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Solid slicing and electrostatic analysis for electrophotographic rapid prototyping machine

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Title:
Solid slicing and electrostatic analysis for electrophotographic rapid prototyping machine
Creator:
Bhaskarapanditha, Sivakumar V
Place of Publication:
[Gainesville, Fla.]
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University of Florida
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English

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Algorithms ( jstor )
Boundary conditions ( jstor )
Electric fields ( jstor )
Electrostatics ( jstor )
Manufacturing ( jstor )
Modeling ( jstor )
Polygons ( jstor )
Software ( jstor )
Three dimensional modeling ( jstor )
Two dimensional modeling ( jstor )
Dissertations, Academic -- Mechanical Engineering -- UF ( lcsh )
Mechanical Engineering thesis, M.S ( lcsh )
electrophotography, electrostatic, element, finite
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government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Summary:
ABSTRACT: The distinctive feature of the rapid prototyping process is that it produces parts layer by layer. The process proceeds by first computing the cross section of the solid model and building these layers physically on top of each other. Electrophotographic solid freeform fabrication (ESFF) is one such method in which the part is built by printing the material layer-by-layer using electrophotography technique. One of the goals of this project is to develop a software which computes the cross sectional image information for printing by implementing a slicing algorithm. In this regard a software called SolidSlicer has been developed which implements the slicing algorithm. Given the solid model, the software can read in the model, slice it and create an image for printing. The software interacts with the ESFF machine and automates the entire ESFF process flow. Another goal of this project is to study the electrophotography process using finite element methods (FEM), so that the insights obtained can be used to improve the fabrication capability of the ESFF test bed. In this regard, an electrostatic analysis module was added to an existing FEA software. This electrostatic module implements the governing equations and various boundary conditions that apply to electrostatics. Using this software, finite element electrostatic analysis was performed to study the electric field distribution in the electrophotographic process. This analysis helps in understanding the electrophotography printing process and the cause of fundamental problems associated with the ESFF process.
Thesis:
Thesis (M.S.)--University of Florida, 2003.
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Includes bibliographical references.
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Title from title page of source document.
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Includes vita.
Statement of Responsibility:
by Sivakumar V Bhaskarapanditha.

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University of Florida
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University of Florida
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Copyright Bhaskarapanditha, Sivakumar V. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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1/1/2004
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029897682 ( ALEPH )
78944798 ( OCLC )

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SOLID SLICING AND ELECTROSTATIC ANALYSIS FOR ELECTROPHOTOGRAPHIC RAPID PROTOTYPING MACHINE By SIVAKUMAR V BHASKARAPANDITHA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

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Copyright 2002 by Sivakumar V Bhaskarapanditha

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Dedicated to Dev

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ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor and the chairman of my thesis committee, Dr. Ashok V. Kumar, for his guidance, encouragement and patience throughout my study. I would like to thank him for his tireless effort in reading several drafts of this thesis. Without all of his kind help this project would never have been completed. I am also grateful to my committee members, Dr. John K. Schueller and Dr. Nagaraj K. Arakare, for their advice, comments and patience in reviewing this thesis. I also thank my group fellows, Mr. Young Nam Hwang and Mr. Jongho Lee, for their kind advice and help regarding programming. Special thanks go to Mr. Anirban Dutta and Mr. Ajay Das for their help in understanding electrostatics. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF FIGURES.........................................................................................................viii ABSTRACT.......................................................................................................................xi CHAPTER 1 INTRODUCTION...........................................................................................................1 Motivation.......................................................................................................................1 Goals and Objectives......................................................................................................2 Outline.............................................................................................................................3 2 SOLID FREEFORM FABRICATION............................................................................4 Solid Freeform Fabrication Techniques..........................................................................4 StereoLithography (SLA).........................................................................................5 Fused Deposition Modeling (FDM).........................................................................5 Laminated Object Manufacturing (LOM)................................................................6 Selective Laser Sintering (SLS)...............................................................................6 Three Dimensional Printing (3D Printing)...............................................................7 Electrophotographic Solid Freeform Fabrication...........................................................9 ESFF Test Bed.........................................................................................................9 Electrophotography system..............................................................................10 Automated build platform................................................................................12 Heating and compaction systems.....................................................................12 ESFF Process Control............................................................................................12 3 SOLID SLICING...........................................................................................................15 Solid Model Representation..........................................................................................15 STL Format....................................................................................................................16 Slicing Algorithm..........................................................................................................17 Polygon Algorithm........................................................................................................21 SolidSlicer.....................................................................................................................25 SolidSlicer Software Commands...........................................................................27 File Opening and Saving Commands.....................................................................29 Positioning Commands..........................................................................................29 v

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Animation and Printing Commands.......................................................................29 ESFF Flow Process.......................................................................................................31 Results...........................................................................................................................31 4 ELECTROSTATICS.....................................................................................................33 Objective.......................................................................................................................33 Introduction to Electrostatics........................................................................................33 Electric Field Strength in the Dielectric of a Capacitor.........................................35 Introduction to Finite Element Analysis.......................................................................37 Finite Element Formulation of Electrostatic Analysis..................................................38 Weak Form.............................................................................................................39 Boundary Conditions.............................................................................................41 Restraints..........................................................................................................42 Loads................................................................................................................43 Symmetry boundary conditions.......................................................................43 Specified value of charge per unit area............................................................43 Specified value of charge per unit volume......................................................43 Constraints.......................................................................................................44 Matrix Equation for Electrostatics.........................................................................45 5 IMPLEMENTATION OF ELECTROSTATICS IN FINITE ELEMENT ANALYSIS SOFTWARE..................................................................................................................47 Objective.......................................................................................................................47 JavaFemViewer.............................................................................................................47 JavaFem Package...................................................................................................48 Electrostatic Element.............................................................................................49 Boundary Conditions.............................................................................................50 Modified elimination method..........................................................................51 Coupled degree of freedom constraints...........................................................52 Viewer Package............................................................................................................54 Pre Processing Classes...........................................................................................54 Post Processing Classes.........................................................................................55 Graphics Classes....................................................................................................55 Finite Element Model classes................................................................................56 JavaFemViewer.............................................................................................................56 Verification Problems...................................................................................................56 Parallel Plate Capacitor Model 1............................................................................56 Parallel Plate Capacitor Model 2............................................................................58 Parallel Plate Capacitor Model 3............................................................................60 6 ANALYSIS AND RESULTS........................................................................................63 Edge Growth Model......................................................................................................64 Problem Statement.................................................................................................64 Finite Element Model.............................................................................................64 vi

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The Pattern Model.........................................................................................................67 Transfer Model..............................................................................................................71 Problem Statement.................................................................................................71 Finite Element Model.............................................................................................71 Effect of Height of Print.........................................................................................73 Effect of Voltage on Diffuse Printing....................................................................76 Effect of Charge Density........................................................................................78 7 CONCLUSIONS AND FUTURE WORK....................................................................79 Conclusions...................................................................................................................79 Future Work..................................................................................................................80 APPENDIX A INPUT FORMAT FOR JAVAFEMVIEWER SOFTWARE.......................................81 B SCENEGRAPH FOR JAVAFEMVIEWER SOFTWARE..........................................84 LIST OF REFERENCES...................................................................................................85 BIOGRAPHICAL SKETCH.............................................................................................87 vii

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LIST OF FIGURES Figure page 2.1 Principle of solid freeform fabrication...........................................................................4 2.2 Schematic of StereoLithography....................................................................................5 2.3 Schematic of FDM.........................................................................................................6 2.4 Schematic of LOM.........................................................................................................7 2.5 Schematic of SLS...........................................................................................................8 2.6 Schematic of 3D printing process..................................................................................8 2.7 ESFF test bed...............................................................................................................10 2.8 Structure of developing system....................................................................................11 2.9 ESFF process flow.......................................................................................................14 3.1 Facet models................................................................................................................16 3.2 Redundancy in the STL file.........................................................................................17 3.3 Slicing algorithm..........................................................................................................19 3.4 The intersection point between the slicing plane and the triangle facet......................20 3.5 Chart diagram for the polygon algorithm....................................................................23 3.6 System of polygons......................................................................................................24 3.7 SolidSlicer class diagram.............................................................................................25 3.8 Object model for the TArray class...............................................................................26 3.9 Object model for the Section class...............................................................................27 3.10 SolidSlicer software...................................................................................................28 3.11 3DPrint action flow chart...........................................................................................30 viii

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3.12 Parts made by ESFF test bed.....................................................................................31 4.1 The capacitor model.....................................................................................................35 4.2 Boundary conditions for 2D electrostatics...................................................................42 4.3 Parallel plate capacitor model with equipotential constraint.......................................44 5.1 Object model for FE_Model class...............................................................................49 5.2 Analysis type hierarchy................................................................................................50 5.3 Flow chart for the coupled degrees of freedom implementation.................................54 5.4 Viewer package class relationships.............................................................................55 5.5 Finite element model for the parallel plate capacitor system......................................57 5.6 Magnitude of electric field for the capacitor model.....................................................58 5.7 Y component of electric field for the capacitor model 2.............................................59 5.8 Parallel plate capacitor model 3...................................................................................60 5.9 Results of model 3, voltage distribution......................................................................61 6.1 Simplified model for development process.................................................................65 6.2 Y component of the electric field for the pattern less model.......................................66 6.3 The y component of electric displacement along the interface....................................67 6.4 Alternately charged and discharged regions for the pattern model.............................68 6.5 Y component of electric displacement for the pattern model......................................69 6.6 Pattern model zoomed in.............................................................................................70 6.7 Y component of electric displacement for the pattern model (zoomed in)..................70 6.8 Transfer between photoconductor drum and the deposition platform.........................71 6.9 Simplified model for transfer model............................................................................72 6.10 The boundary conditions for the transfer model........................................................73 6.11 Y component of electric field for the case h= 0.1mm................................................75 6.12 X component of electric field for h = 0.1 mm...........................................................76 ix

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6.13 Regions of interest in the diffuse print model............................................................77 1.B Scenegraph for the JavaFemViewer software.............................................................84 x

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SOLID SLICING AND ELECTROSTATIC ANALYSIS FOR ELECTROPHOTOGRAPHIC RAPID PROTOTYPING MACHINE By Sivakumar V Bhaskarapanditha May 2003 Chair: Ashok Kumar Major Department: Mechanical and Aerospace Engineering The distinctive feature of the rapid prototyping process is that it produces parts layer by layer. The process proceeds by first computing the cross section of the solid model and building these layers physically on top of each other. Electrophotographic solid freeform fabrication (ESFF) is one such method in which the part is built by printing the material layer by layer using electrophotography technique. One of the goals of this project is to develop a software which computes the cross sectional image information for printing by implementing a slicing algorithm. In this regard a software called SolidSlicer has been developed which implements the slicing algorithm. Given the solid model, the software can read the model, slice it and create an image for printing. The software interacts with the ESFF machine and automates the entire ESFF process flow. Another goal of this project is to study the electrophotography process using finite element methods (FEM), so that the insights obtained can be used to improve the xi

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fabrication capability of the ESFF test bed. In this regard, an electrostatic analysis module was added to an existing FEA software. This electrostatic module implements the governing equations and various boundary conditions that apply to electrostatics. Using this software, finite element electrostatic analysis was performed to study the electric field distribution in the electrophotographic process. This analysis helps in understanding the electrophotography printing process and the cause of fundamental problems associated with the ESFF process. xii

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CHAPTER 1 INTRODUCTION Solid freeform fabrication (SFF) is a group of part fabrication processes that produce three-dimensional shapes from additive formation steps. SFF, also known as rapid prototyping (RP), does not require any part-specific tooling. The three dimensional part is produced from a 3D representation of the solid created in computer aided design (CAD) software. This computer representation is used to compute the cross sectional images, which can then be fed to the control equipment to fabricate the part. SFF entails many different approaches to the method of fabrication. Stereo lithography (SL), selective laser sintering (SLS), laminated object manufacturing (LOM), and fused deposition modeling (FDM) are a few examples that today have commercial machines applying these techniques. The largest impact SFF has had on manufacturing is enhancement of the prototype production process, cutting down by orders of magnitude the time required to iterate through design, fabrication, and testing to reach the final design of a prototype. In this thesis, a novel method for solid freeform fabrication based on electro photography technology is studied and some of the problems associated with this SFF are solved. Motivation Electrophotographic Solid freeform fabrication or ESFF is a novel solid freeform fabrication technique based on electrophotography. Electrophotography, also called Xerography, is the technology used in all laser printers and photocopiers commercially 1

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2 available in the market. The capability to print high-resolution images is an exciting characteristic of electrophotography, which is being tested on the ESFF test-bed for making accurate 3D parts. Electrophotographic solid freeform fabrication is a promising technology in its developmental stages. Past work in this area has proven the feasibility of this technology. Many problems need to be solved before it can become a viable Solid Freeform Fabrication technique. Current work discusses and solves some of these problems based on the understanding of the theory behind the process. One of the problems is to get the cross sectional information from a 3D CAD model. Other problems are in the difficulty in obtaining good quality uniform prints. The problems associated with the spreading of toner near the solid edges (Diffuse Printing) and the excessive deposition of toner powder near the solid edges (Edge Growth) are challenges in obtaining better quality prints. Goals and Objectives The goal of this project is to develop software that can create the cross-sectional images of a solid model, for layer by layer printing using the ESFF and to study the electro photography process using finite element methods (FEM), so that the insights obtained can be used to improve the fabrication capability of the ESFF test bed. In this regard, finite element method is adopted to study the electric field lines in the electrophotographic process. To achieve the above goals, the following objectives were identified.

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3 1) Perform an extensive literature survey for the slicing algorithms and to develop and implement a suitable algorithm for the ESFF process. 2) Design and implement a software that can slice the solid model and print the cross-sectional images using the ESFF printing system. 3) Perform literature survey for the electrostatic analysis using finite element method to understand the various loads and boundary conditions that apply for the electrostatic analysis. 4) Extend the already existing finite element software called JavaFemViewer software, to include the electrostatic analysis capabilities along with the post processing capabilities for better display of the results. 5) Develop finite element models to study the edge growth and diffuse print problems and interpret the results. Outline The topics covered in the rest of this thesis are organized as follows: Chapter 2 discusses the commercially available SFF methods and a brief introduction to electrophotographic solid freeform fabrication process. Chapter 3 discusses the slicing algorithm and the overview of the object oriented SolidSlicer program. Chapter 4 presents problems faced in the ESFF printing and an introduction to finite element formulation of electrostatic analysis. Chapter 5 discusses the object oriented JavaFemViewer software that implements the electrostatic FEA module. Chapter 6 provides a discussion of problems faced in the ESFF process, mathematical models developed to study the problems and interpretation of results of the finite element analysis. And finally Chapter 7 draws conclusions based on the previous chapters and suggests future work in this area.

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CHAPTER 2 SOLID FREEFORM FABRICATION This chapter describes the new approach to manufacturing called Solid Freeform Fabrication (SFF), and briefly describes the commercially successful processes. It then introduces the novel SFF technique called electrophotographic solid freeform fabrication (ESFF) that is in development. Solid Freeform Fabrication Techniques Solid freeform fabrication is a new class of manufacturing technologies characterized by layer-by-layer build up of parts. All SFF techniques have this essential underlying characteristic but differ significantly in the part material and their process parameters such as method of deposition, part orientation, layer thickness, and post process steps such as curing or sintering. As described by Kochan [1], the principle of SFF is shown in figure 2.1. 3D CAD Interface between 3D model and SFF 3D Objects Specific Materials Figure 2.1 Principle of solid freeform fabrication 4

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5 Some of the commercially successful SFF processes are as follows StereoLithography (SLA) Fused Deposition Modeling (FDM) Laminated Object Manufacturing (LOM) Selective Laser Sintering (SLS) Three Dimensional Printing (3D printing) StereoLithography (SLA) StereoLithography uses photo-curable polymers that solidify when exposed to UV rays. The layer is formed by selectively solidifying the resin by UV laser. Dipping the part in the resin and successively solidifying the fresh resin layer on the top, form the 3D part. The schematic of the StereoLithography is shown in figure 2.2. Computer Control System UV Light Z Resin 3D Parts Figure 2.2 Schematic of StereoLithography (US Patent 4,575,330) Fused Deposition Modeling (FDM) Fused deposition modeling uses two nozzles to deposit fused polymers extruded through its tip to the previously deposited layers, which solidifies as soon as coming out. One contains the support material and the other contains the part material that is

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6 deposited upwards on the build platform. The nozzles can traverse horizontally and the platform can move vertically thus building a three dimensional part. Figure 2.3 is a schematic of the FDM system. Figure 2.3 Schematic of FDM (Kochan [1]) Laminated Object Manufacturing (LOM) Laminated object modeling uses laser to cut sheets of material and glue them together to form the part. Precisely cutting the contours of the sheet of material and gluing them together by heat-activated glue create the part. Figure 2.4 is a schematic of the LOM system. Selective Laser Sintering (SLS) Selective Laser Sintering uses laser to sinter the 2D image of the part on the powder. The temperature of the powder layer exposed to the laser beam is raised to the point of sintering just below the melting point to give it enough green strength without actually melting the powder. This process is repeated to form the complete part. The schematic of the process is shown in Figure 2.5.

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7 Figure 2.4 Schematic of LOM (Kochan [1]) Three Dimensional Printing (3D Printing) 3D printing uses inkjet printer to deposit the binder resin on the powder layers to bind the 2D layers of the part. Adding a layer of powder on the top of the bonded powder bed in a piston-cylinder arrangement forms the powder layers. The binder resin bonds the powder and the piston lowers the powder bed for fresh layer of powder. Figure 2.6 depicts the schematic of the 3D printing process.

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8 Figure 2.5 Schematic of SLS (Kochan [1]) Figure 2.6 Schematic of 3D printing process (Kochan [1])

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9 Electrophotographic Solid Freeform Fabrication Electrophotographic Solid Freeform Fabrication (ESFF) is a novel rapid prototype technique under development at the Design and Rapid Prototyping Laboratory in the Department of Mechanical Engineering at the University of Florida. It uses electrophotography to deposit powder, layer by layer, precisely in the part areas on top of previous layer to build the part. Compaction, discharging and fusing of each layer creates a new surface for subsequent deposition. The ESFF test bed was fabricated to investigate the feasibility of the ESFF technique. The ESFF test bed and the process sequence are discussed in the subsequent sections. ESFF Test Bed An ESFF machine was built that enables layer-by-layer deposition of powder using electrophotography technology. A schematic diagram of the system is shown below in Figure 2.7, as given by Zhang [2]. This machine contains many subsystems such as the electrophotographic printing system, automated deposition/build platform, control system, heating and discharging system, and compacting system. The electrophotography printing system provides the delivery engine to deposit the material layer by layer on a build platform. Automated build platform is driven by a 2-axis motion control system. The powder is printed on the platform when it is moving in the x-direction. The platform moves down in z-direction after each print so that subsequent layer can be added. The heating system fuses the material deposited on top of the platform. The compacting system serves two purposes, one is to compress the material deposited on the platform, and the other is to position the platform at the right level so that electrophotography printing system can print on top of

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10 Figure 2.7 ESFF test bed (Zhang [2]) the previous layers. A Galil DMC2030 motion controller controls all these subsystems with a C++ programming interface. The following sections of this chapter will first describe each subsystem in detail, and then give a detailed description of the process sequence of the entire ESFF system. Electrophotography system The Electrophotography printing system is the most important subsystem of the ESFF machine. Its primary responsibility is to deposit the powder in the required shape on the automatic deposition platform. The electrophotography engine consists of a controller, an image formatter, the charging unit, a laser based image projection subsystem, the photoconductor drum and the powder cartridge.

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11 The controller in the electrophotography engine controls the motors and the laser subsystem in synchronization. The image formatter translates the digital data from the PC into the right format for the image projection subsystem. The charging unit negatively charges the surface of the photoconductor drum by direct contact charging, as explained in Schein [3]. The image projection subsystem contains a modulated laser and a rotating polygonal mirror that projects the image line by line on the photoconductor drum. The laser is used to discharge the image areas (discharged area development) forming the latent image on the photoconductor drum. Figure 2.8 shows the powder cartridge with the electrical connections for supplying a negative DC biased AC potential to the developing cylinder or the developer roller with the photoconductor drum grounded. DC Bias AC Bias Photosensitive Dru m Stirre Doctor blade Development roller Figure 2.8 Structure of developing system (courtesy Gokhale [4])

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12 The metal developing cylinder rotates around a fixed magnetic core inside the powder cavity. Magnetic powder is used, which is attracted to the magnetic core inside the cylinder. A rubber blade meters the powder on the developing cylinder to a uniform thickness. The powder is transferred to the discharged image areas of the photoconductor by electrostatic forces. This toner layer is then transferred to the previous layers on the build platform to make the 3D part. Automated build platform Automated deposition platform is an aluminum platform, controlled by a two-axis motion control system. This system drives the platform to the right position with the right speed at the right level. Galil DMC2030 motion controller controls the automated deposition platform, Electrophotography printing system and heating system simultaneously so that it can coordinate and communicate between the different subsystems. Heating and compaction systems After the transfer of powder from the photosensitive drum to the deposition platform, the powder is loose. So for the solid object to achieve the required strength, it is required that the deposited powder be fused and compacted. To fuse the powder, a heating unit was provided after the electrophotography engine as shown in figure 2.7. ESFF process control system synchronizes all these subsystems. The ESFF process control is explained in the next section. ESFF Process Control ESFF fabrication process flow is shown in figure 2.9. The electrophotography engine checks and prepares its subsystems before the start of the ESFF process. The fuser is brought at the right temperature and the servo motion controller initializes the

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13 traversing gear. A datum needs to be set for measuring the deposition thickness or the part height. The servo motion controller raises the build platform under the compaction plate to set the datum. This process is called HOMING or INITIALIZING of the build platform. After setting the datum, the build platform is moved to a fixed position where it waits for a signal from the electrophotography engine. At this point, the cross sectional image of the solid has to be computed using a software. The objective of the software is to read in a CAD model, generate the cross sectional image information at a specified height and control the ESFF process by calling the DMC control programs as shown in figure 2.9. This software should interact with the GALIL controller for the deposition height at which the cross sectional image has to be generated. The algorithms used in computing the cross sectional image and the programming details are explained in chapter 3. Assuming that such a cross sectional image is computed and the printing process is initiated, the build platform moves under the electrophotography engine at the print speed. This results in a formation of a layer of material, in the shape of the cross section, on the build platform. The platform is then moved under the fusing and compaction system to fuse and compact the layer. This entire process is repeated until the 3D part is manufactured. The ESFF process can be summarized as 1) Prepare all the subsystems. 2) Initialize the build platform. 3) Drive the build platform to the correct position and wait for the start signal. 4) Compute the cross sectional image using a software and initiate the printing process.

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14 5) Move the platform under the electrophotography engine at the print speed. 6) Fuse and compact the freshly deposited layer of material. 7) Repeat the process until the entire 3D part is manufactured. Prepare all subsystems Home the platform in both horizontal and vertical directions Move the platform to the right position for prinitng Build platform ready for deposition (STATUS = Ready) Query for START signal from theelectrophotography engine? Move Build platform under the electrophotography engine fordeposition Fuse and compact the powder layer Yes Set the vertical level with respect to the compaction platform andcalculate the deposition height Implemented in DMC andrunning in GALIL Controller No START signal Interact with a Computer Software to ge t the cross sectional image and initiate th e deposition process Figure 2.9 ESFF process flow Next chapter deals with the algorithms used in computing the cross sectional image along with the software implementation of the algorithms.

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CHAPTER 3 SOLID SLICING From chapter 2, it is clear that cross sectional information has to be generated using software, which automates the ESFF process. This chapter summarizes the literature survey for the slicing algorithms, required to generate the cross sectional information, developed for other solid freeform fabrication processes. This is followed by description of the algorithm that is implemented for the ESFF process. A software implementation of the algorithm to automate the ESFF process is described in the rest of the chapter. This software is named SolidSlicer. Solid Model Representation Rapid Prototyping (RP) processes produce parts layer by layer. Hence, the solid model of the part to be fabricated has to be first sliced to create cross-sectional images, using software, before the physical building process can start. The model is sliced by finding the intersection of the solid with the slicing plane. The solid model can be represented in any of the existing formats such as STL, IGES, HPGL, STEP, CT etc. Jacob [5] describes these different formats in some detail. The simplest format is the STL (StereoLithography) format. The STL file, which is a de facto standard, is the most common interface between CAD and Rapid Prototyping systems. 3D Systems developed and published the STL format in 1987 for converting 3D CAD models for use in Stereo Lithography Apparatus (SLA). The following section describes the STL format. 15

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16 STL Format The STL file format is a polyhedral representation of the part with triangular facets. It is generated from a precise CAD model using a process known as tessellation, which generates triangles to approximate the CAD model [6]. The STL file can be in either ASCII or binary format. The size of the ASCII STL format is larger than that of the binary format, but is human readable. In an STL file, triangular facets are described by a set of X, Y and Z coordinates for each of three vertices and a unit normal vector to indicate the side of the facet which is outside the object as shown in the figure 3.1. Figure 3.1 Facet models (courtesy Liao et al [6]) The STL ASCII format as described by Liao et al [6] is given as follows: solid name // the STL filename facet normal n i , n j , n k // n i n j n k are the direction cosines outer loop vertex x 1 y 1 z 1 // the coordinates of vertex 1 vertex x 2 y 2 z 2 // the coordinates of vertex 2 vertex x 3 y 3 z 3 // the coordinates of vertex 3 endloop endfacet endsolid name

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17 The STL file carries a high degree of redundancy because of the duplicate vertices and edges. In figure 3.2 left diagram contains lots of redundancy compared to the original as shown in the diagram on right. STL files usually require unnecessarily high computer storage space as well as more computer resources to process. In addition, the STL format does not provide efficient information storage compared to the higher-level representations such as spline and NURBS. As a result, the STL file contains thousands of triangles to represent a model, whereas in a mathematically precise format, only a few splines or NURBS surfaces may be sufficient to represent the same model. Figure 3.2 Redundancy in the STL file (courtesy Liao et al [6]) Despite these disadvantages, STL is still the most popular solid model representation for rapid prototyping systems. This is because of its absolute simplicity, which makes the slicing algorithms very simple, greatly improving the software speed. Another advantage of the STL format is its independence from specific CAD modeling methods. The slicing algorithm for the solid models represented in STL format is given below. Slicing Algorithm As the SFF process requires the cross section of the solid to build the part, an algorithm has to be developed to section the solid. Many slicing algorithms have been

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18 developed for various SFF processes. For example, Chen et al [7] describes the Direct Slicing from the PowerSHAPE (a different computer representation of 3D models) models, whereas Liao et al [6] describes the top down slicing, bottom up slicing, positive tolerance slicing and negative tolerance slicing. A similar method, which uses the vectorial approach, is described here. The flow chart of the slicing algorithm is given in figure 3. 3. The objective is to find the intersection of the solid model, approximated by triangles (STL), with a plane. The slicing plane is assumed to be perpendicular to the z axis. The z level of the slicing plane is denoted as zz, which is the input for the algorithm. The output of the algorithm is an array of line segments that represents the cross section. If MinZ and MaxZ represent the minimum and maximum z levels of all the triangles in the solid model, the plane intersects the solid model if and only if zz lie between these two limits. If zz is out of bounds then zz is assigned a value of zero and the program is terminated. If zz is within the bounds, then the triangles are checked for intersection with the zz plane. First, each triangle is checked to see if it lies on the zz plane. A triangle lies on the zz plane if the z coordinates of all the three vertices of this triangle are same and is equal to zz. If the triangle lies on the plane then the cross section at this zz level is skipped by incrementing the value of zz by a small number, denoted as z tolerance (1x10 -10 ) in the figure. And the algorithm is called recursively with this new value of zz. Since z tolerance is extremely small, the error due to skipping the cross section is insignificant for all practical purposes. If zz is within the bounds, and the triangles are not in the zz plane, then the triangles are checked for intersection with the zz plane. The slicing plane cuts a triangle if

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19 zz lies between the minimum and maximum z coordinates of that triangle as shown in figure 3.4. Without loss of generality, assuming that z 1 and z 3 are the maximum and minimum z coordinates of the triangle, the plane intersects the triangle if zz lies between z 1 and z 3 . Inputzz Slice i = 0 i < num ofTriangles Check if the ith triangle is in the zzplane zz += Tolerance Find the line of intersection with the plane Add the line segment to thecross section(LineArray) i++ yes No Yes Yes MinZ < zz < MaxZ Yes zz = 0 No Check if the slicing plane intersects the triangle End No No Figure 3.3 Slicing algorithm If the plane intersects the triangle then the line of intersection is found as described below:

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20 P (xp, yp, zp)Q (xq, yq, zq)V1 ( x1, y1, z1)V2 ( x2, y2, z2)V3 ( x3, y3, z3)Slicing planeZ = zz Z-Axis Line of intersection Figure 3.4 The intersection point between the slicing plane and the triangle facet Suppose the position vectors of the vertices of the triangle are v 1 (x 1 , y 1 , z 1 ), v 2 (x 2 , y 2 , z 2 ) and v 3 (x 3 , y 3 , z 3 ). The edges of the triangle, which cut the slicing plane, can be easily found by comparing the z coordinates with that of the cutting plane. Without loss of generality, assuming that the plane cuts the line segments v 1 v 2 and v 1 v 3 (where the AB represents the vector from B to A) at points P (x p , y p , z p ) and Q (x q , y q , z q ), as shown in the figure, the coordinates of these points can be computed from the following equations. The equation of line joining v 1 and v 2 is given by r = u v 1 + (1-u) v 2 (3.1) where u is a parameter that belongs to the set of real numbers. Since P is a point on the line v 1 v 2 and also a point on the plane, we have z p = zz (3.2) z p = uz 1 + (1-u)z 2 (3.3) Solving these two equations, we get

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21 212zzzzzu Substituting this value of u into equation (3.1), we get P = u v 1 + (1-u) v 2 Similarly the point Q is found from the intersection of v 1 v 3 with the slicing plane. Once the points P and Q are computed, these points are then stored as a line segment PQ. Java3D, a java graphics package, has a useful class, called LineArray, for storing such line segments. Thus, after looping through all the triangles, we have the cross section of the solid model at the zz level stored in the LineArray. Some of the SFF processes need the tool path generation after computing the cross section. For example, in the SLA and FDM processes, described in chapter 2, the path for either the UV rays or the nozzle has to be computed for the layer deposition. But since the ESFF process involves printing of an image, there is no need for the tool path generation algorithm. Instead an algorithm to compute an image needs to be developed. This algorithm is described in the Polygon Algorithm. Polygon Algorithm ESFF process require the cross sectional information in the form of an image. This image is then sent for printing, and a layer of material is deposited after the transfer process, as described in chapter 2. The output of the slicing algorithm is an array of line segments in a random order, along the boundary of the cross-section. These line segments can be used to create an image by using a java2d class called Polygon. The Polygon class encapsulates a description of a closed, two-dimensional region within a coordinate space. This region is bounded by an arbitrary number of line segments, each of which is one side of the polygon. Internally, a polygon comprises of a list of (x, y) coordinate pairs,

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22 where each pair defines a vertex of the polygon, and two successive pairs are the endpoints of a line that is a side of the polygon. The first and final pairs of (x, y) points are joined by a line segment that closes the polygon. The input to the Polygon class is ordered coordinates (either clock wise or anti-clockwise) of the vertices of the polygon. Hence an algorithm must be developed to order the randomly ordered coordinate pairs obtained from the slicing algorithm. The following algorithm is developed to sequentially add the points to the polygon. The chart diagram of the polygon algorithm is given in figure 3.5. As shown in the figure the input to the algorithm is the list of lines obtained from the slicing algorithm. The output of the algorithm is an array of polygons that represent the closed contours of the cross section. For example, the algorithm returns an array of three polygons for the cross-section shown in figure 3.6. The details of the algorithm are explained using this. The line segments that are obtained from the slicing algorithm are initially random in order. These can be arranged as follows. Suppose L1 is the first line segment of the list of lines obtained from the slicing algorithm. Then a search is carried out to find a line segment from all the remaining lines, such that, the start point of this line segment is the same as the end point of L1. For this example we get L2 as the connecting line (it can be L8, but it does not matter if it is clockwise or anti-clock wise). This process is continued until the loop is closed. The loop is closed if the start point of the first line of the loop (L1) is same as the end point of the current connecting line (in this case L8). The loop represents one closed polygon. There can be more than one polygon depending on the section. For example, the figure 3.6 consists of three polygons. These polygons are

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23 stored in a polygon list. The algorithm is carried out till all the line segments are exhausted. Inputlist of lines index = 0 index< nLInes i = index + 1 i < nLines Check if the ith line is the nextconnecting line i = i+1 Add the line coordinate to thecurrent polygon Check if the linecloses the polygon nLoops++ Add the polygon tothe loop list Index++ no yes yes yes yes no end No Figure 3.5 Chart diagram for the polygon algorithm

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24 At the end of the algorithm, we have an array of closed polygons representing the closed contours of the cross sectional image. These polygons can be filled in any color and this result in the generation of an image for printing. But there is no way of judging whether the inside of a polygon represents solid area or hole. The solid parts should be in black color, whereas the holes should be in white color, in the cross sectional image generated. To automatically program this capability, XORing of colors is used. XORing of black with black results in white color, and black with white results in black. For example, in figure 3.6, polygon 3 represents a solid area, whereas polygon2 represents a hole. So, Polygon2 which is initially in black will be XORed with the black color of polygon 1 and hence will result in white color for polygon 2. L1L2L3L4L5L6 L7L8L9L10L11L12L13Polygon 1Polygon 2Polygon 3L14L15L16L17 L18L19 Figure 3.6 System of polygons The following sections deal with the software that implements these algorithms. The software is named SolidSlicer.

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25 SolidSlicer The slicing and the polygon algorithms are implemented in the SolidSlicer program. This software is implemented using Java programming language. The object oriented classes defined in this program and the relations between these classes are described below. The main class of the program is the SolidSlicer class. The purpose of this class is to provide the user interface, and to perform all the actions caused by mouse clicks by creating the appropriate classes and calling the appropriate functions. The important class data and functions are given in figure 3.7. SolidSlicer FunctionscreateFrame() // method to create the GUI.readSTLFile() //method to read the STL data..readAssembly() // method read the saved assembly filewriteAssembly() // method to save the assembled STLs.. CommandInterface POlyPanel Draw FieldPanel TArray Figure 3.7 SolidSlicer class diagram As can be seen from the figure, this class contains references to other classes like TArray, Draw, PolyPanel, FieldPanel, and CommandInterface. These classes are explained in detail later in the chapter. SolidSlicer class also contains several important methods, as shown in the figure 3.7. These methods are named based on the function that it has to carry out. For example, the readSTLFile () method reads in the triangle data from

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26 the STL file and stores it in the TArray class. The object model for the TArray class is shown in figure 3.8. TArray int numOfTA // num of STLsTriangularArray[] arrayofTA // to store the multiple STL data Functionsslice() // method to slice the STLs.rotate() //method to rotate the individual STLs..translate() // method to translate the individual STLs..align() // method to align base of all the STLs with the z=0 plane Section Draw Figure 3.8 Object model for the TArray class TArray class is used to store multiple STL file information. It also contains methods to rotate, align and translate the individual STLs. It also contains a reference to the Section class, which implements the algorithms for sectioning the solid. The object model for the section class is shown in figure 3.9. This class contains the important functions that implement the slicing (slice ()) and the polygon (createPolygon ()) algorithms, described earlier in the chapter. This class stores a variable, shown as currentZ, which indicates the position of the cutting plane. It also has references to Draw and PolyPanel classes, to which the generated list of lines and polygons are passed respectively for display purposes.

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27 Section double currentZ // variable to store the slicing plane level Functionsslice() // method to slice the STLs.createPolygon() //method to compute the polygons that represent the //closed contours in the cross section.. PolyPanel Draw Figure 3.9 Object model for the Section class. The Draw class provides the java3D capabilities, which are useful for displaying both the 3D solid model represented by the triangles and the cross sections that are represented by the array of line. The class also provides the functions for zooming, panning, rotating and mouse picking. The PolyPanel class provides the java2D capabilities that are useful to create an image of the polygon that represents the cross section. The polygons are obtained from the polygon algorithm implemented in the section class. The FieldPanel class and the Command interface are auxiliary classes used to efficiently perform the actions caused because of the mouse clicks. It contains input field and different buttons to perform the actions like aborting, pausing, setting the currentZ level etc. The important commands controlled by the SolidSlicer software are described in the next section. SolidSlicer Software Commands A typical environment of the SolidSlicer software is shown in figure 3.10.

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28 Figure 3.10 SolidSlicer software

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29 Different important actions (commands) that can be performed by the software are as follows: File Opening and Saving Commands These commands are found under the File menu. These commands (like open STL, import assembly) are used to read the STL information of the part to be manufactured. The arranged STL files (described below in the positioning commands) can be saved as an assembly file which can be later imported for future purposes. Positioning Commands These commands are found under the Position menu shown in figure 3.10. These commands are used to 1) Align the STL files so that the base of the STL file coincides with the z = 0 plane. 2) Rotate about an axis perpendicular to the z plane. 3) Translate the individual files on the z plane (i.e. XY plane) Using these functions, the STLs can be arranged inside the white square shown in the figure 3.10. This square represents the build platform and is of the same size as that of the build platform. Animation and Printing Commands The Auto animate command, found under the Section menu, is used to check the sections for any defects. This command animates the slicing of the STL files. The Z level of the cutting plane is incremented by thickness of the slice (whose default value equals 20 microns). This animation is carried out in a separate thread so that the animation can be aborted at any time during the execution. The actual printing process is triggered by the 3DPrint command, found under the File menu. The process flow for the 3DPrint command is shown in figure 3.11. As seen from the figure, the first step is to get the current z level of the cutting plane. This value

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30 can be either zero (if starting the printing process for the first time) or the thickness of the part already built, so that the process can be resumed without starting from the beginning. Then initialize (Home) the platform as explained in chapter 2. Slice the STL files to get the cross sectional image (using the slice function in the TArray class). Drive the platform as explained in chapter 2. Deposit a layer of material by printing on the build platform. Compact and fuse the printed layer. Get the thickness of the print and update the currentZ level. Print the information (like the print number, thickness of print, time taken for print) in the log file. Action== 3dprint getCurrentZ() Initialize() currentZ < MaxZ Tarray.slice() drivePlatform() print() getZLevel() Update the log file yes yes End No Figure 3.11 3DPrint action flow chart This printing process can be aborted at any time during the execution. Since the z level of the cutting plane can be set at the time of 3DPrint, the printing process can be resumed.

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31 ESFF Flow Process To build the 3D part using the ESFF machine, steps described below have to be followed. Open the STL files of the parts to be manufactured. Arrange these STLs using the positioning commands. Auto animate the sectioning process to check for the cross sectional images. 3DPrint the part for manufacturing. Results Figure 3.12 Parts made by ESFF test bed

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32 Using this software, a part was fabricated with the ESFF machine. Figure 3.12 shows the picture of the part obtained. This shows the application of this software in automating the ESFF process. A closer inspection of the picture shows a lack of clear edge because of the diffusion of toner powder near the edges. Also the edges are thicker than the solid areas (which is not evident because of compaction, but clearly visible when there is no compaction). These problems affect the dimensional accuracy of the part being manufactured. So these problems have to be solved before ESFF process can be used for practical purposes. To solve these problems, knowledge of the forces acting on the toner particles is essential. These forces can be computed if the electric field distribution is known. Computing the electric field requires a thorough understanding of electrostatics, which is explained in the next chapter.

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CHAPTER 4 ELECTROSTATICS Objective As seen from the previous chapter, a study of electrostatics is necessary to determine the forces on the charged particles, which will provide an insight into the printing process. This can be used in the design of the components for the ESFF engine and to answer some of the issues like the diffuse printing and edge growth problems, explained in chapter 1. The following section deals with an introduction to electrostatics. Introduction to Electrostatics Electrostatics is the study of fields, potentials and forces of system containing any number of stationary charges [8]. The basic law governing the forces between charges is the Coulomb’s law. This law states that the force between two charges q 1 and q 2 is proportional to the charges and inversely proportional to the square of the distance between them. Or mathematically rF1221221rqqK (4.1) Where K is the proportionality constant that depends on the medium in which the particles are located, r 12 is the distance between the particles and r is the unit vector in the direction of line joining q 1 and q 2 . The value of K41 where is the permitivity of the medium. Permitivity of a medium can be expressed as 0 k where k is the dielectric constant or the relative permitivity of the medium and is the permitivity for 0 33

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34 free space. Permitivity of free space has been experimentally found to be equal to farads/meter. 121085.8 0qED.SDD.E The electric field intensity E, at a point in the medium, is defined as the force acting on a unit charge where the unit charge is so small that it does not disturb the properties of the field. Or mathematically, 00qLimF (4.2) Electric Displacement D is defined as E (4.3) The integral form of Gauss law states that the surface integral of the normal component of the electric displacement over a closed surface is equal to the charge enclosed inside the surface. This can be written as qdS The differential form of the Gauss’ Law states that the divergence of the electric displacement at a point is equal to the charge per unit volume at that point. Mathematically Gauss law can be written as: v (4.4) where is the volume charge density (charge per unit volume) at that point. v Electric potential ‘’ at a point in space is the amount of work done to move a point charge from that point to a point of zero potential. Zero potential is arbitrarily sometimes chosen to be at infinity. Mathematically it can be shown that (4.5)

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35 From the above equation it can be seen that, electric field at any point can be determined if is known at every point in space. Hence, by using equation 4.2, force on the charged particle can be computed. Analytical solutions can be obtained for some simple problems as demonstrated in the following example. Electric Field Strength in the Dielectric of a Capacitor Parallel plate capacitor configuration is being investigated with 2 dielectric slabs, as a special case of the n dielectric slabs described by Dutta [9]. These dielectric slabs are sandwiched between two metal plates having the potential V b and V t as shown in figure 4.1. The dielectric slabs have the charge density , relative permitivity and thickness where i varies from 1 to 2. Assuming that the parallel plates have very large linear dimensions as compared to the gap between them, the electric field between them can be considered as one dimensional, by neglecting the fringe effects. Figure 4.1 illustrates such a configuration. i ik id Figure 4.1 The capacitor model Since there is no electric field out of the top and bottom metal plates, the Gauss law for an arbitary control volume, as shown in figure 4.1, gives: 02211 tddb (4.6) Where

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36 b = Charge per unit area at the surface of the bottom electrode t = Charge per unit area at the surface of the top electrode Let the top electrode has a potential V t and the bottom electrode has V b . The electric field as a function of distance x from the bottom plate is 011 K x Ebx where 0 1dx . And 021211 KdxdEbx where d 21dx . (4.7) Integrating equation 4.5 for the one dimensional case, we get EdxV so, dxtbdxEVV0 where d 21dd (4.8) Substituting the value of E x from the equation 4.7 and integrating, we get 221122112102212021210KddKdKdKdKdbtVbV (4.9) or, 21010222212210121221111KdKdKddKdKdtVbVb (4.10) Hence the electric field can be obtained by substituting the value of the surface charge density in the equation (4.7) for the electric field. But these analytical solutions are possible only for simple geometries and simple boundary conditions. For more complex geometries, some numerical approach has to be

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37 adopted to solve for the unknown variables of the problem. Finite element methods have been proved to be very efficient and simple for solving the field problems. This method is described in the following sections. Introduction to Finite Element Analysis The finite element method is a mathematical technique used to obtain numerical solutions of partial differential equations of a wide range of problems. It is the most powerful numerical method available today for the analysis of complex structural, mechanical and electrical systems. The finite element method is used in the areas of solid mechanics, to analyze both linear and nonlinear systems. It is used to solve problems in both static and dynamic systems. Past work has shown the applicability of finite element methods to heat transfer, fluid mechanics, electrostatics, electromagnetism and other field problems. This versatility is the basic reason for the popularity of the method. For many physical problems, which are expressed in terms of partial differential equations, it is difficult to find an analytical solution for any complex geometry with complex boundary conditions. The finite element method addresses this difficulty by breaking the domain into small regions (simple geometric shapes) called finite elements. This process is called finite element discretization. The solution within each element can be represented by a function much simpler than that required for the entire region [10]. The elements are joined together mathematically by enforcing conditions that make each element boundary compatible with each of its neighbors, while satisfying the region’s boundary requirements. This is achieved by assigning the material properties and the governing relationships over these elements and expressing them in terms of unknown values at element corners. The algebraic equations for all elements are assembled to achieve a system of equations for the model as a whole. Solving these

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38 equations gives us the approximate behavior of the continuum. With a sufficiently large number of elements, highly accurate solution can be obtained over the complete region [10]. With the advances in computer technology and CAD systems, complex 3-D geometry can be created and analyzed. In the mean time, commercial finite element softwares have been developed (such as Patran, Nastran, Ansys, Abaqus and I-DEAS) which can perform variety of analyses. An understanding of the basic principles involved in the finite element formulation of the electrostatics is essential to interpret the results obtained from such an analysis. Hence, an extensive literature survey was performed to understand the finite element formulation of electrostatic analysis (along with the boundary conditions). The finite element formulation of the electrostatic analysis is described in the following section. Finite Element Formulation of Electrostatic Analysis The objective is to obtain a finite element formulation for the electrostatic analysis, which reduces the differential equations into a set of linear algebraic equations. For simplicity, finite element formulation of only 2D electrostatics is carried out. It can easily be extended to 3D analysis. For this, consider the equation 4.4 vD. Substituting equation 4.5 in the above equation, we get 0.v (4.11)

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39 The above equation is known as the Poison’s equation for electrostatics. If v =0 and the medium is assumed to be isotropic, i.e. is constant, then the equation 4.11 reduces to the well know Laplace equation, 02 (4.12) In two dimensional case the Poison’s equation can be written as: 000vykyxkx (4.13) with some specified boundary conditions on the boundary S described later in the chapter. Weak Form There are many ways of formulating the finite element method for the equation 4.13, like the Potential Energy or the Rayleigh-Ritz method, Variational method and the Galerkin’s method. Among these methods, Galerkin’s method uses the set of governing equations in the development of an integral form. It is usually presented as one of the weighted residual methods [10]. Using the Galerkin’s approach described by Chandrupatla and Belegundu [10], we have 000dAdAykyxkxAAv (4.14) is constructed from the same basis functions (interpolation functions) as those used for and satisfying =0 wherever the Dirichlet’s (explained below) boundary conditions are applied. From the definitions of the electric displacement we have xkDx0 and ykDy 0 (4.15)

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40 Using the modified divergence theorem [7] and equation 4.15, the equation 4.14 can be written as, AAvSndAdAyykxxkdSD000 (4.16) where D n = D x n x + D y n y = D.n is the normal electric displacement along the unit outward normal, which is specified by boundary conditions. All the boundary conditions related to the electrostatics problem are discussed in the next section. The characteristic of the finite element method is to discretize the entire domain into sub domains of simple geometries (like triangles and quadrilaterals in 2D case). So, discretizing the domain into e elements, we can write, for any function f(x, y), AeAedAyxfdAyxf),(),( (4.17) Assuming some interpolations within each element for the unknown and introducing the isoparametric relations for two dimensional elements, as described by Bathe [11], we can write eTN (4.18) where [N] is the matrix containing the shape functions, and e is a vector of the nodal values. For example, for a three node triangle element if the shape functions are N 1 , N 2 , N 3 then we can write 332211NNN or 321321NNN

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41 In the Galerkin’s method, since we use the same set of interpolations for that are used for, can be written as eTN (4.19) Moreover eTByx (4.20) where yNyNyNxNxNxNB321321 for a triangular element and similarly eTByx (4.21) using these equations, we can write: eeeTeTedABBk0 dAyxyxkdAyykxxkAA000 eeeTeK (4.22) where [K e ] is the element stiffness matrix and is given by eTeedABBkK0 (4.23) Boundary Conditions Let the domain of the problem be as shown in the figure 4.2

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42 ++++++++constant:S0v:S0D:Snd cv +++ Figure 4.2 Boundary conditions for 2D electrostatics. The boundary conditions (BCs) in electrostatics are classified as follows. Restraints These are represented on the boundary S v. These restraints are also called Dirichlet’s boundary conditions. These BCs are specified if a known value of voltage or potential is applied on the surface. Restraints are of two types, homogenous and non-homogenous Dirichlet’s boundary conditions. These are of the type 0 (4.24) where is a constant and is called homogenous if 0 0 is zero and non-homogenous otherwise. Since is specified on the surface is zero on S v , the first integral in equation (4.16) can be written as 0dSDSn for S = S v .

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43 Loads The BCs that are applied on the boundaries other than S v come under the Loads type. These are described below. Symmetry boundary conditions In this type of BCs, across the plane of symmetry, the normal component of electric displacement is zero. This is because; there can be no potential difference across the plane of symmetry, resulting in normal component of electric displacement to be zero. Hence, the first integral in the equation 4.16 becomes 0dSDSn for S = S d Specified value of charge per unit area If a constant charge per unit area is specified on the surface, as given by Gauss law [13] we can write, D n = on that surface. Using equation 4.17 and 4.19, the first integral in equation 4.16 becomes: eedTeTeSndsNdSDsr for S = S (4.25) where (4.26) srdsNd Specified value of charge per unit volume If a charge per unit volume is specified inside the elements then the integral can be written as: dAAv eTeeAvTeAvedANdAr (4.27)

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44 where (4.28) eAvdANr Constraints In addition to the Restraints and Loads there are other types of conditions that constrain the unknown potential . These conditions are called the Constraints. The electric field inside a perfect conductor is zero. Hence, any two points inside the conductor have the same potential. This condition is equipotential constraint. Most of the metals are good conductors of electricity and hence, has to satisfy the equipotential constraint condition. This equipotential constraint can also be called metallic constraint. The fact that the potential is same through out the conductor helps in the reduction of global equations as explained in the following problem. Parallel plate capacitor model with equipotential constraint. Figure 4.3 depicts a parallel plate capacitor model with two dielectrics and a conductor in between. There are two ways of approaching this problem. One is to simply treat the system as consisting of three dielectrics and equations are solved in the limiting case where the dielectric constant of the conductor tends to infinity. A more efficient way, which makes use of the equipotential fact, is described below. Figure 4.3 Parallel plate capacitor model with equipotential constraint

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45 Suppose the potential on the metallic conductor is some unknown value ‘V’. Since V is constant through out the conductor, the model can be reduced to a system of two capacitors each with only one dielectric. Assuming that there is no volume charge in either of the dielectrics, the electric field in the bottom and top capacitors can be written as: E 1 = (V b -V)/ d 1 And E 2 = (V-Vt)/d 2 Also 0 tmtmbb and 0 mtmb (Since it is only an induced charge) where the subscripts b, t, m represent bottom, top, metal respectively. 0 tb And 011 kEb , 022 kEt E 1 k 1 = E 2 k 2 Substituting the expressions for E 1 and E 2 and solving for V, we get 22111122kdkdVkdVkdVtb (4.29) Matrix Equation for Electrostatics From the equations 4.17 through 4.28, the equation 4.16 can be written as 0eeeTeedTeepTeKrr this equation can be written as RKTT (4.23)

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46 where {R} is a vector of loads assembled over all the elements. [K] is the global stiffness matrix which is also assembled over all the elements. and are global vectors of potentials and variation in potentials respectively. Equation 4.23 is true for all satisfying =0 wherever is specified. Hence the equation reduces to EEERK (4.24) where the superscript ‘E’ represents the familiar modifications made to [K] and {R} to handle the Dirichlet’s boundary conditions on S by the elimination approach. This elimination algorithm and the software implementation of finite element electrostatic analysis are discussed in detail, in chapter 5.

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CHAPTER 5 IMPLEMENTATION OF ELECTROSTATICS IN FINITE ELEMENT ANALYSIS SOFTWARE In the previous chapter, finite element formulation of the electrostatic analysis was presented. This chapter deals with the software implementation of the formulation. An object-oriented finite element analysis software has been developed at Design and Rapid Prototyping Laboratory in the department of Mechanical engineering at the University of Florida. This software is called JavaFemViewer. This object-oriented software is written in Java. Objective The objective is to add the electrostatic analysis capability to the JavaFemViewer software. Also, as is the case with other commercial software, implementing the pre and post processing capabilities (applying boundary conditions and displaying the results), greatly improves the applicability of the software. Hence, the objective is also to add these capabilities to the software that allows graphical display of electrostatic analysis results. The addition of these capabilities results in custom software that can be further improved and used for future purposes (like the dynamic simulation of charged particles in electric space). The following sections describe the implementation details. JavaFemViewer JavaFemViewer program is divided into two packages, the Viewer package and the JavaFem package. The Viewer package contains modules for performing tasks like 47

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48 pre and post processing, where as the JavaFem package is an analysis package. The following sections describe these two packages in some depth. JavaFem Package As described by Yu [12], the program architecture, for implementing the object-oriented framework is depicted using the object model in figure 5.1.The framework is designed as an assembly of modules that represent the building blocks of the finite element methodology. The modules were implemented as abstract classes whose member functions serve as the programming interface for this framework. The FE_Model class represents the overall framework and is composed of the modules: Nodes, array of Elements, array of Loads and array of Constraints (Dot in the figure 5.1 represents an array). This class provides the primary programming interface to the finite element model by providing member functions that can be used to access all the data stored in the model. Member functions of this class also provide capability to read a finite element model data from files in a variety of formats as well as to write the model data and results of an analysis to a file. The Nodes class contains nodal data such as the number of nodes in the model and their coordinate positions. The Loads and Constraints classes contain the information about the externally applied boundary conditions in the form of loads, constraints and restraints. The Element class contains the functions to compute the element stiffness matrix and also to compute the desired values like stress or voltage etc at specified points in the element. These functions are useful in plotting of the results. The Element class is composed of abstract subclasses A_Type and Interpolation classes, which are generalizations for all types of analysis and interpolation respectively.

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49 The abstract FE_Solver class contains member functions that implement algorithms that assemble and solve the global equations. This abstract class contains the abstract function solve (), which needs to be implemented in the child classes of this class. The different solvers available are skyline solver (linear static analysis), Eigen solver (modal analysis) and Newmark’s solver (dynamic analysis). The basic steps of the finite element methodology such as computing the element stiffness matrix, assembly of the global stiffness matrix, application of loads and boundary conditions are all accomplished using virtual functions of these abstract classes, as explained by Yu [12]. This ensures that as new elements, analysis capability, load types etc are added to the software, the part of the software that implements the above steps does not have to be modified. FE_Model FE_Solver Nodes Constraints Loads Elements A_Type Interpolation Figure 5.1 Object model for FE_Model class Electrostatic Element The A_Type class is the parent class for Electrostatics, SolidMechanics and HeatTransfer sub classes, as shown in figure 5.2. Each of these classes represents a particular analysis type. All the electrostatic analysis classes should be inherited from the abstract Electrostatics class. Figure 5.2 shows two such classes. EStatic2D class is used for performing electrostatic 2D analysis.

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50 As seen from the figure 5.2, the Estatic2D class implements the abstract function ebdb (), defined in A_Type class. This function is used in computing the element stiffness matrix. The Element class that is made up of EStatic2D class and I-2D class results in an electrostatic 2D element. For example EStatic2D class together with Fr_4node interpolation class gives electrostatic four-node element. A_Type int nmp,ngc; // variables to store the number of material and geometric constantsElement ele; // reference to the element class public abstract void ebdb(); // method to compute the elemental stiffness matrix SolidMechanics Electrostatics HeatTransfer EStatic2D EStatic3D Figure 5.2 Analysis type hierarchy Boundary Conditions The boundary conditions, described in chapter 4 are implemented in the Constraints and the Loads classes. The Constraints class implements the restraints and the equipotential constraints, whereas the Loads class handles all the other boundary conditions, described in chapter 4.

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51 As described in chapter 4, the assembled global equations have to be modified to account for the Dirichlet’s BCs. Following section deals with the algorithm implemented for the modified elimination method. This method is used to reduce the global set of equations by introducing the Dirichlet’s boundary conditions. Modified elimination method Typical boundary conditions that apply to most electrostatic problems are the Dirichlet’s boundary conditions. These boundary conditions were not accounted in the previous version of the program. These BCs are of the form described in (4.23). These BCs can easily be handled by following the modified elimination method; as described below. Consider the equation 4.25, which can be written as: nininnnjninijinnjfffxxxkkkkkkkkk.................................1111111 Suppose the Dirichlet’s BC be x i = a; then the first equation can be written as akfxkinijjjj11,11 Eliminating the i th row and i th column, we have akfakfakfxxxkkkkkkkkkninlilinlnnnjnljlnnj.................................1111ln1111 for il

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52 which is of order n-1. If more than one Dirichlet’s BC is specified then the load vector can similarly be modified to account for these BCs. Coupled degree of freedom constraints The equipotential constraint for the electrostatic analysis is described in chapter 4. Electrostatic analysis is a single degree of freedom problem with electric potential as the unknown degree of freedom. The nodes that are constrained to equipotential constraint have the same unknown value for the electric potential. Coupled degree of freedom constraint is a name given to more general categories of analyses, which can deal with multiple degrees of freedom. The nodes that are constrained to coupled degree of freedom constraint have the same value for the unknown variable for each degree of freedom. This is explained using a specific example as given below. Suppose the set of global equations are 543215432155545352514544434241353433323125242322211514131211fffffxxxxxkkkkkkkkkkkkkkkkkkkkkkkkk and suppose the coupled degrees of freedom are x 2 , x 4 and x 3, x 5 . This implies that x 2 = x 4 and x 3 =x 5 . Adding the second and fourth column and third and fifth column, the equations reduce to

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53 5355532545215143454324442141333533234321312325232242212113151321412111fxkkxkkxkfxkkxkkxkfxkkxkkxkfxkkxkkxkfxkkxkkxk Simple elimination of rows 4 and 5 will result in an asymmetric matrix. To maintain the symmetry of the matrix, the second and fourth and third and fifth rows are added. The system of equations reduces to: 5333555533325452343211513423254543232442442221141213151321412111ffxkkkkxkkkkxkkffxkkkkxkkkkxkkfxkkxkkxk This can be written as [K] m {x} m = {f} m where m denotes that these equations are modified. It can be seen that [K] m is symmetric. Though the algorithm is simple in case of a matrix stored in a two dimensional array, it becomes complex if the matrix is in skyline format. The description of algorithm for the skyline format is shown in the flow chart 5.3. As seen from the flow chart, the input to the algorithm is an array of constrained node numbers that are subjected to couple degrees of freedom constraint. From the flow chart it can be seen that the new number of equations reduces depending on the number of constraint nodes and number of degrees of freedom (ndof) for the problem. And the newMaxK matrix, which stores the positions of the diagonal elements in the skyline format for the reduced matrix can easily be formed. Then the newK matrix in the skyline format is assembled by adding the rows and columns of the constrained equations and

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54 correspondingly changing the load vector. Then the K, f and MaxK matrices are updated to the newK, newF and newMaxK matrices respectively. Inputconstrained nodes newNumofEqu = numofEqu (numCstNodes-1)*dof Compute newMaxK Compute newK K=newKf=newFMaxK=newMaxK Compute newF Figure 5.3 Flow chart for the coupled degrees of freedom implementation Viewer Package The Viewer package implements all the modules that help in pre and post processing. The object model for the Viewer package is shown in figure 5.4. A small description of each of the classes shown in the figure 5.4 is given below. The main class of the program is the Viewer class. It acts as an interface to the program by creating the ViewerFrame object, when the program is executed. ViewerFrame class creates the graphical user interface (GUI) for the program. It is made up of following type of classes. Pre Processing Classes ConstrainFrame, LoadFrame and SolutionFrame classes constitute the pre processing classes. These classes help in the construction of the FE_Model class by

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55 interactively helping the user to define the solution type, analysis type, loads and constraints in the finite element model. Post Processing Classes The ResultFrame class implements the post-processing capabilities. This class controls the various options of the display, like the color mode (either colored or grey scale), deformed mode (in case of solid mechanics). This class also helps in selecting the result type (voltage, field or electric displacement for electrostatic case) for display. Graphics Classes ViewCanvas and SideCanvas classes implement the Java3D graphics capabilities. These classes are similar to the Draw class (explained in chapter 3). These classes help in display, zooming, panning and rotating of the finite element mesh. The ViewCanvas class contains a reference to the MeshDisplay class, as shown in figure 5.4. This MeshDisplay class creates the mesh geometry and passes the geometry to the ViewCanvas class for display purposes. The SideCanvas class helps in the display of the color bar during the post processing. Viewer ViewerFrame ViewerCanvas SideCanvas MeshDisplay SolutionFrame ConstraintFrame LoadFrame FE_Model ResultFrame Figure 5.4 Viewer package class relationships

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56 Finite Element Model classes FE_Model class constitutes the finite element model as described in the earlier sections. Viewer class interacts with this class for the finite element model data. The following section deals with the capabilities of the JavaFemViewer software. JavaFemViewer The input for the program is called an oop file. This file format is given in Appendix A. This file can be created either manually (by typing the node coordinates, solution type, analysis type, loads and constraints) or by using Nastran (commercial FEA software) input file. Nastran input file format is chosen because of its similarity with the oop format. The coordinates of the nodes and element connectivity (which is an extremely tedious process to input manually, in case of a large mesh) can be read easily from the Nastran input file. All the other data like the solution type, analysis type, loads and constraints can interactively be inputted to the program using the capabilities provided by the pre-processing classes. Once the finite element model is created it can be solved and the results of the analysis can be displayed using the post processing capabilities. To verify the validity of the program the following examples were used as verification problems. Verification Problems Parallel Plate Capacitor Model 1 To verify the modified elimination algorithm, a finite element analysis is performed for the model shown in figure 4.1. The dimensions along with boundary conditions for the model are shown in figure 5.5. This model consists of two dielectrics,

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57 sandwiched between two parallel plates. The voltage applied on the plates is V t = 0 Volts and V b = 1000 Volts. 0.005 0.00050.0001Dielectric 2Dielectric 1 Vt = 0Vb=1000 Figure 5.5 Finite element model for the parallel plate capacitor system The model is solved using the JavaFemViewer software. Four-node quadrilateral electrostatic element is used. The dielectric constants k 1 and k 2 are assumed to be equal to 3. The electric field distribution obtained from such an analysis is shown in figure 5.6. Substituting V b -V t =1000, d 1 = 0.0005, d 2 =0.0001, k 1 =3, k 2 = 6 in equation 4.3we get, b = 5.45x10 6 0 and E 1 = 1.81x10 6 . E 2 = 9.09x10 5 . It can be seen that value obtained from the software matches with that of the analytical solution.

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58 Figure 5.6 Magnitude of electric field for the capacitor model Parallel Plate Capacitor Model 2 This model is same as described in Figure 5.5 except that a layer of charge per unit area ‘’ exists at the interface of the dielectrics. This analysis was performed to test the load computation module. The results of the analysis are shown below in figure 5.7.

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59 Figure 5.7 Y component of electric field for the capacitor model 2. Since 1 and 2 (the volume charge densities) are equal to zero and k 1 = k 2 , equation 4.7 can be written as: 011k bE and 022 k bE where E 1 and E 2 are the electric fields inside the dielectrics 1 and 2 respectively and b is the induced charge on the bottom plate. Using equation 4.8, we have dtbEdxVV0

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60 002211)( kdkdbbtbV V (5.1) substituting V b -V t =1000, d 1 = 0.0005, d 2 =0.0001, k 1 = k 2 = 3 and = 0.001 in equation 5.1, we get b = -1.224x10 -4 C. so E 1 = -4.61x10 6 N/C. and E 2 = 3.3057x10 7 N/C. It can be seen from figure 5.7 that the results obtained from the software match with that of the results obtained analytically. Parallel Plate Capacitor Model 3 To verify the equipotential constraint algorithm, a finite element analysis of the model shown in fig 4.3 is performed. The dimensions along with the boundary conditions are shown in figure 5.8. .03 0.004Dielectric 2 Vt = 0 0.006Dielectric 1Vb=1000 Metal Figure 5.8 Parallel plate capacitor model 3 As seen from the figure, the model consists of two dielectric materials with a metallic plate in between. The Dirichlet’s boundary conditions are applied on the top and

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61 bottom plates, as shown in the figure. In addition to these restraints, the equipotential constraints are applied to the model, to account for the metallic plate. This implies that the potential inside the entire metal is same through out. The problem is analyzed using the JavaFemViewer software and the results are shown in figure 5.9. Figure 5.9 Results of model 3, voltage distribution Substituting V b =1000, V t =0, d 1 = 6, d 2 =4, k 1 = 5, k 2 = 2.5 in equation 4.29, we get V = 571.43 volts, where V is the potential on the metallic plate.

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62 The value obtained from the software (from the out put file) is 571.41 volts. Hence, it can be seen that the value of voltage obtained analytically match with that of the value obtained from the software. The next chapter describes the fundamental issues encountered in the ESFF process. It also describes the mathematical models developed for the diffuse printing and the edge growth problems. This is followed by a description of the finite element analysis of these models and interpretation of the results.

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CHAPTER 6 ANALYSIS AND RESULTS As seen from chapter 3, the fundamental problems of the ESFF process, needs to be solved before it can become a viable manufacturing technique. The problems associated with the edge growth and the diffuse printing are challenges in obtaining good quality prints. In this chapter, mathematical models are developed for these problems. Then a finite element analysis is performed on these developed models to understand the cause of the problems. The following models are analyzed to study the cause of the above mentioned problems: Edge growth model Pattern model Transfer model Edge growth model helps in understanding the reason behind excessive deposition of toner near the edges when compared with the solid areas. This model also helps in understanding the development process (described in chapter 2). The pattern model shows the use of patterns in reducing the edge growth effect. The transfer model helps in understanding the transfer process (described in chapter 2). This model also helps to obtain an insight into the reason behind the spreading of toner powder near the solid edges. The following sections describe these models in detail. Each analysis starts with the problem statement, the finite element model along with the boundary conditions and assumptions made in the model followed by discussion of results. 63

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64 Edge Growth Model Problem Statement From the inspection of the prints obtained from the rapid prototype machine, it was noticed that the edges were growing faster than the solid areas (interior part). Following set of analyses were carried out to study the cause of the problem and some patterns were analyzed which help in achieving uniform prints, as described in the following sections. Finite Element Model The actual development process is shown in figure 2.8. As explained in chapter 2, the development process consists of a photoconductor drum, development drum and a toner cartridge. The photoconductor drum has a thin layer of insulation which contains charged and discharged regions (in the shape of cross sectional image). The toner powder is negatively charged with a charge per unit volume of -7.058 C/ m 3 [9]. The developer roller or the development drum is connected to a DC biased AC potential. To study the electric field distribution during development process, a simplified model as shown in figure 6.1 is analyzed. The assumptions made in the model are 1) The photoconductor drum and the development drum are parallel and the curvature effects are not taken into account. 2) The magnetic forces in the development process are not modeled. 3) The DC biased AC potential on the developer drum is modeled as DC potential V equal to -500 Volts. 4) Dynamics of the toner particles are not taken into consideration.

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65 y Char g e p er unit volume Develo p ment Roller ( at V= -500 ) N o Char g e x Photoconductor Drum ( Grounded ) Toner Insulation Figure 6.1 Simplified model for development process As shown in the figure 6.1 the model consists of a grounded photoconductor drum, a layer of insulation, negatively charged toner particles and the developer roller. The boundary conditions for the model are also depicted in figure 6.1. The coordinate system for the model is shown in figure 6.1, with the origin at the center of the insulation. As seen from figure 6.1, the photoconductor drum is grounded where as the voltage on the developer roller is -500 volts. The charge on the toner powder is assumed to be equal to 7.058 C/ m 3 , as given by Schein [3]. To study the edge growth process, the insulation on the positive x direction, has a layer of charge per unit area . The insulation on the negative x direction is discharged. The discharged region represents the place where there is image formation. This can be explained from the results of the analysis. As given by Schein [3], the amount of charge per unit area is assumed to be equal to x10 -4 C/m 2 . In the finite element model that was created, the thickness of the insulation was taken to be 10 microns and that of toner layer as 30 microns. The results of the analysis are shown in figure 6.2.

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66 Figure 6.2 Y component of the electric field for the pattern less model. Figure 6.3 shows the nature of field that exists at the interface. This graph is plotted between the distance of the nodes (in meters) that are along the interface and the Ey ( y-component of electric field in N/C) component. The value of Ey at the x equal to -4.8x10 -3 is equal to 3.6x10 7 , which is a positive value. The origin corresponds to the edge of the solid region. From figure 6.3 it can be seen that there is a positive value of electric field on the discharged regions of the photoconductor insulation. Since the toner particles are negatively charged they move in the opposite direction of the electric field. Hence the development occurs at the discharged regions. The converse is true on the charged

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67 regions of the insulation. The positive y component (figure 6.2) inside the insulation (white color) does not affect the toner deposition as this lies inside the insulation. Edge Growth-1.20E+11 -9.00E+10 -6.00E+10 -3.00E+10 0.00E+00 3.00E+10 6.00E+10 -4.80E-03 -4.10E-03 -3.40E-03 -2.70E-03 -2.00E-03 -1.30E-03 -6.00E-04 1.10E-04 8.70E-04 1.63E-03 2.38E-03 3.14E-03 3.90E-03 4.66E-03 X-Axis ( Node distance from Origin) YAxis (Ey) Figure 6.3 The y component of electric displacement along the interface. This figure also shows the cause of the edge effect. There is a sudden increase in the value of the Ey component near the edge. This results in a greater deposition of the toner powder near the edges. The strong y component at the solid edge clearly indicates the edge growth formation in the development process itself. This is because of the abrupt change from a huge discharged area (solid area) to a charged area. To minimize the edge growth effect patterns are used and the analysis is carried out in the following sections. The Pattern Model As seen from the previous analysis, there is excessive deposition of the toner powder near the edges of the solid area during the development process. This affects the

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68 dimensional accuracy of the part being manufactured. To minimize the edge effect, solid area represented in the form of patterns is printed instead of the entire solid area. The following set of analyses show the use of patterns in minimizing the edge growth effect. In this model the huge discharged area (that represents the image) is broken down to alternate charged and discharged regions, following a pattern. The model depicted in figure 6.4 is studied. Figure 6.4 Alternately charged and discharged regions for the pattern model. As the current software is limited to 2D analysis, only patterns that are parallel to the coordinate axes can be analyzed. The boundary conditions for the model are same as that of the edge growth model, except that this model has an alternate regions of charge per unit area, as shown in figure 6.4. The results of the analysis are given in figure 6.5. From the figure it can be seen that there is an overall increase in the amount of deposition as the electric field strength is greater in the pattern case than that for the edge growth model. The increase in the electric field implies an increase in the force acting on the negatively charged toner powder, which results in an increase in the toner deposition. Also, a comparison of figure 6.5 and figure 6.3 shows that the edge growth is also minimized. So, printing patterns is likely to produce a more uniform print than without the pattern. From the figure 6.5, it is difficult to understand the nature of electric field inside the pattern because of lack of sufficient number of data points. To study the local behavior of electric field within a pattern, a similar analysis with more number of nodes

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69 in the pattern region was performed. The model is shown in figure 6.6. This model is a zoomed in version of figure 6.4 for a single pattern. The boundary conditions are same as for the figure 6.4. The results of the analysis are shown in figure 6.7. The figure is obtained by plotting the y component of electric field with the x distance of the nodes on the interface from the origin. From the figure it can be seen that there is symmetry in the field pattern. Pattern Model-2.00E+11 -1.50E+11 -1.00E+11 -5.00E+10 0.00E+00 5.00E+10 1.00E+11 -4.80E-03 -4.10E-03 -3.40E-03 -2.70E-03 -2.00E-03 -1.30E-03 -6.00E-04 1.10E-04 8.70E-04 1.63E-03 2.38E-03 3.14E-03 3.90E-03 4.66E-03 X-Axis ( Node distsance from Origin) Y Axis (Ey) Figure 6.5 Y component of electric displacement for the pattern model

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70 Figure 6.6 Pattern model zoomed in. Pattern Model (zoomed in)-1.40E+11-1.20E+11-1.00E+11-8.00E+10-6.00E+10-4.00E+10-2.00E+100.00E+002.00E+104.00E+106.00E+10 0.00E+00 2.00E-05 4.00E-05 6.00E-05 8.00E-05 1.00E-04 1.20E-04 1.40E-04 1.60E-04 1.80E-04 2.00E-04 2.20E-04 2.40E-04 2.60E-04 2.80E-04 3.00E-04 3.20E-04 3.40E-04 3.60E-04 3.80E-04 4.00E-04 X-Axis Distance of nodes from origin (m)Y-Axis Ey (N/C) Figure 6.7 Y component of electric displacement for the pattern model (zoomed in). The next section deal with the transfer model that helps in understanding transfer process and gives an insight into the cause of diffuse printing.

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71 Transfer Model Problem Statement From the inspection of the prints obtained from the electro photographic rapid prototyping machine, it was found that there was diffusion of toner powder near the edges of the solid part. This is termed as diffuse printing. From the definition of diffuse printing it is clear that it affects the dimensional accuracy of the part being manufactured. The following sets of analyses were performed to understand the transfer process that gives an insight into the cause of diffuse printing. Finite Element Model The transfer process, as explained in chapter 2, is shown in figure 6.8. In the transfer process, the negatively charged toner particles on the photoconductor drum are transferred to the build platform (deposition platform). A high potential is applied on the build platform to attract the toner powder. As seen from the figure, at any given instance, transfer happens only in a small region shown as the region of interest in the figure. Since the dimensions of area of interest are so small, the actual process can be simplified and modeled, without considering the curvature effects as shown in the figure 6.9. Photosensitive Drum Flexible Deposition Platform Region of Interest Figure 6.8 Transfer between photoconductor drum and the deposition platform.

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72 Plane of Symmetry Photoconductor Drum Air Build Platform Printed Toner Insulation Fresh Toner Figure 6.9 Simplified model for transfer model The model consists of photoconductor drum, insulation, a layer of fresh toner powder, previously deposited toner powder (Printed Toner) and the build platform. The dimensions of the insulation and the fresh toner are assumed to be 20 microns. The air gap between the fresh and the printed toner powder is assumed to be 10 microns. The assumptions made in the model are as follows: 1) Since the region of interest in figure 6.8 is extremely small, the transfer process can be modeled as parallel plate configuration for all practical purposes. 2) The dimensions assumed are only approximate values and these values are used to get an insight into the nature of field distribution and not for exact field values. 3) A 2D model is created assuming that the properties do not change in the direction perpendicular to the plane of figure 6.9. 4) Since the transfer process is symmetric about the plane of symmetry, as shown in figure 6.9, only half the transfer process is modeled.

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73 The boundary conditions for the diffuse print model are shown in figure 6.10. As seen from the figure, the photoconductor drum is grounded whereas a voltage ‘V’ is applied on the build platform. The charge per unit volume in the fresh toner powder is assumed to be equal to 7.058 C/ m 3 . On the exposed parts of the photoconductor insulation, a value of charge per unit area ‘’ is applied, as shown in figure 6.10. The residual charge per unit volume on the printed toner is denoted as v. The parameters that are studied to understand the diffuse printing process are: 1) Height of the printed toner ‘h’ 2) Voltage on the build platform ‘V’ 3) Charge density on the printed toner powder ‘ v ’ Charge per unit volume Ground Charge per unit area h Charge per unit Volume v Voltage V Figure 6.10 The boundary conditions for the transfer model The following sections present the effect of these parameters on the transfer process which gives an insight into to the cause of the diffuse printing. Effect of Height of Print For the model shown in figure 6.9, a voltage V=1000 volts was applied. Charge density in the fresh toner is assumed to be equal to -7.058 C/ m 3 . The charge density on the printed powder is taken as v =0 (printed toner is assumed to be completely discharged). The charge per unit area on the part of photo conductor drum where there is

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74 no toner is assumed to be = -1x10 -4 C/m 2 . The height of the printed toner powder was taken as h=0.1 mm. The results of the analysis are shown in figure 6.11. As seen from the figure there is a positive y component of electric field in the air gap above the printed toner powder. The value of E y , as found from the output file is 2.04x10 5 . The negatively charged toner particles move in the opposite direction of the electric field. Hence, the positive E y results in the transfer of the toner powder onto the previously printed toner powder. From the figure 6.11, it can also be seen that the E y in the regions B is greater than that in the region A. This is because of the negative charge per unit area on the insulation in the region B. So, if there is a negative E x component at the edge of the fresh toner, then the toner particles move in the positive direction and then deposit on the build platform because of the positive E y component. This will result in the diffuse printing. Figure 6.12 shows the E x component.

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75 Figure 6.11 Y component of electric field for the case h= 0.1mm It can be seen from the figure 6.12 that there is sufficiently strong field in the positive x direction near the edge of the fresh toner powder. But the E x component is in the positive direction. The negatively charged toner particles move in the opposite direction of the field. This shows that there should not be any diffuse printing which is contrary to the experimental findings. A similar analysis was carried out for the case of h=0.3mm and the value of E y in the air gap is equal to 9.042x10 3 which is much less than the value obtained for h=0.1mm. This results in decrease in the deposition of the toner powder on the previously printed toner. This is in agreement with the experimental

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76 findings (courtesy Dutta [9]). The value of E x is found to be equal to 2.015x10 7 , which is practically same as the value obtained for the case of h=0.1mm. This shows that there is no significant effect of height on the diffuse printing. Figure 6.12 X component of electric field for h = 0.1 mm. The following sections deals with the effect of charge per unit area and the voltage on the transfer process. Effect of Voltage on Diffuse Printing For the model shown in figure 6.9, a voltage V=5000 volts was applied. Charge density in the fresh toner is assumed to be equal to -7.058 C/ m 3 . The charge density on

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77 the printed powder is taken as v =0 (printed toner is assumed to be completely discharged). The charge per unit area on the part of photo conductor drum where there is no toner is assumed to be = -1x10 -4 C/m 2 . The results for the region of interest as shown in Figure 6.13, are given in table 6.1. Region 1 Region 2 Region 3 Figure 6.13 Regions of interest in the diffuse print model Table 6.1 Electric displacement values for the two cases of voltages V=1000 and V=5000 Dx Dy V=1000 4.977E-06 3.122E-05 Region 1 V=5000 4.980E-06 3.526E-04 V=1000 6.399E-02 1.750E-02 Region 2 V=5000 6.401E-02 1.772E-02 V=1000 1.733E-03 1.139E-02 Region 3 V=5000 1.734E-03 1.159E-02

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78 And the above values are the average values over the regions. As can be seen from the table, the increase in the electric displacement values is not considerable compared to the voltage increase. This analysis shows that increasing the voltage on the build platform does not affect transfer process or the diffuse printing. Effect of Charge Density The above analysis was performed assuming that the printed toner was completely discharged, i.e. assuming that v is equal to zero. In this analysis a value of -1x10 -6 C/m 3 (Dutta [9]) was assumed. The results of the analysis show that there was no significant change in the x and y components of the electric field. So, from the analysis for the diffuse printing it was seen that none of the parameters affect the diffuse printing to a considerable extent. Also, the analysis carried out show a negative result that there should not be any diffuse printing. But this is contrary to the experimental fact. This shows that one of the assumptions made in the analysis is incorrect. From the edge growth model, it is seen that the edges are much thicker than the solid areas during the development process itself. So, this has to be taken into account before modeling the diffuse print problem. The fresh toner in the figure 6.9 should be modeled with a thicker edge. The negative charge per unit area on the photoconductor insulation (figure 6.10) will repel the excessive powder on the edge and result in diffuse printing. Since the printing patterns minimize the edge growth problem, it also reduces the diffuse printing resulting in a better quality prints. The next chapter summarizes the thesis work and suggests some future work that is yet to be implemented to improvise the ESFF process.

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CHAPTER 7 CONCLUSIONS AND FUTURE WORK The current work mostly concentrated on the development of software to automate the ESFF process and to study some of the fundamental problems associated with the ESFF process. The following section deals with the conclusions made from the previous chapters. Conclusions Literature survey was carried out for the slicing algorithm and an algorithm suitable for the ESFF machine was implemented as described in chapter 3. A graphical user interface was developed that implements these algorithms. The software is tested for a number of cases using the ESFF test bed. After a close inspection of the prints obtained from the ESFF test bed, the challenges during printing process (diffuse printing and edge growth problems) were realized. To study the cause of these problems, finite element electrostatic analysis was carried out. In this regard, extensive literature survey was performed on the electrostatic analysis to understand various boundary conditions in electrostatics. Finite element approach was formulated for the electrostatic analysis. Chapter 4 discusses the finite element formulation of electrostatic analysis. This electrostatic analysis module was implemented and added to the JavaFemViewer software. Basic Pre and Post processing capabilities were also added to the software. This software was tested using the various verification problems. The results obtained were compared with the analytical solutions. Mathematical models were developed for the diffuse printing and the edge growth problems. Effect of different parameters on the 79

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80 diffuse printing was studied by performing the finite element electrostatic analysis. Also an insight into the development process was obtained by FEA analysis on the edge growth model described in chapter 6. The problems were shown to be minimized by the use of patterns from the results of the pattern model. The following section suggests some future work for improvement of the ESFF process. Future Work 1) The slicing algorithm was implemented for CAD models represented using STL format. This can be extended to the slicing of models represented in other formats like the boundary representation (BREP), to obtain a more accurate cross sectional information. 2) Finite Element formulation was carried out for only 2D analysis. This can easily be extended to 3D case and can be implemented in the JavaFemViewer software. 3) Only 2D analyses were carried out for the edge growth and diffuse print problems. This can be extended to 3D analyses once the 3D capability is added to the JavaFemViewer software. 4) The printing of the patterns can be tested using the ESFF test bed. The method for printing the patterns using the SolidSlicer program is already implemented. 5) Only static analyses of the charged particles were considered. It can be extended to a dynamic analysis which helps in simulation of charged particle motion. A static analysis can be carried out to obtain the initial conditions which can be fed into the dynamic analysis solver (similar to the Newmark’s method [11]).

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APPENDIX A INPUT FORMAT FOR JAVAFEMVIEWER SOFTWARE Total number of nodes Node number x y z . Number of element group Interpolation type total number of element Element number connectivity mpID gcID . Number of solution Name of solution type Solver Data SkylineSolver : no data NewMarksSolver : line1-mass density, damping coefficient, alpha, delta, time step, total time line2state, value, value (initial condition for displacement) line3state, value, value (initial condition for velocity) line4state, value, value (initial condition for acceleration) (if state is 0, then has constant value 81

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82 state has number, then the number represents the previous solver-no value ) EigenSolver : required number of eigen vector Number of A_Type Name of A_Type Number of associated Egroup EGroupID Number of material property set Material property set Axisymmetric : Young's Modulus, Poisson's Ratio Plane_stress : Young's Modulus, Poisson's Ratio ThreeD_stress : Young's Modulus, Poisson's Ratio, Thermal Coefficient Beam : Young's Modulus Htransf : Thermal Conductivity Number of geometric constant set Geometric constant set Axisymmetric : Thickness Plane_stress : Thickness ThreeD_stress : Thickness Beam : Moment of Inertia, C/s Area of Element, C/s height Htransf : Thickness Number of load type Load type number{1:Concentrated load, 2:Uniform load, 3:Linear load, 4:Thermal load} 1. Concentrated load

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83 Number of loaded node Loaded node number load(ndof) . 2. Uniform load Number of loaded element Loaded element number element type s, t value load(x,y,z) moment(x,y,z) . 3. Linear load Number of loaded element Loaded element number element type s, t value load(x,y,z) moment(x,y,z) load(x,y,z) moment(x,y,z) . 4. Thermal load Reference temperature Present temperature Number of constraint node Node number constraint(ndof)

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APPENDIX B SCENEGRAPH FOR JAVAFEMVIEWER SOFTWARE Figure 1.B Scenegraph for the JavaFemViewer software This figure shows the Scenegraph that is implemented in the ViewCanvas class (chapter 5). As shown in figure, the meshShape Shape3D contains the Geometry node for which the finite element mesh is passed. 84

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LIST OF REFERENCES [1] Kochan D., Solid Freeform Manufacturing: Advanced Rapid Prototyping. Elsevier, Amsterdam, 1993. [2] Zhang Hongxin, "Design and Implementation of a Test Bed for Electrophotographic Solid Freeform Fabrication." MS thesis, University of Florida, 2001 [3] Schein L. B., Electrophotography and Development Physics. Springer, Berlin, 1988. [4] Gokhale Samit, "Study and Implementation of Electrophotographic Solid Freeform Fabrication and Charge Measurement Apparatus." MS thesis, University of Florida, 2001. [5] Gan G.K. Jacob, Chua Chee Kai, Tong Mei, “Development of a new rapid prototyping interface.” Computers in Industry, 39, 1999, pp. 61-70 [6] Y.S. Liao and Y.Y Chiu, “A New Slicing Procedure for Rapid Prototyping Systems.” International Journal of Advanced Manufacturing Technology, 18, 2001, pp. 579-585 [7] X. Chen, C. Wang, X. Ye, Y. Xiao and S. Huang, “Direct Slicing from PowerSHAPE Models for Rapid Prototyping.” International Journal of Advanced Manufacturing Technology, 17, 2001, pp. 543-547. [8] William R. Smythe, Static and Dynamic Electricity. Hemisphere, New York, 1989. [9] Dutta Anirban, "Study and Enhancement of Electro-photographic Solid Freeform Fabrication." MS thesis, University of Florida, 2002. [10] Tirupathi R. Chandrupatla, Ashok D. Belegundu, Introduction to Finite Elements in Engineering. Prentice Hall, Eaglewood Cliffs, 2000. [11] Klaus-Jurgen Bathe, Finite Element Procedures. Prentice Hall, New Delhi, India, 2001. [12] Lichao Yu and Ashok V. Kumar, “An Object-Oriented Modular Framework for Implementing the Finite Element Method.” Computers and Structures, 79, 1998, pp. 919-928. 85

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86 [13] Qiushi Chen and Adalbert Konrad, “A Review of Finite Element Open Boundary Techniques for Static and Quasi-Static Electromagnetic Field Problems.” IEEE Transactions on Magnetics, 33, 1997, pp. 663-676. [14] Jonassen Niels, Electrostatics. Chapman & Hall, New York, 1998. [15] Sergey Polstyanko and Jin-Fa Lee, “Adaptive Finite Element Electrostatic Solver.” IEEE Transactions on Magnetics, 37, Sep 2001, pp. 3120-3124.

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BIOGRAPHICAL SKETCH The author was born on May 20, 1979, in Bangalore, India. In 2000, he graduated with a Bachelor of Technology in the Department of Mechanical Engineering from the Indian Institutes of Technology, IIT Madras, India. He entered the Master of Science program at the University of Florida in Fall 2000. 87