3 (r 0 ;') and 0.8 00 and larger weights for disk with r 0 = 2.0 00 (in blue). Second, the period of 38 h 1 (r 0 ;') varies withr 0 as well, slow for disks with less number of layers (similar weights for adjacent layers) and faster for disks with more layers. • h 2 (r 0 ;') is likely due to the gravity effect. First, it has least influence as indicated in Fig. 2.15. Second, its period does not change withr 0 or the number of layers. Further experimentation can be conducted to verify this hypothesis. • h 3 (r 0 ;') is deemed to be the effect of shape deviation for the same shape built horizon- tally. In another word, h 3 (r 0 ;') is mainly determined by f(x). First, it has the same period withf(x) and base function cos(2'). Second, as expected, the larger the shape deviations in the horizontal plane, the bigger the influence on the same shape built verti- cally. Model improvement through Gaussian process regression Though the error term in model (2.8) is assumed to be noise N(0; 2 ), the shape deviations tend to be spatially correlated. Figure 2.17(a) confirms the spatial correlations by showing the model residuals of four vertically printed half disks. One way to improve the model fitting is to adopt Gaussian process regression (GPR) [95] to model the residual shape deviations: =GP(0;k(;)) + 0 (2.20) whereGP(0;k(;)) is a Gaussian process with kernel functionk(;) and 0 N(0; 2 ). The kernel function adopted in this study is the squared exponential kernel: k(x;x 0 ) = 2 f exp 1 2 2 l jjxx 0 jj 2 (2.21) withx being angle' in the modeling of vertically printed half disks. 39 (a) GPR of residual shape deviation 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −0.005 0.000 0.005 0.010 0.015 0.020 φ Shape Deviation (inch) r 0 = 2.0 in r 0 = 1.5 in r 0 = 0.8 in r 0 = 0.5 in (b) Updated SDG model prediction with GPR Figure 2.17: SDG model prediction with Gaussian process regression of residuals 40 The GPR of model residuals using Matlab function fitrgp produces the result in Fig. 2.17(a) (the thick line). The length scale l = 0:2335 suggests that the predicted values of the Gaussian process do not change slowly (i.e., not smooth) in the interval [0,]. The scaling factor or standard deviation f = 0:0006 is small, which indicates that the predicted values of the Gaussian process will be close to its mean (zero). The residual standard deviation 0 = 0:001, as expected, is smaller than the one in model (2.16). The prediction from the updated SDG model, i.e.,(r 0 )h(r 0 ;') +(r 0 ) +GP(0;k(;)), is given in Fig. 2.17(b). Comparing to Fig. 2.14, the prediction of local shape deviation by the updated SDG model is improved. 2.5.2 Identifying Transfer Functiong(x) for Domes SDG model for 3D shapes The SDG model for general 3D shapes will take the following form: y r 0 (;');;'; = (fg) r 0 (;');;' + = Z 2 Z 1 f 1 ; 2 g 1 ;' 2 d 1 d 2 +GP(0;k(;)) + 0 (2.22) whereGP(0;k(;)) is a 2D Gaussian process and 0 N(0; 2 0). r θ φ x z y r 0 Figure 2.18: Dome shape and four domes printed in a SLA process 41 Since establishing the convolution model for freeform 3D shapes deserves dedicated efforts, this study illustrates the proposed framework using dome shapes with varying sizes (Fig. 2.18) and focus on the extraction of transfer function to understand inter-layer interactions. Enlight- ened by the SDG model (2.8) for half disks and the comment below Eq. (2.8), we propose the SDG model for dome shape as y(r 0 ;;') =(r 0 )h(r 0 ;;') +(r 0 ) + =(r 0 ) Z 2 Z 1 f 1 ; 2 g 1 ;' 2 d 1 d 2 +(r 0 ) +GP(0;k(;)) + 0 (2.23) wheref(;') andg(;') are both normalized basis functions. Note that the SDG model (2.23) for dome shapes contains the SDG model (2.8) for 2D half disks. Model (2.8) can be viewed as a special case of model (2.23) with ==2 and = 3=2. The transfer function g(;') can be viewed as g(;') = ==2 g(') and when it convolves withf(;') in the convolution integralh(r 0 ;;'), the 3D model (2.23) degenerates into a 2D model (2.8). With the result in Eq. (2.4),f(;') in Eq. (2.23) represents the shape deviation of horizontal disks at ', without inter-layer interaction effects. The effect of ' shows on the layer radius, i.e., r 0 sin('). Combining with the in-plane deviation basis function cos(2) used previously, f(;') in Eq. (2.23) is proposed to be f(;') = cos(2) sin(') (2.24) where sin(') reflects the impact of radius change along the build direction. 42 Transfer function identification through deconvolution and model selection via LASSO The deconvolution problem is to identify g(;') given f(;') and measurement data y(r 0 ;;'). Following the rationale for 2D transfer function defined in Eq. (2.6), g(r 0 ;;') for dome shape can be expressed as a combination of 2D Fourier bases, i.e., g(;') = 1 X n=0 1 X m=0 c n;m cos(n + n ) cos(m' +! m ) where n(r 0 ) and m(r 0 ) determine the periods along and ', respectively. n ) and ! m ) are phase variables. Furthermore,g(;') likely has a sparse representation with features selected from a large 2D Fourier bases. Among different regularization methods, LASSO [96] is adopted in this study because it not only reduces model variances but also makes model more interpretable with sparse solutions [97]. Different regularization terms can be added depending on applications. For example, to track anomalies of network traffic volumes, Mardani et al. [98] decompose the traffic flow data into low-rank normal flow, sporadic abnormal flow, and noise by adopting a nuclear norm, L1-norm, and Frobenius norm, respectively. Letg j denote thejth Fourier base with coefficientc j forfg j . For each dome, we conduct feature screening through the LASSO formulation: min C 1 N N X i=1 (y i X j c j fg j ( i ;' i )) 2 + jjCjj 1 (2.25) whereN represents the total number of sampled points on each dome. Four domes with radii of 0:5 00 ; 0:8 00 ; 1:5 00 and 1:8 00 are built in a SLA process. The measure- ment data of dome shape deviation is presented in the SCS (Fig. 2.19). The 0:5 00 ; 0:8 00 and 1:8 00 domes are regarded as the training set, and the 1:5 00 dome is left for model validation. Similar to the phenomenon shown in Fig. 2.7, the deviation patterns of four domes also very with their sizes due to layer buildup and interactions. This poses a challenging issue for model learning 43 because models can make poor prediction of shape accuracy even with simple change of shape sizes. Figure 2.19: Shape deviation measurement of four domes presented in the SCS Significant terms shared across different domes are selected and the resulting set of features (Fourier bases) includes: cos(n 1 + 1 ), cos(n 2 ' + 2 ), and cos(n 1 + 1 ) cos(n 2 ' + 2 ). A further MLE estimation of transfer functiong for individual domes chooses the follow form for g: g(;') = cos(n 1 ')[1 + cos(n 2 + )] (2.26) 44 wheren 1 andn 2 determine the periods along' and, respectively, and is a phase variable. It is interesting to notice that the transfer function is determined not only by the height of layers (defined by cos(n 1 ')), but also by its interaction with shape deviation within that layer (defined by cos(n 2 + )). With the transfer functiong identified in Eq. (2.26), the convolution integralh(r 0 ;;') in Eq.(2.23) becomes h(;') = cos(') cos(n 1 ') 2(n 2 1 1) h 1 n 2 + 2 sin(2 ) 1 n 2 2 sin(2 + ) + 2n 2 n 2 2 4 sin(n 2 + ) + sin(2) i (2.27) SDG model estimation To obtain one unified SDG model for training shapes, we follow the same physics-informed sequential model refinement process as we did for the vertically printed half disks (omitted). The data suggests that - (r 0 ) =a 1 +a 2 r 0 - (r 0 ) =b 1 +b 2 r 0 - n 1 (r 0 ) =c 1 +c 2 r 0 - n 2 and : unknown constants With the training set (three domes), Table 2.6 gives the initial MLE estimation of the SDG model without considering the Gaussian process, i.e., (a 1 +a 2 r 0 )h(r 0 ;;') + (b 1 +b 2 r 0 ) +. The measured shape deviations (response: black point cloud) and the SDG model predictions (blue and red point cloud) are superimposed together in Fig. 2.20. Due to the difficulty of pattern visualization in the 3D space, we plot the shape deviation by point index, which is 45 Table 2.6: Initial MLE estimation without Gaussian process model term Parameters Estimate Standard Error n 2 0.3319 0.00155 3.3897 0.00162 a 1 -0.1137 0.00237 a 2 -0.1683 0.00210 b 1 0.0082 0.00008 b 2 0.0062 0.00007 c 1 0.0551 0.05747 c 2 -0.0324 0.04382 0.0046 0.00002 sorted by' and. So the first quarter from the left corresponds to the 0:5 00 dome, the second quarter represents the shape deviation surface of the 0:8 00 dome, and so on. Only three domes marked in blue are used to train the SDG model, and the red point cloud (the third quarter) is the validation set. Apart from the predicted triangular shape on the upper right corner of each dome, the trends are well captured by the model. To improve the model prediction, the GPR is conducted to capture the spatial correlation in the model residuals, i.e., =GP(0;k(;)) + 0 , which is more critical for 3D shapes. With the square exponential kernel function, the GPR of dome model residuals is conducted with optimized parameter estimates f = 0:0097, l = 0:9886 and 0 = 0:0029. The length scale l = 0:9886 suggests that the predicted values of the Gaussian process vary moderately in the area [0, 2][0,=2]. The scaling factor f = 0:0097 is significantly larger than the residual standard deviation 0 = 0:0029, which indicates spatial correlation and variation is significant for 3D shapes. Figure 2.21 shows the measured shape deviations and the updated SDG model predictions with GPR. The prediction covers most of the measurement point cloud. Figure 2.22 illustrate the predicted shape deviations versus the measurement data. In both figures, the validation outcomes of 1:5 00 dome are as good as the training cases. 46 0 2000 4000 6000 8000 10000 12000 −0.02 −0.01 0.00 0.01 0.02 Point Index Shape Deviation (inch) 0.5−inch dome 0.8−inch dome 1.5−inch dome 1.8−inch dome Figure 2.20: Measured shape deviation (black dots) and the SDG model prediction (blue and red dots) The root mean square errors (RMSE) of the training set and validation set are 0.0036361 00 and 0.0036379 00 , respectively. And the mean absolute errors (MAE) for the two sets are 0.002636 00 and 0.002637 00 for the training and validation sets, respectively. Process insights derived from SDG model for domes We define and analyze the four terms inh(r 0 ;;') as: h 0 (r 0 ;') = cos(') cos[n 1 (r 0 )'] 2[n 1 (r 0 ) 2 1] (2.28) h 1 () = 1 n 2 + 2 sin(2 ) 1 n 2 2 sin(2 + ) (2.29) h 2 () = 2n 2 n 2 2 4 sin(n 2 + ) (2.30) h 3 () = sin(2) (2.31) 47 0 2000 4000 6000 8000 10000 12000 −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 Point Index Shape Deviation (inch) 0.5−inch dome 0.8−inch dome 1.5−inch dome 1.8−inch dome Figure 2.21: Measured shape deviation (black dots) and the updated SDG model prediction with GPR (blue and red dots) Then h(r 0 ;;') =h 0 (r 0 ;')[h 1 () +h 2 () +h 3 ()] Notice that the term h 0 (r 0 ;') is decided by r 0 and ', while the other three terms only depend on. To compare the magnitude of each term, we ploth 1;2;3 () in Fig. 2.23. Three functionsh 1;2;3 () have the same interpretation as those in the half disk case, that is, they represent inter-layer interaction effect, the gravity effect, and the effect of input in-plane shape deviations, respectively. h 3 () is not due to interaction effect because it is the result of convolving in-plane deviation pattern cos(2) inf(;') with the constant one ing(r 0 ;;') = cos(n 1 (r 0 )')[1 + cos(n 2 + )]. Note we separate and'. One difference from the half disk case is that all three effects will be influenced by the height of the layer defined by'. We plot h 0 (r 0 ;') in Fig. 2.24. One interesting fact is that the function values are almost the same when we variater 0 from 0:5 00 to 1:8 00 inh 0 (r 0 ;'). 48 −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 Measured shape deviation (inch) Predicted shape deviation (inch) −0.02 0 0.02 −0.02 0 0.02 −0.02 0 0.02 −0.02 0 0.02 0.5−inch dome 0.8−inch dome 1.5−inch dome 1.8−inch dome Figure 2.22: Final model prediction of shape deviations against the measured shape deviations 0 1 2 3 2 2 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Shape Deviation h 1 h 2 h 3 Figure 2.23:h 1 (),h 2 (), andh 3 () Combine the termh 0 withh 1;2;3 respectively, we visualize the inter-layer interaction effect, gravity effect, and the input in-plane shape effect in Fig. 2.25. The interpretation is consistent with the case for half disks. 49 0 1 8 1 4 3 8 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Shape Deviation Effects across Figure 2.24:h 0 (r 0 ;');r 0 2 [0:5; 1:8] 0 1 2 3 2 2 0 1 8 1 4 3 8 1 2 Shape Deviation (inch) 0.4 0.2 0.0 0.2 0.4 0 1 2 3 2 2 0 1 8 1 4 3 8 1 2 Shape Deviation (inch) 0.4 0.2 0.0 0.2 0.4 0 1 2 3 2 2 0 1 8 1 4 3 8 1 2 Shape Deviation (inch) 0.4 0.2 0.0 0.2 0.4 Figure 2.25:h 0 h 1 (r 0 ;;'),h 0 h 2 (r 0 ; 0 ;'), andh 0 h 3 (r 0 ; 0 ;') 2.6 Summary and Conclusion The Shape Deviation Generator (SDG) developed in this work provides a data-analytical framework to learn geometric measurement data of AM built products. Under a convolution framework, SDG enables a consistent description of 3D shape formation in layer-by-layer fab- rication processes, from horizontally built disks, vertically built disks, to fully 3D domes. This is achieved through proper representation of shape deviation data, modeling of individual layer input, and transfer function derived to capture inter-layer interactions. The physics-informed sequential model estimation and refinement strategy leads to efficient learning and better under- standing of process insights. Effects due to inter-layer interactions, gravity, and deviation of 50 individual layers are separated for guiding further experimentation and validation. Though the initial methodology demonstration applies to simple shapes such as disks and domes, the con- volution framework allows input functions (f ) to take complicated geometries for each layer and to convolute with the transfer function (g) to form complicated 3D shapes. More exciting effort is needed for such an extension and ML4AM in general. 51 Chapter 3 Learning and Predicting Shape Deviations of Smooth and Non-Smooth 3D Geometries through Mathematical Decomposition of AM 3.1 Introduction Due to materials phase changes and high thermal gradients in the layer-wise fabrication processes, final product geometries are often deformed or distorted in AM, especially for non- smooth shapes containing edges and corners [10, 13, 99]. The deviations of 3D shapes from their intended designs can be represented as 2D surfaces in aR 3 space [53, 100, 101], which constitutes a complicated set of data for learning and predicting geometric quality. Patterns of deviation surface data vary with shape geometries, sizes/volumes, materials, and the AM processes [14, 15, 57, 61, 102]. As a motivating example, four domes and three thin walls with half-cylindrical shapes were vertically printed through an AM process using the same material (Fig. 3.1). After collecting the point cloud data on the product surfaces and comparing with the designs, we present their shape deviations as functional surfaces [16] in the spherical coordinate system (SCS) as shown in Fig. 3.2. The deviation profiles of the thin walls are quite different from those of the domes. 52 Thin walls present sharp increments in the deviation around the corners/edges while the devia- tion surfaces of domes are smooth everywhere. Furthermore, the shape deviation profiles and their patterns can vary with the size of the same target shape. (a) (b) Figure 3.1: (a) Domes with 0.5, 0.8, 1.5 and 1.8 inches radii and (b) Thin walls with 0.8, 1.5 and 2.0 inches radii To learn and predict surface deviations of 3D shapes, we face several fundamental chal- lenges: • Heterogeneity. Deviation patterns are affected by multiple factors including geometries, sizes, materials and process parameters. This often produces heterogeneous data even under the same process settings of an AM machine. As observed in Fig. 3.2, thin walls show distinct deviation patterns from domes. Different sizes/volumes would affect not only the magnitude of the deviation surface but also the locations of sharp transitions for the thin walls. Taking the deviation profile as the response, there are three types (a) 0.8-inch Dome (b) 1.8-inch Dome (c) 0.8-inch Thin Wall (d) 2.0-inch Thin Wall Figure 3.2: Shape deviation measurements of two dome and two thin walls with presented in SCS 53 of covariates: (1) location covariates defined with respect to the printing and registration center [14,42]; (2) process parameters such as the type of AM process, printing materials, printing temperature, etc. [43, 44, 101]; and (3) size-and-shape information specified by the designs of AM products [14, 46, 60]. • Limited samples. A key advantage of AM is the easy customization of the printed parts. This one-of-a-kind manufacturing approach limits the applicability of large-sample machine learning methods and classical statistical methods for modeling shape devia- tion. With limited number of training samples and various covariates, it is challenging to establish a reasonable model, estimate its parameters, and validate assumptions on model specifications and parameters, especially for non-parametric models such as random for- est [42] and Dirichlet process [52]. • Spatial correlation within and among shapes. In AM, the layer-by-layer fabrication pro- cess exacerbates the quality issues related to material phase change and heat penetration observed in traditional manufacturing [103–105], leading to stronger spatial correlation among neighboring regions of a product. Joint learning of different 3D shapes presents additional challenge of defining spatial correlation among different geometries. The modeling and control challenges faced in printing 2D shapes are exacerbated for 3D geometries, since deviation patterns can change from layer to layer due to complex process inter-layer interactions, residual stresses and heat dissipation profiles [54–57]. To incorporate the layer-by-layer fabrication mechanism into modeling and learning, Huang et al. [16] pro- posed a convolution learning framework to describe the 3D deviation patterns as the result of the 2D shape deviation of each layer convolved with a layer interaction function that captures the inter-layer interactions. Although this framework was validated with spherical shapes, it still requires fundamental work to predict the complex deviation patterns of smooth and non- smooth shapes together in a consistent modeling framework. Due to the smoothing effect of convolution operation, this approach alone is insufficient for modeling the sharp transitions in 54 the shape deviation profiles of non-smooth polyhedral geometries like the thin wall shape. The main reason to study these particular shapes is that 3D freeform shapes can be approximated as a combination of smooth and non-smooth patches [106]. Understanding these two basic cate- gories of 3D shapes enables 3D freeform shape deviation prediction similar to the significant extension made in the 2D case [13–15]. To tackle these challenges, we mathematically decompose the fabrication of non-smooth 3D shapes into two steps: (1) additively building the smooth base shapes, and (2) subtrac- tively craving out the non-smooth polyhedral shapes. An additive model is proposed to con- nect smooth and non-smooth geometries through the 3D cookie-cutter function with a physic- informed sequentially model estimating strategy. Spatial correlation among regions within a product and across different product geometries is modeled by a Gaussian process (GP) with a novel distance metric integrating both geodesic and geometric information. The remainder of this chapter is organized as follows. Sec. 3.2 introduces the model for mathematical decomposition of shape deviation in AM and discusses its model components. A physics-informed sequential model fitting strategy is proposed to enable efficient and sta- ble parameter estimation in Sec. 3.3. Section 3.4 demonstrates the proposed framework using printed domes and thin wall shapes. Concluding remarks are given in Sec. 3.5. 3.2 A Unified Shape Deviation Modeling Approach for Smooth and Non-Smooth Geometries in AM This section proposes a mathematical decomposition of AM as a general additive model to learn and predict shape deviation surfaces of 3D geometries. Adopting smooth shapes as the model baseline, we propose a new class of cookie-cutter functions to link the shape deviation of 3D smooth and non-smooth convex shapes. Lastly, a GP is employed to capture the spatial correlation with a novel distance metric. 55 (a) (b) Figure 3.3: Mathematical decomposition of AM to additively build the smooth base shape (dome shape outside) and subtractively carve out non-smooth shapes with sharp corners such as (a) thin wall shape and (b) cuboid shape 3.2.1 Mathematical Decomposition of AM through An Additive Model for Shape Deviation Modeling To learn insights from the heterogeneous data (Fig. 3.2), domain knowledge must be incor- porated. As shown in Fig. 3.3, the process of building a 3D shape can be mathematically decomposed into two steps: (1) additive step, which fabricates a smooth base shape that bounds the target geometry, and (2) subtractive step, which removes extra materials to form non-smooth edges and corners. Thus, the smooth and non-smooth 3D shapes are connected through an asso- ciation function that captures the shape deviation caused by the subtractive step. Here, the shape smoothness is defined with respect to the smoothness of curves on the surface of the product design. If any curve along the shape surface is smooth, i.e., there exists a common tangent direction at any point on the surface, then the shape is smooth; otherwise, it is non-smooth. For example, the curved surfaces of spherical and cylindrical shapes are smooth, while the thin wall and cuboid shapes in Fig. 3.3 are non-smooth since any curve crossing the edges would be non-smooth. In general, non-smooth shapes contain edges and corners, which are directly associated with sharp changes in their deviation surfaces. 56 The proposed mathematical decomposition procedure leads to an additive model. Due to the flexibility and interpretability of additive models, they are extensively used in diverse appli- cations such as ecology [107], health care [108,109], machine learning [110], and geomorphol- ogy [111]. For shape deviation at theith location of partj , a general additive model can be defined as y(x i ;s j ;p j ) =(x i ;s j ;p j ) +(x i ;s j ;p j ) +; (3.1) wherey is the shape deviation,x are location covariates,p j are the process parameters or part- independent covariates like materials and process characteristics,s j represents the geometric information or part-dependent covariates, models the mean pattern of the shape deviation, is a zero-mean random field that captures the spatial correlation, and is the measurement error. In the rest of this work, we dropp j for notation simplicity since only one specific printing process is investigated in this work. For 2D shape deviations, Huang et al. [14] considered the regular polygons as being carved out from their circumscribed circles through the additive model (x i ;s j ) = q 1 (x i ) + q 2 (x i ;s j ) + q 3 (x i ;s j ); (3.2) where q 1 describes the shape deviation of disks, q 2 is a 2D cookie-cutter function that links the deviation profiles of circular and polygonal shapes, and q 3 is a high-order term for the remaining pattern. Adopting a similar approach, the mean pattern for 3D shape deviation of a given printing process can be decomposed as (x i ;s j ) = f 1 (x i ) + f 2 (x i ;s j ) + f 3 (x i ;s j ); (3.3) where f 1 describes the deviation profile of the smooth base shape, f 2 is a 3D cookie-cutter function that connects smooth and non-smooth geometries, and f 3 is a high-order term. For 57 example, we can use f 1 to model the shape deviation of the base smooth shape (e.g. domes), f 2 to capture the differences between the base smooth and non-smooth geometries such as thin walls (Fig. 3.3), and f 3 to illustrate the high-order pattern. 3.2.2 Convolution Framework as a Baseline for Smooth Shape Deviation Modeling A convolution learning framework proposed in [16] can be used to identify the baseline function for shape deviation modeling of convex smooth shapes as y(x) = (fg) (x) +(x) +; (3.4) where = fg, f is the input function describing the 2D shape deviation in a horizontal layer,g is the interaction function that models complex layer-to-layer interactions, is a zero- mean GP capturing spatial correlations in the deviation profile, and is the measurement error following a zero-mean normal distribution. The variablex is the spatial location of the points in SCS, i.e.,x = (r;;'), where is the polar angle and' is the azimuth angle. The printed shape of a product is treated as the functional responser(;') and the nominal shape is denoted asr 0 (;'), then shape deviation is defined asy(x) =r(;')r 0 (;'), where2 [0; 2), and '2 [0;=2]. To specify, the first step is to identify the input functionf. As the most common smooth geometries, spherical shapes (as shown in Fig. 3.1(a)) are chosen as the base geometries, where each horizontal layer is a circle. Shape deviation for a circle of radiusr 0 can often be modeled with a few Fourier basis functions due to its geometric simplicity, for example, f(r 0 ;) = c 1 (r 0 ) +c 2 (r 0 ) cos(2) in an SLA process studied by [13]. Using this formulation, Huang et al. [16] modeled the deviation of a dome shape as (r 0 ;;') = 0 (r 0 ) + 1 (r 0 )(fg)(;'): (3.5) 58 The size factorsc 1 (r 0 ) andc 2 (r 0 ) can be absorbed in 0 (r 0 ) and 1 (r 0 ) in Eq. (3.5). Since each layer of a dome shape has radiusr 0 sin', the input function for the domes can be normal- ized, for example, as f(;') = cos(2) sin': (3.6) For the layer interaction function g(x), lasso regression was adopted for model selec- tion [112]: min c 1 N N X i=1 y i X j c j (fg j )( i ;' i ) ! 2 + jjcjj 1 ; (3.7) whereN is the number of sampled points,g j (;') is a 2-D Fourier basis, andc j is the coefficient of the basis functiong j . Significant terms shared among all domes were selected resulting in the layer interaction function g(;') = cos(n 1 ')[1 + cos(n 2 + )]: (3.8) 3.2.3 Association Between Smooth and Non-smooth Geometries: 3D Cookie-Cutter Function The convolution operator alone is insufficient to learn the sharp transitions observed in the deviation profiles around the corners of non-smooth polyhedral shapes (Fig. 3.2). We propose to use a 3D cookie-cutter function to subtractively carve out the deformation profile of non- smooth shapes from that of a smooth baseline shape. For example, a polygon is cut out from its minimum bounding circle for each horizontal layer as shown in Fig. 3.4(a). Similar to [14] and [60], the minimum bounding circle is employed as the smooth base shape since the circumcircle, which passes through all vertices of the polygon, may not exist for an arbitrary polygon. To represent the proposed learning framework as an additive model, we define the basis functions: h 1 (x) = (fg)(x) describing smooth base shape deviation, h 2 (x) being the 3D 59 cookie-cutter function linking the smooth and non-smooth geometries, and a high-order term h 3 (x) for the remaining pattern. Then, the learning framework can be written as (x) = 0 (x) + 3 X j=1 j (x)h j (x) (3.9) where j ;j = 0;:::; 3 represent size-effects,h 1 can be learned from smooth products as in [16], and bothh 2 andh 3 are fully determined by the geometries of AM-fabricated products. Note that because we use convex smooth shapes as the bases, we can infer the shape deviation of convex polyhedra while the shape distortion of concave geometries (e.g., pentagram in 2D) needs to be studied using a concave smooth geometry as the baseline. For the 2D case, [14] applied two candidate 2D cookie-cutter functionsh 2 as the association function to carve out regular polygons from their circumcircles: the square-wave function sq() = signfcos[n( 0 )=2]g; (3.10) and the sawtooth-wave function sw() = ( 0 )MOD(2=n); (3.11) where n is the number of sides and 0 is a phase term to shift the cutting position. These functions allow sharp transitions on the deformation patterns near the corners. A generalization of the sawtooth-wave function was established in [102] as sw() = (# j1 )MOD(# j # j1 ) 2(# j # j1 ) (3.12) for selected angles# j ;j = 1;:::;n. Note that such function is only needed when the interior angle of a corner is less than=6 according to their experimental studies. 60 (a) 0.00 0.25 0.50 0.75 1.00 0 ϑ 1 ϑ 2 π ϑ 3 ϑ 4 2π θ Square Wave (b) 0.00 0.25 0.50 0.75 1.00 0 ϑ 1 ϑ 2 π ϑ 3 ϑ 4 2π θ Sawtooth Wave (c) Figure 3.4: (a) A rectangle cut from its circumcircle and corresponding (b) 2D square-wave cookie-cutter function and (c) 2D sawtooth-wave cookie-cutter function For the 3D case with more complex geometries, we apply the 2D cookie-cutter function in each horizontal layer defined by' in SCS by modifying the frequency of the square-wave or sawtooth-wave function such that the amplitudes alternate at the sharp corners defined by # j (');j = 1;:::;n. Thus, one candidate forh 2 is the 3D square-wave function sq(;') = 1 2 sign sin (1) j+1 # j (') + 1 ; (3.13) and the other alternative is 3D sawtooth-wave function sw(;') = # j1 (') # j (')# j1 (') ; (3.14) for# j1 (')<# j ('); j = 1;:::;n + 1, where# j (');j = 1;:::;n are the polar angles of sharp transitions with# 0 (') = 0 and# n+1 (') = 2. The proposed 3D cookie-cutter functions can be regarded as the stack of 2D cookie-cutter functions over the '-direction, where each layer could have sharp transitions at different angles according to the designed geometry. As the number of corners increases and the polyhedron approaches a sphere, the sharp corners effectively banish andh 2 is approximately constant in both definitions. 61 Figure 3.5: A thin wall fabricated by stacking rectangles As in the motivating example, for a thin wall with a half-cylindrical shape that has radiusr 0 and thicknessw as shown in Fig. 3.5, each horizontal layer is a rectangle with length 2r 0 sin' and width w. To cut out a rectangle defined by the corner points (A 1 ;A 2 ;A 3 ;A 4 ) from its circumcircle as shown in Fig. 3.4(a), we first find the angles of each corner as# 1 ;# 2 ;# 3 and# 4 , then the corresponding 2D square-wave function is shown in Fig. 3.4(b) and 2D sawtooth-wave function is shown in Fig. 3.4(c). Since the rectangle sizes in each horizontal layer defined by ' are different, the sharp transitions for each layer happen at different polar angles, which are purely defined by the geometry of the product. The scatter plots of proposed 3D square-wave and sawtooth-wave cookie-cutter functions are shown in Fig. 3.6. While the same thin wall products are treated as 2D shapes in [16], they are regarded as 3D shapes in this work. Unlike the deviation profiles y(') for 2D cases, shape deviations of 3D non-smooth thin walls show deviation surfacesy(;') in SCS (Fig. 3.2). Furthermore, to predict the shape deviation of these thin walls on the front, back, and curved top surfaces, each thin wall is regarded as the stack of rectangles of different sizes (Fig. 3.5), which requires the 3D cookie-cutter function in Eq. (3.13) and Eq. (3.14). 62 (a) (b) Figure 3.6: (a) 3D square-wave function and (b) 3D sawtooth-wave function for the 0.8-inch thin wall 3.2.4 Spatial Correlation Modeling with a Novel Distance Metric for Het- erogeneous Shape Data To capture the spatial correlation not only among regions of the same product but also across heterogeneous shapes, the random field is assumed to be a zero-mean GP. The squared- exponential kernel is adopted [95] k(x i ;x j ) = exp d(x i ;x j ) 2 ; (3.15) whered(x i ;x j ) is the distance betweenx i andx j . To more accurately describe the spatial correlation between two sample points in theR 3 space, we identify each point in the Cartesian coordinate system, i.e.,x i = (x i ;y i ;z i ) rather than the SCS. The coordinates are scaled to the current coordinate over the maximum in each direction. Note that (0; 0; 0) corresponds to the center of the printing bed, and the maximum height was chosen as to eliminate the effect of different scales in thez direction. 63 Due to the physics involved in generating and measuring the shape deviation, Sun et al. [113] and Castillo et al. [114] pointed out that geodesic distance, which is defined as the shortest distance between points on a 3D surface, is a better measure of the spatial correlation in the same part rather than the Euclidean distance in AM. A thorough review of geodesic paths and distances on the surface of triangle meshes can be found in [115]. Due to the high compu- tational cost of the geodesic distance, we employ the as-rigid-as-possible parameterization to first explore the surface into a 2D plane, and then the point-to-point geodesic distance can be approximated by the corresponding Euclidean distance [116]. However, one challenge is that there is no clear definition of geodesic distance among different parts since such path along the surface does not exist. For two pointsx i andx j lying on the surfaces of two shapes s i and s j , respectively, we propose a new distance metric d(x i ;x j ) = 1 2 h d e (x i ;x 0 i ) +d g (x 0 i ;x j ) +d e (x j ;x 0 j ) +d g (x 0 j ;x i ) i (3.16) where x 0 i is the projection of point x i onto shape s j , i.e., x 0 i and x i have the same angles ( i ;' i ), thenx j andx 0 i are points on the same part with a properly defined geodesic distance d g (x 0 i ;x j ). Similarly, x 0 j andx j are on the line defined by the angles ( j ;' j ) and d e is the standard Euclidean distance. For example, considering the side view of a thin wall and a dome shape as shown in Fig. 3.7, the proposed distance is the combination of four red curves, where the solid curves denoting the geodesic distances d g and the dashed ones are the Euclidean distances d e . By adding the projection distance d e (x 0 j ;x j ) and d e (x 0 i ;x i ), we complete a circuit fromx i tox j and back which ensures thatd is a valid distance measure. Another advantage of Eq. (3.16) is that, if two shapess i ands j are the same, the projection distances are zero, and then the proposed distance is the standard geodesic distance. 64 Figure 3.7: Proposed distance metric betweenx i on shapes i (in blue) andx j on shapes j (in black) 3.3 Sequential Model Estimation Procedure for the United Modeling Framework After specifying each component of the additive model, a sequential model fitting strategy is proposed to efficiently estimate the model parameters, mitigate over-fitting, and transfer the knowledge from smooth base shape to non-smooth polyhedral shapes. In general, the parameter estimation procedure follows a similar strategy to the boosted mod- els [117] of fitting models sequentially based on the residuals of earlier models. As specified in [16], the convolutionh 1 describes the shrinkage of smooth geometries during printing caused by material phase changes, inter-layer interactions, and gravity effects. On the other hand, the deviation profiles of non-smooth shapes exhibit sharp transitions introduced by uneven thermal stresses around the corners [118]. By regarding the non-smooth shape as being cut from a set of smooth patches, we apply the 3D cookie-cutter functionh 2 to capture the sharp transitions and h 3 to model the difference in deviation profiles between smooth and non-smooth geometries. 65 Thus, the smooth shape deviation would be modeled first to build the baseline model, while non-smooth parts are included later to assess the association and remaining pattern. As for, we use GP regression (GPR) on the parametric model residuals in the final step for the following reasons: (1) estimating GP parameters is computationally expensive; (2) the full training set is involved since spatial correlation affects all shapes; (3) due to the flexibility of GP, the main effect could be confounded, which compromises the insights from the parametric model. Algorithm 1 Physics-Informed Sequential Model Fitting Strategy Split dataD intoD 0 (smooth) andD 1 (non-smooth) according to geometry Initialize the param- eters forh 1 Fit the parameters for the modely(x) = 0 (x) + 1 (x)h 1 (x) +;8(y;x)2D 0 Calculate residuals ~ y =y(x) ^ 0 (x) + ^ 1 (x) ^ h 1 (x) ;8(y;x)2D; Fit the parameters for the model ~ y = 2 (x)h 2 (x) + 3 (x)h 3 (x) +;8(y;x)2D 1 Calculate residuals ~ ~ y = ~ y (^ 2 (x)h 2 (x) + ^ 3 (x)h 3 (x));8(y;x)2D; Calculate pairwise distanced(x i ;x j );8x i ;x j 2D Fit a GP for model ~ ~ yN (0;k(d(x i ;x j ))) To be more specific, after dividing the dataD = (y;x) intoD 0 for smooth shapes andD 1 for non-smooth shapes, there are three steps to fit the model sequentially. First, we estimate the parameters in 0 , 1 andh 1 of Eq. (3.9) using the data inD 0 . Since only smooth shapes are involved, the model is reduced toy = 0 + 1 h 1 +. Second, non-smooth shape deviations inD 1 are used to estimate the parameters in 2 and 3 with respect to the residuals ~ y =y(^ 0 +^ 1 ^ h 1 ). Recall thath 2 andh 3 are determined by the geometry, and the model is ~ y = 2 h 2 + 3 (x)h 3 + for the non-smooth shape deviations. Lastly, to model the spatial correlation, GPR techniques are applied to model the residuals ~ ~ y = ~ y (^ 2 h 2 + ^ 3 h 3 ) from all data including both smooth and non-smooth parts, i.e., ~ ~ yN (0;k(;)). By learning these three components sequentially, we can predict the shape deviation of untried products. The proposed parameter estimation strategy is summarized in Algorithm 3.3. 66 Furthermore, if k ;k = 0;:::; 3, have a parametric form, the model parameters can be estimated using the profile likelihood approach. Without loss of generality, assume that k (x) is a basis expansion of the form k (x) = P J k j=1 q j k (x) j k . Then, the model can be expressed as y =H( ) + +; (3.17) where H( ) = q 1 0 (x);:::;q J 0 0 (x);q 1 1 (x)h 1 ( );:::;q J 3 3 (x)h 3 and = 1 0 ;:::; J 0 0 ; 1 1 ;:::; J 3 3 | : The maximum likelihood estimates (MLE) are obtained by solving min 1 2 2 jjyH 1 ( ) ^ 0 jj 2 + n 2 ln( 2 ) s.t. ^ 0 = (H | 1 ( )H 1 ( )) 1 H | 1 ( )y (3.18) wherey is the shape deviation of observations included inD 0 ,n is the total number of points in D 0 ,H 1 is the matrix of the firstJ 0 +J 1 columns ofH, and 0 = ( 0 ; 1 ). For the parameters of 2 and 3 , the MLE is ^ 00 = (H | 2 H 2 ) 1 H | 2 ( )~ y where ~ y is as defined in Algorithm 3.3 for observations inD 1 , H 2 is the matrix of the last J 2 +J 3 columns ofH, and 00 = ( 2 ; 3 ). 67 3.4 Case Study: Shape Deviation Modeling and Estimation for Domes and Thin Walls In this section, we revisit the motivating example in Sec. 3.1 to demonstrate the capability of the proposed learning framework for modeling and predicting the shape deviation patterns of a wide variety of geometries containing both smooth and non-smooth features in AM. Seven parts (Fig. 3.1) were printed through the mask image projection stereolithography (MIP-SLA) process. After the printing process, ROMER absolute arm with RS4 laser scanner is used to collect the measurements as point clouds, which are then registered using the constrained itera- tive closest point algorithm [119] and the printing quality are evaluated by the shape deviation surface [16] as illustrated in Fig. 3.2. We employ all domes and 0.8-inch and 2.0-inch thin walls as the training set and leave the 1.5-inch thin wall as the validation set. To implement the learning framework in Eq. (3.9), the first step is to identify the input functionf. Note that, due to machine repair, the pattern of shape deviation of 2D circular disks changed from what was presented in [13], i.e., the input functionf(;') should have a different pattern and we need to fit the spherical shape model for the new data. Luan and Huang [15] defined the new pattern as f() = cos 2 + 3 1 2[0;) sin(2)1 2[;2) : (3.19) Recall that the form of input function was changed to Eq. (3.6) by multiplying it by sin' because the radius of each layer is r 0 sin' for the dome shape. Similarly, we have the input functionf(;') for spherical shapes as f(;') =f() sin': (3.20) 68 If the radius isr 0 and the thickness isw for the thin wall shape, the circumcircle radius for each horizontal layer' is max fr 0 (;')g = r r 2 0 sin 2 ' + w 2 2 r 0 sin'; (3.21) sincew is much smaller thanr 0 for the thin products. Then, we can compute the convolution explicitly and regard the thin walls as cut from the domes with the same radii, and the same input functionf(;') as in Eq. (3.20) can be applied. For the layer interaction functiong(;'), Eq. (3.8) is used since the printing mechanism is the same and only the shapes and sizes change. Thus,h 1 is fully specified. Next, we need to specify the sharp transition angles # j and n used in the 3D cookie- cutter function h 2 in Eq. (3.13). As shown in Fig. 3.4, n = 4 and the angles are # 1 = arctan(w=(2r 0 sin')), # 2 = # 1 , # 3 = +# 1 , # 4 = 2# 1 according to the geome- try of the thin walls. To capture the arch pattern of thin walls presented in the deviation profiles in Fig. 3.2, we chooseh 3 as h 3 (;') = n sin n 4 1 2[# 1 ;# 2 ) + sin h n 4 () i 1 2[# 3 ;# 4 ) o sin'; (3.22) wheren is the number of sides. For the thin walls, we haven = 4, and whenn!1, this term becomes white noise. Due to the limited number of samples, we follow similar linear assumptions as in [16] to incorporate the size effect. Denoting x = (x;y;z;r 0 ;;'), which contains the location information of Cartesian coordinates and spherical coordinates, the conjectures are 1. n 1 (x) =c 1 +c 2 r 0 , 2. n 2 and are unknown constants, 3. 0 (x) = 0;1 + 0;2 r 0 , 69 0.5 inch 0.8 inch 1.5 inch 1.8 inch 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 −0.02 −0.01 0.00 0.01 0.02 0.03 Point Index Shape Deviation (inch) (a) −0.02 0.00 0.02 −0.02 0.00 0.02 Measured Deviation (inch) Predicted Deviation (inch) (b) Figure 3.8: Measured shape deviation (in gray) and model prediction (in blue) for domes 4. 1 (x) = 1;1 + 1;2 r 0 , 5. 2 (x) = 2;1 + 2;2 h p r 2 0 sin 2 ' +w 2 =4r 0 (;') i 6. 3 (x) = 3;1 + 3;2 r 0 . Note thatn 1 ,n 2 , and are the parameters in the layer interaction functiong as in Eq. (3.8), while i ;i = 0;:::; 3, are the coefficients describing the size effect in Eq. (3.9). The first four conjectures are for the baseline model, and the last two would affect the 3D cookie-cutter function. For simplicity, we assume the layer-to-layer interactions change for different sizes mainly along the'-direction and are related to the number of layers printed, thusn 1 deciding the period over' is assumed to be proportional to the size, whilen 2 and control the period and phase in-direction is assumed to be constant. To achieve better model interpretability, a linear relationship to the size of the product is imposed on the coefficients of 0 ; 1 and 3 . As the coefficient of cookie-cutter term, 2 is proportional to the cutting width, i.e., the difference between the circumcircle radius and polyg- onal shape at each angle, which is p r 2 0 sin 2 ' +w 2 =4r 0 (;'). Under these assumptions, at least two samples of each shape are required to estimate the model parameters. Because a limited number of training samples are usually provided in AM, more complex relationships for require knowledge of the material and process interactions. 70 0.8 inch 1.5 inch 2.0 inch 0 1000 2000 0 1000 2000 0 1000 2000 −0.10 −0.05 0.00 0.05 Point Index Shape Deviation (inch) Deviation Prediction − Training Prediction − Validation (a) −0.10 −0.05 0.00 0.05 −0.10 −0.05 0.00 0.05 Measured Deviation (inch) Predicted Deviation (inch) (b) 0.8 inch 1.5 inch 2.0 inch 0 1000 2000 0 1000 2000 0 1000 2000 −0.10 −0.05 0.00 0.05 Point Index Shape Deviation (inch) Deviation Prediction − Training Prediction − Validation (c) −0.10 −0.05 0.00 0.05 −0.10 −0.05 0.00 0.05 Measured Deviation (inch) Predicted Deviation (inch) (d) Figure 3.9: Measured shape deviation (in gray), training set prediction (in blue) and validation set prediction (in red) for thin walls applying (a and b) square wave and (c and d) sawtooth wave functions The MLE procedure described in Sec. 3.3 is employed to fit the spherical shape deviation model through the mle2 function in R package bbmle with randomized initialization in the parameter space, and the results are given in Table 3.1 and Fig. 3.8. The mean absolute error (MAE) is 0.0048 and the root mean square error (RMSE) is 0.0065. We plot the shape deviation by point index due to the difficulty of comparing model fitting performance in 3D space. The four blocks from the left correspond to the 0.5-inch, 0.8-inch, 1.5-inch, and 1.8-inch domes, respectively. The predictions are close to the actual deviation measurements, except at the upper right corner. Note that the results are different from [16] since the input functionf(;') is changed to Eq. (3.20). The estimates ofc 1 andc 2 are not statistically different from zero and the layer interaction function can be simplified asg(;') = 1 + cos(n 2 + ). Next, the thin walls with radii of 0.8 inch and 2.0 inches are used as the training set, and the 1.5-inch thin wall is left as the validation set. The estimated model parameters for 2 and 3 71 Table 3.1: Parameter estimates and standard error (SE) for the deviation of dome shapes Parameters Estimate SE n 2 0.6462 0.006591 4.0793 0.019610 c 1 0.0034 0.230790 c 2 -0.0016 0.169782 0;1 0.0068 0.000210 0;2 0.0047 0.000166 1;1 0.0063 0.000601 1;2 0.0158 0.000476 0.0065 0.000046 Table 3.2: Parameter estimates and standard error (SE) for cookie-cutter and high-order terms Square Wave Sawtooth Wave Parameters Estimate SE Estimate SE 2;1 -0.0387 0.00106 -0.0041 0.00123 2;2 0.0002 0.00004 -0.0003 0.00004 3;1 0.0124 0.00175 -0.0040 0.00184 3;2 0.0139 0.00074 0.0137 0.00093 are shown in Table 3.2, and the measured deviations versus model predictions are presented in Fig. 3.9 for both square-wave and sawtooth-wave cookie-cutter functions. The model perfor- mance metrics are summarized in Table 3.3. From both figures and performance metrics, the square wave function is better than the sawtooth-wave function for modeling the shape devia- tion of thin walls. Thus, we use the square-wave function for the remainder of the chapter. Since both 2;2 and 3;2 are positive, we know that the effects of the sharp transition and arch pattern on the deviation profile increase with the part’s size. However, there are some remaining spatial patterns to be captured, so the residuals are fitted through GPR with squared-exponential kernel in Eq. (3.15) using Euclidean distance between points through the gam function in R package mgcv. 72 0.5 inch Dome 0.8 inch Dome 1.5 inch Dome 1.8 inch Dome 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 −0.02 −0.01 0.00 0.01 0.02 0.03 Point Index Shape Deviation (inch) (a) 0.8 inch Thin Wall 1.5 inch Thin Wall 2.0 inch Thin Wall 0 1000 2000 0 1000 2000 0 1000 2000 −0.10 −0.05 0.00 0.05 Point Index Shape Deviation (inch) (b) −0.10 −0.05 0.00 0.05 −0.10 −0.05 0.00 0.05 Measured Deviation (inch) Predicted Deviation (inch) (c) Figure 3.10: Measured shape deviation (in gray), training set prediction (in blue) and validation set prediction (in red) after GPR with Euclidean distance Table 3.3: Thin wall model performance applying different cookie-cutter functions Training Validation Cookie-Cutter MAE RMSE MAE RMSE Square Wave 0.0122 0.0160 0.0120 0.0158 Sawtooth Wave 0.0156 0.0200 0.0164 0.0205 The measurements and model predictions are presented in Fig. 3.10 and Table 3.4. For the dome parts, not only the upper right corners in Fig. 3.8 have been offset by GP, the expansion of the larger domes are correctly predicted. Predictions for thin walls are improved from the results in Fig. 3.9, and the performance metrics (MAE and RMSE) are reduced for both training and validation sets. Since the GP combines the location information of all shapes together, and points on the smaller domes are closer to those on the thin walls, their deviation predictions are greatly 73 0.5 inch Dome 0.8 inch Dome 1.5 inch Dome 1.8 inch Dome 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 −0.02 −0.01 0.00 0.01 0.02 0.03 Point Index Shape Deviation (inch) (a) 0.8 inch Thin Wall 1.5 inch Thin Wall 2.0 inch Thin Wall 0 1000 2000 0 1000 2000 0 1000 2000 −0.10 −0.05 0.00 0.05 Point Index Shape Deviation (inch) (b) −0.10 −0.05 0.00 0.05 −0.10 −0.05 0.00 0.05 Measured Deviation (inch) Predicted Deviation (inch) (c) Figure 3.11: Measured shape deviation (in gray), training set prediction (in blue) and validation set prediction (in red) after GPR with the proposed distance Table 3.4: Model performance comparison Training Validation Model MAE RMSE MAE RMSE Parametric 0.0122 0.0160 0.0120 0.0158 Parametric + Euclidean Distance 0.0050 0.0093 0.0107 0.0148 Parametric + Proposed Distance 0.0060 0.0095 0.0087 0.0124 affected by the deviation pattern of thin walls. Thus, the 0.5-inch and 0.8-inch domes show a wider variation of the deviation prediction compared with the 1.5-inch and 1.8-inch domes. This suggests that the spatial correlation among different shapes is overestimated, and we can improve the spatial correlation modeling by modifying the distance metric in the squared- exponential kernel as specified in Sec. 3.2.4. 74 (a) (b) (c) Figure 3.12: (a) Measured shape deviation, (b) predicted shape deviation, and (c) residuals for the 1.5-inch thin wall After computing the proposed geodesic distance among all shapes and estimating the spatial correlation with the mKrig function in R package fields, the updated model prediction and per- formance are shown in Fig. 3.11 and Table 3.4. With the proposed distance metric considering both geographic and geometric information, the GP accurately captures the spatial correlation. Though the deviations of the first few points in domes are overestimated, predictions cover most of the actual deviation measurements in the training set. With slightly worse training set performance, the performance on the validation set improved around 20% using the proposed distance metric. The measured shape deviation, predicted deviation surface, and residuals of the final model with GPR using the proposed distance metric are presented in Fig. 3.12. 3.5 Conclusions This work establishes a unified learning framework for shape deviation modeling of smooth and non-smooth 3D geometries in AM. The AM process is mathematically decomposed into two stages to tackle the heterogeneous deviation surfaces. In stage one, a base smooth shape is additively build, while in stage two non-smooth shapes complex geometries are subtractively carved out. A unified predictive shape accuracy model is developed to combine the baseline 75 deviation of smooth shapes, sharp transitions caused by the subtractive step, and the remain- ing spatial correlation. The previously proposed convolution framework serves as the baseline model for smooth shapes. The proposed 3D cookie-cutter function effectively captures the unique shape deviation pattern of sharp corners of non-smooth convex shapes and enable joint learning of smooth and non-smooth geometries. A novel distance measure is proposed to model the spatial correlation among heterogeneous shapes by combining the local geodesic distance between points in the same 3D object and the euclidean distance between points projected across 3D objects. A case study shows that the unified model can successfully predict the shape deviation of convex smooth (domes) and non-smooth (thin walls) shapes. Since 3D freeform shapes can be approximated as a combination of smooth patches and sharp corners, the proposed learning framework builds a foundation to further extend this modeling approach to predict the quality of 3D freeform shapes. Furthermore, the strategy of connecting engineering surface data through decomposition of data generation processes can be adopted in other domains for engineering- informed learning of heterogeneous data. 76 Chapter 4 A Shape Registration Methodology for Geometric Deviation Correction in AM 4.1 Introduction Though significant research efforts have been taken to develop prescriptive models to reduce shape deviation of AM products through physics-based and data-driven methods, the efficacy of these models depends on the proper shape registration or alignment of the product measurement to its design. Incorrect alignment can generate deviation patterns that are not present in the manufactured parts, undermining any modeling and compensation work. As a motivating example, consider the registration of tilted thin walls as shown in Fig. 4.1. There are two main reasons to investigate thin-wall structures. First, compared to other solid shapes, thin walls are more prone to the bending effect. Second, assuming the distortions induced by the printing process are much smaller than the designated curvature, the tilted thin wall provides an approximation of the global distortion when compared with a straight wall. The bottom sections of the parts within the red boxes are of high quality, while the top presents significant bending. A consistent registration method separates the conforming and noncon- forming sections. In contrast, an inconsistent registration may shift these thin walls to the right side (along they-direction) by minimizing the overall mismatch between the product and the design. Moreover, the magnitude of the shift will increase with the level of the bending effect. Although the bottom portions of two thin walls have no distortion, excessive deviations are created by the registration process, and the deviation of Part B is more significant than Part A. 77 Design compensation [53,120] or process adjustment based on the inconsistent registration can worsen the overall accuracy. (a) Consistent registration (b) Inconsistent registration Figure 4.1: Quality assessment for two tilted thin walls with measurement in black and design in blue. Bottom portions within the red boxes have no distortion. Achieving consistent and accurate quality assessments for deviation correction in AM faces three significant challenges. First, local shape deformation can only be quantified after the reg- istration process. Second, potential global distortion or displacement like shifting, warping, and twisting, especially for thin-wall structures [121, 122], requires careful assessment and sepa- ration since it may not be compensated by changing the product design and expects printing process adjustment and optimization. Lastly, surface roughness needs to be distinguished from the overall distortion. Shape registration has long been studied in various fields, such as geography, medical imaging, and computer vision. Denoting the measured point cloud of m points as M = f(x i ;y i ;z i );i = 1;:::;mg and the design as D, both in the Cartesian coordinate system, the shape registration problem can be formulated as min T E(T (M);D) (4.1) 78 where T is the transformation that aligns the measurement to the design and E is the error function quantifying the mismatch between them. For quality assessment and correction in manufacturing, a rigid transformation functionT is more suitable since it preserves the product geometry [123]. Based on whether the transformation functionT is restricted to rigid transformation or isom- etry, which preserves the point-to-point Euclidean distance, the methods can be categorized into rigid and non-rigid registration approaches. Though the latter is more flexible in finding the point correspondence [116, 123], a rigid transformation function is more suitable for quality assessment in manufacturing applications since it preserves the product geometry. A compre- hensive comparison of these two approaches can be found in Tam et al. [124]. Depending on whether the landmarks are selected and applied to compute the registration errorE, registration techniques can also be classified into landmark-based and landmark-free methods. In the first category, several points are selected as references to estimate the unknown transformation between two or more coordinate systems. Local geometric features are often calculated and used to estimate the point correspondence. For example, Chua and Jarvis [125] proposed to represent each point by the distance from all points in the neighborhood to the tangent plane. Wang et al. [126] adopted least-square conformal maps to uniquely represent 3D shapes as 2D images. However, a great amount of landmark-based methods require extensive human efforts and are subjective to the operator. Ground control points are often adopted in stereo matching [127]. Corners of simple geometries are manually selected in [128] and [129] as the landmark for registration. In addition, the distinct features, especially the areas with high curvatures like corners, are prone to thermal stress and thus often deformed in AM and are not suitable to be selected as estimate the correspondence. Moreover, to model and compensate for the shape deviation, point-to-point correspondence is hard to estimate and not critical. Instead, establishing a quality assessment that reflects the reproducible manufacturing defects is more favorable. 79 For the landmark-free registration methods, two major categories are contour-based and region-based registrations. One the most popular contour-based methods is the iterative closest point (ICP) algorithm [130], which estimates a rigid transformation that minimizes the point- to-point Euclidean distance. Different variants exist to enable better initialization [131, 132], robust registration [133, 134], faster computation speed [135, 136], object deformation [116, 137] and so forth. A throughout overview of ICP and its variants can be found in [138] and [139]. While in the region-based approaches, V oronoi diagram [140], medial axis [141, 142] and medial surface [143, 144] is widely employed. Since deviations of shape boundaries are of interest in AM, contour-base methods are more suitable. However, one major drawback of them is that inconsistent registration results, as shown in Fig. 4.1, introduce shape deviations patterns that are not caused by the manufacturing process to the analysis. This chapter proposes a new shape registration method to obtain consistent alignments that are robust to global distortion by sequentially constraining the six degrees of freedom (DoFs) of a rigid transformation. After segmenting and aligning the ground points, three DoFs controlling the actual height of each point are estimated. Then the reliable layers of the measurement data is filtered through a statistical control chart, where the monitoring statistics are selected based on a decomposition of shape deviation in AM. A final alignment can be achieved by the constrained ICP method. The proposed method has the advantage of facilitating the localization and correction of printing failures. The rest of the chapter is organized as follows. Section 4.2 introduces the proposed reg- istration methodology for geometric deviation correction in AM. After clarifying the devia- tion decomposition framework, a three-step registration procedure is described. The simula- tion study is conducted in Sec. 4.3 to validate the proposed registration approach. Section 4.4 demonstrates the proposed methodology with a tilted thin wall fabricated through an FDM pro- cess. Conclusions are given in Sec. 4.5. 80 4.2 Proposed Methodology This section proposes a new shape registration method by sequentially constraining six DOFs of a rigid transformation to evaluate the deviation of AM fabricated products. A sta- tistical control chart is applied to monitor the components of a novel deviation decomposition. Then, the portion of the measured point cloud with minimal distortion is selected for alignment with the design. 4.2.1 Shape Deviation Decomposition The sources of geometric error between a printed product from its intended shape in an AM system can be classified as process-induced error such as machine axis displacement and material-induced error including thermal shrinkage and residual stress [145]. While the former causes layer displacement, the latter is related to part shrinkage or expansion. The main target of this work is to reveal the true local deformation in the presence of global distortion, sur- face roughness, and measurement error. To model these effects, the overall shape deviation is decomposed as = + + +; (4.2) where – is the shape deviation describing how the measurement varies from the design. In this work, the geometric deviation is parameterized as (x;z) = y(x;z)y 0 (x;z), where (x;y;z) 2 M and (x;y 0 ;z) 2 D. Essentially, we define the shape deviation as the directional difference alongy-axis. For thin-walled structures, the deviation patterns on both sides of the structure are similar and only one of them needs to be considered. – is the global distortion including shift and rotation of layers, which is mostly incurred by machine-related errors [145]. For example, the displacement of the nozzle in an FDM 81 process after a few layers could cause the upper portion of the product to be shifted or twisted. – is the local deformation due to material-induced errors including material phase change, thermal gradients, and inter-layer interactions [34, 118]. While the global distortion can be regarded as the result of shifting or rotating design layers, local deformation cannot be replicated by layer-wise rigid transformations. – is the surface roughness, which quantifies the directional variation of the fabricated surface. In FDM, the major causes of poor surface finish are build plate variation, extruder variation, and build plate temperature fluctuation. represents the measurement error, which is assumed to be normally distributed with mean zero. 4.2.2 Overview of the Proposed Registration Method In order to establish accurate and consistent registration results as a foundation of deviation modeling and correction, the following two assumptions are proposed. (1) The part can be stabilized on a flat surface. (2) The bottom portion of the part has minimal global distortion. The first assumption requires the product to adhere to the building platform and the bottom surface to stay flat after fabrication. Both detachment during the printing process and bend due to the residual stress cannot be compensated by changing the design in AM. On the other hand, distorted initial layers lead to error accumulation and ultimately to printing failure. A reliable bottom portion provides the foundation to build upper layers successfully. 82 To align the measurement towards the design, the estimated rigid transformationT can be parameterized by an orthogonal rotation matrixR and a translation vectort as T (x) =Rx +t: (4.3) To preserve the orientation of measured points, reflections are excluded by imposing detjRj = 1 so that the size and shape information is retained by T (xjR;t). Rigid transfor- mations have six DoFs inR 3 , i.e., t x ;t y ;t z ;r x ;r y , and r z corresponding to translations along x-, y- and z-axes and rotations over x-, y- and z-axes, respectively. These variables can be classified into two groups: (t x ;t y ;r z ) controlling in-plane movement and (t z ;r x ;r y ) deciding z-coordinates of the measurement. The proposed shape registration approach contains three steps as shown in Fig. 4.2 to sequentially estimate the translation and rotation variables. First, points on the ground are seg- mented from the product surface; then, a transformation matrix is estimated to set the ground to horizontal with normal towardsz-direction and the height equal to zero, which fixes (t z ;r x ;r y ). Next, a layer-by-layer data selection strategy based on the statistical quality monitoring method is proposed to select regions with minimal global distortion. The last step is to align the selected portion of the product to the corresponding design through a constrained ICP algorithm to esti- mate (t x ;t y ;r z ). 4.2.3 Ground Points Segmentation and Alignment As shown in Fig. 4.3, when a scanned point cloud includes points on the ground like the building platform or the desk where the measurement is conducted, they need to be segmented from the product measurement. By aligning the ground points, three DoFs, i.e.,t z ;r x andr y , are determined. Any transformations involving these parameters would either detach the bottom surface from the ground or make the bottom surface incorrectly lean towards one direction, thus not horizontal. One popular choice for the segmentation is the cloth simulation [146], which 83 Ground Points Segmentation Off-Ground Points Ground Points Transformation Matrix Horizontal Off- Ground Points Sliced Layers Layer i Quality Monitoring Yes No In Control? i = i + 1 Filtered Data Final Alignment Measurement Design Start End Step 1 Step 2 Step 3 Figure 4.2: Proposed shape registration methodology 84 Figure 4.3: Points on the surface (in black) are segmented from ground points with the fitted plane (in green). separates ground and non-ground measurements by analyzing the interaction between simulated cloth nodes and the measured surface. Fig. 4.3 shows the segmentation of a thin wall from the points on the ground. Under assumption (1), all ground points lie on the same plane. Thus, probabilistic algo- rithms like random sample consensus [147] can be employed to fit a plane to them. A rigid transformation to make this plane horizontal atz = 0 is computed and applied to the off-ground points. Then the product bottom is horizontal and the actual z-coordinates of the points are obtained. Another advantage of adopting ground points is that most registration methods, including ICP, require a global estimation of the transformation matrix and are sensitive to such initial- ization. The transformation making the ground points horizontal at the origin offers a stable and consistent initialization. Moreover, the estimations oft z ;r x andr y are robust to the product shape deviation and noisy measurement. 85 4.2.4 Control Chart for Layer Selection Under the assumptions presented in Sec. 4.2.2, printing quality can be assessed layer-by- layer from the bottom. When translations along x- and y-directions and rotation over z-axis are the only variables considered, the registration problem degenerates to the 2D case. Each layer of the measurement can be projected to the horizontal cross-sections, i.e., xy-planes at different heights. However, 2D shape registration methods cannot be applied for two reasons: (1) Different layers could correspond to unique shapes determined by the intended design; (2) All layers must have the same transformation matrix. A series of geometry and size invariant statistics is proposed to monitor the layer-by-layer quality of the fabricated part. Then statistical quality control methods [148] are adopted for monitoring the changes in quality of the printed layers. Based on the deviation decomposition in Sec. 4.2.1, to evaluate the local shape deformation, other components, i.e., global distortion and surface roughness, need to be identified by assum- ing the measurement error as the random noise. Considering the unknown local deformation, monitoring the center of each layer is more robust than tracking all points on the boundary. Note that, assuming the layers facing thez-direction, a persistent shift of the centroids implies the presence of a bending effect. Sorting vertices (x i ;y i );i = 1; 2;::: of each layer in the counterclockwise order, the layer centroid (c x ;c y ) can be calculated as [149] c x = 1 6A n X i=1 (x i +x i+1 )(x i y i+1 x i+1 y i ); c y = 1 6A n X i=1 (y i +y i+1 )(x i y i+1 x i+1 y i ); (4.4) whereA = 1 2 P n i=1 (x i y i+1 x i+1 y i ) is the area of the polygon. 86 Since Eq. (4.4) works for both measurement and design with proper sampling like uniform sampling [150] to get (c x M ;c y M ) and (c x D ;c y D ), respectively. Then the translation of the centroid can be calculated as t x =c x M c x D ; t y =c y M c y D : (4.5) Another way for the layers to be distorted is the twisting effect, i.e., the upper layers might be curled. To compute the rotation overz-axis, measurements with high curvatures are excluded since the corners or complicated areas are often deformed due to high residual stress [151]. Thenr z is estimated using the rest points by solving the following optimization problem. min 0 @ x cos y sin x sin +y cos 1 A 0 @ x 0 y 0 1 A 2 s.t. arctan x cos y sin x sin +y cos = arctan y 0 x 0 (x 0 ;y 0 )2D (4.6) Given the designD, each point after the optimal rotation should be as close as possible to the corresponding point on the design, which has the same angle as the rotated point. For the surface roughness, roughness averageR a is computed as [152] R a = 1 n n X i=1 jy i yj; (4.7) where y is the mean deviation over a small neighborhood in the measurement. If any of the monitoring statisticst x ;t y ;r z , andR a changes, there is a critical printing error like printing bed displacement, unwanted extruder movement, or building temperature fluctua- tion. In contrast, having all these statistics in control suggests that the printing process is stable and the quality is reproducible. 87 In order to detect global distortion fast, multivariate exponentially weighted moving average (MEWMA) control chart [153] is adopted to monitor the proposed statistics as Z i =X i + (1)Z i1 ; (4.8) whereX i is the vector of monitoring statistics for theith layer,Z i is the cumulative weighted average withZ 0 = 0, and smoothing parameter2 (0; 1]. An out-of-control signal is triggered if the statistics 2 i = Z | i 1 Z i Z i is greater than the upper control limit h, which is selected to achieve a desired average number of layers before detection [154]. More details on MEWMA can be found in [148]. One crucial step before applying the control chart method is to estimate the in-control parameters 0 and . According to the second assumption, the bottom few layers can be regarded as the in-control process. Thus, if signal 2 i <h, there is no apparent global distortion, and surface roughness stays the same level; otherwise, the printing process is out-of-control, and layers fabricated afterward are unreliable. Note that fault diagnosis methods can be applied to determine which variable contributes the most to the out-of-control signals. 4.2.5 Final Alignment by Constrained ICP Based on the in-control data, ICP constraining to translation along the x- and y-axes and rotation over thez-axis is used to align the bottom part of the measurement to the corresponding design. Then, the resulting translation matrix is applied to the out-of-control points to assess the quality of the whole part. 88 4.3 Simulation Study Numerical studies are conducted to validate the proposed registration methodology under three types of global distortion patterns observed when fabricating thin walls in AM.In the sim- ulation process, it is assumed that the bottom of the part is already aligned with the horizontal plane atz = 0. The point clouds for thin walls with 3 mm thickness, 30 mm height, 80 mm width, and 0.5 mm layer thickness are generated. For simplicity, each layer of the thin wall is shifted toward they-direction by a predefined amount in the left column of Fig. 4.4. A small noise N(0; 0:1 2 ) is added to mimic the measurement error. Then the MEWMA control chart is adopted to filter the reliable layers, and the results are shown in the middle column. To detect the shift of one standard deviation when monitoring four variables, the parameters are set to be = 0:105 andh = 15:26 to achieveARL 0 = 500 andARL 1 = 14:6 [154]. The first row is the uniform bending towards one side, which can occur due to the gravity effect since a slight deviation in the lower layer could be accumulated and cause bending in the upper layers. The second type corresponds to a bent-and-recovered thin wall due to the stresses from the top side like pressing from the extruder or excess materials deposited. The last one is non-uniform bending, which is frequent in AM. Since the corners cool down quicker than the middle of the products, they often bend towards the middle, and the whole part is warped. Note that the first 20 layers, corresponding to the lower 10 mm in the bottom, have zero or little deviation as shown in the brighter color and thus are used as the in-control process. Small shifts are added from the 21st layer when z is greater than 10 mm. For the uniform bending effect, the out-of-control signal is identified at layer 23, which is 1.5 mm higher than the ground truth. Since the shift is abrupt and significant for the bent-and-recovered pattern, it is detected immediately at layer 21. The non-uniform bending is more complicated, where the bottom portion used as the in-control process is also slightly tilted. However, the out-of-control signal is triggered shortly at the 22nd layer. The results show that the proposed data filtering approach is robust to a minor violations of the second assumption, i.e., the bottom portion is not distorted. 89 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1.0 (a) Deviation of uniform bending effect 23 24 25 26 27 0 50 100 150 0 20 40 60 Layer Index Plotting Statistic (b) MEWMA of uniform bending effect 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1.0 (c) Estimation of uniform bending effect 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1.0 (d) Deviation of bent-and-recovered effect 21 0 50 100 150 0 20 40 60 Height (mm) Plotting Statistic (e) MEWMA of bent-and-recovered effect 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1.0 (f) Estimation of bent-and- recovered effect 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1.0 (g) Deviation of non-uniform bend- ing effect 22 0 50 100 150 0 20 40 60 Height (mm) Plotting Statistic (h) MEWMA of non-uniform bend- ing effect 0 10 20 30 −40 −20 0 20 40 x (mm) z (mm) Δ (mm) 0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 (i) Estimation of non-uniform bend- ing effect Figure 4.4: Simulated thin walls with different types of global distortion Lastly, the maximum difference between actual and estimated deviations is less than 0.1 mm introduced by the measurement error. 90 4.4 Experimental Validation In this section, a tilted thin wall is fabricated and measured to validate the proposed shape registration method, where no ground truth on the manufacturing-induced deviations is pro- vided. Specifically, the measurement is aligned to different designs through both the proposed method and the ICP algorithm. The estimated shape deviation patterns are compared and vali- dated through a compensated thin wall based on the proposed methodology. 4.4.1 FDM Fabricated Tilted Thin Wall To validate the proposed shape registration method, a solid titled thin wall with 50 mm length, 3 mm width, 60 mm height, and 0.2 mm layer thickness as shown in Fig. 4.5 is printed. A fifth-generation MakerBot Replicator FDM printer is used to fabricate the thin wall with MakerBot PLA filament. The part is positioned at the center of the printing bed, and a flat base or raft is printed beneath the wall to ensure a flat bottom surface sticking to the build plate. The product is then scanned using a Hexagon ROMER Absolute Arm 7325 SEI with the build-in RS4 laser scanner, which collects the point cloud data with an accuracy of 80 μm. Though the shape registration method is validated through the FDM fabricated part, it is not restricted to the FDM process or any specific printing system. As long as the two assumptions in Sec. 4.2.2 are satisfied, this registration procedure can be adopted to accurately reveal the true shape deviation. Following the proposed registration process, ground points are first segmented as shown in Fig. 4.3 since the raft layer can be regarded as horizontal; otherwise, the points on the table or build plate can be adopted. By setting the bottom surface to horizontal withz = 0,t z ;r x , and r y of a rigid transformation are determined. 91 Figure 4.5: The tilted thin wall fabricated through FDM 4.4.2 Registration to the Straight Thin Wall In order to evaluate the proposed registration methodology, the tilted thin wall is first aligned to a straight wall design. Since the wall is designed to be tilted, there is some knowledge about the actual deviation. Though the printing process incurs other deviations, the revealed shape deviation should be close to the design. After aligning the ground points, the second step is to construct the control chart. The MEWMA chart is built based on the first ten layers, and layer thickness is set to be 0.5 mm for simplicity. Then the bottom 5 mm is considered in-control. The established MEWMA chart is shown in Fig. 4.6. Since global distortion in the upper portion of the thin wall is detected at the 18th layer, only the first 17 layers are reliable and thus selected to run constrained ICP registration to estimatet x ;t y , andr z . Shape deviation profiles after registration using the proposed method and conventional ICP are shown in Fig. 4.7 (b) and (c), respectively. Compared with the design deviation in Fig. 4.7 (a), it is evident that the distortion of the thin wall is more accurately revealed through the proposed method since the bending effect is unveiled accurately. The ICP algorithm incorrectly moves and rotates the measurement to minimize the overall mismatch between the measurement and the design. Thus, the middle of the wall registered by ICP shows shrinkage in green, while the proposed algorithm correctly exhibits the curl in the upper portion of the thin wall in red. However, other than the design 92 18 19 20 21 0 50 100 150 0 25 50 75 100 125 Layer Index Plotting Statistic Figure 4.6: MEWMA control chart for the thin wall, where the red line is the control limit, and indices show detected out-of-control signals. 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 (a) Design deviation 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 (b) Estimated deviation 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8 −1.0 (c) Estimated deviation by ICP Figure 4.7: Shape deviations comparing to the straight thin wall bending effect, processing-incurred shape deviations appear at the upper portion of the thin wall, thus it is tilted more than the design. Moreover, the spatters suggests an insignificant level of surface roughness. In the next step, the printed thin wall is registered to the titled wall design to uncover the true deviation of the manufacturing system. 4.4.3 Registration to the Tilted Thin Wall Setting the tilted thin wall to be the design, the measurement can still be registered by the proposed method and the ICP algorithm. Unlike Sec. 4.4.2, there is no knowledge of the 93 16 17 18 19 20 21 22 0 50 100 150 0 25 50 75 100 125 Layer Index Plotting Statistic (a) MEWMA control chart 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 (b) Estimated deviation 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 0.2 0.1 0.0 −0.1 (c) Estimated deviation by ICP Figure 4.8: Shape deviations comparing to the tilted thin wall shape deviation in this case. The registration results are shown in Fig. 4.8. Since the wall bends more than designed, the out-of-control process caused by global distortion is captured quickly by the MEWMA control chart. Compared to the deviation recovered by the proposed method (Fig. 4.8(b)), ICP states that there is almost no deviation since Fig. 4.8(c) is around zero everywhere. In order to validate that the proposed registration method can reveal the actual deviation incurred, a naive compensation plan is established by adding the detected deviation to the CAD design. For example, according to Fig. 4.8(b), there is an expansion of 0.4 mm at ofz = 25 mm. Then, the design is shortened by 0.4 mm to counteract the expansion. It should be noted that better compensation methods based on the parametric model could achieve superior results [13]. However, the compensation here aims to illustrate the existence of shape deviations and the reliability of the proposed registration method. The compensated thin wall is registered again with the result shown in Fig. 4.9. The process is out-of-control after layer 42 due to the layer roughness. According to the shape deviation decomposition proposed in Sec. 4.2.1, however, it is hard to correct surface roughness related to the printing process like the layer thickness or machine vibration. By comparing Fig. 4.9(b) to Fig. 4.8(b), the bending effect is successfully compensated by the naive compensation plan since most of the area shows no deviation in green. Also, the ICP algorithm detects no difference between the tilted thin wall before and after compensation, which indicates its inconsistency. In 94 42 43 44 46 50 58 59 60 61 62 63 64 65 66 67 0 50 100 150 0 25 50 75 100 125 Layer Index Plotting Statistic (a) MEWMA control chart 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 0.2 0.1 0.0 −0.1 −0.2 (b) Estimated deviation 0 20 40 60 −20 −10 0 10 20 x (mm) z (mm) Δ (mm) 0.2 0.1 0.0 −0.1 −0.2 (c) Estimated deviation by ICP Figure 4.9: Shape deviations of the compensated thin wall contrast, the proposed shape registration algorithm unveils the actual shape deviation incurred by the AM process and can be corrected by changing the design. 4.5 Conclusions This work proposes a new shape registration method for robust fabrication accuracy assess- ment and shape deviation correction in AM. The methodology sequentially estimates the param- eters in a rigid transformation that aligns the measurement data to its intended shape and reveals the true geometric deviation. By exploiting the ground points as a reference, parameters decid- ingz-coordinates are estimated. Then a statistical control chart is adopted to filter the reliable data with minimal distortion according to the monitoring statistics selected by the proposed deviation decomposition. Simulation and experimental studies show that the proposed method can identify the process-induced shape deviation effectively and consistently. Future research will expand the work on the consistent registration of various shapes. 95 Chapter 5 Small-Sample Learning of 3D Printed Thin-Wall Structures Using Printing Primitives 5.1 Introduction Due to the one-of-a-kind nature of AM and the trend of personalized manufacturing, a very small volume of parts is often fabricated with a large variety of 3D geometries [13, 46]. The mismatch between the large design variety and the small sample size for each design results in a challenging small-sample learning problem in AM quality control. This study aims at methodology extension to thin-wall structures, a challenging category of geometries in AM. The geometric shape deviation of AM built thin walls is not only caused by material shrinkage but also warping due to thermal stresses [155]. The proposed small-sample learning approach introduces the concept of printing primitives, which describes the fabrication of geometric primitives through a convolution modeling of AM. It considers the physical stack- up of primitives in the layer-by-layer printing process. As shown in Fig. 5.1, building the thin- wall structures can be regarded as stacking beams layer upon layer. By approximating each beam with a few geometric primitives, the whole product is decomposed into many pieces or blocks. Thus, deviation of the thin wall structure can be described as the physical accumulation of primitive deviations. 96 Figure 5.1: Fabrication of a curved wall by stacking curved beams The remainder of this chapter is organized as follows. Section 5.2 reviews the geometric primitives in the literature and proposes the printing primitive, a fabrication-aware representa- tion of geometric primitives. A small-sample learning method based on printing primitives is developed to predict deviation of thin-wall structures. In Sec. 5.3, joint modeling of curved beams and walls of different sizes is investigated. Conclusion and future work are discussed in Sec. 5.4. 5.2 A Small-Sample Learning Strategy for Thin-Wall Struc- tures Using Printing Primitives In this section, the definitions of geometric primitive and printing primitive are specified. Then, their deviation modeling strategies in 2D and 3D spaces are discussed. To stack up the printing primitives, a layer interaction function is adopted in the convolution learning frame- work and estimated through a tensor basis. 97 5.2.1 Geometric Primitives and Printing Primitives Many efforts have been devoted to the geometric primitives in computer vision, pattern recognition, and CAD systems for shape representation, detection, registration, segmentation, and reconstruction. Xia et al. [156] classified them into two groups: shape primitives and struc- ture primitives. For 2D geometries, shape primitives include segments of straight and curved lines, while structure primitives contain skeletons and 2D outlines. In 3D cases, both surfaces (e.g., planar and spherical patches) and volumetric shapes (e.g., cubes, cylinders, spheres, cones, and tori) are considered shape primitives, and structure primitives include 3D edges. Compared to shape primitives, structure primitives constitute object profiles. However, learning and predicting quality based on purely geometric primitives cannot cap- ture complications of the physical fabrication of 3D objects. In order to accurately predict the build quality of thin-wall geometries, we mathematically decompose the AM process. The first step is to decompose the part into 2D beams. Each beam as a 2D shape can be further decom- posed into 2D shape primitives, including straight and curved line segments. By introducing location and size covariates for a small sample of beams, the beam deviations can be learned and predicted. Since a thin wall can be viewed as a physical stack-up of beams, a convolution learning framework established in [16] can be applied. It captures the error accumulation of beams through a layer interaction function. In the same manner, the 3D printing primitives, or the stack-up of 2D shape primitives, can be established through the convolution learning framework. For example, to learn the shape deviation of a curved wall in Fig. 5.1, circular sectors are adopted as 2D geometric primitives. By segmenting each beam into multiple pieces or blocks, the deviation of each circular sector is estimated from a few beam products or the bottoms of thin walls, where less error accumulation across layers happens. After building the predictive model for the deviation of 2D shape primitives and beams, the deviation of the thin wall can be learned by stacking the beam deviations through the convolution framework. Notice that, for 98 complex geometries, more types of geometric primitives need to be included for an accurate approximation of the products. 5.2.2 Deviation Modeling for Thin-Wall Structures Using Printing Prim- itives in AM Deviation modeling of 2D geometric primitives: For 2D geometries, shape deviation in each horizontal layer can be defined in the polar coordinate system as [13] () =() 0 (); (5.1) where2 [0; 2) is the azimuthal angle, defines the actual shape of the layer as a function of , and 0 corresponds to the designed shape. Assume a shape is segmented into n blocks and approximated by m types of geometric primitives, then it can be presented as () = m X i=1 n X j=1 i (# j )1 2[# j ;# j+1 ) ; (5.2) where # j 2 [0; 2);8j = 1;:::;m are the cutting angles for each block with # m+1 = 2. Essentially, the shape is approximated by multiple connected sectors and the angle within each block or geometric primitive is from 0 to# j+1 # j . Representing the design 0 similar to Eq. (5.2) introduces a primitive-based shape deviation as () = m X i=1 n X j=1 i (# j )1 2[# j ;# j+1 ) ; (5.3) where i demonstrates the deviation pattern of the type-i primitive, which can be modeled as i (# j ) =f i (# j ) +; (5.4) 99 where is the measurement error. By substituting Eq. (5.4) into Eq. (5.3), the 2D primitive-based deviation model is () = m X i=1 n X j=1 f i (# j )1 2[# j ;# j+1 ) +: (5.5) Deviation modeling of 3D printing primitives: For 3D cases, the convolution learning framework [16] is applied to stack up the 2D geometric primitives to 3D printing primi- tives. Defining the shape deviation of 3D geometries in cylindrical coordinate system as (;z) =(;z) 0 (;z), the shape deviation of the whole product can be modeled as (;z) = 0 + 1 (fg)(;z) +; (5.6) wheref demonstrates the in-plane shape deviation of 2D shapes,g describes the layer interac- tions or error accumulation across layers, and is measurement error. Bothf andg functions are normalized so that the scale and shift effects are controls by parameters 0 and 1 . By replacing thef function in Eq. (5.6) with the 2D primitive-based model in Eq. (5.5), the 3D primitive-based deviation learning framework is (;z) = 0 + m X i=1 n X j=1 i (f i g)(# j ;z)1 2[# j ;# j+1 ) +; (5.7) whereg describes the layer interaction of shape deviations. 5.2.3 Tensor Basis Expansion for the Layer Interaction of Thin Walls Though parametric forms of the layer interaction functiong were proposed and estimated in [16] and [17], a tensor basis expansion provides more flexibility when the training data for 2D geometric primitives is limited. 100 Assume that the interaction effect along and z directions is separable, i.e., g(;z) = g () ~ g z (z), then the convolution in Eq. (5.7) can be written explicitly as (f i g)(;z) = Z 2 0 f i (t)g 1 (t)dt Z z 0 g 2 ()d : (5.8) Since R z 0 g 2 ()d is still a function ofz, we denote it asg z (z). Substituting Eq. (5.8) into Eq. (5.7) gives (;z) = m X i=1 n X j=1 i Z # j+1 # j f i (t)g (t)dt g z (z) + 0 +; (5.9) where g () = n X k=1 b k () k (5.10) and g z (z) = nz X k=1 b z k (z) z k (5.11) are basis expansions, i.e.,b k andb z k are basis functions like cubic splines. Essentially, the effect of interlayer interactions or error accumulation is modeled by the tensor basis expansion. 5.3 Case Study for Methodology Validation In this section, two curved beams and thin walls are fabricated through the fused deposition modeling (FDM) process to demonstrate the capability of the proposed primitive-based small- sample learning strategy. 5.3.1 AM Experiments and Observations As shown in Fig. 5.2, two curved beams and thin walls with 100 mm length and 3 mm thickness are fabricated through the 5th generation MakerBot Replicator desktop printer, which 101 Figure 5.2: Two curved beams and thin walls with different radii fabricated in a FDM process builds 3D objects with PLA filaments through the FDM process. The heights of thin walls are 30 mm, which are ten times the heights of curved beams, i.e., the beams are of 3 mm height. Thus, the curved walls can be regarded as stacks of curved beams. As illustrated in the middle of Fig. 5.2, the two curved walls can be considered as one-third and one-sixth of cylindrical walls with different radii. The curvature or radius can classify the four products into 60-degree parts (in black) with 100 mm radii and 120-degree parts (in blue) with 57.7 mm radii. The layer thickness is set to be 0.2 mm, and 95% in-fill is selected to produce almost solid shapes. Four parts are placed in the middle of the building platform and printed separately to reduce the impact of spatial repeatability of the AM system. The Hexagon ROMER Absolute Arm with the RS4 laser scanner is used to collect the point cloud data on the product surfaces with a measurement error of 0.08 mm. The measurement data are registered to the designs using the method proposed in [119], which implements a restricted iterative closest point algorithm [130] for stable identification of shape deviations. Due to the small thickness of both beams and thin walls, the deviation on both sides are similar. Thus, only the front side shape deviation is considered. As defined in Sec. 5.2.2, the 102 (a) 60-Degree Curved Beam (b) 120-Degree Curved Beam (c) 60-Degree Curved Wall (d) 120-Degree Curved Wall Figure 5.3: Shape deviation profiles of two curved beams and two thin walls deviation profiles of two curved beams and thin walls are shown in Fig. 5.3. Note that two curved beams have different domains in the polar coordinate system since they correspond to different portions of a full circle. While the front side of 120-degree beam has radius of 54.7 mm, the radius of the 60-degree beam is 97 mm. Different deviation patterns can be observed for beams with different radii, and the pieces with larger for both beams are more noisy than the other sides due to the surface roughness. Similar phenomenons can be found in the thin walls in both Fig. 5.2 and Fig. 5.3 (c) and (d). 103 5.3.2 Initial Shape Deviation Modeling through Equally-Spaced Geomet- ric Primitives For the curved beams in Fig. 5.2, circular sectors as 2D geometric primitives are enough to represent the shapes. Since only one type of primitives is needed, we havem = 1, and the model in Eq. (5.5) is reduced to (jr 0 ) = n X j=1 f(# j jr 0 ;# j )1 2[# j ;# j+1 ) +: (5.12) Here, we write the covariates explicitly to emphasize that the deviation pattern of each circular sector is related to the size effectr 0 and the location effect# j . Thus, distinct deviation patterns are expected for the curved beams with different radii at different locations. To learn the shape deviation of the circular primitives, the curved beams are first segmented to blocks of equal sizes, i.e., # j+1 # j = # 0 = 15 . Due to the simplicity of the deviation pattern observed in Fig. 5.3 (a) and (b), a single cosine function is used to model the primitive deviation as f(# j jr 0 ;# j ) = 0 + 1 cos[n(# j ) + ]: (5.13) The following three conjectures are proposed for the parameters: (1) 0 = 0;1 + 0;2 r 0 + 0;3 # j , (2) 1 = 1;1 + 1;2 r 0 + 1;3 # j , (3) = 1 + 2 # j . Assuming both 0 and 1 to be functions ofr 0 and# j , magnitude and mean shift of the devi- ation pattern are decided by the size and location of each primitive. While period or frequency of the pattern is assumed to be a constant shared among all blocks, the phase is a function of the location effect and does not change with different sizes. The estimated parameters and their standard errors are provided in Table 5.1. Model prediction results are shown in Fig. 5.4. 104 Table 5.1: Initial parameter estimates for curved beams deviations Parameter Estimate Standard Error 0;1 1.1191 0.015723 0;2 -0.0066 0.000288 0;3 -0.3743 0.006268 1;1 1.2446 0.018769 1;2 -0.0067 0.000317 1;3 -0.4836 0.008296 n 1.5605 0.001371 1 2.2287 0.047491 2 0.9647 0.027708 0.0484 0.001015 (a) (b) Figure 5.4: (a) Initial model prediction of curved beams, where points are the measured devia- tion and curved line segments are the model predictions. (b) Actual and predicted shape devia- tion of two curved beams. Blocks of shape primitives are represented in different colors. The model performance is satisfactory since all parameters are significant at the 0.001 level. The root mean square error (RMSE) is 0.0484 mm, and the mean absolute error (MAE) is 0.0368. Other than the two boundary blocks of the 120-degree curved beam, i.e., block 1 in dark blue and block 8 in dark red, the model predictions accurately capture the mean trend of the shape deviation within each block. Model performance for the 60-degree beam is superior to the 120-degree beam since all four blocks are present in both parts, then the location effect can be better estimated with replicates. 105 (a) (b) Figure 5.5: (a) Smoothed model prediction of curved beams, where points are the measured deviation and curved lines are the model predictions. (b) Actual and predicted shape deviations of two curved beams with 60-degree beam deviations in black and 120-degree beam deviations in blue. 5.3.3 Shape Deviation Model Refinement and Extension to Printing Prim- itives of Thin Walls One drawback of the model described in Sec. 5.3.2 is that the model predictions in different blocks are not connected, while the actual shape deviation profile of any product is continuous. Thus, the model predictions for the curved beams need to be smoothed. Assume that the size effect 0 is a random effect and apply cubic spline functions over to connect and smooth the pieces of shape primitives. The final model estimation is presented in Fig. 5.5 with RMSE and MAE being 0.0547 and 0.0408, respectively. The model perfor- mance is slightly worse to preserve the smoothness of the predictions across different sections. Although the end of the 60-degree beam with large has significant noise, the model prediction is reasonable. By connecting the neighbor blocks, better model performance is achieved for the two ends of the 120-degree beam. The actual-versus-fitted plot in Fig. 5.5 (b) also indi- cates a good model performance with around 0.1 mm difference, which is at the same level of measurement system error. Selecting 20 cubic splines forg and 5 cubic splines forg z in Eq. (5.9), the model prediction is shown in Fig. 5.6, with RMSE being 0.1049 and MAE being 0.0806. Comparing the model 106 (a) 60-Degree Curved Wall (b) 120-Degree Curved Wall (c) Figure 5.6: (a) and (b) Primitive-based convolution model prediction of two thin walls. (b) Actual and predicted shape deviation of two curved walls with 60-degree wall deviations in black and 120-degree wall deviations in blue. predictions in Fig. 5.6 (a) and (b) to the actual shape deviations in Fig. 5.3 (c) and (d), we can find that the deviation patterns of both curved walls are well-captured. The concentration of points around the 45-degree line in Fig. 5.6 (c) also indicates the great model performance of the proposed primitive-based deviation learning methodology. 5.4 Conclusion This work proposed a primitive-based shape deviation learning framework to tackle the mismatch between the large design variety and the small sample size in AM. By dividing each training sample into multiple pieces of printing primitives, a small number of samples are trans- formed into a large number of primitives to ensure a stable model estimation. Combined with the convolution learning framework, the established deviation modeling strategy can accurately capture the deviation pattern of curved thin-wall structures. One limitation is that the spatial correlation is not considered in this work. In the future, more complicated thin walls will be included to demonstrate the capability of the proposed methodology, and Gaussian process regression will be adopted to capture the spatial correlation. 107 Chapter 6 Discussion and Future Work As one vital component of Industrial 4.0 and cyber-physical systems, AM promises a future of producing highly personalized products with high efficiency at low cost. Despite the capabil- ity of achieving complexity-free manufacturing, AM faces critical challenges in quality control, especially the geometric shape accuracy. Due to the one-of-a-kind nature of AM, a small num- ber of parts are often fabricated with various geometries. To address the mismatch between product quantity and shape variety, a prescriptive modeling strategy to learn and predict shape deviations of heterogeneous 3D geometries has been proposed by incorporating the process and geometry information. There are four main tasks of this dissertation research: (i) Engineering-informed convolution modeling of shape deviation in smooth 3D geometries, (ii) joint learning of heterogeneous shape deviation patterns, including smooth and non-smooth 3D geometries, (iii) robust shape registration for error compensation of thin-wall structures, and (iv) expanding small sample size of thin walls through 3D printing primitives. Consistent with the previous prescriptive modeling works in 2D, the first task establishes an engineering-informed ML framework for 3D shape deviation modeling by characterizing the layer-by-layer AM process through a mathematical integration and capturing the interlayer interactions through a transfer function. Process insights are derived by decomposing the pre- dictive model into additive components and associating each term with the physical understand- ing of the error generation mechanism. For heterogeneous deviation surfaces caused by different geometries and sizes of the prod- ucts, mathematical decomposition of the manufacturing process is proposed in the second task. The manufacturing process of a non-smooth target shape is divided into two steps. First, a 108 smooth geometry circumscribing the target shape is produced; then, each layer of the non- smooth geometry is carved out from the corresponding layer of the smooth shape. 3D cookie- cutter function is developed to capture the association of smooth and non-smooth shape devia- tions. And a hybrid distance measure is proposed to accurately capture the spatial correlation of the shape deviation. To effectively evaluate the shape deviation and to help the error compensation in AM, a robust shape registration method is proposed in the third task, which sequentially constraints the six DoFs of a rigid transformation to align the measured point clouds with the designed shape. After segmenting and aligning the ground points, layers are regarded as individual observations feeding into a statistical control chart, where the monitoring statistics are estimated based on a novel deviation decomposition scheme. This proposed method separates the non-conforming portion of the product for error compensation. The mismatch between the large shape variety and the small sample size is exaggerated for the deviation modeling of freeform shapes. The idea of printing primitives is proposed in the fourth task to expand the sample size by considering a shape as the composition of multiple printing primitives with covariates like locations, rotations, and sizes. Considering the 3D printing primitives as the stack-up of 2D geometric primitives, including straight lines, circular segments, and corners, brings the opportunity of applying the convolution framework to describe the error-generating mechanism. Then the challenging small-sample learning problem can be addressed by segmenting and approximating freeform shapes with a few types of 3D printing primitives, which is similar to the dimension reduction idea. The initial study utilizes the circular beams as the printing primitives to construct the curved walls, where the shape deviation can be modeled using the convolution framework proposed in the first task. This work shows the potential to be extended to 3D freeform shapes. To summarize, this dissertation establishes a novel prescriptive learning framework for mod- eling and predicting the shape deviations of 3D geometries from a limited number of samples in AM, which can be applied to a generic 3D printing process to evaluate and compensate for the 109 geometric inaccuracy. The developed methodologies highly contribute to the knowledge base of ML for advanced manufacturing. In the future, I am planning to extend this dissertation research and explore the following two aspects: • Shape deviation modeling for freeform 3D geometries. Though the fourth task shows the preliminary work to model the shape deviation of curved thin walls by the stack-up of curved beams, further studies must be conducted to extend the modeling framework to fully 3D freeform geometries. First, we need to segment and approximate freeform 3D geometries with the set printing primitives. Both the CAD design and measurement data need to be considered to achieve the "most informative" segmentation, which is related to the predictive model with multiple covariates such as sizes and locations. Sec- ond, reconstruction of the complete deviation surface from primitive patches needs to be established. The main challenge is how to connect the deviation patches of differ- ent primitives and when sharp transitions could happen. Last, this dissertation research mainly focuses on the mean pattern of the shape deviation, while the spatial correlation has not been specifically studied. Distinct correlation structures are expected for different primitives or shapes, and the convolution framework may also be applied to model the spatial correlation of 3D geometries. • Process qualification for AM. Besides the product quality assessment, modeling, and compensation, the process- or system-level qualification is crucial for quality control. Research is needed to find an optimal design of artifacts for process capability analysis and system deviation assessment, making the automatic calibration of the printing sys- tem feasible. Suitable information metrics and optimality criteria for selecting printing primitives need to be defined. Then, a minimal set of complex geometries can be fab- ricated to estimate the shape deviation considering the covariates and assess the spatial repeatability of the platform. 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Abstract (if available)

Abstract As a revolutionary technology, additive manufacturing (AM) or three-dimensional (3D) printing enables producing of personalized products with highly complex geometries through layer-by-layer fabrication using various materials, including metals, ceramics, polymers, and their composites, hybrid, or functionally graded materials. Unlike traditional subtracting manufacturing methods, such as milling, machining, carving, and shaping, AM has the potential to build extremely complex geometries with high efficiency and low material waste. However, one major barrier to the broader adoption of AM techniques is geometric shape inaccuracy.

However, due to the vast spectrum of processes, high variety of geometries, and low volume of samples, accuracy control has been a daunting task for researchers and practitioners. In this dissertation research, an engineering-informed machine learning (ML) methodology is proposed to learn and predict the shape deviation of 3D shapes from a limited number of fabricated products. The ultimate goal is to overcome the bottleneck of inadequate shape accuracy and to enable the compensation or calibration for general 3D printing systems with a few testing artifacts.

To achieve this objective, the first task is to establish a statistical modeling framework to integrate the knowledge of in-plane and out-of-plane shape deviation modeling and to learn 3D shape deviation patterns. By characterizing the layer-by-layer manufacturing process through a mathematical integration and capturing the interlayer interactions through a transfer function, a convolution learning framework is established to describe the error accumulation mechanism in AM. Experimental validations are conducted with the out-of-plane shape deviation of vertical-printed half-disks and 3D shape deviation of domes printed in the stereolithography process. Process insights are derived to relate the model components to the input shape effect, layer interaction effect, and gravity effect.

However, due to the smoothing effect of convolution operation, the proposed convolution learning framework cannot be applied to non-smooth 3D geometries since their deviation patterns contain sharp transitions at the edges and corners. To further extend the model to a wider category of 3D geometries, the production of a non-smooth geometry is decomposed into two steps. First, a smooth 3D shape circumscribing the target geometry is produced. Then, each layer of the non-smooth geometry is "carved" out from the corresponding layer of the smooth shape. 3D cookie-cutter function is proposed to capture the association of smooth and non-smooth shape deviations. A unified model for heterogeneous 3D geometries is established and validated through a sequential model estimation of the domes, as the stack of circular shapes, and thin walls of half-cylindrical shape, as the stack of rectangles.

Before extending the methodology to 3D freeform geometries, a new shape registration strategy is proposed for effective product qualification and error correction of AM fabricated parts, because the efficacy of learning from shape deviation data relies on proper alignment between the printed product and its intended design. Robust to distortion of the products, a novel shape deviation decomposition is proposed, and statistical control chart is applied to filter the reliable portion of the data to sequentially constrain rigid transformation parameters to reveal the true deformation of the product.

The last task is to model and predict the shape deviation of 3D freeform shapes from a limited number of training products. To address the small-sample learning problem, a set of printing primitives is employed to approximate and represent freeform 3D geometries. Following a similar idea to dimension reduction, the complex target geometry can be segmented or represented by a few types of printing primitives. Thus, it is sufficient to learn the deviation patterns of these printing primitives to model and predict any untried freeform shapes. One choice of the primitives would include flat surfaces, spherical patches, and corners, which fall into the category of smooth or non-smooth shapes that have been studied.

These four tasks build upon each other to establish a unified learning and prediction methodology for shape deviations of arbitrary untried geometries from a limited number of 3D printed products. This would greatly help to relieve the burden of geometric shape inaccuracy in AM. Moreover, this strategy can be applied to any AM process with all kinds of materials and printing mechanisms, though transfer learning methods would be desired to enable efficient learning across multiple printing processes.

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Asset Metadata

Creator Wang, Yuanxiang (author)

Core Title Fabrication-aware machine learning for accuracy control in additive manufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2022-08

Publication Date 07/22/2022

Defense Date 05/04/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag additive manufacturing,machine learning,OAI-PMH Harvest,quality control,shape deviation

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Huang, Qiang (committee chair), Carlsson, John Gunnar (committee member), Chen, Yong (committee member)

Creator Email yuanxian@usc.edu,yuanxiangwang94@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111373920

Unique identifier UC111373920

Legacy Identifier etd-WangYuanxi-10929

Document Type Dissertation

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Type texts

Source 20220722-usctheses-batch-961 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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