University of Southern California Dissertations and Theses

Statistical modeling and machine learning for shape accuracy control in additive manufacturing

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Statistical modeling and machine learning for shape accuracy control in additive manufacturing

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Content STATISTICAL MODELING AND MACHINE LEARNING FOR SHAPE ACCURACY CONTROL IN ADDITIVE MANUFACTURING by He Luan 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) May 2018 Copyright 2018 He Luan Acknowledgments First, I would like to express my most sincere gratitude to my advisor, Dr. Qiang Huang for his kindly help, support and encouragement during my Ph.D. study. He was always there whenever I encountered difficulties. I really appreciate his intelligent guidance, invaluable comments, and detailed training during the whole dissertation work. This dissertation would not have been possible without him. I would like to express my gratitude to Dr. Yong Chen, Dr. Meisam Razaviyayn, Dr. Arman Sabbaghi and Dr. Jun Zeng for serving in my dissertation committee, and Dr. Berok Khoshnevis, Dr. Wei Wu and Dr. Phebe Vayanou for serving in my qualifying exam commit- tee. All your constructive comments and suggestions made this dissertation better. I would thank all my collaborators, Dr. Arman Sabbaghi, Dr. Marco Grasso and Dr. Bianca M. Colosimo, for their support and instructions, especially Dr. Arman Sabbaghi, who is always ready to give you kindly instruction and encouragement. IwouldalsothankDr. JunZeng, mymentorduringmyinternshipatHPLabs, forbringing me into the industrial research. His amazing guidance and impressive attitude toward research alwaysinspiremetobebetter. ThankallmycollaboratorsatHPLabs,includingSamStodder, Jordi Roca, David Murphy, and Thomas Paula for supporting me finishing my collaborative research work in this dissertation. Also, my Ph.D. study could not have been so unforgettable without all my colleagues including Yuan Jin, Yanqing Duanmu, Zhengyu Zhang, Yuanxiang Wang, Ye Wang and Andi Wang. ii Finally, I would like to give my deepest thanks to my parents, who are there to encourage me, calm me down and cheer for me all the time. Their tireless love supports me, their only child, to cross the Pacific ocean and pursue my dream. My last thanks go to my boyfriend, Yawen Liu. I can’t imagine how my Ph.D. life would be without you. iii Contents Acknowledgments ii List of Tables vii List of Figures ix Abstract xiii 1 Introduction 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Additive Manufacturing and Potentials . . . . . . . . . . . . . . . . . . 2 1.1.2 Research Challenges on Shape Accuracy Control in Additive Manufac- turing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 State of the Art on Shape Accuracy Control in AM Research . . . . . . . . . . 7 1.2.1 Physical Modeling based Approaches . . . . . . . . . . . . . . . . . . . 7 1.2.2 Statistical Modeling based Approaches . . . . . . . . . . . . . . . . . . 8 1.2.3 Process planning adjustment and calibration . . . . . . . . . . . . . . . 14 1.3 Research Tasks and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Prescriptive Modeling and Compensation of In-plane Shape Deviation for 3D Printed Freeform Products 20 2.1 Predictive Modeling of Polyhedron Products – Previous Work . . . . . . . . . 21 2.2 Two Plausible Strategies for Prescriptive Methodology Extension to Freeform Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Strategy A: Polygon Approximation with Local Compensation (PALC) 23 2.2.2 Strategy B: Circular Approximation with Selective Cornering (CASC) . 24 2.2.3 RelatedApproachesofPolygonApproximationtoFreeformandMethod Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Strategy Evaluation and Selection through Experimental Investigation . . . . . 25 2.4 Prescriptive Modeling of Freeform deviation through Circular Approximation with Selective Cornering (CASC) . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 Prescriptive Modeling of Freeform deviation via CASC . . . . . . . . . 29 2.4.2 Prescriptive Model Estimation based on Limited Trial Shapes . . . . . 34 2.4.3 Prescriptive Model Validation through Experimentation . . . . . . . . . 36 iv 2.4.4 Prescriptive Compensation and Experimental Validation . . . . . . . . 37 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Statistical Process Control of In-Plane Shape Deviation for Additive Man- ufacturing 41 3.1 Prescriptive Monitoring of Deviation From Shape to Shape . . . . . . . . . . . 42 3.1.1 A Prescriptive SPC Scheme for Shape Deviation Monitoring in AM . . 42 3.1.2 Prediction of Shape Deviation of A New Product . . . . . . . . . . . . 45 3.1.3 Control Chart to Monitor Shape Deviation . . . . . . . . . . . . . . . . 46 3.1.4 Process Capability Index C p for AM Processes . . . . . . . . . . . . . . 47 3.1.5 Estimation of μ η and σ η . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Monitoring Stereolithography Process: Methodology Demonstration . . . . . . 48 3.2.1 Experiments to Obtain Training Data and Prescriptive Model Estab- lishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 Monitoring Statistic η and Phase-I Control Charting . . . . . . . . . . 48 3.2.3 Phase-II Control Charting and Validation . . . . . . . . . . . . . . . . 51 3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Prescriptive Data-Analytical Modeling of Selective Laser Melting Processes for Accuracy Improvement 53 4.1 Review of Existing SLM Literature for Geometric Accuracy Control . . . . . . 54 4.2 Proposed Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Prescriptive Shape Deviation Modeling for Freeform Shapes in SLM with Sur- face Roughness Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.1 Shape Deviation Experimentation and Initial Analysis of Shape Devia- tion for SLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3.2 Filtering Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.3 Initial Model Building and Validation by Filtering Surface Roughness . 65 4.4 Laser Beam Positioning Error Prediction and Elimination with Error Equiva- lence Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.1 Location Effect Experimentation . . . . . . . . . . . . . . . . . . . . . 68 4.4.2 Predictive Modeling of Laser Beam Positioning Error . . . . . . . . . . 69 4.4.3 Transforming Positioning Error into the Equivalent Amount of Shape Design Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5 Analysis and Modeling of Additional Location Effects . . . . . . . . . . . . . . 73 4.5.1 Estimation of Location Effect Term x 0 (s) at Different Locations . . . . 74 4.5.2 Predictive Modeling of Location Effect Term x 0 (s) . . . . . . . . . . . . 76 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5 Data-Driven Modeling of Thermal Physics in Powder Bed Fusion 3D Print- ing Using Deep Learning 79 5.1 Introduction and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Challenge: Complex Process Physics and the Need for Quantitatively Accurate Process Modeling and Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 81 v 5.2.1 Introduction to HP’s Multi Jet Fusion . . . . . . . . . . . . . . . . . . 82 5.2.2 Challenge: End-Part Quality . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 Our Method: Apply Deep Learning to Achieve Quantitative Predictive Tool . 88 5.3.1 Deep Learning and Advantages . . . . . . . . . . . . . . . . . . . . . . 88 5.3.2 Key Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.3 Workload Estimate and Performance Modeling . . . . . . . . . . . . . 92 5.4 Application1: One Step Ahead Fusing Layer Thermal Behavior Prediction . . 94 5.4.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.2 Scalable Solution Accounting for Bed-Level Inference . . . . . . . . . . 97 5.4.3 Architecture Learning Experience . . . . . . . . . . . . . . . . . . . . . 98 5.4.4 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.5 Application2: Learning Heterogenous Thermal Diffusivity as Deep Neural Net- work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5.1 Physical Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5.2 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5.3 Architecture Learning Experience . . . . . . . . . . . . . . . . . . . . . 111 5.5.4 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.1 Extension to Other Techniques . . . . . . . . . . . . . . . . . . . . . . 119 6 Discussion and Future Extensions 120 Reference List 125 vi List of Tables 1.1 SummaryandcomparisonofcommonAdditiveManufacturingtechniquesdefined by ASTM (Standard (2012)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Summary of Design Of Experiments (DOE) methods for process setting opti- mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Summary of empirical process modeling methods for part quality prediction . 11 1.4 Summary of feedback control methodologies in AM . . . . . . . . . . . . . . . 15 2.1 MIP-SLA process settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Estimated parameters for cylinder model . . . . . . . . . . . . . . . . . . . . . 35 2.3 Estimated parameters for polyhedron model . . . . . . . . . . . . . . . . . . . 35 2.4 Compensation performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Design of Training Sample Products . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Monitoring statistic η i of training data . . . . . . . . . . . . . . . . . . . . . . 49 3.3 EWMA control limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Monitoring statistic η i of validation data . . . . . . . . . . . . . . . . . . . . . 51 4.1 The specific parameters of the SLM process . . . . . . . . . . . . . . . . . . . 59 4.2 Summary of posterior draws in cylindrical basis model . . . . . . . . . . . . . 60 4.3 Summary of posterior draws for estimating freeform model . . . . . . . . . . . 60 4.4 Surface roughness indexes for cylinder with r 0 = 5mm . . . . . . . . . . . . . 64 4.5 Summary of posterior draws in cylindrical model . . . . . . . . . . . . . . . . . 65 vii 4.6 Summary of posterior draws for estimating freeform model . . . . . . . . . . . 66 4.7 Summary of model fitting for positioning errors in x− and y− directions with 95% confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.8 Goodness of fitting for positioning error in x− and y− directions . . . . . . . . 71 4.9 Summary of model fitting for location effect term x 0 (s) with 95% confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.10 Goodness of fitting for location effect term x 0 (s) . . . . . . . . . . . . . . . . . 76 4.11 Fitted vs predicted x 0 for the three cylinders in Figure 4.2 (unit: mm) . . . . . 77 5.1 Operation computation in fusing layer heat map prediction, covering one patch of the input. (Note that the first two CNN units occur once for each sequence of the Conv-LSTM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Performance quantification of fusing layer thermal prediction . . . . . . . . . . 105 5.3 Performance quantification of thermal diffusivity DNN . . . . . . . . . . . . . 114 viii List of Figures 1.1 Generalized procedure of 3D printing (Huang et al. (2015)) . . . . . . . . . . . 2 1.2 Schematic diagram of the three techniques . . . . . . . . . . . . . . . . . . . . 5 1.3 Relationship of the four tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Modeling extension from cylinder (Huang et al. (2015)), polyhedron (Huang et al. (2014b)) to freeform (this work) . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Strategy of polygon approximation with local compensation (PALC) (solid line: freeform shape; dash line: approximated polygon shape; shadow area: amount of compensation to approach the freeform from the polygon shape) . . . . . . 23 2.3 Strategy of circular approximation with selective cornering (CASC) (solid arcs: actual radii of sectors; solid line segments: polygons sides) . . . . . . . . . . . 24 2.4 Printed freeform shape and corresponding similar polyhedron . . . . . . . . . . 26 2.5 deviation profiles of convex freeform and corresponding polygon approximated 27 2.6 Polygon approximation of freeform shapes . . . . . . . . . . . . . . . . . . . . 31 2.7 deviation and prediction profiles of simple trial shapes . . . . . . . . . . . . . . 34 2.8 deviation and prediction profiles for concave freeform shape . . . . . . . . . . . 37 2.9 Freeform shape deviation profiles: before (blue lines) and after (red lines) com- pensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.10 Trend of freeform shape deviation profiles: before and after compensation . . . 39 3.1 Training data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ix 3.2 Statistic for AM process monitoring . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Absolute shape area deviation (Huang (2016)) . . . . . . . . . . . . . . . . . . 46 3.4 Normal Q-Q plot of sample data η i . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 phase-I EWMA control chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 phase-II EWMA control chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1 Flow chart of proposed strategy for accuracy improvement . . . . . . . . . . . 56 4.2 Design of shape deviation experimentation on a single plate (unit: mm) . . . . 59 4.3 Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by cylindrical basis model g 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by freeform model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of the polyhedron shapes by freeform model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 Deviation profile (dots) and prediction (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of 12mm freeform shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.7 Measured surface roughness profile for cylinder with r 0 = 5mm . . . . . . . . . 63 4.8 Illustration of roughness influence in PCS . . . . . . . . . . . . . . . . . . . . . 64 4.9 Actual deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by cylindrical basis model g 1 while filtering surface roughness . 66 4.10 Actual deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by freeform model while filtering surface roughness . . . . . . . 66 x 4.11 Actual deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of the polyhedron shapes by freeform model while filtering surface roughness . . . . . 67 4.12 Actual deviation profile (dots) and prediction (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of 12mm freeform shape while filtering surface roughness . . . . . . . . . . . . . 67 4.13 Experimental grid with 9× 9 small cylinders in location effect experimentation 69 4.14 Patternofthelaserbeampositioningerror. Eachpointdenotesthedesiredposi- tion of cylinder center. Measured and predicted positioning errors are denoted by solid and dashed arrows, respectively . . . . . . . . . . . . . . . . . . . . . 70 4.15 Measurement (points) and prediction profiles (surface) of the laser beam posi- tioning error (unit: mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.16 Model fitting residual plots of the laser beam positioning error (unit: mm) . . 71 4.17 Center fitting standard deviation of each cylinder (unit: mm) . . . . . . . . . . 72 4.18 Equivalent deviation profiles of three cylinders . . . . . . . . . . . . . . . . . . 74 4.19 Deviation profiles of the 9 cylinders in the top row of the grid in location effect experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.20 Model fitting and residual plots of the location effect term x 0 (s) (unit: mm) . 77 5.1 BasicelementsofHP’sMultiJetFusionprocess(HPInc.(2017))(blackdroplets: fusing agents, blue droplets: detailing agents) . . . . . . . . . . . . . . . . . . 83 5.2 HP’s Multi Jet Fusion printing principle . . . . . . . . . . . . . . . . . . . . . 83 5.3 Layer energy process variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Fusing Science in Multi Jet Fusion . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Voxel level thermal history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.6 Voxel level machine instruction from HP Jet Fusion 3D 4200 Printer . . . . . . 91 5.7 Close to voxel level thermal sensing from FLIR camera . . . . . . . . . . . . . 92 5.8 Hybrid Spatiotemporal Model architecture for fusing layer thermal prediction . 95 xi 5.9 Compact Spatiotemporal Model architecture for fusing layer thermal prediction 96 5.10 Sample patch prediction results . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.11 Sample build-bed prediction result generated from patch level solution . . . . . 103 5.12 Sample build-bed prediction result generated from build-bed level solution . . 104 5.13 Deep neural network architecture for thermal diffusivity model . . . . . . . . . 108 5.14 The architecture of the inception modules . . . . . . . . . . . . . . . . . . . . 110 5.15 Prediction results from sample 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.16 Prediction results from sample 1 – continued . . . . . . . . . . . . . . . . . . . 116 5.17 Prediction results from sample 2 – continued . . . . . . . . . . . . . . . . . . . 117 5.18 Prediction results from sample 2 – continued . . . . . . . . . . . . . . . . . . . 118 xii Abstract Additive manufacturing (AM), or three-dimensional (3D) printing, refers to a new class of technologies associated with the direct fabrication of physical products from Computer-Aided Design (CAD) models by a layered manufacturing process. It has been widely recognized as a disruptive technology with the potential to fundamentally change the nature of future manufacturing, and the changes can amount to a third industrial revolution. Despite the vigorous development of different 3D printing techniques, the end-part quality of3Dprintinghowever, isstillnotcomparabletotraditionalmanufacturingwhichcontinuesto be one ofthe most significantissues in adoption. As an essential aspect of end-partquality, the shape accuracy still requires better control. Therefore, development of quantitative models that could predict process behaviors or end-part shape deviation is fundamental to inform both part design and process control. Developing the quantitative models and achieving high and consistent shape accuracy in AM is a challenging task. Those challenges include three aspects: (i) process physics is complex and more fundamental process knowledge is required to enable more precision control, but they are not yet fully understood; (ii) quality of the same product vary in different machines due to variations of machine properties, making the generic quality prediction and improvement pretty difficult; (iii) low volume and high geometric complexity due to frequent change of product geometries in discrete printing process challenges the modeling. There are a great amount of research efforts on the shape accuracy control of AM. Yet the accuracy enhancements are not yet satisfactory. xiii In this dissertation, we put forth several statistical modeling and machine learning based methods to improve the shape accuracy of 3D printed parts from two perspectives: shape deviation modeling and process behavior modeling. The shape deviation modeling predicts shape deviation directly from the design without considering the process specifications, while the process behavior modeling involves process information. Our ultimate objective is to build both product quality and process behavior models to guide the shape accuracy improvement. To achieve this objective, we include four tasks. Builtuponourpreviousprescriptivemodelingandoptimalcompensationstudyforcylinder and polyhedron shapes, the first task develops circular approximation with selective cornering (CASC) strategy to extend the basic models to freeform shapes. Polynomial approximation is first adopted to approximate an arbitrary shape to a polygon with large number of sides. Then each side is approximated by an arc, to which the cylindrical base is added. The corners with actual sharp transitions are then selected to add cookie-cutter function. Experimental investigationusingprojection-basedstereolithography(SLA)processsuccessfullyvalidatesthe proposed methodology and the optimal compensation. Then the second task extends the prescriptive modeling and compensation of shape devi- ation to the field of statistical process control (SPC) scheme for shape to shape deviation monitoring. We put forth the statistic and capability index for AM process monitoring. Then we adopt Exponentially Weighted Moving Average (EWMA) control chart for shape to shape monitoring. Only a limited number of test shapes are required to establish control limits. Experimental investigation is still conducted on projection-based SLA process. Thethirdtaskfurtherextendstheframeworktoselectivelasermelting(SLM)process. The framework extension decouples different error sources and achieves a comprehensive model to predict shape deviations. Build upon our prescriptive modeling framework, the non-negligible surface roughness is filtered out from the geometric deviation profile to guarantee confident prediction. We establish spatial models to quantify laser beam positioning error in both x− and y− directions. Then we estimate and model the other machine-dependent location xiv effects. Both the above two effects are transfered into equivalent shape design error following equivalent error concept and compensated along with geometric deviation. Different from the first three tasks, which build general models without considering process parameters, our last task looks into the process behaviors and adopts HP’s Multi Jet Fusion (MJF) as a study case. We apply deep learning to achieve fast and accurate thermal behavior prediction and knowledge discovery. With the physical insight, we establish two deep neural networks (DNN). The first network predicts fusing layer thermal behavior from the principle energy driver and heat transfer. The other one learns the thermal diffusivity as deep neural network. The experiments report promising model performance in both prediction quality (e.g., accuracy) and the computational cost. xv Chapter 1 Introduction Thisdissertationfocusesontheshapeaccuracycontrolofadditivemanufacturing(AM).As a promising technology, additive manufacturing faces the crucial bottleneck of shape accuracy. Numerous efforts base on either physical model based or statistical model based methods have been put on enhancing the shape accuracy. The objective of this dissertation is to propose statistical modeling and machine learning based methods to improve the shape accuracy of 3D printed parts from different perspec- tives. It includes prescriptive modeling, monitoring, compensation and shape-independent error analysis of in-plane geometric deviation for freeform 3D printed products in projection- based stereolithography (SLA) and selective laser melting (SLM) processes. Besides, we build deep neural networks (DNN) as predictive tools to predict the build bed thermal behavior and discuss a specific application to HP’s Multi Jet Fusion (MJF) technique. It should be noted that in this dissertation, prescriptive modeling means the prediction of product devi- ation for new and untried categories of shapes, while traditional predictive modeling usual makes prediction within its experimental categories. Following this chapter, we would like to introduce the general background of AM, the research challenges for AM shape accuracy control and our motivation first. Then we sum- marize the literature review on related research works. Our objective and key research tasks are presented in the end. 1 1.1 Background and Motivation 1.1.1 Additive Manufacturing and Potentials Additive manufacturing (AM), or three-dimensional (3D) printing, refers to a new class of technologies associated with the direct fabrication of physical products from CAD models by a layered manufacturing process. Unlike traditional manufacturing techniques, which mostly rely on the removal of material by methods such as cutting or drilling, AM adds material layer by layer directly following the CAD model. This direct digital manufacturing technique enables the promise for products with complex geometry or material compositions, which would dramatically reduce the cost on materials and time (Beaman et al. (2014); Beyer (2014); Gibson et al. (2009); Hilton & Jacobs (2000); Melchels et al. (2010); Campbell et al. (2011)). Fig. 1.1 illustrates the general procedure of AM, where a 3D digital prototype of the product is first built by CAD tools (such as AutoCAD, Inventor and SolidWorks, etc). A specialized program slices the CAD model into cross sections, which are sent to the AM machine to construct each layer in a sequence physically. Figure 1.1: Generalized procedure of 3D printing (Huang et al. (2015)) The revolutionary change of “additive” manufacturing from the traditional “subtractive” manufacturing frees the production from tools, molds as well as the complicated cutting, milling and drilling post-processing procedures. Dramatic advantages come along with the revolutionary change. First, the layer-by-layer additive methods simplified the manufacturing 2 of complex shapes, making the production of complex design almost the same as simpley design (“geometrical complexity for free”). Therefore, it allows the building of products with complex geometry, cavities even lattice structures which are highly restricted in traditional manufacturing. Meanwhile, getting rid of the molds and post-processing reduces the fixed costs of molds and the heavily wastes of raw material in subtraction. Finally, it enables the customization flexibility, making it flexible to rapidly produce low cost customized items to meet user’s specific needs. Despite the high potential, the unique properties of AM inevitably lead to some limita- tions. The relatively small build bed and long building time makes it not suitable for volume production. The current 3D printing techniques mainly produce short runs or one-of-a-kind parts. In addition, different AM techniques always require specific material, leading to the restricted material options. More important, the layer-by-layer fabrication process and the complicated physical process (e.g, melting, curing and phase change) may limit the end-part quality (e.g, surface property, strength and shape accuracy) as well as the yield of good parts, especially for cheap domestic 3D printers. Better understanding of the underlying process physics, better quantitative modeling and control of both the process & end-part quality are herein highly required. A variety of AM techniques with different printing process, material as well as industrial applications have been proposed and rapidly developed during past decades. A summary and comparison of several common AM techniques is given in Table 1.1. This dissertation involves three AM techniques: SLA, SLM and MJF. Figure 1.2(a) intro- duces the schematic diagram of the projection-based SLA technique, which we employ in first two tasks. In layered printing, the sliced 2D geometry is converted into a mask image, through which the lamp energy is projected onto the liquid resin. The mask will block the blank area, remaining only the part area receive enough energy to solidify. The SLM technique shown in Figure 1.2(b) is adopted in our third task. In each layer, it first recoats a thin layer of metal powder, then a high energy leaser beam will scan the surface line by line and cure the layer 3 Table 1.1: Summary and comparison of common Additive Manufacturing techniques defined by ASTM (Standard (2012)) Techniques Principle Material Solidification Fused Deposition Modeling (FDM) material extrusion plastic filament N/A Stereolithography (SLA) photopolymerization liquid photopolymer cured with laser or lamp Digital Light Processing (DLP) photopolymerization liquid photopolymer cured with projector Selective Laser Sintering (SLS) powder bed fusion plastic powder fused with laser Selective Laser Melting (SLM) powder bed fusion metal powder fused with laser Electron Beam Melting (EBM) powder bed fusion metal powder fused with electron beam Material Jetting material jetting liquid photopolymer cured with UV light Binder Jetting binder jetting powdered gypsum, sand, metal N/A Multi Jet Fusion (MJF) powder bed fusion plastic powder fused with lamp energy driven by agents according to the sliced 2D geometry. HP’s MJF technique (Figure 1.2(c)) is also a powder based technique, but it doesn’t use the laser. Instead, it uses the chemical compound called fusing agent, which will absorb energy from lamps and drive the powder fusion. The liquid agents are dropped by a carriage passing through the build bed, which enables the quick printing. Our last task will implement on MJF. 1.1.2 Research Challenges on Shape Accuracy Control in Additive Manufacturing Shape accuracy of parts is essential to meet required standards of manufacturing. The geometric accuracy of 3D printing however, is still not comparable to traditional manufac- turing which continues to be one of the most significant issues in adoption. Challenges that 4 (a) Projection-based SLA (Xu & Chen (2015)) (b) SLM (Zhang et al. (2014)) (c) HP’s MJF (HPInc. (2017)) Figure 1.2: Schematic diagram of the three techniques blocked the geometric quality improvement of AM products could be summarized into three aspects: (1) Process physics is complex and more fundamental process knowledge is required to enable more precision control, but they are not yet fully understood Many unique properties of AM process including build bed thermal physics, material phase-changing, non-uniform and nonlinear shape shrinkage, machine setting, boundary effects of shape complexity and interlayer bonding contribute together to the geometric deviationofprintedproducts. Evenworse, someofthesefactorsinteractwitheachother. In fusion based techniques, the thermal and material behavior determines the end part quality directly, yet they are complicated and not yet fully understood. Progress in 3D 5 print quality is, at least in part, impacted by the fact that the process physics is complex and more fundamental process knowledge is required to enable more precision control. (2) Quality of the same product vary in different machines due to variations of machine properties, making the generic and system-level quality improvement approach pretty difficult The variations in materials, machines, processing techniques, and processing conditions among different AM machines raise the challenge that a design model successfully con- structed by a specific machine may not be manufactured with satisfactory quality by another machine. On the one hand, trail-and-error calibration and post-processing with machine tools are in many cases still required to meet design specifications, significantly negating the time and cost benefits of direct digital manufacturing. On the other hand, the variations limit many quality improvement approaches to specific types of material and machine, making the generic and system-level quality improvement approach pretty difficult. (3) Low volume and high geometric complexity due to frequent change of product geometries in discrete printing process challenges the geometric modeling In contrast to mass production, AM processes fabricate products with high shape com- plexity at low volume, which leads to disparate training data with small sample size from one product shape to another. To address these challenges, the bulk of current AM research is devoted to the novel development of CAD and process planning methods, statistical and first principles based modeling, materials, machines and novel techniques suitable for AM (Bourell et al. (2009)). In this dissertation, a statistical modeling, monitoring and compensation framework is first proposed and illustrated for in-plane shape accuracy control. We hope to predict the actual shape-dependent and shape-independent deviation and add compensation to the original CAD design in advance to reduce the deviation. Besides that, we apply deep neural networks to 6 build quantitatively material behavior models and thermal predictive models to enable the close-loop thermal control in HP’s Multi Jet Fusion technique. 1.2 State of the Art on Shape Accuracy Control in AM Research Generally, we summarize the research efforts on shape accuracy control of AM into three categories: (1) Physical modeling based approaches (2) Statistical modeling based approaches (3) Process planning adjustment and calibration 1.2.1 Physical Modeling based Approaches Dimensional defects of AM come from complicated error sources, especially the physical processes during material solidification. First principles based modeling and simulation is a powerful tool to understand the underlying physical processes, quantify the influence of process variables as well as predict the component mechanical performance (such as residual stress and shape accuracy) (Francois et al. (2017); Bourell et al. (2009)). The investigated relationships will instruct the process parameter optimization to improve product quality. At very first stage, Williams & Deckard (1998) recognized the need for process modeling and simulation on SLS process. Then first principle models with 3D finite element simula- tion were widely researched. Bugeda Miguel Cervera & Lombera (1999); Dong et al. (2009) considered thermal and sintering phenomena to simulate the phase transformation in SLS process. The thermal model in Dai et al. (2004) investigated the transient temperature dis- tribution and effects of substrate preheating during laser densification of dental powder bed. 7 Hussein et al. (2013) investigated the temperature and stress fields in SLM process. Cou- pled thermo-mechanical models incorporating specific boundary conditions and temperature dependent material properties are developed by Zaeh & Branner (2010) to identify the heat impact on residual stresses and deviations.Hodge et al. (2014) proposed a thermo-mechanical model integrating powder melting, phase change, and thermo-mechanical properties. Recently, the series of works by King et al. (2015a,b); Kamath (2016); Khairallah et al. (2016); Francois et al. (2017) suggest a framework of modeling and simulation to improve the part quality. Their framework contains two steps. The first step uses physical model and finite element methods (FEM) to gain insight into the physical processes. The modeling and simulation covers both powder scale and part scale. The powder scale model investigates the influence of process characteristics such as laser beam, powder and melt pool thermal. With the data from powder scale model, the part scale model aims to predict the manufactured properties such as shape accuracy, residual stress and local effective material properties. The second step employs data mining and statistical inference to extract useful information from both simulations and validation experiments. The extracted relationships provide insight guidelines for process parameters optimization and uncertainties evaluation. Although the understanding of physical insight is rather helpful, high-fidelity first principle models are costly to build and requires in-depth understanding of underlying complex process. Improvingpartaccuracybasedpurelyonsuchsimulationapproachesisfarfrombeingeffective, and seldom used in practice (Bourell et al. (2009)). 1.2.2 Statistical Modeling based Approaches Based on different objectives, we could further divide the statistical modeling based approaches into three categories. The process modeling based process optimization aims to improve the part quality by optimizing the process parameters. Differently, the geometric modeling based design compensation predicts the shape deviation directly from CAD design and add compensation accordingly. While the last category, online monitoring and feedback 8 control, builds statistical models to monitor specific process characteristics then correct the controlling parameters. Process Modeling based Process Optimization It is widely acknowledged that process settings in AM are very influential in the part accuracy. Consequently, adopting design of experiments (DOE) or statistical methods for process parameters tuning and optimization became an important research direction. To achieve the optimal settings, the relationship between significant process parameters and part quality characteristics (such as shape accuracy, surface roughness and part strength) becomes the research focus (Bikas et al. (2016)). Basing on analytical methods the method- ologies could be categorized into two portions: design of experiments (DOE) methods and statistical process modeling methods. Among DOE methods, Taguchi method, full factorial experiment as well as response surface methodology (RSM) are commonly chosen for parameter tuning and optimization. Taguchi method, a powerful tool for tunning and optimizing control factors on performance output is the most popular one. For example, Zhou et al. (2000) conducted Taguchi experi- mentaldesignforexperimentsinSLAmachine. Relationshipbetweenpartshapeaccuracyand process parameters as well as optimal settings are obtained through response surface method- ology and analysis of variance (ANOVA) table. Similar research efforts are summarized in Table 1.2. Lots of efforts are also paid on the empirical process models to predict the relationship between performance characteristics and process parameters (Garg et al. (2014)). Comparing to the first principle models which requires deep physical understanding, statistical models show advantages in good prediction with incomplete information. Early on, simple regression models are adopted to build prediction models. For instance, Reddy et al. (2007) built linear regression models for the relationship between process parameters and surface roughness, inter-road bond strength as well as inter-layer bond strength separately for FDM process. 9 Table 1.2: Summary of Design Of Experiments (DOE) methods for process setting optimiza- tion Research Works Quality characteristic Design of Experiments Technique Zhou et al. (2000) shape accuracy Taguchi method, RSM, ANOVA SLA Lee et al. (2005); Campanelli et al. (2007) shape accuracy Taguchi method, ANOVA SLA Lynn-Charney & Rosen (2000) shape accuracy Response surface modeling SLA Onuh & Hon (2001) distortion, flatness Taguchi method SLA Lee et al. (2005); Moza et al. (2015) shape accuracy Taguchi method, ANOVA FDM Sood et al. (2009, 2010) shape accuracy Grey Taguchi method, ANOVA FDM Anitha et al. (2001) surface finish Taguchi method, ANOVA FDM Galantucci et al. (2009) surface roughness Full factorial design, Main effect plot FDM But in many cases, the large number of process parameters, unknown variables and incom- plete understanding of complex phenomena make it hard and inappropriate to use regression models. Artificial neural network (ANN) offers a good alternative approach, and it was then widely adopted. For instance, Sood et al. (2012) adopted both quadratic regression and artifi- cial neural network (ANN) to model the non-linear relationship between five process parame- ters and part compressive strength in FDM process. Comparing with validation experiments, ANN outperforms the regression model in prediction. A summary of empirical process mod- eling methods can be found in Table 1.3. Recently, Sun et al. (2017) for the first time proposed the methods of modeling both quantitative (shape accuracy) and qualitative (binary indicator of surface roughness) (QQ) response with both offline and in situ process variables simultaneously. The functional QQ model is proved to outperform the separate QQ response, which indicates a new interesting research direction. 10 Table 1.3: Summary of empirical process modeling methods for part quality prediction Research Works Quality characteristic Empirical Models Technique Wang et al. (1996) 2D shrinkage linear regression on each parameter SLA Reddy et al. (2007) inter-road, inter-layer bond strength, surface roughness linear regression FDM Lee et al. (2001) shape accuracy artificial neural network SLA Sood et al. (2010) shape accuracy ANN with back propagation FDM Sood et al. (2010, 2012) part strength quadratic regression, artificial neural network FDM Boschetto et al. (2013) surface roughness feed-forward (FF) neural networks FDM Mozaffari et al. (2013) clad height, melt pool depth ANN and fuzzy logic SLM Part shape accuracy could indeed be improved by better parameter settings, yet the com- plex error sources still bring challenges. To some extent, these approaches are mostly limited to the scope of a family of products, specific types of material and machine, and process planning methods. Geometric Modeling based Design Compensation All other categories of methodologies will more or less change the 3D printing machines. A great alternative is the error compensation approach, which predicts the geometric deviation and then add optimal compensation to CAD design before printing to approach the desired shape accuracy. The compensation aims to eliminate the predicted deviation while printing. So an accurate geometric prediction model and an optimal strategy to derive compensation are two crucial tasks in this category of methodology. Early on, machine manufacturers used shrinkage compensation factor (SCF) to automat- ically add compensations. For example, Dao et al. (1999) constructed the SCF for a FDM machine. The SCF is defined as the scope of the linear relationship between mean error and nominal dimensions for x and y direction. Works in Tong et al. (2003, 2008) first put forward 11 the machine parametric error model to evaluate the part shape accuracy and model the para- metric error functions. The negative values of predicted errors will be added to CAD design directly to compensate the product deviation. Cajal et al. (2013, 2016) further extended this error compensation method through employing conical sockets for probe measurement. How- ever, one obvious disadvantage is that the shrinkage compensation factor is usually simple, uniform and not optimal. In the meanwhile, benchmark artifacts are always required and the establishment of SCE library is time consuming. Some works tried to put forth more accurate geometric model and optimal compensation. Wang (1999) used linear models to formulate the shrinkage and beam offset of SLS machine in x, y and z directions separately. Compensation values in each direction is calculated as X g = 1 1 +s X d −b, whereX g is the shape after compensation,X d is the designed shape,s and b are predicted shrinkage and beam offset. Senthilkumaran et al. (2008) modeled the percentage shrinkage in x, y and z directions as a linear function of scan length separately in SLS process. Senthilkumaran et al. (2009) made improvement by adopting nonlinear function. But they narrowed their focus on x and y direction only. For a new shape, the compensation value Δc for desired lengthL c is calculated with predicted percentage shrinkage as: Δc = SL c 1−S since Δc =SL =S(L c + Δc). However, considering x, y and z direction separately is complicated. It also ignores the interaction and integration. Some efforts employ learning methods such as artificial neural network (ANN) for more accurate predictive model. For instance, Noriega et al. (2013) defined the distance between parallel facesD in cube as quality characteristic, and adopted ANN to model the relationship between actual D and designed D as well as part orientation angle θ. Optimal D in design is derived through prediction model following a numerical optimization algorithm. Although ANN provides a better model, their quality characteristic is shape dependent. The series of works by Huang and co-authors adopted polar coordinate system to represent in-plane deviation which greatly decoupled the complexity. Good prescriptive shape deviation 12 model and corresponding optimal compensation have been proposed and validated in Xu et al. (2013); Huang et al. (2015, 2014a,b); Sabbaghi et al. (2014); Sabbaghi & Huang (2016); Sabbaghi et al. (2017); Luan & Huang (2015, 2017); Luan et al. (2017) for projection-based SLA process, Song et al. (2014); Wang et al. (2017) for FDM process and Luan et al. (2018) for SLM process. Out-of-plane models and compensation are also demonstrated in Jin et al. (2015, 2016a,b). Modeling the geometric shape deviation directly from design gets rid of the demanding for understanding the underlying process or modifying the process parameters. But this may also limit the shape accuracy improvement capability and the model robustness. Online monitoring and feedback control Online process monitoring is critical in guaranteeing the normal process conditions. And comparing to all the above offline methods, online feedback control strategy achieves the dynamic real-time adjustment while printing, which shows huge advantages in guaranteeing productaccuracy. Therefore,ithasreceivedincreasingattentionandbecomeacriticalresearch area. Although statistical process monitoring has been well studied and applied, limited efforts has been put on monitoring of AM process. Some efforts are put on monitoring of process status. Fang et al. (1998) defined the overfill and underfill in the interior region as the defect in FDM process. Then the process signature, which is the grayscale representative of a road in a given region is online monitored via image processing. Rao et al. (2015); Wu et al. (2016) adopted statistical models for online monitoring part process status from in situ sensor data. A nonparametric Bayesian Dirichlet process (DP) mixture model is established in Rao et al. (2015) to classify the process into three status from the independent non-Gaussian sensor signals. And Wu et al. (2016) builds hidden semi-Markov model (HSMM) for status classification from sensor signals. 13 Efforts are also put on the monitoring of shape deviation. Work in Colosimo et al. (2008) put forth statistical profile monitoring, which combines a spatial autoregressive regression (SARX) model with control charting, to monitor the roundness of cylinders. Colosimo et al. (2014) further extended it to surface deviation monitoring via Gaussian process models. But their works lack of a clear path to handle shape complexity and limited data faced in AM. The feedback control in AM developed quickly in recent decades. To build the feed- back control system, sensors are firstly required to monitor important quality characteristics. According to the difference of monitored characteristics, the approaches could be divided into process variables based and geometry accuracy based. In the first category, process variables such as melt pool geometry and temperature in SLM are usually monitored and stabilized by adjusting machine parameters (such as laser power and scan speed). This is because their behaviors are directly correlated with the part quality (Tapia & Elwany (2014); Reutzel & Nassar (2015)). The other category directly monitors the important geometry, and the process is adjusted to achieve the desired geometry. The summary of related works could be found in Table 1.4. In spite of the promising merits, these methods are still more or less dependent on spe- cific geometric shapes and processes. In the meanwhile, complicated and expensive sensors and equipment setups with high quality are required. And process uncertainties will further complicate the control problem. 1.2.3 Process planning adjustment and calibration Process planning is the crucial bridge between CAD design and actual manufacturing. Although process planning will be automatically finished by software, a more intelligent pro- cess planning has the potential to greatly improve the part accuracy (Kulkarni et al. (2000)). Zhou et al. (2009) first proposed the mask image planning method for quality improvement inmaskprojectionbasedAMprocess. Thebasicideaisthatsinceeachpixelhassomefuzziness or image blurring, if we could intelligently manipulating pixels’ gray scale values (0-255) 14 Table 1.4: Summary of feedback control methodologies in AM Research Works Sensor Monitored Characteris- tic Controlling Parameter Control Algorithm Hu & Kovacevic (2003) CCD camera melt pool size laser power PID control Bi et al. (2013) germanium photodiode melt pool temperature laser power PID control Song et al. (2014) dual-color pyrometer melt pool temperature laser power dynamic state-space model based predictive control Hua & Choi (2005) clad height laser power fuzzy logic-based controller Fathi et al. (2007) clad height scanning speed feed forward PID control Cohen & Lipson (2010) geometric measure- ments overall product geometry droplet scale, locations of subsequent droplets greedy geometric feedback (GGF) control Cheng & Jafari (2008) 3D imaging system road defect roller speed fuzzy dynamics model considering the layer’s correlation Lu et al. (2014, 2015) height sensor height profile next layer droplet pattern height predictive model based control algorithm instead of simply setting white or black, the part deviation could be reduced. Their approach contains three steps. Geometric heuristic, which is a rule to set gray scale values is adopted to initialize the image gray scale. Then two optimization models, one for minimizing pixel blending errors and one for maximizing separation of boundary pixels with different values, are used successively to optimize the gray scale values. Zhou & Chen (2012) improved their method with the basic idea that the part accuracy could be further improved by allowing the projection image to move its XY coordinates. Therefore, multiple mask images with slightly moved XY coordinates rather one image are projected in sequence during the total exposure time for each layer. Similar two-stage heuristic optimization process is conducted for image planning. 15 Insteadofonlymodifyingthegrayscalevalues, themaskimageplanningmethodpresented by Xu & Chen (2015) divided the mask image in each layer into multiple smaller disconnected images and projected these images in sequence. The multiple smaller curing regions greatly eliminate the shrinkage caused by large areas due to density difference between liquid and solid, temperature increase and internal stress. Besides the mask image planning, Pan et al. (2012) put forth a process planning method to address the stair-stepping effect due to layer-to-layer manufacturing and achieved high surface smoothness. The key idea is to form liquid meniscus at the corners when cured layers emerge from the liquid. The shape of meniscus to the given curved surfaces is determined by a process optimization model. For FDM machine, Cho et al. (2003) proposed a feature based design (FBD) approach for modeling complex components with local composition control (LCC), converting the features into visible 3D presentation first. Then an algorithm is proposed to identify the necessary boundary droplets to be printed and the amount of their proportional deflections during printing process to guarantee the accuracy of shape boundary. Those methods are mechanism specified. New findings and achievements are hard to be utilized in another mechanism due to the principle discrepancy. Some of them are computa- tionally and mechanically complicated and will lead to the increase of printing time. 1.3 Research Tasks and Objectives The objective of this dissertation is to propose statistical modeling and machine learning based methods to improve the shape accuracy of 3D printed parts from different perspectives. The four proposed tasks are expected to improve the shape accuracy of products printed by different techniques either through prescriptive shape deviation modeling or process behavior modeling. Offline prescriptive shape deviation modeling and compensation for cylindrical and polyhe- dral shapes in projection-based SLA process have been developed in previous work by Huang 16 and co-authors (Huang et al. (2015, 2014b); Sabbaghi et al. (2017)). The first three tasks aim to extend the generic prescriptive methods to freeform shapes, online monitoring and SLM process. The last task put forth a new research insight that applies deep learning to uncover the process physical behavior underlying and adopt the HP Multi Jet Fusion as a study case. (1) Prescriptive modeling and compensation of in-plane shape deviation for 3D printed freeform products The goal of this task is to achieve a breakthrough in predicting shape accuracy of freeform products with a prescriptive learning strategy independent of geometric and process complexities. We aim to extend previous methodology from cylindrical and polyhedral shapes to freeform shapes. To accomplish this extension, we proposed and evaluated two candidate strategies: (i) polygon approximation with local compensation (PALC) and (ii) circular approximation with selective cornering (CASC). We adopted merge based polygon approximation to fulfill the polygon approximation of freeform in both of the two strategies. CASC strategy were demonstrated to be better than PALC. With CASC strategy, we could capture the basic features of new freeform shapes then derive the deviation prediction directly from CAD design. (2) Statistical process control of in-plane shape deviation for additive manufacturing This task aims to put forth a new prescriptive online monitoring scheme free of shape complexities. Whenanewanddifferentshapeisbuilt,themonitoringschemeisexpected to decide whether the process is in-control or out-of-control. In our monitoring scheme, we put forth the statistic and capability index for AM process monitoring. Then the EWMAcontrolchartwasadoptedforshapetoshapemonitoring. Onlyalimitednumber of test shapes will be required to establish prescriptive process control limits for new products. The strategy is extension of the prescriptive modeling and compensation of shape deviation in task one to the field of SPC. The prescriptive online monitoring scheme provides an important guideline for online feedback control. 17 (3) Prescriptive data-analytical modeling of selective laser melting processes for accuracy improvement This task aims to extend our prescriptive modeling and compensation methodology to SLM processes. Due to higher accuracy requirement for metal parts, surface roughness, laser beam positioning error, and part location effect can all be coupled with shape accu- racy of SLM build products. Bring surface roughness into consideration, our prescrip- tive modeling approach is adopted to minimize geometrical deviations in SLM process through compensating CAD models. An error decomposition and compensation scheme is developed to decouple the influence from different error components and to reduce the shape deviations caused by part geometrical deviation, laser beam positioning error and other location effects simultaneously via an integrated modeling and compensation framework. (4) Data-driven modeling of thermal physics in powder bed fusion 3D printing using deep learning This task aims to apply deep learning to generate quantitatively meaningful material behavior models and thermal predictive models for powder bed fusion based techniques. And we use HP’s Multi Jet Fusion as a study case. In powder bed fusion based tech- niques, end part functional quality depends upon each voxel that forms a part expe- riencing a similar thermal history to minimize functional irregularities such as shape deformation. To enable closed-loop voxel level thermal control, we propose a deep neu- ral network model to predict the fusing layer thermal behavior at voxel level and a deep neural network architecture to learn the thermal diffusivity as a deep neural network. The promising results demonstrate the ability of deep learning to provide predictive models that run both sufficiently accurate and sufficiently fast to enable embedding the models into the firmware for runtime closed-loop thermal control. It also demonstrates the knowledge discovery potential of deep learning to uncover the underlying thermal physics. 18 Figure 1.3: Relationship of the four tasks Figure 1.3 summarizes the relationship of the above four tasks. The first task extends the predictive shape deviation modeling framework from simple shapes (Huang et al. (2015, 2014b)) to freeform shapes. It also provides modeling and framework bases for the second and third tasks. The second tasks extends the offine shape deviation modeling to online shape deviation monitoring. And the last task extends the prescriptive modeling framework from SLA process to SLM process. These three tasks builds geometric shape deviation models from product quality data. They predict the shape deviation from product design directly without considering the process information such as parameters. We agree that in the application to specific process or machine, considering the closed-loop control and involving the process variables will be very beneficial, so our last task looks into the process information and tries to model the process behavior. We will discuss a specific application in HP’s Multi Jet Fusion 3D printing technique. We build deep neural networks to not only predict the layer by layer thermal behavior which could enable the closed loop thermal control, but also infer the underlying physical knowledge. 19 Chapter 2 Prescriptive Modeling and Compensation of In-plane Shape Deviation for 3D Printed Freeform Products Additive manufacturing has the characteristics of extremely high product varieties and low product volumes. Achieving high and consistent shape accuracy in additive manufacturing is therefore a daunting task because new products may not be tried before, especially for the fabrication products with complex arbitrary freeform shapes. Research on solid freeform fabrication (SFF) (Beaman et al. (1997); Cho et al. (2003); Cohen & Lipson (2010)) have been focusing more on design, process planning, finite element modeling (FEM), materials, processes, and machines. Yet these methods are more or less dependent on specific geometric shapes and processes. Therefore in this chapter, we present a novel prescriptive methodology independent of geo- metric and process complexities to predict and control product in-plane geometric errors. The prescriptive model bases on a limited number of trial cases with simple shapes. We will first formulatetwoplausiblestrategiesthatmakemethodologyextensionfromcylindricalandpoly- hedron shapes to arbitrary freeform. Then we evaluate the two proposed strategies through experimentation and analysis. The detailed methodology development based on the selected circular approximation with selective cornering (CASC) strategy is also presented. Finally 20 we discuss the experimental validation of the developed methodology as well as compensation plan, then summary the conclusion. 2.1 Predictive Modeling of Polyhedron Products – Pre- vious Work In a series of work (Huang et al. (2015); Xu et al. (2013); Huang et al. (2014b); Wang et al. (2017); Huang (2016); Sabbaghi et al. (2014)), Huang and co-authors intend to develop a new prescriptive modeling strategy with the ability of learning from a limited number of tested shapes and deriving compensation plans for new and untried products. This new alternative strategyismotivatedbythefactthatAMhastobuildproductswithhugevarietiesandlimited quantities. It is desirable to establish a methodology independent of shape complexity and specific AM processes. Huang et al. (2015) first establishes a generic and physically consistent approach to model and predict in-plane (x−y plane) shape deviation along product boundary and derive optimal compensation plans. The essence of this new modeling approach is to decouple the geometric shape complexity from the deviation modeling by transforming in- plane geometric errors from the Cartesian Coordinate System into a functional profile defined in the Polar Coordinate System (PCS). Huang et al. (2014b) extends the work in Huang et al. (2015) from cylindrical shape to polyhedrons. As shown in Fig.2.1, the key idea of connecting the cylindrical shape model to polyhedron models is to treat an in-plane polygon as being cut from its circumcircle. The so-called cookie-cutter function is added to the cylindrical basis model in order to carve out the polygon shape from a circumcircle. The cookie-cutter function is proposed to be a periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values. Two cookie-cutter functions, i.e., square wave model and sawtooth wave model, have been adopted in Huang et al. (2014b), where experiments have shown that the 21 in-plane deviation can be significantly reduced for both cylinder and polyhedron shapes by implementing the generic optimal compensation algorithm developed in Huang et al. (2015). Figure 2.1: Modeling extension from cylinder (Huang et al. (2015)), polyhedron (Huang et al. (2014b)) to freeform (this work) However, the ultimate breakthrough is to extend the methodology from cylinder and poly- hedron to freeform shapes. Research on solid freeform fabrication (SFF) (Beaman et al. (1997)) have been focusing more on design, process planning, finite element modeling (FEM), materials, processes, and machines. For instance, the work in Cho et al. (2003) proposes a feature-based design approach for modeling complex components with local composition con- trol. To ensure shape accuracy of freeform products, an algorithm is developed to calculate the correct amount of droplets along the boundary and the amount of proportional deflections. The work in Cohen & Lipson (2010) puts forward a closed-loop feedback control strategy to improve geometric quality by making real-time decision regarding droplet scale and locations of subsequent droplets. Yet these methods are more or less dependent on specific geometric shapes and processes. The goal of this work is to achieve a breakthrough in predicting shape accuracy of freeform products with a prescriptive learning strategy independent of geometric and process complex- ities. We aim to extend our previous methodology from cylindrical and polyhedron shapes to arbitrary freeform shapes (as shown in Fig. 2.1). 22 2.2 Two Plausible Strategies for Prescriptive Method- ology Extension to Freeform Shapes The first challenging issue for methodology extension to freeform shapes is the functional representation of the in-plane geometric shape error for arbitrary shapes. A close-form rep- resentation is essential for implementing the optimal compensation policy derived in Huang et al. (2015). Below we discuss the formulation of two possible strategies for representing the in-plane error of freeform products. These two strategies will be evaluated in detail in Section 2.3. 2.2.1 Strategy A: Polygon Approximation with Local Compensa- tion (PALC) One observation is that any in-plane 2D freeform shape can be approximated by a polygon (Fig. 2.2). Since we have predictive in-plane shape deviation models established for cylindrical and polyhedron shapes (Huang et al. (2015); Sabbaghi et al. (2014); Huang et al. (2014b); Wang et al. (2017)), one intuitive idea is therefore to approximate an arbitrary freeform shape by a polygon first and then improve the shape deviation model of that polygon by compensation. This line of thinking results in the first strategy: polygon approximation with local compensation (PALC). Figure 2.2: Strategy of polygon approximation with local compensation (PALC) (solid line: freeform shape; dash line: approximated polygon shape; shadow area: amount of compensa- tion to approach the freeform from the polygon shape) 23 Denote the in-plane error model for the fitted polygon as f(θ,r(θ)) defined in the Polar Coordinate System (PCS) (Different polygonal approximation approaches will be discussed shortly.). At angle θ, the approximation error is denoted as x(θ), which is the amount of compensation to be applied to improve prediction. The predictive in-plane shape deviation model for the freeform can thus be derived as f(θ,r(θ) +x(θ)). In this way, the previous in-plane deviation modeling for polyhedrons can be extended to arbitrary freeform shapes. 2.2.2 StrategyB:CircularApproximationwithSelectiveCornering (CASC) An alternative strategy, which is not that intuitive, is based on the observation that any in-plane 2D freeform shape can be approximated by the addition of a series of sectors with different radii (Fig. 2.3). To accommodate the potential sharp transitions or corners between adjacent sectors, we properly select corners and impose a cookie-cutter function proposed in Huang et al. (2014b) to the cylindrical base functions. This second strategy is called circular approximation with selective cornering (CASC). Figure 2.3: Strategy of circular approximation with selective cornering (CASC) (solid arcs: actual radii of sectors; solid line segments: polygons sides) The first step of CASC strategy is to obtain a series of sectors with different r i (θ i ) atθ i in the PCS (Fig. 2.3). A natural approach to deriver i (θ i ) is to fit a polygon with a large number of sides to the freeform shape. The deviation of each sector can be predicted by g(θ i ,r i (θ i )), 24 a modification of the cylindrical base model in Huang et al. (2015) as a section of a circle (see details in Section IV). The second step of CASC is to properly select corners to catch the transition points along the boundary of the freeform shape. The good approximation in step 1 of CASC results in a large number of sectors. Yet the sharp transition points along the freeform can be less than the number of sectors. We will present a method to design the cookie-cutter function to catch the transition points from the CAD design. Therefore, the two-step CASC strategy presents another path to predict the in-plane shape deviation of a freeform product. 2.2.3 Related Approaches of Polygon Approximation to Freeform and Method Selection It can be seen that polygon approximation to an in-plane freeform shape is an important step in both strategies. The related research has been widely reported in Image Processing, Computer Vision and Pattern Reorganization with methods classified into three groups: (1) split based approaches (Dinesh et al. (2005); Yin (2004)); (2) merge based approaches (Wu & Leou (1993); Pikaz & others (1995)); and (3) dynamic programming based optimal approx- imation (Perez & Vidal (1994); Salotti (2001)). Both the split and merge based approaches demonstrate good approximation of convex and concave freeform. Not a focus of this research, the merge based approach is chosen for polygon approximation in our study. 2.3 Strategy Evaluation and Selection through Experi- mental Investigation To evaluate the two plausible strategies proposed in Section II, we conduct experimental investigation in order to make the final selection. Our null hypothesis is that the intuitive PALC strategy works better than CASC. If this hypothesisistrue,themaindeviationpatternofafreeformshouldbecapturedbythedeviation 25 pattern of a polygon that approximates the freeform (cf. Fig. 2.2). Otherwise, CASC strategy has to be considered. (a) Freeform with circumcircle radius = 2 00 (b) Approximated similar polyhedron with circumcircle radius = 1 00 Figure 2.4: Printed freeform shape and corresponding similar polyhedron To test this hypothesis, we build one freeform product and one polyhedron with its in- plane polygon shape approximating the in-plane freeform shape. As shown in Fig. 2.4, the freeform product takes arbitrary shape with “circumcircle" radius being 2 00 . Here the radius of minimum circumcircle that contains a part shape is used to determine its size. Using the merge based approach, a polygon is generated to approximate the freeform shape. Using the same polygon shape at a smaller size (circumcircle radius = 1 00 ), a polyhedron is built as well. All test parts have the same thickness of 0.25 00 . It should be noted that size of the polygon is not a concern to us because our previous studies (Huang et al. (2014b)) show that the deviation pattern of a polygon does not vary with its size, though the magnitude differs. To facilitate the identification of the orientation of test parts during or after the build- ing process, a non-symmetric sunk cross with line thickness of 0.02 00 is designed on the top surface. The 3D CAD models are exported to STL format files and then sent to the Mask Image Projection SLA (MIP-SLA) machine. The MIP-SLA platform in this experiment is the ULTRA R machine from EnvisionTec. Specification of the manufacturing process is shown in Table 2.1. All test parts are measured using a Micro-Vu precision machine. Same measurement procedure is followed for each test part to reduce errors. For convenience and consistency, the 26 Table 2.1: MIP-SLA process settings Thickness of the product 0.25 00 Thickness of each layer 0.00197 00 Resolution of the mask 1920× 1200 Dimension of each pixel 0.005 00 Illuminating time of each layer 7s Average waiting time between layers 15s Type of the resin SI500 center of the cross is chosen as the origin of measurement coordinate system in the Micro-Vu machine. Boundary profile is fitted using splines in the metrology software associated with the machine. For the two printed parts shown in Fig. 2.4, their deviation profiles defined in the PCS are shown as blue solid lines in Fig. 2.5. Comparing these two deviation profiles, their basic deviation patterns show significant discrepancies in two aspects: (1) The trends of deviation profiles do not match, and (2) The polygon deviation profile has more sharp deviation transitions and the positions of sharp changes are inconsistent with its freeform counterpart. (a) deviation profile (blue solid line) and model pre- diction(reddashline)ofconvexfreeformshapewith circumcircle radius = 2 00 (b) deviation profile of approximated convex poly- gon with circumcircle radius = 1 00 Figure 2.5: deviation profiles of convex freeform and corresponding polygon approximated The immediate conclusion from the experimental investigation suggests that our null hypothesis is not true, i.e., the intuitive PALC strategy will not work well. We believe major reasons causing the discrepancies can be as follows. 27 • Comparing the two parts in the Fig. 2.4, the amount of approximation errors vary with location θ. This different level of approximation accuracy can alter the trend of deviation profile for the approximated polygon. • Although increasing the number of sides can improve the approximation, a more serious issue is that vertices in approximated polygon and its corresponding freeform don’t match. It is unavoidable that a vertex can be identified by the merged based method at a smooth segment of the freeform. Additionally chances exist that a line segment passing through a vertex of the freeform can be fitted for the polygon. Both these two reasons lead to vertex mismatch. Therefore, if we directly use the shape deviation model for polygons to represent the major shape deviation pattern of the freeform, large errors can be introduced. Compensation for sharp changes could be introduced at wrong locations and real sharp transitions of the freeform shape may be missed. Based on experimental result and analysis, CASC is deemed to be more suitable strategy for methodology extension. 2.4 Prescriptive Modeling of Freeform deviation through Circular Approximation with Selective Cornering (CASC) The experimental disapproval of PALC strategy raises critical issues for CASC strategy: i How to present cylindrical basic function appropriately to capture the correct trend pattern of freeform deviation? ii How to detect actual vertices or sharp corners in freeform by adding cookie-cutter func- tion? 28 2.4.1 Prescriptive Modeling of Freeform deviation via CASC It is worthwhile to first review the predictive model developed in Huang et al. (2014b) for in-plane polygon shapes. Denote f P (θ,r(θ)) as the deviation profile of a polygon defined in the PCS. It is generally formulated as f P (θ,r(θ)) =g 1 (θ,r(θ)) +g 2 (θ,r(θ)) + (2.1) where g 1 depicts the deviation profile of cylindrical shapes, g 2 is the cookie-cutter function trimming the cylindrical base, and represents the un-modeled term. Since the PALC strategy of directly applyingf P (θ,r(θ) +x(θ)) to freeform prediction has been invalidated by experiments and analysis, we need to modify the modelf P (θ,r(θ)) in Eq. (2.1) so that it is consistent with strategy CASC. According to CASC (Fig. 2.3), a large number of sectors with different radiir i (θ i ) atθ i will be adopted to capture the curvature and trend of freeform deviation profiles. This suggests that, rather than using a single and whole cylinder base, we need to modify g 1 (θ,r(θ)) as a combination of cylindrical bases with each of them only representing a portion or a sector, e.g., g 1 (θ i ,r i (θ)) for the ith sector falling between (θ i−1 ,θ i ). Anexampleofcylindricalbasismodelg 1 (θ,r(θ))hasbeenestablishedforMIP-SLAprocess in Huang et al. (2015): g 1 (θ,r(θ)) =x 0 +β 0 (r 0 +x 0 ) a +β 1 (r 0 +x 0 ) b cos(2θ) (2.2) where cos(2θ) term represents the dominant deviation pattern, x 0 is a constant effect of over or under exposure for the MIP-SLA process, which is equivalent to a default compensationx 0 applied to original CAD model. However, as has been mentioned in Huang et al. (2014b), the MIP-SLA machine settings were changed after the experimentation in Huang et al. (2015). Experiments on cylinders are therefore conducted again and new deviation profiles clearly show the difference between 29 deviation of the upper and lower half of the cylinders. For the upper half (θ = 0∼π) of the product, the cylindrical basis model g 1 (θ,r(θ)) is modified as: g 1 (θ,r(θ)) =x u 0 +β u 0 (r 0 +x u 0 ) a u +β u 1 (r 0 +x u 0 ) b u cos(2(θ +π/8)) (2.3) For the lower half (θ =π∼ 2π) of the product, the cylindrical basis model is modified as: g 1 (θ,r(θ)) =x l 0 +β l 0 (r 0 +x l 0 ) a l +β l 1 (r 0 +x l 0 ) b l (− sin(2θ)) (2.4) Note that superscripts u andl indicate parameters corresponding to upper and lower half of the product, respectively. Following the CASC strategy, the upper cylindrical basis function in Eq. (2.3) is general- ized for freeform shapes as: g F 1 (θ,r(θ)) =x u 0 +β u 0 (r i (θ i ) +x u 0 ) a u +β u 1 (r i (θ i ) +x u 0 ) b u cos(2(θ +π/8)) for θ i−1 ≤θ<θ i , 1≤i≤n u , θ 0 =θ n (2.5) And the lower cylindrical basis function in Eq. (2.4) is generalized for freeform shapes as: g F 1 (θ,r(θ)) =x l 0 +β l 0 (r i (θ i ) +x l 0 ) a l +β l 1 (r i (θ i ) +x l 0 ) b l (− sin(2θ)) for θ i−1 ≤θ<θ i , n u + 1≤i≤n, (2.6) wheren is the number of sides of the approximated polygon, n u is the number of sides of the upper half polygon (θ from 0 to π). In order to obtainn,n u andθ i in Eqs. (2.5) and (2.6), we adopt the merge based approach for polygon approximation. By linear scanning of a digital curve, merged based approach uses 30 twothresholdsregardingdistanceandareatodeterminewhethereachscannedpointisavertex of the approximated polygon. n is the number of all the selected vertices. When thresholds are small enough, the freeform boundary can be precisely approximated by a polygon with large number of sides (see in Fig. 2.6). θ i can be obtained numerically as one outcome of the approximation procedure. In the approximation procedure, the boundary points are stored as digital coordinates. Therefore the angle θ i in the PCS could be easily derived once the ith vertex is determined. For large n, the radius of each sector can be approximated by the distance from ith vertex to the origin of circumcircle, which is denoted as r i (θ i ) (as shown in Fig. 2.3). Figure 2.6 shows that both convex and concave freeform shapes can be well approximated. (a) Convex freeform with n = 104, m = 11 (b) Concave freeform with n = 88, m = 0 Figure 2.6: Polygon approximation of freeform shapes With the generalized cylindrical basis model, the second issue to be addressed for CASC strategy is the proper determination of vertices or corners with sharp transitions along the boundary of a freeform shape. The cookie-cutter model or g 2 (θ,r(θ)) in Eq. (2.2) is first developed in Huang et al. (2014b) to capture the sharp transitions of polygon shapes. g 2 (θ,r(θ)) =β 2 (r 0 +x 0 ) α cookie.cutter(θ−φ 0 ) (2.7) One example of cookie-cutter functions is the square wave model: sign[cos(n(θ−φ 0 )/2)] (2.8) 31 where n is the number of sides of a polygon and φ 0 is a phase variable to shift the cutting position in the PCS. The sawtooth cookie-cutter model is another alternative: saw.tooth(θ−φ 0 ) = (θ−φ 0 ) MOD (2π/n) (2.9) where x MOD y = remainder of (x/y). These two cookie-cutter functions apply to regular polygons well where the transitions of the square wave or sawtooth wave can be easily identified. For a freeform shape, however, we alreadypointoutinSectionIIIthattheverticesoftheapproximatedpolygonscannotcorrectly describe the transition points of the freeform boundary. In addition, the fitted polygons from merged based approach can be irregular in general. We therefore have to develop a method to design the cookie-cutter function to catch the transition points of the freeform from its CAD design. Notice that when an interior angle γ i of the fitted polygon (Fig. 2.3) is close to π or |γ i −π|≤ δ critical , the corresponding ith vertex positioned at θ i is not likely to produce a sharp transition in the deviation profile at location θ i . Since each boundary point is stored as coordinates, the interior angle γ i could be derived accordingly. Here δ critical is a threshold value, e.g., δ critical is less than (1/6)π from the experimental studies in Huang et al. (2014b). Based on this observation and proposed criterion, only vertices with sharp transitions in the fitted polygon will be selected for the cookie-cutter function to alternate the function amplitude. Let m be the number of selected vertices for the cookie-cutter function with m n (the small circles in Fig. 2.6 show m vertices finally selected), and the angle of m vertices be: ϑ k , k = 1, 2,...,m. Both m and ϑ k are obtained by the polygon approximation procedure. Note that the cookie-cutter will only be applied to sectors whose vertices have 32 sharp transitions. Then the square wave cookie-cutter function of Eq. (2.8) is extended for a freeform shape as: g F 2 (θ,r(θ)) =                            β 2 (r j (θ j ) +x 0 ) α sign[cos( (2 + (−1) j )πθ 2θ j )] if θ j−1 ≤θ<θ j , 1≤j≤n, θ 0 =θ n , θ j =ϑ k , 1≤k≤m 0, otherwise (2.10) where the term (2 + (−1) j ) is adopted to guarantee that sign(·) changes between−1 and +1 alternatively to build the cookie-cutter function. The sawtooth wave cookie-cutter model in Eq. (2.9) is also extended as: g F 2 (θ,r(θ)) =β 2 (r j (θ j ) +x 0 ) α saw.tooth(θ) =                            β 2 (r j (θ j ) +x 0 ) α π(θ−θ j−1 ) MOD (θ j −θ j−1 ) 2(θ j −θ j−1 ) if θ j−1 ≤θ<θ j , 1≤j≤n, θ 0 =θ n , θ j =ϑ k , 1≤k≤m 0, otherwise (2.11) We still adopt the sawtooth cookie-cutter function proposed in Huang et al. (2014b). Note that not all freeform shapes have sharp transitions required for cookie-cutter functions. For example, the freeform in Fig. 2.6(b) doesn’t contain sharp transition points, and the cookie- cutter function is not needed in prediction. With the generalized cylindrical basis model and extended cookie-cutter model, the pre- criptive learning model f F (θ,r(θ)) for freeform deviation is extended from the polyhedron model in Eq. (2.1) as f F (θ,r(θ)) =g F 1 (θ,r(θ)) +g F 2 (θ,r(θ)) + (2.12) 33 2.4.2 PrescriptiveModelEstimationbasedonLimitedTrialShapes Model (2.12) is deemed to be prescriptive if its model parameters can be estimated based on limited number of trial shapes and it can be applied to predict the deviation of a freeform shape given the product CAD design. This section illustrates model parameter estimation based on six test parts: three cylindrical and three polyhedron shapes in various sizes. (a) deviationprofiles(red,blueandgreenlines)and prediction (smooth dark lines) of three cylinders (b) deviation profiles (red,blue and green lines) and prediction (dark lines) of three polyhedrons Figure 2.7: deviation and prediction profiles of simple trial shapes Three cylinders are fabricated with circumcircle radii 0.5 00 , 1.5 00 and 3 00 , respectively. Figure 2.7(a) shows their deviation profiles. The maximum likelihood estimation (MLE) of cylindrical basis model is given in Table. 2.2, while the predicted cylindrical deviation profiles are shown as smooth lines in Fig. 2.7(a). Note that we assume ∼ N(0,σ 2 ) and the initial parameter values for the numerical MLE estimation are chosen as β u 0 =β l 0 = 0.01, a u = a l = 0.5, β u 1 = β l 1 = 0.001, b u = b l = 0.5, x u 0 = x l 0 =−0.01, which are based on our previous analysis in Huang et al. (2015). Three regular polyhedrons, that is, a 2 00 by 2 00 cube, a pentagon with circumcircle radius 1 00 , and a pentagon with circumcircle radius 3 00 , are fabricated to establish the in-plane polygon model. Figure 2.7(b) shows the in-plane shape deviation profiles of these polyhedrons. Since the polyhedron is regarded as being trimmed from a cylinder, the cylindrical part in polyhedron model should be consistent with the cylindrical model. Therefore, we plug the six fitted parameters x u 0 , x l 0 , a u , a l , b u , and b l in the cylindrical model to polyhedron model 34 Table 2.2: Estimated parameters for cylinder model Model of the upper half cylinder Parameter Estimated value Standard deviation β u 0 0.0076 9.593× 10 −5 a u 0.7878 0.0069 β u 1 0.0015 1.670× 10 −5 b u -0.1043 0.0125 x u 0 -0.0056 9.398× 10 −5 Model of the lower half cylinder Parameter Estimated value Standard deviation β l 0 0.0312 8.060× 10 −4 a l 0.2198 0.0054 β l 1 0.0018 1.804× 10 −5 b l 0.7110 0.0096 x l 0 -0.0264 7.931× 10 −4 directly. The MLE parameter estimation of remaining parameters is shown in Table 2.3, and the smooth black lines in Fig. 2.7(b) show the predicted shape deviation profiles of three polyhedrons. The initial parameter values for numerical iteration are set as: β u 0 =β l 0 = 0.01, β u 1 =β l 1 = 0.01, β u 2 =β l 2 = 0.001, α u =α l = 1 (Huang et al. (2014b)). Table 2.3: Estimated parameters for polyhedron model Model of the upper half polyhedron Parameter Estimated value Standard deviation β u 0 0.0058 1.313× 10 −5 β u 1 0.0011 2.117× 10 −5 β u 2 0.0027 2.870× 10 −5 α u 0.8778 0.0094 Model of the upper half polyhedron Parameter Estimated value Standard deviation β l 0 0.0291 2.141× 10 −5 β l 1 0.0012 1.004× 10 −5 β l 2 0.0009 2.712× 10 −5 α l 1.4067 0.0250 35 2.4.3 Prescriptive Model Validation through Experimentation Themodel(2.12)isobtainedbylearningfromalimitednumberoftestedshapes(cylinders, cubes, andpentagons)inSection2.4.2. Themodelisdeemedtobeprescriptiveifitcanpredict new and untried products. To validate the prescriptive power of model (2.12) for arbitrary shapes, we build one convex and one concave freeform shape with circumcircle radius 2 00 , which are shown in Figs. 2.4(a) and 2.8(a), respectively. We will compare the prediction of model (2.12) to the measured shape deviation of two freeform shapes. To obtain model prediction of freeform shapes, we first plug the estimated parameters for polyhedron model in Section 2.4.2 into the generalized cylindrical basis model (2.5) and (2.6), and the extended cookie-cutter model (2.11). Secondly, following CASC strategy, the polygon approximation is applied to the freeform CAD models to obtain n, m, θ i , r i γ i and ϑ k (Fig. 2.6), i = 1,...,n, k = 1,...,m. They are plugged into the extended freefrom model to predict the deviation for each sector then combined together to form the prediction. In merge based approach, the distance threshold is 3 pixel, the area threshold is 10 pixel. The threshold in catching transition points is δ critical = π/6, which is not a strict constraint. For the convex freeform (Fig. 2.4(a)), the approximated number of sidesn = 104, and the number of vertices selected in cookie-cutter function m = 11 (Fig. 2.6(a)), which demonstrates that all the true vertices are founded without mistake. The concave freeform shape (Fig. 2.8(a)) has parameters n = 88 and m = 0 (Fig. 2.6(b)). The smooth shape has no transition point selected. The observed deviation profiles of the two freeform parts are shown as the solid blue lines in Figs. 2.5(a) and 2.8(b), respectively. The predicted deviation profiles for two freeform shapes using model (2.12) are represented as red dash lines in each figure. Considering the limited data used for model fitting and errors involved in freeform shape measurement (e.g., curve registration and boundary approximation in measurement), the prescriptive prediction of model (2.12) is remarkably close to the measurement. The encouraging result suggests: 36 (a) Concave freeform shape with cir- cumcircle radius = 2 00 (b) deviation profile (blue line) and model predic- tion (red dash line) of concave freeform with cir- cumcircle radius = 2 00 Figure 2.8: deviation and prediction profiles for concave freeform shape • Our methodology of predicting in-plane error of AM built product is generic, which can be directly extended from cylinder and polyhedrons to freeform shapes; • Our methodology has the capability of predicting new and untried products by learning from a limited number of tested shapes As a result, the in-plane shape deviation profile of any arbitrary freeform can be derived directly from CAD design. This provides a great opportunity to implement the optimal compensation policy established in Huang et al. (2015) to improve the shape accuracy of AM built products. It should be mentioned that parameters in Table 2.3 are estimated from experimentation with regular polyhedrons, while most of the approximated polygons are irregular. Our model prediction accuracy can be greatly improved if irregular polyhedron data are available for model learning. This indirectly demonstrates the robustness of model (2.12). 2.4.4 Prescriptive Compensation and Experimental Validation One direct way to take advantage of the prescriptive model (2.12) is to improve printing accuracy by compensating the CAD design model of an arbitrary freeform shape even before fabricating the product. Work in Tong et al. (2003, 2008) first put forward the machine 37 parametric error model to evaluate the part shape accuracy and model the parametric error functions. The negative values of predicted errors will be added to CAD design directly to compensate the product deviation. Cajal et al. (2013, 2016) further extent this error compen- sation method through employing conical sockets for probe measurement. The compensation strategy, however, is not optimal. We first establish the optimal compensation policy for 2D shapes in Huang et al. (2015) and 3D shapes in Huang (2016). Based on the result in Huang et al. (2015), the optimal amount of compensation x ∗ (θ) for 2D freeform shape is: x ∗ (θ) =− g F 1 (θ,r(θ)) +g F 2 (θ,r(θ)) 1 +g F 0 1 (θ,r(θ)) +g F 0 2 (θ,r(θ)) (2.13) Therefore, with the freeform model and its input information, the optimal compensation for each approximated sector is calculated then combined together to achieve the whole com- pensation plan. (a) deviation profile of convex freeform shape: before and after compensation (b) deviation profile of concave freeform shape: before and after compensation Figure 2.9: Freeform shape deviation profiles: before (blue lines) and after (red lines) com- pensation Two new experiments are conducted to fabricate two freeform products by modifying orig- inal freeform CAD designs based on model (2.13). Figure 2.9 compares the deviation profiles of two freeform products before (blue lines) and after (red lines) compensation implemented. We also generate Fig. 2.10 by removing the sharp spikes in Fig. 2.9. 38 (a) Main trend of deviation profile of convex freeform shape (b) Main trend of deviation profile of concave freeform shape Figure 2.10: Trend of freeform shape deviation profiles: before and after compensation It is clear from these two figures that the whole deviation profiles after compensation are more centering around zero line. Table 2.4 summarizes the compensation performance. The average deviation is reduced nearly 60% and the absolute average deviation is reduced nearly 50%. Therefore, on average the deviation reduction is remarkable considering the limited data used to predict the complicated freeform shapes. Table 2.4: Compensation performance Convex freeform shape deviation Before After Reduction Average 0.0053 00 0.0029 00 45% Absolute average 0.0055 00 0.0033 00 40% Concave freeform shape deviation Before After Reduction Average 0.0038 00 0.0015 00 61% Absolute average 0.0061 00 0.0031 00 49% 2.5 Conclusion Aiming at improving shape accuracy of AM built freeform products, this work makes a breakthrough by establishing a generic and prescriptive methodology to predict the in-plane (x−y plane)shapedeviationofarbitraryshapes. Builduponourpreviouspredictivemodelfor 39 cylinder and polyhedron shapes, we propose circular approximation with selective cornering (CASC) strategy to extend the polyhedron model to arbitrary freeform shapes. This strategy essentially approximates a freeform with a series of circular sectors with different radii first and then further improve the approximation by imposing a generalized cookie-cutter function. The series of circular sectors are modeled as generalized cylindrical basis functions. The prescriptive model is estimated/calibrated based on a limited number of trials shapes including cylinders, cubes, and pentagons. Experimental observations of both convex and concave freeform shapes match the model prediction well. It demonstrates the prescriptive capability of predicting new and untried products by learning from a limited number of tested shapes. As a result, the in-plane shape deviation profile of any arbitrary freeform can be derived directly from CAD design. To take advantage of prescriptive power of the developed model, we apply optimal com- pensation policy to freeform shapes. The shape accuracy of compensated freeform products improves by 50% on average. This experimental validation further indicates that the devel- oped prescriptive methodology has the capability of predicting new and untried products by learning from a limited number of tested shapes. 40 Chapter 3 Statistical Process Control of In-Plane Shape Deviation for Additive Manufacturing Online monitoring and feedback control has become a critical area for AM part accuracy enhancement. In contrast to mass production, AM processes fabricate products with high shape complexity at extremely low volume, which leads to disparate training data with small sample size. Furthermore, constant change of product designs demands the capability of monitoring the process of building new products not being tried before. Traditional Statistical Process Control (SPC) (Montgomery (2009)) therefore can hardly be applied directly. Existing process monitoring approaches developed for AM can be viewed as predictive processmonitoringinthesensethattrainingsamplesfromthesamepopulationarecollectedto monitor future production of the similar products. For one-of-a-kind additive manufacturing, we believe that a new category of prescriptive process monitoring methods are required to monitor quality of new and untried categories of product shapes. Therefore in this chapter, we put forth a new prescriptive online SPC monitoring scheme free of shape complexities. The strategy is extension of the shape deviation modeling in last chapter to the field of SPC. Only a limited number of test shapes are required to establish pre- scriptive process control limits for new products. Monitoring statistic is proposed to quantify shape deviation both for ideal and realistic AM processes. EWMA is adopted for individual product monitoring. The methodology is demonstrated using actual projection-based SLA process. Summary and conclusion is provided in the end. 41 3.1 Prescriptive Monitoring of Deviation From Shape to Shape For simplicity of representation, examples of product shapes are two-dimensional (2D). In real applications, a thin product or a section of a product with small thickness can be approximated with a 2D shape where only in-plane geometric error is of major concern. Training data: The training data can be illustrated by Fig. 3.1, where three circles with various sizes and three polygons are required to built and measured. Assumption: The training products are fabricated under the same process settings with the same materials. The process is relatively repeatable and deemed as normal condition. Note that normal condition does not require zero shape deviation. Objective of process monitoring: When a new and different shape is built, the monitoring scheme is expected to decide whether the process is in-control or out-of-control. Figure 3.1: Training data Challenge: Clearly, the challenge of developing a SPC scheme resides in data disparity, limited sample size, and complicated data structure. In addition, decision on new products requires the identification of inherent connection among the data. 3.1.1 A Prescriptive SPC Scheme for Shape Deviation Monitoring in AM Under the general modeling framework of control charts, the critical step is to identify the statistic for monitoring. In AM, training data illustrated in Fig. 3.1 needs a proper 42 transformation in order to derive the control chart model. To illustrate the rationale of the proposed statistics, we will first discuss an ideal AM process without systematic shape deviation patterns. We will then derive the monitoring statistic for realistic AM processes often containing systematic deviation patterns varying from shape to shape. Figure 3.2: Statistic for AM process monitoring Suppose an ideal AM process builds a circle and square shape shown in Fig. 3.2. With the presence of only natural process variation, the built products in solid curves should be contained within narrow envelops. Quantities such as roundness or cylindricity are not con- sidered here because these measures tend to be shape-dependent, which limits the ability of quantifying a different shape. Oneintuitivemeasureofshapedeviationacrossdifferentshapesisthepercentageofshrink- age/expansion η, which can be defined as η = |ΔS| Actual S Nominal (3.1) where S Nominal represents the nominal in-plane surface area of the product and|ΔS| Actual is the measured, absolute change of surface area from the design. Extension to 3D case naturally follows, which is not discussed here. Under stable process condition and the same materials, we could expect that products with various shape would have similar percentages of shrinkage. In practice, AM industry often uses it as a benchmark value for different materials. The training data in Fig. 3.1 can 43 be transformed into|ΔS|/S and individual control charts such as EWMA can be applied for process monitoring. In practice, however, current AM machines often demonstrates systematic, shape- dependent deviation patterns, even key process variables are within control. In another word, AMmachinesundernormalconditionswillhavesystematicdeviationpatterns. Directlymoni- toring|ΔS|/S couldenduplargefalsealarmrate. Thefocusofthisworkisthereforetomodify |ΔS|/S in Eq. (3.1) for realistic AM processes. The modified monitoring statistic has two variations. In the first case we remove the expected or predicted amount of surface deviation|ΔS| Predicted from the measured|ΔS| Actual . η = |ΔS| Actual −|ΔS| Predicted S Nominal (3.2) The logic justification of this proposal is that the process is deemed in control if the actual amount of deviation is close to the expected one. Since the expected deviation pattern is removed from the measurement, the AM process is equivalent to an ideal process and the false alarm rate can be reduced. ThefirstproposalinEq. (3.2)ispassivecontrolstrategyinthatsensethatactualdeviation cannot be reduced. To remedy this problem, we propose the second variation of monitoring statistic: η = |ΔS| Actual −|ΔS| Compensated S Nominal (3.3) where|ΔS| Compensated represents the predicted, absolute surface deviation after deviation com- pensation. If the shape deviation of a new product can be predicted and compensated before- hand, the percentage of shrinkage/expansion will be as small as the ideal AM process under normal condition. The false alarm rate of individual control charts can also be reduced. 44 3.1.2 Prediction of Shape Deviation of A New Product The critical element of both proposals, however, is the prescriptive model for predicting the shape deviation of a new product based on training data illustrated in Fig. 3.1. We adopt the prescriptive shape deviation modeling framework developed in Huang et al. (2015); Xu et al. (2013); Huang et al. (2014b); Sabbaghi et al. (2017); Luan & Huang (2015, 2017); Wang et al. (2017). Since the framework has been detailed introduced in Section 2.4.1, here we will not repeat. Denote Δr(θ,r 0 (θ)) as the in-plane deviation profile of a product shape in the PCS: Δr(θ,r 0 (θ)) =r(θ,r 0 (θ))−r 0 (θ) (3.4) where r 0 (θ) is the nominal design shape in PCS. The prescriptive shape deviation model for freeform shape is formulated as: Δr(θ,r 0 (θ)) =f(θ,r 0 (θ)) +(θ) =g 1 (θ,r 0 (θ)) +g 2 (θ,r 0 (θ)) +(θ) (3.5) where g 1 depicts the generalized cylindrical basis function, g 2 is the cookie-cutter function, (θ) represents the unmodeled term. With limited number of test shapes, dramatic in-plane accuracy improvement has been achieved through compensation for cylindrical shape (90%), polyhedrons and freeform shapes (> 50%) (Huang et al. (2015, 2014b); Wang et al. (2017); Jin et al. (2016a); Luan & Huang (2017)). With the prescriptive model,|ΔS| Predicted in Eq. (3.2), according to Huang (2016), can be derived as |ΔS| Predicted = Z 2π 0 r 0 (θ)|f(θ,r 0 (θ))|dθ (3.6) |ΔS| Compensated in Eq. (3.3), according to Huang (2016), can be derived as |ΔS| Compensated = Z 2π 0 r 0 (θ) f(θ,r 0 (θ) +x(θ)) +x(θ) dθ (3.7) 45 where x(θ) denotes the amount of compensation applied to CAD design to reduce shape deviation. The absolute shape area deviation with and without compensation can be illustrated by Fig. 3.3. r(θ) S θ! o r 0 (θ) (a) No compensation r(θ) S θ! o x(θ) r 0 (θ) f%(θ,)r 0 (θ))+ x(θ)) (b) After compensation Figure 3.3: Absolute shape area deviation (Huang (2016)) Undertheminimumareadeviationcriteria, theoptimalamountofcompensationis(Huang et al. (2015); Huang (2016)) x ∗ (θ) =− f(θ,r 0 (θ)) 1 +f 0 (θ,r 0 (θ)) (3.8) |ΔS| Compensated in Eq. (3.3) can be simplified as |ΔS| Compensated = Z 2π 0 r 0 (θ)|(θ)|dθ (3.9) Clearly,|ΔS| Compensated in ideal case approaches to zero. Lastly, S Nominal can be simply computed as S Nominal = Z 2π 0 r(θ)dθ 3.1.3 Control Chart to Monitor Shape Deviation With individual measurement of AM built products and the objective to detect small process shift, we adopt EWMA control chart to monitor the proposed statistic of percentage of shrinkage η defined in Eq. (3.2) or Eq. (3.3). 46 For a sequence of AM built products and their η i numbered in a chronicle order, the EWMA statistics is z i =λη i + (1−λ)z i−1 , z 0 = 0; with the corresponding control limits: UCL =μ η +Lσ η s λ 2−λ [1− (1−λ) 2i ] CL =μ η LCL =μ η −Lσ η s λ 2−λ [1− (1−λ) 2i ] (3.10) whereμ η andσ η are mean and standard deviation ofη i observed from training data, 0<λ< 1 is a constant and L is the width of the control limits. 3.1.4 Process Capability Index C p for AM Processes The proposed monitoring statistic η enables the adoption of the widely accepted process capability index C p in AM processes. Due to the nature of η, i.e., the percentage of shrink- age/expansion, it is meaningful to only specify a USL (Upper Specification Limit), e.g., 2%. The C p for AM processes can be defined as C p = USL−μ η 3σ η (3.11) where μ η and σ η represents the mean and standard deviation of statistic η. 3.1.5 Estimation of μ η and σ η For training data with n products, e.g., n = 5, μ η can be estimated as the average of η i , i = 1, 2,...,n. σ η can be estimated through moving ranges MR i =|η i −η i−1 |, and σ η is σ η = MR d 2 (3.12) 47 where MR denotes the average of moving ranges and d 2 takes the value of 1.128 since we obtain the range using two consecutive observations. 3.2 Monitoring Stereolithography Process: Methodol- ogy Demonstration In this Section we will demonstrate the proposed prescriptive SPC scheme to monitor an actual projection-based SLA process. 3.2.1 ExperimentstoObtainTrainingDataandPrescriptiveModel Establishment As introduced in Section 2, three cylinders and three polyhedrons are fabricated in a MIP- SLA machine to validate the prescriptive model for cylinder and polyhedron shapes. Design of training sample products are summarized in Table 3.1 and Fig. 2.7 shows their deviation profiles. All test parts have the same height of 0.25in. Table 3.1: Design of Training Sample Products Cross-section shape Circumcircle radius Circle r = 0.5 00 ; 1.5 00 ; 3 00 Square r = 2 00 / √ 2 Regular Pentagon r = 1 00 ; 3 00 The specification of prescriptive deviation model in Eq. (3.5) are detailed in Section 2.4.1. Specific models for could be find in Eqs. 2.3, 2.4, 2.7, 2.8, 2.9. 3.2.2 Monitoring Statistic η and Phase-I Control Charting Since all the six training parts in Table 3.1 are fabricated without compensation, Eq. (3.2) is adopted to derive the monitoring statistic η (as shown in Table 3.2) 48 Table 3.2: Monitoring statistic η i of training data Cross-section shape |ΔS| Actual |ΔS| Predicted η (×10 −4 ) 0.5 00 circle 0.00396 0.00318 2.5057 1.5 00 circle 0.05988 0.05839 1.5814 3 00 circle 0.24503 0.24175 1.7407 2 00 / √ 2 square 0.04391 0.03830 7.9578 1 00 pentagon 0.01092 0.01355 4.8185 3 00 pentagon 0.21370 0.21120 1.5328 The Normal Q-Q plot of the sample dataη i is analyzed and shown in Fig. 3.4. It indicates that the normality assumption reasonably holds. ● ● ● ● ● ● −1.0 −0.5 0.0 0.5 1.0 2e−04 3e−04 4e−04 5e−04 6e−04 7e−04 8e−04 Normal Q−Q Plot Theoretical Quantiles Sample Quantiles Figure 3.4: Normal Q-Q plot of sample data η i The estimated mean and standard deviation are: μ η = 3.3563× 10 −4 , and σ η = 2.4337× 10 −4 . For the EWMA control chart with λ = 0.1 and L = 2.7, The average run length (ARL) has ARL 0 ≈ 500 and ARL 1 ≈ 10.3 for detecting 1σ mean shift. The corresponding phase-I EWMA control chart EWMA control limits following Eq. (3.10) is shown in Table. 3.3 and Fig. 3.5: 49 Table 3.3: EWMA control limits No. Cross-section shape UCL (×10 −4 ) CL (×10 −4 ) LCL (×10 −4 ) 1 0.5 00 circle 4.0132 3.3562 2.6991 2 1.5 00 circle 4.2402 3.3562 2.4721 3 3 00 circle 4.3880 3.3562 2.3243 4 2 00 / √ 2 square 4.4938 3.3562 2.2185 5 1 00 pentagon 4.5727 3.3562 2.1396 6 3 00 pentagon 4.6331 3.3562 2.0792 ● ● ● ● ● ● 1 2 3 4 5 6 0.00020 0.00025 0.00030 0.00035 0.00040 0.00045 EWMA control chart x EWMA Figure 3.5: phase-I EWMA control chart If the Upper Specification Limit (USL) for the percentage of shrinkage/expansion is 1%, the process capability index following Eq. (3.11) is C p =13.23716. As a comparison, if compensation is applied and quality measure in Eq. (3.1) is adopted, the mean and standard deviation of η without compensation are: μ η = 6.9905× 10 −3 , σ η = 5.9972× 10 −3 . Then the process capability index following Eq. (3.11) is: C p = 0.1672709. The comparison of twoC p indicates that the optimal compensation plan owns the potential to greatly improve the ability of a process to meet the specification. 50 3.2.3 Phase-II Control Charting and Validation In order to test the detection ability of SPC scheme for AM processes, four parts printed under different conditions are employed to simulate the occurrence of process shift. The first part is a 3 00 dodecagon with compensation. Since this is an untried new shape, the compensation is derived directly from the prescriptive model for polyhedron validated in Huang et al. (2014b). Although our compensation has reduced on average 75% deviation (Huang et al. (2014b)), there are still some deviations and patterns remain, indicating that our model could be further improved with this new dataset. The second part is a 1 00 pentagon with compensation. But the compensation is wrongly added, i.e., the compensation plan is not optimal. The third part is a 1 by 1 00 square, but this square is the inner boundary of a square cavity. Since the material deviation mechanism is definitely different for inner boundary, its deviation pattern is also different. The last part is a 2 by 2 00 square printed in a different process condition. As we mentioned in Huang et al. (2014b), the MIP-SLA machine settings were changed after the experiments in Huang et al. (2015). Therefore, the part produced before machine repair have a different shape of deviation profile. The monitoring statistic η of these 4 parts are shown in Table 3.4. Table 3.4: Monitoring statistic η i of validation data Cross-section shape |ΔS| Actual |ΔS| Predicted η (×10 −3 ) 3 00 dodecagon 0.07282 0 3.9575 with compensation 1 00 pentagon 0.01267 0 2.3245 with compensation 1 00 / √ 2 square 0.02729 0.00351 6.7521 inner 2 00 / √ 2 square 0.05461 0.03812 2.3446 before repair Adding those 4 data obtained under “abnormal” printing processes, the EWMA control chart is shown in Fig. 3.6. The EWMA chart control correctly detects the changes. 51 ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 EWMA control chart x EWMA Figure 3.6: phase-II EWMA control chart 3.3 Conclusion This study first time presents a statistical process control (SPC) scheme for additive man- ufacturing (AM). Process capability index is also established to quantify the capability of AM processes. The proposed statistic for monitoring product shape deviation is consistent with industrial practice, i.e., the percentage of shrinkage/expansion. The requirement for train- ing data is minimal and prescriptive model is adopted to predict the deviation of new and untried product shapes. This monitoring statistic incorporates the predicted deviation, which enables the monitoring of new product in one-of-a-kind manufacturing environments. Real AM process experiments show the promise of applying the proposed SPC scheme. It should be noted that the proposed monitoring statistic only reflects the overall deviation of a product and there might be other alternatives to catch local deviation conditions. More future work is necessary to establish comprehensive SPC methods for AM. 52 Chapter 4 Prescriptive Data-Analytical Modeling of Selective Laser Melting Processes for Accuracy Improvement Metal Additive Manufacturing (AM) market has been growing at an impressive rate in recent years. Among various technologies suitable to produce metal parts on a layer-by-layer basis, powder bed fusion processes are attracting an increasing industrial interest thanks to their ability to produce parts with very complex and customized shapes while maintaining high mechanical performance. The study in this chapter focuses on the laser-based powder bed fusion process, a.k.a. selective laser melting (SLM), where a laser is used to locally melt the fine metal powder along a pre-defined scanning path (Gibson et al. (2010)). Despite a wide literature devoted to the study of the final part quality and the factors that affect the process accuracy, currently a comprehensive causal model is hard to be established due the process complexity. A data-driven predictive approach is therefore a promising alter- native to improve the geometric accuracy for SLM processes. This chapter intends to extend the shape deviation modeling framework we proposed in Chapter 2 to the SLM process by solving three new challenges: • Surface roughness coupled with shape deviation profile: When SLM built parts are small in size, the shape deviation error may be comparable in magnitude to the surface rough- ness, cause large uncertainty in the determination of the actual shape profile. 53 • Laser beam positioning error: Unlike the projection based AM processes, the position accuracy of laser beam in SLM may vary with location, leading to location-dependent positioning error. • Additional machine-dependent local effects: further location-based effects on the shape deviation profile may be introduced by specific properties of the SLM system used to fabricatethepart. Theycanbecausedbynon-uniforminertgasflowswithinthebuilding chamber or other inaccuracies of the equipment used for chamber environmental control. Thereforeinthischapter, anerrordecompositionandcompensationschemeisdevelopedto decouple the influence from different error components to reduce the shape deviations caused by part geometrical deviation, laser beam positioning error and other location effects simul- taneously via an integrated modeling and compensation framework. After the general idea of the integrated strategy, we will first illustrate the prescriptive modeling considering surface roughness. Then we will introduce the experimentation, analysis and prediction method for laser beam positioning error and machine-dependent location effects, respectively. 4.1 Review of Existing SLM Literature for Geometric Accuracy Control SLM technology has demonstrated its capability to produce functional parts for highly regulated industrial sectors such as aerospace and healthcare. These promising potentials have motivated a number of studies devoted to address quality related issues arising from SLM processes, e.g., geometric and surface defects, poor dimensional accuracy, porosity, residual stresses, cracks and delamination, balling phenomena, microstructural inhomogeneities and impurities. Comprehensive reviews of these defects, including their root causes and final impacts on part quality and mechanical properties can be found in Sames et al. (2016); Olakanmi et al. (2015); Mani et al. (2015); Tapia & Elwany (2014); Spears & Gold (2016); Everton et al. (2016); Grasso & Colosimo (2017). 54 As a major quality issue, dimensional and geometric deviations in SLM have been further classified into: i) shrinkage and oversize effects, ii) warping and curling, iii) bad part contours and dross formation at down-facing surfaces, and iv) super-elevated edges. A major objective of the existing SLM literature is to relate process information with these quality defects for the selection of optimized process parameter settings and scanning strategies (please refer to Mani et al. (2015) and references therein). Indeed, process parameters determine the melt pool size and stability, which directly affects the geometry of the track and, consequently, the size of the solidified part together with the shape of its local features. For example, both local and global shrinkages in SLM are reported and investigated in Thomas (2009); Sharrat (2015), while in some applications, over-size effects are observed (Abd-Elghany & Bourell (2012)). Proper optical system alignment, mirror quality, perfection of the f−θ lens, and right local profiles of the laser beam have been attributed to in-plane geometrical accuracy as well (Moylan et al. (2014)). Foster et al. (2015) points out that a mis-calibrated system may result in parts with inaccurate final dimensions. In addition, an elliptical distortion of the laser spot in SLM may occur near the edges of the baseplate due to high scanner deflection angles, which inflates the local dimensional and geometrical inaccuracy, depending on the part location on the baseplate (Foster et al. (2015)). Warping effects are believed to be influenced by heat dissipation mechanisms and the development of thermal stresses during the build (Sharrat (2015)). A non-uniform ther- mal expansion and part contraction lead to the so-called curling phenomenon (Gibson et al. (2010)). Almabrouk Mousa & Almabrouk Mousa (2016) shows that this phenomenon is usu- ally associated to an uneven shrinkage between the top and the bottom of overhanging areas. Indeed, it consists of a combination of shrinking and warping effects that yield a curved profile of down-facing surfaces intended to be flat. This geometric distortion can be mitigated by applying powder bed pre-heating or by properly setting process parameters depending on the geometry and size of the scanned feature (Sharrat (2015)). In the presence of down-facing surfaces, dross formation and bad contours are known to be caused by a lack or an improper 55 design of supporting structures (Kruth et al. (2007)). Kruth et al. (2007) also proposes a feedback control method to mitigate this kind of defects by adapting in real-time the laser power based on in-situ melt pool area measurements. Super-elevated edges represent an additional kind of out-of-plane geometrical distortion. They consist of elevated ridges of the solidified material, which affect not only the final quality of the part, but they can also induce the propagation of further defects due to possible inter- ferences with the powder recoating system. Yasa et al. (2009) discusses the effect of process parameters and scanning strategies on the flatness of scanned surfaces and the generation of super-elevated edges. 4.2 Proposed Strategy Our goal is to improve the geometrical accuracy by reducing shape deviations. Issues arising from SLM such as surface roughness, laser beam positioning error, and other location effects, have not been considered in our previous work for SLA processes (Huang et al. (2015, 2014b); Sabbaghi et al. (2017); Luan & Huang (2017)). Therefore, we propose a strategy to extend our work of shape accuracy control to metal-based SLM processes. (a) Modeling procedure (b) Implementation proce- dure Figure 4.1: Flow chart of proposed strategy for accuracy improvement 56 Figure 4.1(a) illustrates the modeling procedures based on SLM experimentation. Two sets of experiments are required: (1) location effect experimentation for investigating the laser beam positioning error and other location effects; and (2) shape deviation experimentation for establishing. Metrology effort includes surface roughness measurement and CMM (coordinate measurement machine) measurement of geometrical shapes. This modeling procedure involves three components: • Prescriptive shape deviation modeling for freeform shapes in SLM process with surface roughness influence: The actual measured shape deviation Δr(θ) at any angle θ could be represented as: Δr(θ) =f(·) + θ , wheref(·) is the shape deviation model and error term θ ∼N(0,σ 2 ). The prescriptive model f(·) will be initially established with a few test cases from shape deviation experiments (Fig. 4.1(a)). To filter out the influence of surfaceroughnessandachievebetterpredictionofshapedeviations, wesplitthevariance of error term into σ 2 = σ 2 1 +σ 2 2 where σ 2 1 and σ 2 2 are related to shape deviations and surface roughness, respectively. The surface roughness related termσ 2 will be measured and assumed to be consistent among shapes, while σ 1 has to be estimated. • Laser beam positioning error modeling using error equivalence concept: Similar to FDM extruders (Song et al. (2014); Wang et al. (2017)), laser beam during printing process may deviate from its intended position (x,y) by error (e x (x,y),e y (x,y)). To predict the laser beam positioning errors, we adopt the error equivalence concept (Wang et al. (2005); Wang & Huang (2006, 2007); Sabbaghi & Huang (2016)), by transforming the positioning error into the equivalent amount of shape design error. The transformed positioning error will be readily integrated into the shape deviation prediction model. • Location effect modeling to capture other machine-related location effects: The machine- dependent location effect is approximated as a fixed effect at each location to change the shape. In the same logic thinking, we adopt the error equivalence concept and use a location-dependent term x 0 (s) to capture the machine-dependent location effect 57 at location s. Termed as the location effect term, it is represented as the equivalent amount of shape design error as well, which can be easily integrated into the finalized shape prediction model. Similar treatment can be found for “over-exposure" effect in SLA processes (Huang et al. (2015)). Once the shape deviation model is established, accuracy prediction and compensation before printing a new part can follow the procedure in Fig. 4.1(b). 4.3 Prescriptive Shape Deviation Modeling for Freeform Shapes in SLM with Surface Rough- ness Influence 4.3.1 Shape Deviation Experimentation and Initial Analysis of Shape Deviation for SLM We adopt the in-plane shape deviation modeling and compensation strategies for freeform 3D printed products in SLA processes established and validated in Huang et al. (2015, 2014a,b); Sabbaghi et al. (2015, 2017); Luan & Huang (2015, 2017). Since the framework has been detailed introduced in Section 2.4.1, here we will not repeat. The design of shape deviation experimentation is shown in Fig. 4.2. All six test parts are produced on the same baseplate by using a Renishaw AM250 SLM system. Among them, three cylinders, one square and one pentagon are adopted to estimate the parameters in Eq. 2.12 (generalized for freeform shapes from Eq. 2.1). The freeform shape is used to test the prescriptive power of the established model, i.e., using only a few simple test parts to predict shape deviation of freeform products. A gas-atomized 18Ni (300) maraging alloy powder supplied by Sandvik Osprey LTD (Neath, UK) with average particle size of 35μm is used. Default process parameters for 58 Figure 4.2: Design of shape deviation experimentation on a single plate (unit: mm) this kind of metal powder are applied, which are shown in Table 4.1. The height of all the parts is 5mm. The shape profile measurements for in-plane error evaluation are performed at a fixed height of 4.5mm above the baseplate by using a Zeiss Prismo 5 VAST MPS HTG Coordinate Measuring Machine (CMM) equipped with a 1mm radius probe. Table 4.1: The specific parameters of the SLM process Variable value laser power P 200W exposure time t 104μs point distance along the track d p 65μm hatch distance d h 80μm layer thickness z 50μm scanning strategy meandering The three cylinders with radii 20mm, 10mm and 5mm are applied to estimate the cylin- drical basis model Eq. 2.2 with cookie-cutter function g 2 = 0. Bayesian model estimation via Markov Chain Monte Carlo (MCMC) is given in Table 4.2. The deviation and predic- tion profiles are shown in Figure 4.3. Note that we assume the priors a∼ N(1, 2 2 ), b∼ N(1, 1 2 ), log(x 0 )∼N(0, 1 2 ) and place flat priors on β 0 , β 1 , and log(σ). The square and pentagon with circumcircle radii 13 √ 2mm and 15mm respectively and the three cylinders are then pooled together to estimate the complete freeform model Eq. 2.12 59 Table 4.2: Summary of posterior draws in cylindrical basis model Mean SD 2.5% Median 97.5% β 0 −0.003623 0.001137 −0.005463 −0.003824 −0.001186 β 1 −0.001065 4.3919× 10 −4 −0.002040 −0.001018 −0.000339 a 0.626926 0.104855 0.4950065 0.597999 0.915121 b 0.841573 0.157791 0.5731525 0.831657 1.202621 x 0 0.008658 0.002045 0.004202 0.009035 0.012160 σ 0.014855 3.0791× 10 −4 0.014281 0.014861 0.015475 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 Deviation Profile and Model Fit of Large Circle Shape θ Deformation (mm) (a) 20mm cylinder 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 Deviation Profile and Model Fit of Medium Circle Shape θ Deformation (mm) (b) 10mm cylinder 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 Deviation Profile and Model Fit of Small Circle Shape θ Deformation (mm) (c) 5mm cylinder Figure 4.3: Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by cylindrical basis model g 1 (generalized for freeform shapes from Eq. 2.1). To specify a model robust to both cylinder and polyhedron shapes, we keep the parameters a, b and x 0 in Table 4.2. The parameter estimation with same Bayesian procedure as well as the prediction profiles are given in Table 4.3 and Figure 4.5. We assume prior α∼ N(1, 1 2 ) and place flat priors on β 0 , β 1 , β 2 , and log(σ). Table 4.3: Summary of posterior draws for estimating freeform model Mean SD 2.5% Median 97.5% β 0 −0.002653 1.1062× 10 −4 −0.002861 −0.002651 −0.002444 β 1 −6.9131× 10 −4 7.4398× 10 −5 −8.3113× 10 −4 −6.9054× 10 −4 −5.4787× 10 −4 β 2 −5.5581× 10 −4 2.7009× 10 −4 −0.001174 −5.2043× 10 −4 −1.1727× 10 −4 α 1.436602 0.212887 1.088236 1.415376 1.998083 σ 0.020724 3.3094× 10 −4 0.020086 0.020726 0.021376 60 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 Deviation Profile and Model Fit of Large Circle Shape θ Deformation (mm) (a) 20mm cylinder 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 0.04 Deviation Profile and Model Fit of Medium Circle Shape θ Deformation (mm) (b) 10mm cylinder 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 0.04 Deviation Profile and Model Fit of Small Circle Shape θ Deformation (mm) (c) 5mm cylinder Figure 4.4: Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by freeform model 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Pentagon Shape θ Deformation (mm) (a) 15mm pentagon 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Square Shape θ Deformation (mm) (b) 13 √ 2mm square Figure 4.5: Deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of the polyhedron shapes by freeform model Note that the cylindrical base model Eq. 2.2 for SLM process doesn’t need to separate upper and lower like in Eqs. 2.3 and 2.4. Its generalization for freeform shapes no longer need the two separate models in Eqs. 2.5 and 2.6. Instead, we just simply adopt Eq. 4.1. 61 g 1 (θ,r(θ)) =x 0 +β 0 (r i (θ i ) +x 0 ) a +β 1 (r i (θ i ) +x 0 ) b cos(2θ) for θ i−1 ≤θ<θ i , 1≤i≤n, θ 0 =θ n (4.1) Applying the estimated parameters to the generalized freeform model (Eqs. 2.12, 4.1 and 2.11), we derive the prediction profile of the freeform with circumcircle radius 12mm in Figure 4.6. Here circumcircle means the smallest circle that contains the freeform shape. 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 0.04 Deviation Profile and Prediction of Freeform Shape θ Deformation (mm) Figure 4.6: Deviation profile (dots) and prediction (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of 12mm freeform shape The prediction results capture the main trend of shape deviation well. However, since the relatively small size of experimental parts leads to small shape deviation, it is necessary to verify whether the actual geometric deviation is significant in the test parts. The null hypothesis is that there is no geometric shape deviation in the 6 experimental parts (Figure 4.2), the Bayesian posterior intervals are used. Figures 4.4, 4.5 and 4.6 indicate 95% predictive intervals that contain the zero line, meaning the null hypothesis cannot be rejected. A major reason of this conclusion is the low signal to noise ratio (S/N), where the relatively large “noises" come from surface roughness, laser beam positioning error and measurement error. From experimental analysis, the standard deviation of surface roughness is around 62 14μm, the magnitude of laser beam positioning error is±25μm and the magnitude of CMM measurement error is±2μm. Those errors are of similar magnitude as the shape deviations of parts with small size. Since it is important to conduct independent assessment of shape deviation by excluding surface roughness, we will use this set of experimental data to demonstrate the proposed methods. For parts with larger size, the developed methods and procedures can be applied in the same manner. 4.3.2 Filtering Surface Roughness Surface roughness in SLM is determined by different factors (Strano et al. (2013)). The out-of-plane roughness is dominated by the stair-step effect, which mainly depends on the powder size and the layer thickness. In-plane roughness, the one of interest in this study, is mainly affected by process parameters, the material properties and the powder size. An as- build average roughness,R a in the order of 8-12μm was reported for SLM of steel powders by using default process parameters (Strano et al. (2013); Wang et al. (2016)). Depending on the surface functionality, post-process treatments and machining operations are usually applied to comply with final product specifications. However, to determine the in-plane geometric error of as-build parts, the effect of surface roughness needs to be quantified and taken into account. In our experiment, the roughness measurement is conducted on the circumference of the cylinder with radius 5mm, assuming that surface roughness is not influenced by the part size. The roughness profile and measured roughness indexes are reported in Figure 4.7 and Table 4.4. Figure 4.7: Measured surface roughness profile for cylinder with r 0 = 5mm 63 Table 4.4: Surface roughness indexes for cylinder with r 0 = 5mm R a R q R z R max 10.9μm 14.2μm 60.2μm 80.5μm where R a = 1 n n P i=1 |y i | is the arithmetic average of absolute values, R q = s 1 n n P i=1 y 2 i is the root mean squared value, R z is the average maximum height of the profile, R max is the maximum roughness depth and y i is the roughness value at ith point. Comparing the measured deviation (dots in Figure 4.3) and the roughness measurement (Figure 4.7 and Table 4.4), the surface roughness is non-negligible. The mixture of roughness dramatically increases the variance of deviation profile, leading to much wide predictive inter- vals. Therefore, the variance caused by surface roughness needs to be filtered out to achieve actual predictive intervals on shape deviations. Surface roughness and high frequency components of shape deviations are hard to be distinguished. Considering their distinct characteristics and mechanisms, we introduce two independent components in model error term θ ∼N(0,σ 2 ) with σ 2 =σ 2 1 +σ 2 2 . Here σ 2 1 and σ 2 2 are related to shape deviations and surface roughness, respectively. We set σ 2 equal to R q (R q = 14.2μm in our test part) and leave σ 1 for model estimation from the data. Figure 4.8: Illustration of roughness influence in PCS One critical issue is that surface roughness is evaluated along the direction normal to the surface, while shape deviations of non-cylindrical surfaces are represented as the deviation 64 along the radial direction under a PCS. Figure 4.8 shows these two directions are not necessar- ily parallel to each other. Denotey v (ϕ) as the measured roughness along the direction normal to the surface. Its impact on shape deviation along the radial direction is y r (ϕ) = y v (ϕ) sinϕ whereϕ is the angle with reference parallel to the nominal surface. Translation fromθ toϕ is shown in Eq. 4.2 which is edge-dependent. Therefore in deviation modeling, σ 2 should vary with ϕ and follow σ 2 (ϕ) = σ 2 sinϕ and ϕ =θ−φ 0 + (n− 2)π 2n + ( 2π n )(θ−φ 0 )INT ( 2π n ), θ∈ [0, 2π] (4.2) 4.3.3 Initial Model Building and Validation by Filtering Surface Roughness In this section, we re-establish the shape deviation model by filtering the surface roughness σ 2 . We use the same dataset and model estimation procedure introduced in Section 4.3.1. The Bayesian model estimation of cylindrical basis model Eq. 2.2 and the whole freeform model Eq. 2.12 are listed in Tables 4.5 and 4.6, separately. Their corresponding prediction profiles are shown in Figure 4.9 and 4.11, while the 2.5%, mean and 97.5% posterior quantiles consider σ 1 only. Apparently, considering surface roughness in model fitting greatly improve the prediction power for shape deviations. Table 4.5: Summary of posterior draws in cylindrical model Mean SD 2.5% Median 97.5% β 0 −0.002118 6.2380× 10 −4 −0.003185 −0.002157 −0.000851 β 1 −0.001067 4.2940× 10 −4 −0.002067 −0.001034 −0.000382 a 0.763537 0.098170 0.624365 0.743364 1.012372 b 0.841486 0.156348 0.587817 0.830261 1.160492 x 0 0.006113 0.001427 0.003129 0.006126 0.008806 σ 1 0.002675 0.001813 6.8206× 10 −4 0.002923 0.005676 65 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Large Circle Shape θ Deformation (mm) (a) 20mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Medium Circle Shape θ Deformation (mm) (b) 10mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Small Circle Shape θ Deformation (mm) (c) 5mm cylinder Figure 4.9: Actual deviation (dots) and prediction profiles (solid lines denote posterior means, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by cylin- drical basis model g 1 while filtering surface roughness Table 4.6: Summary of posterior draws for estimating freeform model Mean SD 2.5% Median 97.5% β 0 −0.001614 6.9346× 10 −5 −0.001752 −0.001612 −0.001490 β 1 −7.0420× 10 −4 6.5672× 10 −5 −8.2621× 10 −4 −7.0442× 10 −4 −5.7230× 10 −4 β 2 −5.5815× 10 −4 3.5792× 10 −4 −0.001193 −4.6014× 10 −4 −1.0681× 10 −4 α 1.45006 0.295512 1.042592 1.426695 1.987236 σ 1 0.010975 4.9288× 10 −4 0.009997 0.010974 0.011985 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Large Circle Shape θ Deformation (mm) (a) 20mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Medium Circle Shape θ Deformation (mm) (b) 10mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Small Circle Shape θ Deformation (mm) (c) 5mm cylinder Figure4.10: Actualdeviation(dots)andpredictionprofiles(solidlinesdenoteposteriormeans, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of three cylinders by freeform model while filtering surface roughness Applying the estimated parameters to the generalized freeform model (Eqs. 2.12, 4.1, 2.11), the predicted shape deviation profile of the freeform shape after filtering the surface roughness is shown in Figure 4.12. 66 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of Pentagon Shape θ Deformation (mm) (a) 15mm pentagon 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Deviation Profile and Model Fit of square Shape θ Deformation (mm) (b) 13 √ 2mm square Figure4.11: Actualdeviation(dots)andpredictionprofiles(solidlinesdenoteposteriormeans, and dashed lines denote the 2.5% and 97.5% posterior quantiles) of the polyhedron shapes by freeform model while filtering surface roughness 0 1 2 3 4 5 6 −0.04 −0.02 0.00 0.02 0.04 Deviation Profile and Prediction of Freeform Shape θ Deformation (mm) Figure 4.12: Actual deviation profile (dots) and prediction (solid lines denote posterior means, anddashedlinesdenotethe 2.5%and 97.5%posteriorquantiles)of 12mmfreeformshapewhile filtering surface roughness Figures 4.10, 4.11 and 4.12 still indicate 95% predictive intervals that contain the zero line, meaning the null hypothesis of no geometric shape deviation still cannot be rejected. 67 The major reason is that besides surface roughness, other “noises” such as laser beam posi- tioning error and measurement error are still relatively large comparing to the small size of experimental parts, making the signal to noise ratio (S/N) still relatively low. 4.4 Laser Beam Positioning Error Prediction and Elim- ination with Error Equivalence Concept Quality of AM built parts can vary with spatial locations on a building plate. Laser beam positioning error and other machine-dependent location effects are two influential sources contributing to spatial variations. In this section, we will first introduce the location-effect experimentation to investigate these two error sources. Laser beam positioning error will be modeled and integrated into shape deviation prediction. Other location effects will be analyzed in the next Section. 4.4.1 Location Effect Experimentation To gain insight into the laser positioning accuracy, a regular grid plate with 9× 9 small cylinders are printed by the same SLM machine (as shown in Figure 4.13). The radius of each cylinder is 2.5mm and the height is 5mm. The cylinders are produced by using the same process settings and same metal powder described in Section 4.3.1. The same CMM system with 1mm radius probe is used to measure the circumference profile of cylinders at the height of 4.5mm above the baseplate. This experimentation is employed to model and predict the two location-dependent errors. The laser beam positioning error is benchmarked by the difference between the actual center of each cylinder and its nominal center, while the machine-dependent location effect at each cylinder location is estimated from each deviation profile. Sincenoextramarkerisaddedtolocatethepartcenter, thecenterofeachcylinderneedsto be identified via numerical approximation. A linear least square based circle fitting algorithm 68 Figure 4.13: Experimental grid with 9× 9 small cylinders in location effect experimentation is adopted here to estimate the actual cylinder center according to CMM measurement data (Coope (1993)). This algorithm is not sensitive to outliers and provides a quite reliable center estimation for the regular geometries. 4.4.2 Predictive Modeling of Laser Beam Positioning Error In the location effect experimentation, the cylinders are so small that their positions are highly influenced by the laser beam positions. Therefore, the actual center of each cylinder could be benchmarked by its nominal position to represent the laser beam positioning error at different locations. The pattern of positioning error within the building area is shown in Figure 4.14. It shows that the actual laser beam positions tend to be closer to plate center, which is consistent with the analysis of extruder position errors in FDM process (Song et al. (2014); Wang et al. (2017)). The measured laser beam positioning error along x− and y− directions separately are shown as the points in Figure 4.15. We adopt a 2nd order polynomial model under the 69 Figure 4.14: Pattern of the laser beam positioning error. Each point denotes the desired position of cylinder center. Measured and predicted positioning errors are denoted by solid and dashed arrows, respectively assumption of lack of spatial correlation of the residual in Eq. 4.3 to model the laser beam positioning error in x− and y− directions separately. Z(x,y) =p 00 +p 10 ∗x +p 01 ∗y +p 20 ∗x 2 +p 11 ∗x∗y +p 02 ∗y 2 +,∼N(0,σ) (4.3) where Z(x,y) represents either e x (x,y) or e y (x,y). Model estimation as well as 95% confidence interval are listed in Table 4.7. To verify the existence of laser beam positioning error in both directions, F test with 5 and 75 degrees of freedom is conducted. The small p−value for F test indicates the existence of positioning error in both directions. We adopt the sum of square error (SSE) and adjusted R-square to quantify the goodness of fitting, which could be found in Table 4.8. Corresponding modeling fitting plots and residual plots are shown in Figure 4.15 and Figure 4.16. The dashed arrows in Figure 4.14 show the prediction of positioning error. Since in our experiment, the centers are identified by numerical approximation (Coope (1993)), the measured laser beam positioning error at each cylinder center may contain two 70 Table 4.7: Summary of model fitting for positioning errors inx− andy− directions with 95% confidence interval e x (x,y) e y (x,y) Mean 2.5% 97.5% Mean 2.5% 97.5% p 00 2.359×10 −4 -0.003575 0.004047 0.00671 0.004665 0.008756 p 10 -2.497×10 −4 -2.93×10 −4 -2.065×10 −4 -8.804×10 −6 -3.201e×10 −5 1.44×10 −5 p 01 -1.453×10 −4 -1.886×10 −4 -1.021×10 −4 -2.722×10 −4 -2.954×10 −4 -2.49×10 −4 p 20 -1.02×10 −6 -2.081×10 −6 3.961×10 −8 7.069×10 −7 1.38×10 −7 1.276×10 −6 p 11 -9.501×10 −7 -1.880×10 −6 -1.995×10 −8 -9.936×10 −8 -5.986×10 −7 3.999×10 −7 p 02 1.333×10 −6 2.733×10 −7 2.393×10 −6 5.038×10 −7 -6.516×10 −8 1.073×10 −6 Table 4.8: Goodness of fitting for positioning error in x− and y− directions e x (x,y) e y (x,y) SSE 0.00618 0.00178 Adjusted R-square 0.6997 0.8732 p-value of F test < 2.2×10 −16 < 2.2×10 −16 (a) x− direction (b) y− direction Figure 4.15: Measurement (points) and prediction profiles (surface) of the laser beam posi- tioning error (unit: mm) (a) x− direction (b) y− direction Figure 4.16: Model fitting residual plots of the laser beam positioning error (unit: mm) 71 portions: the center fitting error and actual positioning error. Consequently, the center fitting error needs to be quantified. Here we adopted a numerical method to estimate the standard deviation of center fitting in both x− and y− directions. For each small cylinder, the measurement data contains around 200 points along the boundary. We randomly pick 160 points for center fitting and repeat 50 times. Figure 4.17 presents the standard deviation of the 50 estimations for each cylinder. Comparing to the measured positioning error in Figure 4.15, the center fitting error is negligible. So we will not consider the center fitting error when modeling laser beam positioning error. (a) x− direction (b) y− direction Figure 4.17: Center fitting standard deviation of each cylinder (unit: mm) 4.4.3 Transforming Positioning Error into the Equivalent Amount of Shape Design Error In this sub-section we aim to model the impact of laser beam positioning errors on shape deviation. Similar to the extruder positioning error in FDM process (Wang et al. (2017); Song et al. (2014)), the proposed strategy of controlling laser beam positioning error is to transform it into the equivalent amount of shape design error. After the transformation, our compensation framework could be able to compensate the positioning error along with the shape deviations. 72 Laser beam during printing process may deviate from its intended position (x,y) by error (e x (x,y),e y (x,y)). As a result, the actual point position (x 0 ,y 0 ) could be presented as:      x 0 =x +e x (x,y) y 0 =y +e y (x,y) (4.4) The error equivalence concept (Wang et al. (2005); Wang & Huang (2006, 2007); Sabbaghi & Huang (2016)) is adopted to transform positioning error into the equivalent amount of shape design error by: (i) predicting the actual laser beam position (x 0 ,y 0 ) with Eq. 4.4; (ii) transforming (x 0 ,y 0 ) into PCS as r 0 (θ), then calculating deviation error as Δr(θ) = r(θ)−r 0 (θ). Here r(θ) and Δr(θ) represent the actual size and deviation along radius direction at angle θ respectively. Our compensation framework is built to compensate Δr(θ), which contains both shape deviation and positioning error. Applying the transformed positioning error to the three cylinders in shape deviation exper- imentation, the equivalent amount of deviation profile containing both actual shape deviation and positioning error is shown in Figure 4.18. Since the null hypothesis of no geometric shape deviation cannot be rejected in Section 3, here we will not re-estimate the shape deviation model again. 4.5 Analysis and Modeling of Additional Location Effects Different SLM systems may exhibit different location effects on the resulting geometrical accuracy. This can be caused by difference in the chamber equipment and environmental control configurations. As an example, non uniform inert gas flows may lead to non-uniform 73 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Equivalent Deviation Profile of Large Circle Shape θ Deformation (mm) (a) 20mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Equivalent Deviation Profile of Medium Circle Shape θ Deformation (mm) (b) 10mm cylinder 0 1 2 3 4 5 6 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 Equivalent Deviation Profile of Small Circle Shape θ Deformation (mm) (c) 5mm cylinder Figure 4.18: Equivalent deviation profiles of three cylinders shape deviation profiles within the building area. In this study, the existence of additional location effects is approximated as a fixed effect at each location. We adopt the error equiv- alence concept and use a location-dependent term x 0 (s) to capture this effect at location s. Here we use the position of part centers = (x,y) to represent the part location. Termed as the “location effect term”, it is represented as the equivalent amount of shape design error, which is integrated as a term into the finalized shape prediction model Eq. 2.1. Similar treatment can be found for “over-exposure” effect in SLA processes (Huang et al. (2015)). 4.5.1 Estimation of Location Effect Term x 0 (s) at Different Loca- tions To gain insight of any possible machine-dependent location effect, the actual deviation profiles of the small cylinders in location effect experiment (Figure 4.13) are analyzed first. For instance, Figure 4.19 presents the deviation profiles of the 9 cylinders in the top row. Comparing all the deviation profiles, two conclusions could be derived: (i) deviation profiles of those cylinders share the same pattern with our validated shape deviation model; (ii) the deviation magnitudes are slightly different especially in some columns. We adopt the cylindrical basis model Eq. 2.2 and the three cylinders (r = 20mm, 10mm, and 5mm) in shape deviation experiment plus the 81 small cylinders (r = 2.5mm) in location 74 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 1 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 2 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 3 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 4 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 5 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 6 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 7 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 8 θ Deformation (mm) 0 1 2 3 4 5 6 −0.04 0.00 0.04 0.08 row 1 col 9 θ Deformation (mm) Figure 4.19: Deviation profiles of the 9 cylinders in the top row of the grid in location effect experimentation effect experiment to estimate x 0 (s) at different locations. To gain an estimation consistent with the previously validated model, we fix all the parameters in Table 4.5 except x 0 and σ 1 , then conduct same Bayesian procedures to estimate the 84 independent x 0 simultaneously. The estimated x 0 (s) at the 81 locations in location effect experiment is shown as the points in Figure 4.20(a), which verifies that a more severe location effect, possibly cause by a non- uniform gas flow indeed exist at the right top area of the plate. 75 4.5.2 Predictive Modeling of Location Effect Term x 0 (s) The 81x 0 estimated by the small cylinders in the location effect experiment are employed to fit the predictive model forx 0 (s). The three larger cylinders in shape deviation experiment remain for validation. We still employ a 2nd order polynomial model to predict x 0 (s) at any position s = (x,y), which could be formulated as: x 0 (s) =p 00 +p 10 ∗x +p 01 ∗y +p 20 ∗x 2 +p 11 ∗x∗y +p 02 ∗y 2 +,∼N(0,σ) (4.5) Model estimation as well as 95% confidence interval are listed in Table 4.9. Thep-value of F test with 5 and 75 degrees of freedom (shown in Table 4.10) further illustrate the existence of location effect. The sum of square error (SSE) and adjusted R-square are shown in Table 4.10 to quantify the goodness of fitting. Figure 4.20 presents the corresponding modeling fitting plots and residual plots. Table4.9: Summaryofmodelfittingforlocationeffecttermx 0 (s)with95%confidenceinterval x 0 (s) Mean 2.5% 97.5% p 00 0.02502 0.02406 0.02597 p 10 1.416×10 −4 1.308×10 −4 1.525×10 −4 p 01 6.331×10 −5 5.244×10 −5 7.419×10 −5 p 20 -1.257×10 −6 -1.524×10 −6 -9.906×10 −7 p 11 8.097×10 −7 5.758×10 −7 1.044×10 −6 p 02 -3.916×10 −7 -6.582×10 −7 -1.25×10 −7 Table 4.10: Goodness of fitting for location effect term x 0 (s) x 0 (s) SSE 0.0003908 Adjusted R-square 0.9221 p-value of F test < 2.2×10 −16 Validation analysis is conducted on the three cylinders in the shape deviation experiment (Figure 4.2). The original x 0 fitted when estimating shape deviation model in Section 3.4, the new x 0 fitted when estimating x 0 (s) at different locations in Section 5.1 as well as the x 0 76 (a) Estimated x 0 (s) (points) and prediction pro- file (surface) (b) Model fitting residual plot Figure 4.20: Model fitting and residual plots of the location effect term x 0 (s) (unit: mm) predicted by the established model in Eq. 4.5 are compared in Table 4.11. The desired center position s = (x,y) of each cylinder is also listed in the table. Table 4.11: Fitted vs predicted x 0 for the three cylinders in Figure 4.2 (unit: mm) cylinder radius (mm) x 0 fitted in shape deviation modeling x 0 fitted when estimating x 0 (s) x 0 predicted by the estimated model x y 20 0.006113 0.007339 0.0348 55 72.5 10 0.006113 0.008653 0.0175 25 -87.5 5 0.006113 0.004668 0.0184 75 -67.5 The fittedx 0 are similar no matter being treated as a constant or independent parameters. However, the predictedx 0 is relatively larger than the actual estimatedx 0 . The reason mainly comes from the lack of data. Although the location effect term x 0 is fixed among different parts, it has a relative higher influence on the small parts. Therefore, if all the parts are very small, the estimation process will pay more weights on x 0 (s), leading to some bias. To avoid the bias, multiple parts with different size should be printed at each location and estimated together. 4.6 Conclusion Facing new challenges in the final part quality improvement of SLM process, this work proposes a prescriptive data-analytical modeling approach that could decouple different error 77 sources and achieve a comprehensive model to predict shape deviations. Build upon our pre- scriptive in-plane shape deviation modeling and compensation framework for freeform shapes in stereolithography process, this work achieves methodological extension from the following three aspects. First, the non-negligible surface roughness is filtered out from the geometric deviation profile to guarantee confident part deviation prediction. Second, we establish spatial models to quantify laser beam positioning error in both x− andy− directions. Last, we esti- mate a machine-dependent location effect and build a predictive model. Both the laser beam positioning error and this additional location effect are transfered into equivalent shape design error following equivalent error concept and compensated along with geometric deviation. The prescriptive shape deviation model is established by using only a limited number of benchmark geometries and validated on a freeform product. The analysis and modeling of the location-dependent errors including laser beam positioning error and machine-dependent location effects are conducted on a 9× 9 regular cylinder grid. The modeling analysis demon- strates the existence of both errors in SLM process. The established spatial predictive models illustrate the goodness of fitting. Though the proposed data-analytical black-box modeling framework can be applicable to different SLM processes, further experimentation and analysis is needed to investigate the SLM process performance when building larger products with more complicated shapes. 78 Chapter 5 Data-Driven Modeling of Thermal Physics in Powder Bed Fusion 3D Printing Using Deep Learning Despite the promising advantages, the accuracy of the product quality data based prescrip- tive shape deviation modeling is sometimes limited because it is an offline modeling schema and ignores the process information. 3D printing relies on complicated physical process no matter which principle it follows. For instance, in the powder bed fusion based technique (e.g, SLS, SLM, EBM and MJF), the material goes through a complicated thermal journery involving powder melting, phase change, fusion and so on. The process thermal and material behaviors are directly related to the end-part quality. Significant process variables, for exam- ple, laser beam and scanning speed in laser based techniques, powder, as well as the fusing agents in MJF, are essential to determine the process behaviors. Consequently, to achieve better shape accuracy control in specific technique, a better control over the specific process variables and characteristics are important. An accurate and high speed process behavior prediction model is therefore highly demanded. Currently, the first principles based models are dominant, but they still require more process knowledge. And the first principles based prediction is too slow to satisfy the online feedback control. In this chapter, we will look into the powder bed fusion based techniques and explore the quantitative process behavior models which could enable the feedback control of the precess thermal behavior. This chapter will analyze HP’s Multi Jet Fusion (MJF) technique as an example. We aims to discover quantitatively accurate material behavior models and 79 thermal predictive models using in-situ machine generated data including both sensing data and comprehensive machine instructions generated by printer firmware pipeline. This chapter starts with introducing our research problem, and framing our deep learning architecture, and followed by two concrete research outcomes to exemplify the use and performance of our architecture. First example is predictive thermal models to allow “look ahead” of the layers to be constructed in an in-situ environment. This model needs to be both sufficiently accurate and sufficiently fast to enable embedding the models into the firmware for runtime closed- loop thermal control. The second example lies in scientific discovery of heterogenous material properties in an in-situ environment. 5.1 Introduction and Contribution In the powder bed fusion 3D printing techniques, the material goes through a complicated thermal journey involving powder melting, phase change, fusion and so on. The process thermal and material behaviors directly relate to and reflect the end-part quality. Despite the researchefforts,thereisnoprioreffortonquantitativemodelsthatallowtransientpredictionof thermal behavior at print process resolution (voxel-by-voxel in space, layer-by-layer in time). First principles based approaches are dominant in the related research, but it is difficult for powder bed fusion 3D printing techniques to rely on such methods in near future for quantitatively accurate predictions due to either the high computational workload or the complicated heterogenous material behaviors that are not yet known. Meanwhile, limited efforts exist on applying deep learning approaches to thermal physics. Today deep learning predominately is applied to computer vision and speech recognition. This dissertation for the first time put forward quantitative modeling strategies to apply deep learning to thermal physics. The basic idea is to apply deep learning to build thermal predictive models and material behavioral models. Those models will not only provide us an accurate thermal prediction, but they also will help to uncover the complicated physical 80 parametersandproperties. Inaddition, theaccuratethermalpredictivetoolwillenableclosed- loop thermal control at voxel level to ensure the end part quality. The domain knowledge, the thermal physics, allows us to introduce physics-oriented insights to drive model designing and setting. Our modeling approach successfully built a quantitative model for predicting the transient fusing layer thermal behavior with print process resolution. Besides that, our deep learning architecture successfully built out a quantitative model that learns thermal diffusivity as a deep neural network and predict the heat flux as well thermal behavior over time. Just like 3D printing research at large, within HP’s 3D printing R&D, multiple different modeling approaches are simultaneously developed and deployed. The first-principle based approaches (signified by various finite element models) have been providing great insights to inform research however they cannot provide quantitatively accurate predictions accounts for both part geometry complexity and process physics complexity. Experimental-measurement- based modeling approaches have been providing excellent quantitative predictions for fusing sciences that have been applied to inform printer control to date. The preliminary results from our deep neural network model have been able to demonstrate much better accuracy than empirical models (in certain test case order of magnitude improvement). And the methodol- ogy demonstrated here are broadly applicable to other powder bed fusion based 3D printing techniques. 5.2 Challenge: Complex Process Physics and the Need for Quantitatively Accurate Process Modeling and Prediction In the powder bed fusion 3D printing techniques (e.g., SLS, SLM, EBM and MJF), end- part functional quality depends on each voxel that forms this part experiencing a similar thermal history to minimize any functional irregularities such as built-in thermal stress, which may result in warpage. Additionally, to ensure multiple parts out of the same “lot” (i.e., 81 build volume) have similar functional behavior (e.g., ultimate tensile strength, elongation at break), it is useful to ensure each part goes through a similar thermal journey. The control of transient thermal behavior is essential to ensure the end part quality and thus yield. To realize closed-loop voxel level thermal control we require a quantitative model that allows transient prediction of thermal behavior with print process resolution. This is fundamentally challenging because the fusing science underlying, is very complex, involves multiple domains of sciences, and is not well understood. 5.2.1 Introduction to HP’s Multi Jet Fusion Inthisdissertation,wewillanalyzeHPâĂŹsMultiJetFusionasanexampletodemonstrate our approach. HP’s MJF is a powder bed fusion based technology but does not use lasers. Instead, HP invents the chemical compound called fusing agent, which will absorb energy from lamps and drive powder fusion. It uses droplets as an agent delivery vehicle enabling the application of HP’s famed scalable printing architecture that allows precision delivery of droplets of pico-liter resolution with lightning speed. Working as the energy source, two types of lamps are assembled in the printer. Several top lamps are equipped on top of the building bed, aiming to heat the powder bed to a certain temperature uniformly. Another type of lamps, which are called fusing lamps, are placed on the printing and fusing carriage. The objective of fusing lamps is to provide enough energy to fuse the powder selectively. Figure 5.1 shows the printing process of one layer. After printing the past layer, the build bed will first move down a little bit. Then the y-moving material recoater will recoat the build bed area evenly with a thin layer (70 - 120 micron) of material powder. Next, a x-moving printing and fusing carriage with an HP Thermal Inkjet (printhead) array and energy source (fusing lamps) passes over the build bed forward and backward to drive the layered printing. The forward pass will jet fusing agents to where particles need to be selectively molten. At the same time, the jetted fusing agents will absorb the fusing lamp energy, increase the powder temperature, and drive the powder being fused together. The backward pass will jet detailing 82 agent around the contours to cool down then improve part resolution. The detailing agents are mainlywater, whoseevaporationwilltakeawayenergytherebyreducingthetemperature. The adoption of detailing agents could effectively avoid over-cure due to high powder temperature. A schematic of MJF printing architecture and a snapshot of layer printing can be found in Figure5.2. Formoredetails, pleaserefertoHPMultiJetFusiontechnicalwhitepaper(HPInc. (2017)). The material adopted in the current Multi Jet Fusion 3D printing machine is the polymer. Figure 5.1: Basic elements of HP’s Multi Jet Fusion process (HPInc. (2017)) (black droplets: fusing agents, blue droplets: detailing agents) (a) Schematic of MJF synchronous printing architecture (HPInc. (2017)) (b) Model fitting residual plot Figure 5.2: HP’s Multi Jet Fusion printing principle Consequently, the operating principle of HP’s Multi Jet Fusion 3D printing technique is projecting thermal energy over a build area, then rely on selectively depositing agents 83 (droplets) at pixel level to enable control over voxel-level energy deposition, which eventually triggers phase change and solidifies voxels selectively. 5.2.2 Challenge: End-Part Quality With fusing science as the operating principle, the thermal energy and material thermal behavior play an essential role in the end-part dimensional accuracy. As shown in Figure 5.3, the fusing window for good parts is tight, which requires accurate thermal control at the voxel level. Figure 5.3: Layer energy process variation Theclosed-loopcontrolatvoxellevelholdsthepromisetoimproveend-partqualitygreatly. However, to enable closed-loop control, we require the quantitatively accurate thermal pre- dictive tool that allows transient prediction of thermal behavior with print process resolution. Derivingsuchaquantitativemodelisnotaneasytask. Thedifficultycomesfromthreeaspects: (1) complexity from physics; (2) high computational workload, and (3) high-resolution phys- ical sensing. 84 Complexity from Physics: Fusing Science The operating principle underlying Multi Jet Fusion, the fusing science, is very compli- cated. The diagram in Figure 5.4 shows the components and their corresponding relationships in the fusing science. Generally speaking, fusing science mainly covers four components: (1) printer hardware; (2) printer firmware; (3) thermal science, and (4) material science. Figure 5.4: Fusing Science in Multi Jet Fusion Printer hardware enables the printing process. Two examples of the thermal behavior related hardware are lamps and fans. As the essential energy source, there are two sets of lamps inside MJF: the top lamps that control the build bed temperature and the fusing lamps in the printing and fusing carriage that aim to fuse the material. The lamp design of the hardware directly determines the energy input. In the printing process, the air flow above the build bed results in thermal convection, which will significantly influence the partâĂŹs temperature distribution. Therefore, two fans are designed above the build bed to ensure uniform air flow. 85 The firmware provides the data pipeline (a.k.a. machine instructions). Machine instruc- tions give a detailed instruction on how to print each layer in the z-direction, which includes the shape and the agent (fusing and detailing) distribution maps. The agent distribution maps decide where to put what amount of energy to trigger the part fuse. In other words, it determines how to transfer the lamp energy input into the energy absorption at each voxel. The thermal science is about the voxel-level thermal physics, which includes complicated factors such as voxel energy absorption/loss, in-layer thermal diffusion (spatial), cross-layer thermal diffusion (tempo), phase change, non-discriminative flux (e.g. convection, radiation), non-discriminative sourcing (e.g. spread, top lamp), conduction, and so on. The material science or the polymer physics studies the material (polymer in MJF) behav- iors such as the material structure, strain, phase change, etc. Figure 5.5 shows the voxel level thermal and material behavior during the layer printing process. At different printing stage, temperature or cooling process (post-process), the voxel material behaviors may be very differ- ent, making it extremely hard to provide accurate material parameters for the first principles based modeling. At different printing stage, temperature or cooling process (post-process), the voxel material behaviors may be very different. For example, the loose powder contains high amount of crystal structure, which will break down to amorphous after being melted. And in the solidification, shorter cooling time will result in lower crystallinity while longer cooling time will lead to higher crystallinity. Those different material structures perform dif- ferent properties, making it extremely hard to provide accurate material parameters for the first principles based modeling. Furthermore, thermal science and material science are highly coupled together to influence the underlying physics, making it even harder to fully understand the fusing science. We lack quantitative knowledge about how material behaves which is required as input for multi- physics simulation. For instance, the thermal diffusivity in process at voxel level is expected to be not only anisotropic but also phase dependent (from loose powder to melting pool and phases in between). Different phase possesses different material properties. In other words, we 86 Figure 5.5: Voxel level thermal history are fundamentally dealing with heterogenous material. Thermal diffusivity, the ability for this voxel to absorb/disperse heat from/to its neighbors, is not only the function of the properties of this voxel but also the function of the properties of its neighbors. On the other hand, the impact of the neighbors is antistrophic rather than the same. However, in most of the current first principles based modeling, thermal diffusivity is simply treated as a scalar which is the function of temperature. So a quantitative thermal diffusivity model is much-needed. Every component plays an essential role in the printing process and is closely related to others. The mechanical engineering provides the energy input, making it possible for fus- ing/detailing agents to transfer energy into heat. The firmware output, the machine instruc- tion, especiallythefusing/detailingagentsmapswilldrivethevoxellevelthermalandmaterial behavior. The thermal science and material science are highly coupled together to influence the end-part quality. Therefore, a transient thermal predictive tool is important to enable the closed-loop control of agent distribution maps (energy driver). 87 High Computational Workload & High-Resolution Physical Sensing In HP Jet Fusion 3D 4200 Printer, the build volume is 406× 305× 406 mm (16× 12× 16) in, which indicates we have a relatively huge build bed. In the future, even larger build bed size is in need. Meanwhile, the size of fusing/ detailing agent distribution maps is 2496×1872 pixels (156 dpi), indicating that the firmware generates high-resolution distribution maps. However, since printing technology is fast, the printing time for one layer is around 10 seconds, which could only afford limited time (1-2 seconds) for thermal prediction computation. The requirement for quick prediction raises the challenge for the computational workload. High resolution physical thermal sensing is crucial in investigating the actual thermal behavior, predictive model training and validating prediction accuracy. High resolution and high-speed IR thermal sensing is a challenging task (Tapia & Elwany (2014)). 5.3 Our Method: Apply Deep Learning to Achieve Quantitative Predictive Tool This chapter launches a machine learning based research initiative to apply sensing data and data pipeline outputs to generate quantitatively meaningful predictive models. We develop a deep learning based thermal modeling framework that could both model the com- plicated material behavioral models and predict the thermal behavior at voxel level. Our goal is to provide predictive models that run both accurate enough and fast enough to enable runtime closed-loop thermal control in Multi Jet Fusion first, then adopt the trained models to uncover the unknown physical properties better. 5.3.1 Deep Learning and Advantages As a remarkable class of techniques, deep learning is applied to more and more domains of science. Unlike the conventional machine learning algorithms, which may need careful engineering and domain expertise to design and extract feature representations, deep neural 88 network could automatically extract the features at different abstraction levels. The feature learning (representation learning) alleviates the dependency on complicated domain science (thermal physics). Meanwhile, representation learning has turned out to be good at discover- ing intricate structures in high-dimensional data (LeCun et al. (2015)), thus also providing a huge potential to uncover the complicated physical properties underlying thermal physics. The local features extracted by a CNN have information about local spatial correlation. In our thermal prediction problem, the part shape, boundary as well as thermal coupling interaction among neighboring parts are all represented as spatial correlation. Therefore, deep learning could learn different kinds of spatial correlations, alleviating us from designing different models to handle different features manually. Numerous works have successfully adopted deep learning in image related areas such as image recognition (Krizhevsky et al. (2012)), image super-resolution (Dong et al. (2016)), and video prediction (Srivastava et al. (2015); Mathieu et al. (2015); Xingjian et al. (2015); Patraucean et al. (2015); Jia et al. (2016); Finn et al. (2016); Xue et al. (2016)). Although the thermal temperature distribution maps and agent distribution maps are not natural images, they are still 2D images with local features such as part shape, boundaries, and part-to-part thermal couplings. Meanwhile, as discussed in Section 5.2.2, the thermal flux is not only within layers but also between layers, making our temperature maps 3D sequential images with both spatial and temporal information. Recurrent neural networks (RNN) are widely adopted to model sequential correlations. It can capture temporal correlation between inputs from different timesteps. First proposed in Hochreiter & Schmidhuber (1997), the long short-term memory (LSTM) recurrent net- work made a huge breakthrough by enabling the learning of long-term dependencies. A cell state controlled by three gates is added to implicitly store the long-term history information. Such gates allow the network to learn whether to keep information from previous timesteps for future prediction, thus helping to solve the vanishing gradient problem. Ranzato et al. (2014) proposes an RNN based generative model for video prediction. Srivastava et al. (2015) 89 puts forward an LSTM encoder-decoder framework which uses an encoder LSTM model to learn a representation vector from a sequence of video frames and another decoder LSTM model to predict future frame sequence from the representation vector. Xingjian et al. (2015); Patraucean et al. (2015) replaces the fully connected layers in LSTM by a convolutional layer which further considers the spatial correlation. The Conv-LSTM proposed in Xingjian et al. (2015) perfectly fits our physical intuition that thermal diffusion is not only within layer (spatial) but also cross-layer (temporal). Besides the general purposed video prediction, enabling motion prediction from video prediction is another related research area. Jia et al. (2016) proposed dynamic filter networks which aim to dynamically generate the filtering operations to extract the specific motion pattern for a given video. The underlying intuition is that different video frames may contain different motion patterns, and we should predict future frames with parameters adapted to the motion pattern within a video. A similar idea could be found in the action-conditioned video prediction model in Finn et al. (2016). Their pixel transformation models aim to predict the motions from previous frames. Xue et al. (2016) also models the motions as convolutional kernels which are then applied to the feature maps of images at multiple scales to predict the next frame. These works point out huge potentials for deep learning to uncover the multi- physics material behavioral models and achieve better thermal prediction. For instance, in thermal physics, the heat flux is exactly the heat motion. Just like the velocity in object motion, thermal diffusivity is a motion pattern which could be extracted and modeled from temperature and agent distribution (energy driver) map sequences. Inspired by all above, we apply deep learning to fusion fusing science, with the hope that its expressive power will allow us to model the complexity of fusing science adequately. 90 5.3.2 Key Datasets Beforedetailingthedeeplearningarchitecture, itisimportanttointroducethekeydatasets that enable our deep learning based thermal modeling. The two datasets we employed here are the voxel level machine instruction and close to voxel level thermal sensing. Key Dataset 1: Voxel Level Machine Instruction As shown in Figure 5.6, Multi Jet Fusion print data pipeline takes 3D models (3D mesh and/or stack of 2D vector slices) as input, accounts for the process physics, and generates per-layer machine instructions. That is, the shape map and agent distribution at voxel level. The fusing agents which control the in-take thermal intensity and the detailing agents which control the out-take thermal intensity collectively is the primary driver for writing system to control energy deposition at voxel level. Figure 5.6: Voxel level machine instruction from HP Jet Fusion 3D 4200 Printer These voxel-level agent distribution maps for each agent are the only control for writing system to manipulate energy at voxel-level resolution. They are particularly interesting to us because our fundamental goal is to eventually feed the prediction back to optimize these voxel-level energy control. Our methods in this paper can readily expand to multiple different agent distribution maps corresponding to different abilities to absorb or remove thermal energies and different print modes where more than one distribution map may be used for each agent. 91 Key Dataset 2: Close to Voxel Level Thermal Sensing At HP Labs, FLIR sensing cameras are used in Multi Jet Fusion R&D to capture bed- level temperature distribution. It provides the voxel-level ground truth of the thermal science. One example of the FLIR image is shown in Figure 5.7. It records the build bed temperature distribution per layer in HP Jet Fusion 3D 4200 Printer. The sensing results will need a corrective procedure to overcome the distortion introduced by the sensing device. Different types of thermal sensing devices can be used. Figure 5.7: Close to voxel level thermal sensing from FLIR camera Within this context of thermal physics prediction, we refer to voxels as “thermal voxels”. The size of a thermal voxel is defined as the minimum that is thermally meaningful. It is expected to be substantially larger than 42 micron (i.e., 600 dpi). 5.3.3 Workload Estimate and Performance Modeling A core motivation of this work is to reduce the computational cost of the modeling of thermal behavior to a level where it is possible to utilize this model directly in the print pipeline. This translates into a per layer runtime requirement for prediction of transient thermal behavior of well below the cycle time of a layer, which is currently around 11 secs. While current first principles based simulations are far too compute intensive to serve this role, a deep neural network (DNN) based model is capable of learning precisely which features of the detailing and fusing agent distribution maps and the thermal image data are germane to 92 the prediction and hence should have a computational footprint that is reasonable to execute on a GPU within the printer. To validate this assumption, we take an analytical approach. We sum all the operations in the computational graph to determine the total number of operations that would be needed to execute it and then compare to the compute capabilities of modern hardware. Summing of these operations would yield a lower bound on the execution time as it does not consider overhead. Operations are counted in Multiple-Accumulate (MAC) units, as this is a common element in the data pipeline of a GPU. Table 5.1 below gives a summary of the operations calculated for the Compact Spatiotemporal architecture shown in Figure 5.9 below. Table 5.1: Operation computation in fusing layer heat map prediction, covering one patch of theinput. (NotethatthefirsttwoCNNunitsoccuronceforeachsequenceoftheConv-LSTM) Layer type Computation (MAC) CNN 2D (detailing / fusing distribution maps) 28,640,000.0 CNN 2D (heat map) 20,320,000.0 Conv-LSTM (sequence length 30) 7745,280,000.0 CNN (decode) 14,000,000.0 Total MAC 9,228,080,000.0 While the computation for the CNN portions of the network is straight forward, as is the basic structure of the ConvLSTM, it is worth noting that the computation for two of the activation functions are less straightforward. The sigmoid and tanh functions utilized in the ConvLTSM have data dependent execution time. As we do not have a reliable distribution of the input values to these functions during prediction, we choose a nominal MAC value for these two functions. We can see from the table that the computational load for a single patch is 7.5GFlops. A NvidiaTitanXMaxwellcardhasapeakexecutionrateof 4.6Tflops. Ifweassumeautilization at run time of 50%, this translates into 3.3ms per patch. This is of course only the network execution time. There will be significant overhead for data copy during execution, but this execution time places predictions well below the layer cycle time. 93 We measured the performance of the network on a TitanX Maxwell card to validate the computation. The average patch execution time was 18.11ms. As anticipated however, mem- ory copy accounts for 14.3ms of that time, yielding an execution time for the graph operations of 3.81ms, which is close to our prediction from analysis of the graph. 5.4 Application1: One Step Ahead Fusing Layer Ther- mal Behavior Prediction This application applies deep learning to build a quantitative model that allows transient prediction of fusing layer thermal behavior with print process resolution. Such quantitative model is important to enable voxel level closed-loop thermal energy control therefore improve part accuracy and yield. 5.4.1 Architecture Rooted in our insight into the MJF thermal physics, particularly the voxel-level thermal influencers to the fusing layer, we develop a deep neural network model. There are mainly two voxel-level thermal influencers for fusing layer thermal behavior. One is the energy absorp- tion/loss driven by the agent distribution maps. The other is the spatiotemporal heat flux from underneath layers. Our network architecture has two variations. Figure 5.8 presents the first architecture, which is called Hybrid Spatiotemporal model. In recognition to the primary role of agent dis- tribution maps of the current layer, a convolutional neural network model is first constructed to generate the quantitative predictive model for the thermal component attributed to the agent distribution maps. In recognition of the voxel-level thermal coupling among different layers, i.e., for a voxel at the current layer, if there exists temperature difference between this voxel and the surrounding voxels belonging to previous layers, there will be heat flux generated about this voxel. With 94 Figure 5.8: Hybrid Spatiotemporal Model architecture for fusing layer thermal prediction that, a recurrent neural network (RNN) is used to generate a quantitative model for the thermal component attributed to the heat fluxed from previous layers. This recurrent neural network (in current implementation, we use LSTM) principally accounts for the inter-layer effects. On top of that, a convolutional neural network is coupled to LSTM cells (which is called Conv-LSTM) at each layer to account for the thermal diffusion within this layer (Xingjian et al. (2015)). The predicted feature maps from the above two thermal components are then synthesized by a convolutional neural network to account for its different level of contribution to the final prediction. The second architecture variation is shown in Figure 5.9, which is called Compact Spa- tiotemporal Model. Instead of treating the prediction from agent distribution maps as a separate component, we concatenate it into the input of the recurrent neural network. To predict the heat map of the fusing layer, the fusing layer agent distribution maps and the actually measured heat map of the last buried layer will pass through the corresponding 95 encoders, separately. Then the generated features are concatenated as the input to a Conv- LSTM model. The output at current timestamp, passing through a decoder, is the prediction. The encoder and decoder are still convolutional neural networks. This variation of architec- ture enables us to pass the information of agent distribution maps through time to predict future layers. Since the agent distribution maps may indicate the phases of each voxel (from loose powder to melting pool and phases in between), involving the sequential information of agent distribution maps could potentially learn how heat would flux better. Figure 5.9: Compact Spatiotemporal Model architecture for fusing layer thermal prediction No matter in which architecture, our tests have validated that the expressive power of Conv-LSTM will be able to simulate, quantitatively, the transient 3-D thermal diffusion since we are accounting for inter-layer coupling (z-) and spatial diffusion (x- and y-). This model should be able to simulate a given layer’s thermal physics later when given a snapshot of the thermal behavior of this layer as input at an earlier time. In other words, not only we 96 simulate the thermal diffusion with previous layers but also layers deposited on top of this layer (“future layers”). With that in mind, one architectural choice is to use the thermal sensing data for any given layer when it was fusing layer as input, and the model will capture and generate its future thermal behavior. A future variation of this implementation is, deleting the thermal sensing data and only adopt current layer agent distribution maps as input. In such a way, the thermal sensing data does not become part of the model, rather appearing only as ground truth to compute the loss function to optimize the model during training. In other words, once model is trained, the prediction is completely independent of the thermal sensing dataset. It enables much smaller model size for real deployment. 5.4.2 Scalable Solution Accounting for Bed-Level Inference Thetrainingcostfordeepneuralnetworksisknowntobeveryhigh. Ontheotherhand, we are driving towards bigger print bed and higher voxel resolution. In other words, to make this work practical, it is necessary to devise a computationally scalable approach. We proposed two approaches: patch level solution and build-bed level solution. The patch level solution accomplishes this by partitioning each layer of the build bed into a group of patches, which become the fundamental training instances. Note that the patch size is tuned for computational efficiency not thermal physics. It is necessary to account for patch- to-patch interaction because we care about effects that are larger than the patch size such as large parts, build bed location variations, and so on. From the thermal physics perspective, the patch-to-patch interaction is the thermal diffusive effects between patches. That is, the thermal flux due to the temperature difference between neighboring patches drives thermal energy penetrating into neighboring patches from the boundary. To account for this, we devise “thermal stripe” surrounding all the sides of a patch. The width of the thermal stripe is computed based on the thermal diffusivity and the layer production time. In prediction, the build bed level input is split into patches with neighboring patches overlapped by the 97 width of the “thermal stripes”, allowing thermal coupling among patches. Then the predicted patches are merged as their overlapping “thermal stripes” to form build bed level prediction. The good performance in predicting effects beyond patch size such as larger parts and bed location difference also illustrates the effectiveness of our solution. The build-bed level solution is more straightforward. We apply the full build-bed agent distribution and heat maps directly. But we adopt numerical interpolation to downscale the input before feeding it into the architecture and upscale the output. Then the output at the original size is compared with the ground truth for loss function. This solution keeps both the initial high-resolution image and the suitable training sample size. 5.4.3 Architecture Learning Experience A good many of model training experiments and analysis are done here to arrive our final deep neural network (DNN) architecture. We start with the patch level solution (split the images into 100× 100 patches) and simple DNNs. Since we are dealing with one channel temperature maps instead of the natural images widely adopted in deep learning, small image size and simple architecture is easier to make sure our DNN converges well and help us to obtain the basic ideas about how the DNN will learn the thermal behavior. We start with training a CNN to learn the heat generated from agent distribution maps, and a Conv-LSTM network to learn the heat transfered from previous layers, separately. The initial network design is motivated by the similar video prediction and whether prediction problems regarding what architecture is suitable to learn the long term spatiotemporal rela- tionships. To enable the model training, we invented an entropy based algorithm to select the valid subsets suitable for those two models. We tuned the hyper-parameters and archi- tecture on both models to get the basic idea about how the model and hyper-parameters performance. The hyper-parameters here include the number of convolutional layers in CNN model, encoder model and decoder model (Figure 5.8), kernel size and depth in each con- volutional layers, learning rate and batch size, etc. We also settle down some important 98 deep neural network settings during the experiment and analysis, including the optimization method, loss function, activation function, model initializer, data normalization, dropout, and batch normalization. The two separate models finally converge well and we obtained many important under- standings. For instance, the Conv-LSTM outperforms the purely CNN and predicts obviously more smooth heat maps, illustrating the importance of incorporating recurrent neural net- works to pass temporal information. An important finding is that deeper neural network and deeper kernels not necessary lead to better prediction. This is possibly because our problem is the pixel prediction problem and the output has the same size as input. The huge complexity underlying the input-output relationship brings huge challenge in converging DNN which are “too complicated”. Meanwhile, keeping the original image size in each convolutional or LSTM layer achieved significantly better performance than down-sampling first then up-sampling. With the good prediction results from the two separate models, we build another CNN to synthesis the results from the above two models as the final layer thermal prediction. The training of final synthesis CNN implements the whole architecture in Figure 5.8 but fixes all the parameters in the spatial CNN and Conv-LSTM as pre-trained parameters. The whole dataset is adopted for the synthesis model training. Training the three parts of models in Figure 5.8 separately gave us pretty impressive pre- diction results. Then we improved the architecture by adopting the whole dataset to train the whole architecture simultaneously. The basic thinking here is that training the whole architecture simultaneously could feed more information while estimating each parameter and learn the relative contribution of each component better. We made further modification by feeding the multi-layer prediction feature maps of the spatial CNN and Conv-LSTM models to the final synthesis CNN instead of feeding the single layer “predictions”. This is because the multi-layer feature maps reserved more original information, and training the whole archi- tecture simultaneously no longer requires the intermediate results have single layer prediction. 99 We eventually settle down this architecture as our final Hybrid Spatiotemporal DNN archi- tecture. The next consideration is whether the hybrid network is necessary. In the hybrid network, only fusing layer agent distribution maps are involved in thermal prediction. From the deep learning perspective, it is more natural to also pass the agent distribution maps of previous layers along with the heat maps. The agent distribution maps could to some extent reflect the material phase at each voxel, which would potentially provide more comprehensive infor- mation when learning the fusing layer thermal behavior. Following this logic, we proposed the Compact Spatiotemporal DNN architecture in Figure 5.9. The patch level solution could achieve quite accurate prediction, but the splitting and assembling of patches are sometimes awkward. Further thoughts are put here to propose a more global solution, resulting in our build level solution. The performance comparison and use-cases of these two training solutions are discussed in section 5.4.4. 5.4.4 Experiments and Results Our architecture is tested on HP Jet Fusion 3D 4200 Printer with real thermal sensing dataset from FLIR T650sc IR camera. The thermal maps captured by FLIR camera are corrected for distortion by geometric image transformation to match the fusing/ detailing agent distribution maps as a pair. The resolution of FLIR thermal map after correction is 600× 460× 1 pixel while the resolution of agent distribution maps are 600× 460× 2 pixel with two channels represent fusing and detailing agent maps, separately. With this resolution, each pixel corresponds to 0.41mm in the real print bed. In other words, the print resolution for closed-loop thermal control our current predictive model could achieve is 0.41mm, which could capture the signal well. The experiments try both patch level and build-bed level solutions. In the patch level solution, we splitthe thermal sensingmaps and agentdistribution maps to sequence of patches by grids. 8680 patch sequences are generated with sequence length 30 and patch size 100×100 100 (100× 100× 2 for agent distribution maps). Then we randomly split the patch sequences into training, validation and test datasets, which contain 6300, 1050, 1330 patch sequences, respectively. In the build-bed level solution, we adopt the bilinear interpolation to scale down the original input by 4 and feed the 150× 115 images into architecture. The outputs are scaled up to 600× 460 as ultimate predictions. Our experiment collects 248 layer sequences with sequence length 30. The size of training, validation and test datasets are 180, 30 and 38, respectively. Our model is one-step ahead prediction. Namely, we adopt the ground truth at current layer as input to predict the heat map of next layer. We adopt L2 norm as loss function and calculate the loss from the 11th image in the sequence. This is because the first 10 images may havenâĂŹt learned enough heat flux information. And the outputs from last 10 images are used as prediction. In other words, with each sequence input, our model predicts 10 successive layers. The experiments use Google TensorFlow as the deep learning framework and two Nvidia GP100 GPUs (memory 16GB× 2) with CUDA 9.0, cuDNN 7.0 as computing infrastructure. We implement both the Hybrid Spatiotemporal (Figure 5.8) and Compact Spatiotemporal architecture (Figure 5.9) with the same number of parameters. The two implementations achieve similar prediction results, we will only show the result from Compact Spatiotempo- ral Model here. Figure 5.10 presents two sample prediction sequences from the patch level training. Besides the prediction and ground truth, we also present the fusing agent (FA) and detailing agent (DA) distribution maps for reference. One sample full build bed prediction sequence generated by patch level solution with thermal stripes = 10 pixels is shown in Figure 5.11. Another build-bed prediction sequence directly generated from the build-bed level train- ing is shown in Figure 5.12. Note that all the predictions come from the test dataset. These results indicate that our model could predict the thermal behavior and fine details pretty well, which demonstrate the prediction power of learning from fusing/ detailing agent distribution maps and previous layers. 101 (a) Prediction example 1 (b) Prediction example 2 Figure 5.10: Sample patch prediction results 102 Figure 5.11: Sample build-bed prediction result generated from patch level solution 103 Figure 5.12: Sample build-bed prediction result generated from build-bed level solution 104 We employ two accuracy metrics for quantification: mean square error (MSE) and mean structural similarity (MSSIM). First proposed in Wang et al. (2004), the MSSIM is a percep- tion based model that considers three measurements: luminance, contract, and structure. It tries to mimic people’s perception to compare whether the structure of two images are similar or not. The range of structural similarity is -1 to 1, the closer to 1, the more accurate your prediction is. Detailed quantitative performance measurements are listed in Table 5.2. The accuracy metrics from both patch level and build-bed level solutions indicate promising performance, which demonstrates much better accuracy than empirical models. The patch level solution reserves the fine details in training. It also reserves the extension flexibility when the build bed becomes larger or thermal sensing with higher resolution is available. The build-bed level training method maintains the advantage that it is easier and more lightweight to apply. Compact Spatiotemporal Model outperforms the Hybrid Spatiotemporal Model, demonstrat- ing the efficiency of passing agent distribution maps along with time. Considering that the per-layer production time is in order of 10 seconds, the less than 0.1 second order of per-layer prediction time cost shown here gives us reasonable hope that we may be able to integrate this as a run-time prediction-correction step. Table 5.2: Performance quantification of fusing layer thermal prediction Model Training solution Training time (hrs) MSE MSSIM Build bed prediction time (sec) Hybrid Spatiotemporal Patch level 236.4 3.34 0.95 0.07 /layer Build-bed level 67.0 2.81 0.94 0.02 /layer Compact Spatiotemporal Patch level 167.0 2.35 0.95 0.07 /layer Build-bed level 104.1 2.64 0.94 0.02 /layer 105 5.5 Application2: Learning Heterogenous Thermal Dif- fusivity as Deep Neural Network Aquantitativemodelthatallowsthelearningofthermaldiffusivityhasmanyimportantuse cases from transient thermal behavior prediction to uncovering the process physics. It could enable learning how heat flux and the prediction of more comprehensive thermal behavior such as the thermal behavior of the buried layers along with the time. On the other hand, the better understanding of thermal diffusivity could help to uncover the heat transfer better and has the potential to build the gray-box models with first principle laws. However, as described in Section 5.2.2, the thermal diffusivity is difficult to learn due to the complicated antistrophic properties. We proposed a deep neural network to model thermal diffusivity, and then predict the heat flux as well as thermal behavior based on the predicted diffusivity kernel. The dataset used here is the area thermal sensing data at build layer level (e.g., FLIR). 5.5.1 Physical Insight Our deep neural network architecture is rooted in the understanding of the thermal dif- fusion process. Here we will introduce the physical thermal diffusion laws under the single homogenous material first. Under this situation, the heat equation is: ∂ ∂x (k ∂T ∂x ) + ∂ ∂y (k ∂T ∂y ) + ∂ ∂z (k ∂T ∂z ) =ρc ∂T ∂t (5.1) where k is the material thermal conductivity, ρ is the material density, c is the material heat capacity, T is the temperature, t is time and (x,y,z) represents the position. k, ρ and c are simply treated as a constant under this simplest case. Then the rate of temperature change is: 106 ∂T ∂t = k ρc ∇ 2 T =α∇ 2 T (5.2) where α = k ρc is the thermal diffusivity. Under the actual situation, we are dealing with heterogenous anisotropic material. The material conductivity k varies with material temperature, orientation, and phase (material) properties. Similarly, the material density ρ and heat capacity c are also related to the phase properties. Consequently, the thermal diffusivity α should no longer be simply treated as a constant. Instead, it is a multi-dimensional function that varies with material temperature, orientation and material phase. It is extremely hard to direct build the causal model for this function, so we intend to build the properly designed deep nerual network to learn this diffusivity function. In this work, we propose a DNN architecture that learns the thermal diffusivity in z- direction. We record the actual hap map of the fusing layer (k-th layer) at its last timestamp. Then at the beginning of next layer printing, we record a sequence of heat maps with specific timestep after recoating fresh powder and before jetting fusing and detailing agents. During this period, the dominant influencer of the fusing layer powder thermal behavior is the heat fluxed from underneath layers along the z-direction. This enables us to learn how much heat underneath flux to the fusing layer after timestep dt driven by the temperature difference between fusing layer ((k + 1)-th layer) and one layer underneath (k-th layer). As shown in Eq. 5.2, if the timestep dt is small enough, the prediction of continuous layer temperature change could be discretized and our model will be able to predict each discretized timestep and form a sequential thermal prediction. To quantify the magnitude of dt, we adopted the experimentally measured part thermal diffusivity α and the printing layer thickness for calculation and believedt = 0.1s should be the order of magnitude for layer heat flux. 107 5.5.2 Architecture From the physical insight, we create the deep neural network in Figure 5.13. We concate- nate last buried layer agent distribution maps, last buried layer heat map and fusing layer heat map at current timestamp t as the input to the thermal diffusivity DNN. We adopt the two layers’ heat maps because temperature varies the thermal diffusivity. The agent distribution maps are included to reflect the voxel phase (from loose powder to melting pool and phases in between), which will indicate different material properties thus influencing the thermal diffu- sivity. Only the last buried layer agent distribution maps are included because currently the fusing layer is just powder. Figure 5.13: Deep neural network architecture for thermal diffusivity model We implement a Google inception module (Szegedy et al. (2015)) based convolutional neural network to train the thermal diffusivity as a deep neural network, which is rooted in the fact that the thermal diffusivity should be a function of both the temperature, and material properties of a voxel itself and its neighbors. The input will pass through three 108 inception modules and one convolution layer to compress the thermal transfer information into a multi-dimensional kernel. Extrapolated from our prior experiences in modeling material thermal properties, we believe the intrinsic material properties and interaction among them can be captured by a compressed kernel by alleviating geometrical complexity encoded in the heat map. With that, we created an element thereafter referred to as the thermal diffusivity kernel, as vehicle to carry this compressed information that would be predominantly material specific. Thermal diffusion is transient energy “motion” signified by heat flux as its “velocity”. This kernel is designed to vary as a function of time such that when convolved with the temperature dif- ferential maps it can capture the transient thermal diffusive behavior. Related applications could be found in computer vision (Jia et al. (2016); Finn et al. (2016); Xue et al. (2016)), where dynamic convolutional kernels have been used successfully to capture movements of physical objects. Figure 5.14 introduces the detailed architecture of the three inception networks. We first adoptthenaiveinceptionmodulewithout 1×1dimensionalityreductionduetothelowdimen- sion of input. Then two same inception modules as shown in Figure 5.14 are implemented successively. The inception modules use stride 2 to reduce the dimension of features. A final convolution layer with stride 2 is put at the end to predict the thermal diffusivity. 109 Figure 5.14: The architecture of the inception modules We further hypothesize the dimension of the thermal diffusivity kernel reflects the physical or environmental parameters that affect the heat transfer. We draw analogy with dimensional analysis (π theorem). Each dimension of kernel corresponds to one type of thermal distri- bution pattern, which may represent a different relevant physical attribute. The dimension (kernel size and depth) is tuned as a hyperparameter. Convolving the inter-layer temperature difference by the predicted diffusivity kernels, we generate multi-layer intermediate results where each result potentially represents the impact of one physical factor. The kernels con- volve the inter-layer temperature difference because temperature difference drives the heat diffusion. The multi-layer results need to be merged as the single layer heat flux prediction. There- fore, we adopt another convolutional neural network to generate multi-layer masks from the diffusivity kernels. The masks have the identical size to the intermediate results and contain the weight of different layers, i.e., the relative significance of different physical factors. With a similar idea as image segmentation, the masks try to isolate different physical factors to indi- catewhichphysicalpropertiesarepresentatwhichlocation. FollowingthesimilarideainFinn 110 et al. (2016), we apply the channel-wise softmax to the final convolutional layer to make sure that the channel weights sum to 1 at each pixel position. We tried several model structures for mask generation including deconvolutional and up-sampling convolutional neural network. The up-sampling CNN functions similarly to de-convolutional NN. It adopts numerical inter- polation to upscale the input into the desired dimension first, then uses another convolutional layer to tune the up-sampling better. Currently, we adopt the up-sampling CNN since it overcomes the checkboard artifacts in the de-convolution. Additionally, to pass more input information to mask generation, we add three skip connects to concatenate the three inter- mediate convolution feature maps with deconvolutional layer inputs. As indicated in Figure 5.14, we use the first 1× 1 convolutional layer output in inception module for previous layer’s skip connection considering the high dimension of inception module output. Layer-wise compositing the intermediate results and the masks, the model predicts the heat fluxed to the fusing layer after timestep dt. Adding this predicted difference to the heat map at last timestamp t, the prediction of heat map at next timestamp t +dt is derived. In the model training, this predicted heat map will be compared to the ground truth to calculate the loss. 5.5.3 Architecture Learning Experience The initial architecture design is inspired by the motion prediction problem in video pre- diction (Finn et al. (2016); Jia et al. (2016); Xue et al. (2016)). Similar as the motion of physical objects, the thermal flux is the motion of heat driven by neighbor voxel tempera- ture difference. Therefore, learning the thermal diffusivity model is learning the pattern of thermal motion bases on the specific temperature and material input. Following this idea, we initially designed the architecture with thermal diffusivity DNN, mask generation DNN and final composition as we discussed in section 5.5.2. Further experiments and analysis didn’t change the basic architecture structure, but we greatly improved the prediction performance by improving the detailed architecture. In the 111 thermal diffusivity prediction network, we start with the simple convolutional layers and adopt the max pooling for dimension reduction. However, the experiments indicate that reducing image size by using larger stride (2 or 3) is better than pooling. And smaller stride with more convolutional layers outperforms larger stride with less convolutional layers. The possible reason is that both the pooling and large stride lose too much information. More impressive improvement is achieved by replacing the convolutional layers by Google inception modules (Figure 5.14). The Google inception modules achieved better performance with less parameters comparing to the initial convolutional layers. The “more intelligent” structure in Google inception provides the possibility to explore intrinsic structures better. The model performance is also slighted improved by adding agent distribution maps to the input. This is also expected since the agent distribution maps to some extent reflects material phase information. The mask generation network started with a sequence of deconvolutional layers, which is the fundamental idea to up-sample the images. But then we replaced the deconvolutional lay- ers by the up-sampling plus convolutional layers, since the deconvolutional layers show check- board artifacts in our experiments and the up-sampling convolution overcomes the artifacts to some extents. Further model improvement are achieved by adding the skip connections, which directly concatenate the output of inception layers to the corresponding up-sampling convolutional layers. Better model performance is expected since skip connections pass more information from the original input, including the phase or material properties (shown in agent distribution maps) and the temperature distribution, to the mask generation. One significant challenge of the thermal diffusivity DNN that distinguish itself from tra- ditional deep learning problems is that we need to not only achieve good final prediction, but also make sure all the intermediate results make sense. Analysis on the diffusivity ker- nels, masks, intermediate convolution results and predicted heat flux are needed after each experiment. Therefore a good many of experiments are done here to test and improve the detailed model structure and finally arrive the architecture as we discussed above. In the 112 meanwhile, the hyper-parameters and network settings including the size and depth of diffu- sivity kernel, kernel size and depth in each convolutional layer, learning rate, loss function, data normalization and model initializer are carefully tuned by experiments. 5.5.4 Experiments and Results We still do the experiment on HP Jet Fusion 3D 4200 Printer with real thermal sensing dataset from FLIR T650sc IR camera. We record the layer heat map at the rate of 10 frames/sec. In the frame sequence of each printed layer, around 22 frames are collected before jetting agents. We randomly select 14 successive frames in each layer. Each sequence is put together with the last buried layer heat map at its last timestamp and the last buried layer agent distribution maps for model training. In other words, the 14 heat maps in each sequence correspond to same last buried layer heat map and agent distribution maps. In our experiment, we adopt dt = 0.3s considering the randomness in printing and IR thermal measurement. Therefore in each sequence, the heat map at each timestamp is used to predict the heat map 3 frames later (0.3s after). The resolution of FLIR thermal map after correction is 450× 500× 1 pixel while the resolution of agent distribution maps is 450× 500× 2 pixel with two channels represent fusing and detailing agent, separately. With this resolution, each pixel corresponds to 0.49mm in the real print bed. The experiments use Google TensorFlow as the deep learning framework and two Nvidia GPUs (memory 16GB×2) with CUDA 9.0, cuDNN 7.0 as computing infrastructure. We split the maps to 100× 100 patches for training and adopt the entropy to choose patch sequences with part printed as valid training sequences. We record 2,891 layers from two builds, and generate 34,933 patch sequences (14× 100× 100) and corresponding 34,933 patches for last buriedlayerheatmapand34,933patches(twochannels)forlastburiedlayeragentdistribution maps. The patch sequences are randomly split into training, validation and test datasets with 113 25,200, 4,200 and 5,533 sequences each. We tuned the diffusivity kernel size and depth as hyper-parameter, and finally adopt 5× 5 kernel with depth 3 (i.e., 5× 5× 3 kernels). Figure 5.15, 5.16, 5.18 and 5.17 present two sample sequences of prediction. Using dt = 0.3s, our model is able to predict the last 11 frames in each sequence. Figure 5.15(a) and 5.17(a) present one layer’s temperature involvement over time. The first row is 11 successive frames from the FLIR thermal sensing. The second row presents the predicted heat fluxed to this layer at each timestamp from its underneath layers after the time interval of dt = 0.3s. The third row is the predicted heat map after dt = 0.3s. And the last row is the ground truth for reference. In the meanwhile, we also present some intermediate results including the predicted thermal diffusivity kernels, intermediate convolution results, and masks. Those intermediates results have 3 images at each time-step due to the diffusivity kernel depth. The last buried layer heat map and fusing/ detailing distribution maps are also put for reference. The results indicate that two kernels learned some information so that two different thermal motion patterns (convolution results) are generated. The masks form just like the image segmentation, indicating the relative weighs among different patterns. We still employ mean square error (MSE) and mean structural similarity (MSSIM) as accuracy metrics and the quantifications are shown in Table 5.3. Table 5.3: Performance quantification of thermal diffusivity DNN Mean square error (MSE) Mean structural similarity (MSSIM) 0.37 0.72 114 (a) Predicted heat flux, predicted heat map and ground truth (b) Predicted thermal diffusivity kernels (c) Intermediate convolution results Figure 5.15: Prediction results from sample 1 115 (a) Gnereted masks (b) Last buried layer heat map and fusing/ detailing distribution maps Figure 5.16: Prediction results from sample 1 – continued 116 (a) Predicted heat flux, predicted heat map and ground truth (b) Predicted thermal diffusivity kernels (c) Intermediate convolution results Figure 5.17: Prediction results from sample 2 – continued 117 (a) Gnereted masks (b) Last buried layer heat map and fusing/ detailing distribution maps Figure 5.18: Prediction results from sample 2 – continued 5.6 Conclusion This paper for the first time implements deep learning to model the thermal behaviors in 3D printing. We proposed a recurrent neural network based deep neural network that integrates both energy driver and spatiotemporal thermal flux to predict fusing layer thermal distribution. Our deep neural network models achieve significantly higher fidelity than the traditionalexperimentbasedthermalprediction. Wealsoproposedadeepneuralnetworkthat could learn the thermal diffusivity as a deep neural network. The results not only provide thermal diffusivity model and predictions, but also give promising thermal prediction along with time. The two applications reported in this paper prove out the great potential in applying deep learning to fusing science, and more research efforts have to be employed. 118 Moreover, thisworkisanovelattemptthatappliesdeeplearningtobrandnewapplications andcomplicateddomainscience. Itdemonstratesthatbesidesthetraditionalapplicationareas such as computer vision, audio processing, and natural language processing, deep learning still shows expressive auto-learning power in other applications. Furthermore, it inspires us with the potential to apply deep learning in physical domain science. The learning power of deep learning could alleviate the constraints from complicated unknown physical models and achieve better prediction. And the domain science could, in turn, provide abundant intuition in building and training deep learning models. 5.6.1 Extension to Other Techniques The methodology proposed in this paper is demonstrated on but not restricted to HP’s Multi Jet Fusion. The deep learning based methodology applies to other powder bed fusion 3D printing techniques such as SLS and SLM. SLS or SLM technique relies on the laser beams that fuse the powder (plastic or metal) line by line to form the end-part, therefore faces the similar build bed thermal problem. Similar IR thermal sensing could be adopted as ground truth. But the other input, the energy driver becomes the laser parameters including laser power, beam profile, scan speed, scan overlap, toolpath and so on. The quantitative models provide the guideline about how to adjust the laser and toolpath for quality improvement. Besides the powder bed fusion techniques, our methodology also applies to the photopoly- merization based techniques such as SLA and DLP. The material thermal behavior during photopolymerization also dominates the end-part quality. Very similar to the agent distribu- tion maps in MJF, the UV images or projection masks in SLA or DLP control the pixel level light intensity, therefore becomes the voxel level energy driver. The UV images or projection masks, together with thermal sensing, are the input to our quantitative models, that enables the feedback control over voxel level light intensity via UV images or projection masks. 119 Chapter 6 Discussion and Future Extensions Additive manufacturing, or 3D printing, has received worldwide attention during the last several decades. This direct digital manufacturing technology enables the promise of prod- ucts with complex geometry or material compositions, which would dramatically reduce the cost of materials and time. Shape accuracy of parts is essential to meet required standards of manufacturing. The shape accuracy of 3D printing however, is still not comparable to traditional manufacturing which continues to be one of the most significant issues in adop- tion. Achieving higher shape accuracy requires better understanding, prediction and control at both process level and part level. The development of quantitative models that could pre- dict shape accuracy or process behaviors becomes fundamental to inform both part design and process control, therefore improve shape accuracy. In this dissertation, we are devoted to apply statistical modeling and machine learning to improve the shape accuracy from two perspectives: prescriptive shape deviation modeling and process behavior modeling. Our fun- damental research goal is to build both prescriptive shape deviation model based and process behavior model based approaches to improve part shape accuracy. To achieve our research goals, this dissertation analyzed four research tasks step by step. They are: (i) prescriptive modeling and compensation of in-plane shape deviation for 3D printed freeform products, (ii) statistical process control of in-plane shape deviation for addi- tive manufacturing, (iii) prescriptive data-analytical modeling of selective laser melting pro- cesses for accuracy improvement and (iv) data-driven modeling of thermal physics in powder bed fusion 3D printing using deep learning. The first three tasks build general models to predict the shape deviation directly from product quality data then compensate the part design. As the base of those three tasks, 120 the first task proposed a prescriptive in-plane shape deviation modeling and compensation strategy for freeform shapes. The main contribution here is that the prescriptive model is estimated based on a limited number of simple shapes and trials but could be applied directly to predict new and untried freeform shapes through our methodology. This task successfully extents the shape deviation modeling framework first proposed in Huang et al. (2015, 2014a,b) from simple cylindrical and polyhedron shapes to freeform shapes. The second task adopts the prescriptive modeling as a crucial component to build a statistical process control (SPC) schema for additive manufacturing. By integrating the prescriptive model, we proposed a new monitoring statistics that enables the monitoring of shape deviations from shape to shape. This task extents our offline shape deviation modeling framework to online monitoring. Those two tasks are validated on the projection-based SLA process. The third task demonstrates the extendibility of our methodology by extending it into SLM process. We analyzed the additional challenges in SLM process and proposed a prescriptive model based integrated strategy that could reduce all the additional errors along with shape deviation. A promising contribution here is that all the three new challenges we analyzed in this task are not unique in SLM process, instead, they are commonly existed in other 3D printing techniques. Therefore, our integrated strategy proposed a flexible strategy that is easily adopted to handle similar error sources in broader 3D printing techniques. The above three tasks are easy to implement. They build general models from product quality data without considering process information. We agree that in the application to specific process or machine, involving the process information and enabling closed-loop control will be very beneficial. In the powder bed fusion based techniques, such as SLM, SLS and HP’s MJF, the part shape accuracy is directly related to the build bed thermal behavior. Consequently, our last task looks into the process in powder bed fusion based techniques and models the thermal behavior. We for the first time applied deep learning to quantitatively predict the fusing layer thermal behavior, which enables the closed loop thermal control. We also for the first build te thermal diffusivity of heterogenous anisotropic material as a deep 121 neural network, which proved out feasibility and great potential to apply deep learning to discover and enrich domain science. To summarize, our work in this dissertation contributes to both generic prescriptive shape deviation modeling and process specific thermal behavior modeling. The prescriptive shape deviation modeling provides a flexible generic framework that is easy to implement in different printing techniques to improve shape accuracy. It doesn’t require deep process knowledge and understanding, nor change any process variables. The thermal behavior modeling for the first time applies deep learning to both provide accurate and high speed thermal behavior and discover the thermal physics domain knowledge. Both of them are important perspectives to improve the shape accuracy. In the future, we may extend the dissertation work for more comprehensive shape accuracy improvement in both two perspectives. Possible direction include but are not limited to the following. • Data-analytical modeling integrating different error sources: First it is important to further research on the SLM process performance with addi- tional experimentation and analysis. More builds with larger parts and more compli- cated shapes are needed to further investigate the deviation patterns. For example, how the shape deviation change with part size? Will the line-by-line laser melting bring new shape deviation patterns to complex shapes (particularly sharp corners and sharp curves)? Will the part orientation influence the deviation pattern? In the meanwhile, further experimentation, analysis and validation about location related effects are neces- sary. For instance, how much of the laser beam positioning error pattern and additional location effects pattern are repeatable? Is there any significant random effect that needs additional consideration? Achieving these are difficult considering the high-complex error sources and experiment costs. Elaborate Design of Experiments are therefore needed here. But this further research is believed to be very beneficial since on the 122 one hand, our integrated shape accuracy improvement approach could be more com- prehensively validated or enhanced. On the other hand, the better understanding and modeling could enable more flexible methodology extension to other printing techniques such as SLS and FDM. • Prescriptive shape deviation modeling for 3D freeform shapes: This dissertation analyzed and modeled the 2D cross section shape deviation, further shape deviation modeling for the full three-dimensional freeform geometry is necessary and essential. To achieve this, a reliable 3D shape measurement equipment is firstly needed. 3D scanner is a good candidate, but careful considerations are needed here for better alignment between measurement and nominal design. We need to make sure the measurement along radius direction is accurate and the alignment doesn’t bring in significant additional errors. To predict the 3D shape deviation, extending the in-plane shape deviation model proposed in this work is a good start. The challenge here is that the layer-layer interaction and z-direction deviation will also influence the in-plane shape deviation. A possible approach is still adopting our in-plane model to layer-wise predicttheshapedeviationalongz-direction, butaddingtheazimuthalangleinspherical coordinate system (SCS) as an additional parameter in our model. In other words, the in-plane shape deviation in each layer is azimuthal angle dependent to represent the z-direction influence. • Apply deep learning to process thermal physics: The inaugural attempt in the last task demonstrates impressive potentials, more works could be done here. For the fusing layer prediction, we are implementing the fusing layer thermal prediction model to closed-loop thermal control for life testing and eventual productization. For the thermal diffusivity modeling, besides the z-direction thermal diffusivity we reported in this dissertation, future works are needed to build the x-y direction thermal diffusivity DNN. The intuitive idea is to record the period of time after 123 jetting detailing agents at each layer, during which the x-y direction thermal diffusivity dominates the thermal behavior, then apply the same DNN. Inthemeanwhile, wewillneedtoembedthelearnedthermaldiffusivityDNNtothefirst- principle model based high fidelity computational physics simulation engine to create gray box predictive tool. Such gray box predictive tool is essential and valuable since on the one hand, it is the direct way to verify whether the diffusivity DNN we learned is physically meaningful. A candidate verification method is to apply both the gray box model and the white box model (purely first-principle model) to simulate the final part geometry, then compare them with the measured geometry of truly printed parts. On the other hand, the gray box model owns the advantages from both the black box model (DNN) and white box model, which is widely believed to be able to provide more comprehensive, reliable and robust prediction. To successfully embed our DNN to the simulation engine, more experiments and analysis are needed to improve the thermal diffusivity neural network further. 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Abstract (if available)

Abstract Additive manufacturing (AM), or three-dimensional (3D) printing, refers to a new class of technologies associated with the direct fabrication of physical products from Computer-Aided Design (CAD) models by a layered manufacturing process. It has been widely recognized as a disruptive technology with the potential to fundamentally change the nature of future manufacturing, and the changes can amount to a third industrial revolution. ❧ Despite the vigorous development of different 3D printing techniques, the end-part quality of 3D printing however, is still not comparable to traditional manufacturing which continues to be one of the most significant issues in adoption. As an essential aspect of end-part quality, the shape accuracy still requires better control. Therefore, development of quantitative models that could predict process behaviors or end-part shape deviation is fundamental to inform both part design and process control. ❧ Developing the quantitative models and achieving high and consistent shape accuracy in AM is a challenging task. Those challenges include three aspects: (i) process physics is complex and more fundamental process knowledge is required to enable more precision control, but they are not yet fully understood

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Creator Luan, He (author)

Core Title Statistical modeling and machine learning for shape accuracy control in additive manufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 07/09/2018

Defense Date 04/27/2018

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

Tag 3D printing,additive manufacturing,machine learning,OAI-PMH Harvest,shape accuracy control,statistical modeling

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Huang, Qiang (committee chair)

Creator Email hluan@usc.edu,qingyur@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-510290

Unique identifier UC11266847

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Legacy Identifier etd-LuanHe-6379.pdf

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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