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A network approach for thermo-electrical modelling : from IC interconnects to textile composites Chiun-Shen, Liao 2010-09-16

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A NETWORK APPROACH FORTHERMO-ELECTRICAL MODELLING:FROM IC INTERCONNECTS TO TEXTILECOMPOSITESbyCHIUN-SHEN LIAOB.Sc., Chung Cheng Institute of Technology, Taiwan, 1990M.Sc., Lehigh University, PA, United States, 1996A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(OKANAGAN)August 2010c© Chiun-Shen Liao, 2010AbstractSimulations of the temperature distribution in regular IC interconnect networks and textilecomposites are achieved by means of an analytical-symbolic approach. Analytical heatingsolutions along each interconnect can provide accurate solutions with far fewer nodes thannumerical solutions. To simulate the case of textile composite, the textile composite is mod-elled by a network of interconnects. The necessary input information is contained in netlistfiles, similar to the SPICE (Simulation Program with Integrated Circuit Emphasis) input for-mat. Analytical solutions to the heat equation along each interconnect can provide accuracyand require the minimum number of symbolic network nodes. The LU decomposition of thesymbolic network scales as the cube of the number of nodes. Multiple evaluations, includ-ing iterating temperature-dependent thermal conductivity to achieve a self-consistent solution,scale linearly with the number of nodes and hardly affect the total solution time. Memory con-sumption, CPU time, and solutions of the new network calculation method compare favorablyto a finite element analysis using ABAQUS.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Summary of Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Network-Based Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Basic Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Thermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Electrical Analogy for Thermal Conduction . . . . . . . . . . . . . . . 142.2 Interconnect Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16iiiTable of Contents2.2.1 Textile Fibre Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Network-Based Simulation Concept . . . . . . . . . . . . . . . . . . . . . . . 212.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Therminator3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 The Analytical Solution of Networked Elements . . . . . . . . . . . . . . . . 233.1.1 Resistor-Capacitor Network . . . . . . . . . . . . . . . . . . . . . . . 233.1.2 The Format of Input Files . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Symbolic Network Analysis Solution . . . . . . . . . . . . . . . . . . . . . . 283.3 Iterative and Multiphysics Evaluation . . . . . . . . . . . . . . . . . . . . . . 323.3.1 On Convergence of Non-linear Problems in Therminator3D . . . . . . 353.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Basic Physical Model of Finite Element Analysis . . . . . . . . . . . . . . . . 414.1.1 Mesh Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Simulation with ABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Applications and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1 Basic Layout Examples: Verification of Therminator3D . . . . . . . . . . . . 485.2 IC Interconnects Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3 Non-Crimp Fabric Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Woven Fabric Composite Network . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Large-Scale Network Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 Evaluation of Effective Thermal Conductivity . . . . . . . . . . . . . . . . . . 805.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81ivTable of Contents6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86AppendicesA The Input Parameters of Weave Program . . . . . . . . . . . . . . . . . . . . . . 96B Modified Netlist and Coupling Data for Simulations . . . . . . . . . . . . . . . . 99C Using DOE in the Evaluation of Effective Thermal Conductivity of the Fabric withT3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102vList of Tables3.1 Therminator3D input file format and typical values. . . . . . . . . . . . . . . . 275.1 Material properties used in simulations adjusted by narrow-line effects. . . . . 525.2 Material properties used in simulations. . . . . . . . . . . . . . . . . . . . . . 525.3 Comparison of CPU time and memory required by Therminator3D and ABAQUS 575.4 Comparison of Therminator3D and ABAQUS CPU times for the heat analysisof a 4x4 metal non-crimp structure. . . . . . . . . . . . . . . . . . . . . . . . . 655.5 Comparison of CPU time and memory required by Therminator3D and ABAQUSon woven fabric materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.6 Comparison of CPU time required by Therminator3D and ABAQUS on LargeScale network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75C.1 Study factors and their levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104C.2 ANOVA results and effect estimates from minitab. . . . . . . . . . . . . . . . . 104viList of Figures1.1 An example of temperature-aware ASIC design flow. . . . . . . . . . . . . . . 52.1 Heat Transfer along a bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Diagrams of thermal resistance. (a) Series thermal resistance (b) Parallel ther-mal resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Resistor segment of interconnects . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Heat conservation model and capacitances . . . . . . . . . . . . . . . . . . . . 203.1 A net-based interconnect model with vias shown with an ‘X’. . . . . . . . . . . 243.2 A two-layer interconnect grid with vias showing approximate segmentation toconnected resistors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 EMR methodology diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Typical lateral thermal conductance model in IC Interconnects. . . . . . . . . . 283.5 Lateral thermal conductance for woven case (shown by the capacitor symbols) . 293.6 The distribution of problem matrix A (the 20 columns and 20 rows case withvia between each row and column). . . . . . . . . . . . . . . . . . . . . . . . . 313.7 The flowchart of Therminator3D thermo-electrical simulation . . . . . . . . . . 343.8 Fixed point iteration for a general function g(x) for the four cases of interest. . 363.9 Sample circuit 1 for Therminator3D convergence verification . . . . . . . . . . 373.10 Sample circuit 2 for Therminator3D convergence verification with infinitely-long resistors and substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39viiList of Figures4.1 Typical models of Finite Element Method (a) stitched unit cell of a NCF, (b) aclosed-form RVE of the same fabric. . . . . . . . . . . . . . . . . . . . . . . . 434.2 Typical finite element shapes and mesh points in one through three dimensions. 444.3 Comparison between the finite element and experimental results in a coupledthermo-electrical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 The layout of one wire simulation. . . . . . . . . . . . . . . . . . . . . . . . . 505.2 The temperature distribution of one wire simulation. . . . . . . . . . . . . . . 505.3 The layout of three-wire simulation . . . . . . . . . . . . . . . . . . . . . . . 515.4 The temperature distribution of three-wire simulation. . . . . . . . . . . . . . . 515.5 Four-by-four, two-layer grid with vias showing approximate segmentation intoconnected resistors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.6 Boundary conditions achieved by equivalent circuit source and resistors. . . . . 555.7 Temperature-dependent thermal and electrical conductivities of copper. . . . . 555.8 Interconnect temperature distribution for the network shown in Figure 5.5 . . . 565.9 Comparison of column 3 temperature distribution in Figure 5.5, calculated byTherminator3D and ABAQUS. . . . . . . . . . . . . . . . . . . . . . . . . . . 585.10 Interconnect temperature distribution for the network shown in Figure 5.5 withvias removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.11 Comparison of column 3 temperatures shown in Figure 5.5 with vias removed . 605.12 Comparison of column 4 temperature distribution in Figure 5.5 with vias re-moved, calculated by Therminator3D and ABAQUS. . . . . . . . . . . . . . . 605.13 An NCF represented by a two-layer interconnect grid with vias showing ap-proximate segmentation into connected resistors. . . . . . . . . . . . . . . . . 625.14 Interconnect temperature distribution for the network shown in Figure 5.13 . . 635.15 The row 3 temperature trajectory for the network in Figure 5.13 . . . . . . . . 645.16 A two-layer interconnect grid layout with vias and Joule heat generated oninterconnect 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66viiiList of Figures5.17 T3D implementation of NCF layout in Figure 5.16 . . . . . . . . . . . . . . . . 675.18 ABAQUS implementation of NCF layout in Figure 5.16 . . . . . . . . . . . . . 675.19 The comparison of all 8 wires between ABAQUS and Therminator3D withvias and heated wire 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.20 A woven fabric composites represented by a two-layer interconnect grid withvias showing approximate segmentation into connected resistors. . . . . . . . . 705.21 Yarn temperature distribution for the network shown in Figure 5.20, computedby Therminator3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.22 Yarn temperature distribution for the network shown in Figure 5.20, computedby ABAQUS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.23 The comparison of all 8 yarns between ABAQUS and Therminator3D on wo-ven case without vias and with heated yarn 5 . . . . . . . . . . . . . . . . . . . 735.24 The comparison of all 8 yarns between ABAQUS and Therminator3D on wo-ven case without vias and with heated yarn 7. . . . . . . . . . . . . . . . . . . 745.25 The layout of a 20 (columns) by 20 (rows) large scale network with vias (X)with a heated wire (red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.26 The layout of a 20 (columns) by 20 (rows) large scale network without viasand with a heated wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.27 The simulation results of Therminator3D for a 20x20 net. (a) Layout as Figure5.25. (b) Layout as Figure 5.26. . . . . . . . . . . . . . . . . . . . . . . . . . 785.28 The simulation results of ABAQUS for a 20x20 net. (a) Layout as Figure 5.25.(b) Layout as Figure 5.26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.29 The multiple boundary condition simulation results by Therminator3D for a18x18 net. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.30 Cause-effect diagram used in the DOE study. . . . . . . . . . . . . . . . . . . 81C.1 The layout used in the DOE study. . . . . . . . . . . . . . . . . . . . . . . . . 103C.2 Main effects plot for Ke f f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105ixList of FiguresC.3 Interaction effects plot for Ke f f . . . . . . . . . . . . . . . . . . . . . . . . . . 106xList of AcronymsAcronyms Definitions3D Three-Dimensional3DIC Three-dimensional Integrated CircuitANOVA Analysis of VarianceASIC Application-Specific Integrated CircuitBC Boundary ConditionCIR Channel Impulse ResponseCAD Computer-Aided DesignDC Direct CurrentDOE Design of ExperimentEMR Electromigration ReliabilityFE Finite-ElementFEA Finite-Element AnalysisFEM Finite Element MethodIC Integrated CircuitITRS International Technology Roadmap for SemiconductorsMTTF Mean Time to FailureNCF Non-Crimp FabricPCB Printed Circuit BoardsRC Resistor-CapacitanceRVE Representative Volume ElementxiList of AcronymsCFD Computational Fluid DynamicsSPICE Simulation Program with Integrated Circuit EmphasisTCAD Technology CADT3D Therminator3DVLSI Very-Large-Scale IntegrationULSI Ultra-Large-Scale IntegrationxiiAcknowledgmentsI am deeply grateful to many individuals and organizations. Without their support, this projectcould not be completed.First and the most, I would like to thank my supervisors, Dr. Andrew H. Labun and Dr.Abbas S. Milani, for their understanding, encouragement, support, enthusiasm, and guidance.Their helpful suggestions and patient conduction have gone through all my works in the pasttwo years and made this thesis possible. I will continue to be influenced by their rigorousscholarship, clarity in thinking, and professional integrity.I would also like to express my thanks to Dr. Deborah Roberts and the following advisorycommittee members: Dr. Peter Hallschmid, and Dr. Ahmad Rteil. Thanks also go to Dr.Edmond Cretu for his willingness to serve as my external examiner. I highly appreciate theirvaluable time and constructive comments on my thesis.I would like to thank my dear peers Nick Kuan-Hsiang Huang, Xian Jin, Xuegui Song,Ning Wang, and Yeyuan Xiao for all of their assistance, insight, encouragement and supportduring my two-year studied at the University of British Columbia Okanagan.Finally, special thanks go to my mother and my family for their patience, understanding andsupport over all these years. In particular, I wish to acknowledge my dear wife, Wei-Wei Yu,for giving invaluable and heartfelt support. Without their constant support and encouragement,all my achievements would not be possible.xiiiDedicationThis thesis is dedicated to my family . . .for all of their love and endless support.xivChapter 1IntroductionIn this chapter, background and thesis organization will be presented. The first part of thischapter introduces the background and motivation. Related literature papers are reviewed inthe second part. The third part illustrates the research goals of this study. The scope andorganization of the thesis are described in the final section of the chapter.1.1 Background and MotivationThis thesis applies a new fast network-based simulation methodology to solve coupled thermo-electrical problems. The solution algorithm was originally developed to calculate the tempera-ture distributions of interconnects due to Joule heating in microchips. This algorithm demon-strates a very fast calculation on large scale layouts. To extend the benefit of this algorithm,this thesis proposes to use the same algorithm to textile composite materials whose geometryis also network-based. The following paragraphs will illustrate the background and motivationof this research starting from interconnects of microchips to textile composite materials. Also,each chapter in this thesis is roughly divided into two applications: interconnects and textilecomposite materials.The International Technology Roadmap for Semiconductors (ITRS) illustrates that model-ing and control of thermal reliability mechanisms in circuit interconnects is one of the diffi-cult challenges in the semiconductor industry for the next decade [1]. The 2007 InternationalTechnology Roadmap calls for new computer-aided design (CAD) tools that would be usedas inputs to predict interconnect resistance as it increases due to temperature variation based11.1. Background and Motivationon wire length/current, and “to calculate local operating temperature, [including] the effects ofJoule heating in the circuit and elsewhere” [[1] p. 39]. Since the interconnects are the majorcomponents to connect with other components in a circuit, accurate temperature estimation ofinterconnect is very important in CAD verification of circuit behaviour and reliability.Previous studies on interconnects have focused on the electromigration reliability (EMR)verification fields with specific technologies. Electromigration verification is a prediction andfunctional evaluation on integrated circuit (IC). In the worst case, the electromigration ef-fect could cause the connection to eventually disconnect and result in failure of IC. In thereal world, ICs rarely fail due to electromigration since all designs and simulations are madeusing worst-case temperature assumptions. Those temperature assumptions prevent and re-duce the possibilities of failure due to electromigration effects. This kind of design is called“electromigration-aware design”. To evaluate the electromigration reliablity of a wire, the fol-lowing Black equation [2] is used to estimate the “Mean Time to Failure (MTTF)” of wires,MT T F = A(J−n)eEa/kT, (1.1)where A is a constant based on the cross-sectional area of the interconnect, J is the currentdensity (amperes per square meter), n is a scaling factor, Ea is the activation energy in eV, k isthe Boltzmann’s constant (eV K−1), and T is the absolute temperature in K.It states that current density and temperature are the main factors that affect electromigra-tion. The temperature is changing during operation of systems. It is a challenge to preciselypredict the temperature distributions and effects of MTTF on electromigration. Most electromi-gration verification studies on the reliability of a specific interconnect technology (e.g., [3, 4])are based on uniform temperature experiments . The temperatures of interconnects are of con-cern because of the low-thermal-conductivity dielectric, high-resistivity conductor, and highinterconnects stacks. All these factors tend to increase the temperatures of interconnects. Elec-tromigration reliability verification of high-performance integrated circuits requires a detailedinterconnect temperature simulation capability to accurately identify problematic interconnect21.1. Background and Motivationlayout configurations and operating conditions. Those new CAD tools developed on the 2007ITRS Roadmap could precisely predict the current behaviours and EMR reliability with thetemperature increased by the Joule heating. There is no need for worst case assumptions onthe temperature after the components of interconnect temperature simulations have been addedproperly. This thesis proposes a rapid symbolic, analytical, and network based approach forfull-chip interconnects reliability verification that calculates the detailed temperature distribu-tion along each interconnect. This very fast simulation tool is called “Therminator3D” (T3D).Most of the layouts of textile composite materials are similar to the layouts of intercon-nects in integrated circuits, except for those angular woven materials. I could apply the fastspeed simulation and network-based idea of “Therminator3D” into the simulations of textilecomposite materials starting from temperature estimations. Therminator3D has the potentialto extend to different mechanical properties.For textile material applications, composite materials are used widely in a broad range ofindustries to improve design properties such as lower density with higher stiffness and strength,lower effective thermal conductivities to block the heat, higher effective thermal conductivitiesto detect the real temperature instantly, etc. Composite materials can reach the desired mechan-ical properties by changing the reinforcement layouts, reducing the need for extra materials andfabrications.In textile materials which can be knitted, woven or braided, the fibre yarns and matrixmaterials are consolidated during manufacturing. A number of composite components in theaerospace, automotive, marine and construction industries are built of woven textile preforms.They can give excellent mechanical properties with very low weights, but their complex weav-ing fabrication processes occasionally make them expensive for particular composite manu-facturers. As an alternative, non-crimp fabrics (NCFs) have proven to provide fairly goodmechanical properties while maintaining low weight and providing cost effective solutionsthrough their simplified fabrication processes. NCF composites are fabricated from preformswith multiple layers of straight fibre bundles that can be oriented in different directions and31.2. Literature Reviewstitched together by a (warp) knitting procedure. The previous study by Dransfield et al. (1998)showed that out-of-plane fracture toughness and damage tolerance properties of these materialsare also improved through the stitching process [5].The layout of NCF materials is similar to that of printed circuit boards (PCB). The archi-tecture is layer by layer and wire-crossed fabrication. For woven materials, the layout is notidentical but similar to printed circuit boards. The woven structure can be interpreted into thesame layouts of interconnects with suitable twisted (angular) coupling values. The simulationof full-layout textile material properties in commercial CAD/FEM tools including the layoutdrawing, CPU time, and memories consumes several resources. The proposed method here isto employ “Therminator3D” for textile materials to reach more rapid multi-physics simulationswith low CPU time and memory.1.2 Literature ReviewConsideration of a complete physical interconnect model includes the configuration, such asthe set of layouts and materials, their microstructure and interfaces (e.g. Cu, Ta, and SiNx), andthe dimensions of interconnects; and operating conditions, such as local temperature, drivingcurrents, heat and electrical flux, and stress, etc. This complexity and the specific characteris-tics of individual process technologies require detailed Technology CAD (TCAD), includingFinite Element Analysis (FEA), to predict reliability for specific interconnect configurations[6]. For circuit design, however, it is necessary to have a CAD solution, and it can be analyzedbased on a network representation extracted from layout and process technology information.There are several models to predict the temperature of interconnects and apply the result toSystem-on-Chip in order to obtain more precise simulation results.Huang et al. (2004) proposed a ”compact thermal modeling for temperature-aware design”model [7]. They used the idea of “temperature-aware design” with a model reduction conceptapplied on every step of the application-specific integrated circuit (ASIC) design flow, as shown41.2. Literature Reviewin Figure 1.1. This design reveals the importance of temperature estimates on every stage ofCAD design. Without temperature evaluation on each stage, the function of ASIC will notperform as designed due to dubious temperatures. For thermal estimation of interconnects,they used a compact thermal model to estimate the temperature of each node in the RC circuitthat corresponds to a block. Then they treated the heat dissipation of each block as a currentsource, “hotspot”, connected to the corresponding node. This model shows good benefits for awide range of applications on the temperature simulation and model reduction evaluations.Figure 1.1: An example of temperature-aware ASIC design flow (Cited from [7]).Franzon el at. (2008) presented a CAD tool for a three-dimensional IC (3DIC) [8]. Theydeveloped a thermal extraction and modeling flow which provide sufficient solutions for 3DIC design. The thermal model included a factor for the effect of local metallization, that canmodify the thermal conductivity and anisotropic thermal conductivity. They also used modelreduction to determine the interest ports of their model. After modeling the layout, they ex-tracted the thermal information of the layout, then exported the extracted data to third-partythermal analysis software such as Pro-ENGINEER Mechanica. The third-party thermal analy-sis software performs the steady state thermal analysis. Their concepts of model reduction andlayout extraction are similar to the ones used in this study and very efficient for performing the51.2. Literature Reviewthermal analysis. However, they did not develop an integrated simulation program to avoid themigration inconsistency between extraction softwares, circuit analysis software and thermalanalysis software and, thus, to attain a faster and more practical simulation tool.Alam et al. (2003) proposed that operating conditions may be determined by rapid circuitsimulation and that a reliable prediction based on prior and detailed TCAD analysis can thenbe made by combining configuration and operating information using a simpler, parameterizedmodel [9–11]. They used their SySRel, a CAD tool, for interconnect tree-based reliability as-sessment and thermal-aware cell-based extraction. However, they also use third-party thermalanalysis software - ANSYS, a Finite Element Analysis solver, to verify and simulate the three-dimensional full-chip steady state and transient thermal behaviours. It could have some datainconsistency as the previous CAD tool.Walkey et al. (2004) used controlled sources to simulate a compact, netlist-based represen-tation of the thermal coupling problem [12]. In these two technologies of controlled source,sub-circuits separated self-heating and coupling effects or terms from the original circuits, toavoid a complex systems solution.There are several studies that used netlist-based methods to extract thermal and electricalnetlists and then performed thermo-electrical temperature simulation, including self-heating[13, 14]. Sz´ekely et al. (1997) used a relaxation method with coupled thermal and electricalsolvers [15]. They extracted a thermal and electrical netlists based on layout information ofthe circuit. Once the electrical and thermal netlists were extracted, their self-consistent electro-thermal simulation was executed. This concept of coupled thermal and electrical simulation issimilar to the implementation of Therminator3D. However, their simulation required longer runtime and computing resources, and was economically expensive since they used simulator forintegrated semiconductor structures by simultaneous iteration (SISSSI) under Cadence DFWII.Most small- and medium-size companies could not afford to use this system.Joule heating is the heat generated by current in IC circuits. It could cause the failureof circuits without treating. Shen (1999) studied the analysis of Joule heating in multilevel61.2. Literature Reviewinterconnects with two dimensional six-level interconnect stack. He did not mention the simu-lation method. However, he pointed out the interconnect of Joule heating became a reliabilityproblem [16].This research applies a new network approach, Therminator3D, to interconnect tempera-ture estimation [17]. This research compares the relative accuracy, speed, and memory re-quirements of T3D to the ABAQUS FEA software, and demonstrates the feasibility of thisCAD method for thermal and reliability analysis of large circuits. I also extend the conceptof interconnect simulation to the thermal properties of textile composites materials. In thisthesis, I only apply this algorithm to fully coupled thermo-electrical simulations and the futuregoal is to initiate other applications such as estimating mechanical properties using the samealgorithm.There are numerous works that are dedicated to mechanical, thermal and other physicalproperties of woven and non-woven fabrics. Bibo, Hogg and Kemp (1997) studied the elasticproperties of several non-crimp fabric materials under tension and compression [18]. Theycompared the elastic properties of four kinds of uni-directional composite materials and foundthe NCF materials could provide competitive advantages. Drapier and Wisnom (1999) inves-tigated the interlaminar shear behaviour of non-crimp-fabric-based composites [19]. Tessitoreand Riccio (2006) performed Finite Element Method (FEM) modelling for biaxial extension ofnon-crimp fabric composite materials [20]. Hind, Robitaille and Raizenne (2006) conductednumerical studies on the effective thermal conductivity of a plain weave [21]. Nordlund et al.(2004) [22] and Lundstrom (2006) [23] have performed both numerical and experiential stud-ies on the permeability of NCFs. Ning and Chou (1995), using a closed-form representation ofthe transverse thermal conductivity, derived a thermal resistance network [24]; Xu and Wirtz(2002), employing a simple-to-fabricate woven mesh, described the in-plane effective thermalconductivity of two-dimensional bonded-laminate plain-weave screen laminates [25]. Peng,Lu and Balendra (2004) studied the FE simulation of the blanking of electrically heated engi-neering materials [26]. They also used thermo-electrical simulation with FEA implementation71.2. Literature Reviewscheme.Tiwari, Basu and Biswas (2009) employed a new simulation of thermal and electric fieldevolution during spark plasma sintering [27]. They conducted a different thermal conductivitypowder compact on the die and punch surface of spark plasma sintering by a fully coupledthermo-electrical finite element analysis using ABAQUS and MATLAB. Their simulation re-sults from ABAQUS revealed that maximum surface temperature is attained at the punch re-gion. It indicates that a good finite element analysis could help users to find the critical points.Ishikawa and Chou (1982) studied stiffness and strength behaviour of woven fabric com-posites [28]. Peng and Cao (2005) studied a continuum mechanics-based non-orthogonalconstitutive model for woven composite fabrics [29]. Li et al. (2007) proposed a finite ele-ment model of woven fabric composite PCB to predict the bending behaviours of PCB [30].They built four shell-element layers in ABAQUS. Then they used the stiffness factors, suchas Young’s modulus, Poisson’s ratios, and shear modulus into non-linear equation systems andperformed the numerical simulation for a three-point bending test. They compared and verifiedthe results of experiments with the results of ABAQUS simulation. They concluded that thehigh fibre bundles in the filling direction of woven fibres could enhance the stiffness of themulti-layer PCBs .The majority of the numerical studies on the mechanical and/or thermal analysis of fabricshave been based on the meso-level FEA analysis of unit cells representing repetitive (network)patterns in the fabrics. In practice, however, fabric unit cells are not perfectly repetitive, owingto uncertainties during fabric fabrication processes and/or unsymmetrical loading conditionsfor a given problem. They can require performing analyses on larger representative volumes.As the size of the problem scales, the solution complexity and the simulation run time canpose a problem. In addition, the selection of the position of a unit cell and the associatedboundary conditions can become highly important. (Therminator3D is capable of simulatingtemperature distributions of an entire thermal network rapidly and accurately, without relyingon the simplified unit cell descriptions.) Therminator3D applies network simulation concepts81.3. Summary of Research Goalswith technological parameters extracted from CAD layout descriptions to the entire network(similar to FE models). A detailed description of the method in its original format for thermo-electrical analysis is presented in Chapter 3.1.3 Summary of Research GoalsIn this work, I propose not only to perform IC interconnects temperature estimation, but alsoto treat thermal problems of NCFs and woven materials using the same concept of net-basedcalculations by Therminator3D which was originally developed for IC interconnect systems.An auxiliary program is used to create netlist files to describe the thermal (resistor) networkproperties of biaxial NCFs and woven fabrics, including the thermal conductance of fabricsegments. It will also create additional, virtual segments around the edges of the fabric toimpose appropriate boundary conditions.The eventual benefit of my study is to use the same modified interconnect net-based methodto rapidly estimate the temperature distribution on big complicated IC interconnects, as wellas the textile materials including NCF and woven types. Another major benefit of the methodwill be that it relies on a symbolic, semi-analytical, and net-based approach that can reducethe computational time and enhance the accuracy of the results at full resolution. Comparativestudies with a commercial finite element software (ABAQUS) will be conducted.My contributions of this study are:1. Simulative results of Therminator3D are comparative to those of ABAQUS.2. Application of textile materials.3. Description of additional heat conduction behaviour, Glattwist, for woven materials.4. Demonstration of Large-scale network simulations using Therminator3D.5. Three-dimensional model could be easily represented by one-dimensional model withcorresponded lateral thermal conductance.91.4. Thesis Organization1.4 Thesis OrganizationThis thesis consists of six chapters. Chapter 2 starts from the thermo-electrical modellingtheory with a basic physical model and introduces the network based simulation. In Chapter 3,a new network calculation method “Therminator3D” is presented with detailed algorithms andsolvers. Chapter 4 discusses the finite element analysis and describes the calculation of thermo-electrical simulations. Chapter 5 presents applications and comparisons on IC interconnectsand textile composites in Manhattan layout. Computational cost and accuracy are compared toFE approach in the last section of this chapter. Chapter 6 concludes the thesis and lists my maincontributions and results. In addition, some future work is proposed for further applications ofthe method to other mechanical properties.10Chapter 2Network-Based SimulationIn this chapter, the fundamental concept of network-based simulation is presented. The firstpart of this chapter emphasizes the basic physical model and describes the supported theorems.The second part of this chapter introduces the interconnect model, and textile fabric model ininterconnect form. The detailed network-Based Simulation is introduced in the third part ofthe chapter. A summary is given in the last section of the chapter.2.1 Basic Physical ModelOperating temperature is an important issue in both IC interconnects and textile material com-ponents. One of the greatest challenges of the temperature calculation is the quick and full-scale simulation.In ICs, the temperature calculation is involved in all levels. The temperature is varied fromthe interaction of Joule heating, thermal conduction, and the coupling of thermal and electricaleffects. The purpose of temperature prediction for integrated circuits is to find the operatingtemperature of a whole system, perform the simulation closest to real situation, and prohibit thefailure of the system. In a circuit, embedded thermal and electrical conductive interconnects arenot only to provide low power consumption but also to conduct the heat out to the environment.A good layout of interconnects will maintain the temperature close to the designated operatingtemperature in order to keep the system functioning normally.For textile materials, the same issues are expressed, but the applications are not limitedto lower temperature under critical failure temperature. There are several different purposes112.1. Basic Physical Modelusing textile materials compared to IC interconnects. For example, with so called “electricaltextile materials” in medical applications, the embedded wires could transfer accurate real-timetemperature data to the control unit and /or keep the body warm in a critical circumstance bychanging the thermal conductivity. For earth friendly design, it could keep the temperature ofa living space within comfortable level without the use of air-conditioning by blocking loss ofheat by designing with low-thermal-conductivity matrix materials.In this thesis, I focus on the electro-thermal application only. Thus, the basic physicalmodels in my application include thermal conduction, Joule self-heating, thermal resistors,and the coupling of electrical and thermal effects as discussed below.2.1.1 Thermal ModelHeat transfer processes are classified into three types: heat conduction, heat convection, andheat radiation. Heat convection is a heat transfer problem due to a flowing fluid or gas. Sincethis research is focused on the IC interconnect and textile materials below their melting points,I do not include the effect of heat convection. Heat radiation is the emission of energy throughan external space, material or source. In this research, heat radiation is not directly consid-ered, however it can be conveniently modelled as an arbitrary input (boundary condition) flux.Heat conduction is the interaction of energy within the materials. The heat transfer rate canbe dependent on the thermal property of the material which is called “thermal conductivity”.Fourier’s law is the main concept of heat conduction. This is the main effect that I discuss andsimulate in this research.A general form of the heat equation is [31]:1α∂T∂t = ∇2T + qk, (2.1)where the thermal diffusivity α = k/ρCp (m2/s), T is temperature function (Kelvin oK), kis thermal conductivity (W/m-K), ρ is mass density (kg/m3), q is the volume heat flux (W/m3),122.1. Basic Physical Modeland Cp is the specific heat (J/Kg-K).Basically, the function of temperature in a material has two factors – position with coordi-nates and time (t). For steady state, the thermal system is balanced. The temperature does notchange while time is changing. For a one-dimensional problem, I consider the heat flow alonga bar with two different temperatures on the ends: Ta and Tb as shown in Figure 2.1, the totalheat transfer rate Q (W) could be described as the following equation:Q =−kA(Tb−Ta)L =−kAdTdx , (2.2)where k is the thermal conductivity (W/m-K), the negative sign indicates that the directionof the heat flux is opposite to the temperature gradient, T is temperature in Kelvin (K), and Ais the area of bar cross-section.The heat transfer per unit area could be defined as qA = Q/A (W/m2), then the one-dimensional steady-state heat transfer conduction equation becomes:qA =−kdTdx , (2.3)where qA is the surface heat flux, which is the heat transfer rate from a high temperature regionto a low temperature region. The unit of qA is W/m2Figure 2.1: Heat Transfer along a bar132.1. Basic Physical Model2.1.2 Electrical Analogy for Thermal ConductionFor the electrical analog with heat conduction, the thermal conduction could be addressed as acurrent flow driven by the difference of voltages with thermal resistance. For a homogeneousbar, the analog of Q is the current I, and the voltage difference is analogous to the temperaturedifference (Tb−Ta):Q = (Tb−Ta)Req, (2.4)where Req is the thermal resistance to the conduction through the bar, and is equal to L/(kA) .For the series thermal resistance such as two different material bars connected along theirlengths, shown in Figure 2.2 (a), the total heat transfer rate is given by:Q = (Tb−Ta)Req= (Tb−Ta)R1 +R2. (2.5)For the parallel thermal resistance such as a square bar surrounded with resin (matrix)material, shown in Figure 2.2 (b), the heat transfer rate is given by:Q = (Tb−Ta)Req, 1Req= 1R1+ 1R2. (2.6)In electrical circuits, Ohm’s law states that the current is directly proportional to the poten-tial difference across the two points, and inversely proportional to the resistance between them.The equation isI = VR , (2.7)where V is the relative voltage across the conductance and its unit is volts; I is the currentthrough the conductance in units of amperes, and R is the resistance of the conductor in units ofohms and the inverse of electrical conductance. Ohm’s law also states that the R is independentof the current. Equation (2.7) appears in a form similar to equation (2.4). Because I want toextract the thermal and electrical netlist information with a layout extraction tool which cannot142.1. Basic Physical ModelFigure 2.2: Diagrams of thermal resistance. (a) Series thermal resistance (b) Parallel thermalresistance152.2. Interconnect Modelundergo thermal extraction, I need to interpret the relationship between thermal and electricalfields by the concept of resistors. While I consider the Joule heating, described below, thethermal and electrical properties of materials will be changed corresponding to temperature.In electrical components, there is heat generation in a resistor due to the current Joule self-heating. The Joule heating equation for a uniform bar/wire can be expressed asΦl = I2rmsRl, (2.8)Φl is the Joule heat per unit length, Rl is the electrical resistance of the bar per unit length,Irms is the time-averaged root mean square (rms) current.The longitudinal thermal conductivity klong can be related to the electrical conductivity σand temperature, as stated by the well-known Wiedemann-Franz law:klong = λT σ => λ = klong/(Tσ). (2.9)where λ is a Lorenz constant, σ is electrical conductivity, and T is the absolute tempera-ture of the resistor [32]. The Wiedemann-Franz law is named after Gustav Wiedemann andRudolph Franz. In 1853, they reported that different metals have approximately the samevalue of klong/σ at the same temperature. In 1872, Ludvig Lorenz discovered the λ which isthe proportionality of klong/σ with temperature. By applying the Wiedemann-Franz law withLorenz’s constant, I can derive the value of thermal conductivity from the electrical conduc-tivity extracted by TCAD tools. The material used in interconnects and textile layout in thisthesis is copper. The Lorenz constant is 2.23X10−8(W −Ω−K−2) for copper at 0oC [33].2.2 Interconnect ModelConsidering the interconnects of Very-Large-Scale Integration (VLSI) devices, the temperaturedistribution of interconnects is a subject of interest as VLSI devices become denser and low-162.2. Interconnect Modelthermal-conductivity dielectrics with reduced thermal conductivity are introduced. In currentCMOS technologies, estimating interconnects temperature from the layout and a model of theswitching behaviour of the circuits has become an important part of electromigration reliabilityverification. For detailed temperature estimation of interconnects, interconnect is divided intorectangular “resistor” segments (Figure 2.3) paired with capacitance by a layout extraction tool.The layout extraction tool extracts the information of net topology, resistance and capacitanceinformation, and coordinates of interconnects. This extracted information of nets from a layoutis then relabeled to match the electrical circuit. The extraction method may merge the resistorsto an order-reduced model. For example, Clement et al. extracted the information of resistorsand capacitors from layout using CAD tools (e.g., REX) [34] [35]. The capacitors then arereplaced by corresponding current sources which are set to capacitances multiplied by voltageswing divided by the cycle time. This strategy could directly solve the node voltages and branchcurrents by standard nodal analysis technologies [36]. I used the concept of RC network withextraction information in my interconnect simulations by the Therminator3D program.Figure 2.3: Resistor segment of interconnectsIn a general interconnect layout, segmentation occurs at vias and at changes in width, junc-tions, and corners. Each segment has a resistance Rn given by the following formula:Rn = Ln/knAn , (2.10)172.2. Interconnect Modelwhere L is its length (m), k is its conductivity (Ω−m), and A is its cross-sectional area (m2).Consider a segment of length ∆x on the thermal resistor network as shown in Figure 2.4.Its heat conservation can be written [17]mCp ∂T∂t = Q(x+∆x)−Q(x)+Φl(x)∆x− f(x)∆x. (2.11)The longitudinal heat transfer rate, Q, is opposite to the increasing direction-x and obeysEquation (2.2); Glong is a longitudinal thermal conductance found by integrating thermal con-ductivity klong over the resistor cross-section. The lateral heat rate per unit length (W/m) iswritten as f(x) = Glat(T(x)−T re f ) with the lateral thermal conductance Glat representing thediffusion of heat flux from the sides of the resistor with unit length, through a dielectric withthermal conductivity klat, to a uniform reference temperature T re f . Parasitic interconnect ca-pacitance C is analogous to Glat and both may be calculated using the same technique. Thesegment mass is m and Cp denotes the specific heat. The Joule self-heat per unit length Φl(W/m) generated by a current in the circuit is Φl = I2rmsRl.In a steady-state case, T(x,t) is a function of x only, and when ∆x → 0, Equation (2.11)becomes a second order ordinary differential equation:Φl =−Glongd2T/dx2 +Glat(T −T re f ). (2.12)The solution on each resistor segment with boundary conditions T(x = 0) = T0and T(x =L) = TL, and defining a constant termT ∞ = ΦlGlat +T re f , (2.13)182.2. Interconnect Modelis then given by [17]T (x) =parenleftbiggTL−T ∞ −(T0−T ∞)cosh(ξ L)sinh(ξ L)parenrightbiggsinh(ξ x)+(T0−T ∞)cosh(ξ x)+T ∞, (2.14)where ξ =radicalbigGlat/Glong. For specific resistors such as vias for which Glat=0, the solutionbecomesT (x) =−I2rmsRx22Glong +parenleftbiggTL−T0L +ΦlL2Glongparenrightbiggx+T0. (2.15)Equation (2.14) and (2.15)have been applied to thermal estimation of multilevel intercon-nects [37]. In Therminator3D, electrical and thermal resistances are taken as constants withina resistor given a temperature. However, each resistor has its own values, depending on itsaverage temperature.The network numerical solutions are iterated over temperature and current until a self-consistent state (equilibrium) is achieved.2.2.1 Textile Fibre ModelFor the analysis of textile material, the solutions of the one-dimensional heat equation arederived on fibre segment resistors and on vias that represent the contact of overlapping fibres.The networks that are formed by these resistors and vias are treated in the same way as theelectrical resistors that are used in metal interconnect EMR analysis [34].The textile fibre model can be interpreted as an equivalent interconnect RC model. Thestarting point for the textile fibre model examples will be a steady-state and a NCF fabric withno heat source. Thus, the following assumptions are made: (a) steady state:∂T /∂t =0; (b)one dimensionality: the heat is flowing in only one coordinate direction along the length of theresistor. (c) no heat source: there are no heat sources within the system. Then equation (2.1)192.2. Interconnect ModelFigure 2.4: Heat conservation model and capacitances (a) Heat conservation within a differen-tial slice of a one-dimensional thermal resistor accounts for longitudinal flux, lateral flux, andJoule self-heat. (b) Horizontal resistors have Glatand Glong, while (c) via resistors only haveGlong.202.3. Network-Based Simulation Conceptturns to:∂ 2T∂x2 = 0, (2.16)andT(x) = (Tl −T0) xL +T . (2.17)Results of simulation for this case with a fixed temperature boundary conditions will beshown as the first example in Section 5.3.The second step was to apply heat source to a specific wire and consider a model in steady-state. Such, the whole model of NCF material will look like the interconnect model presentedin Section 2.2. Results of simulation for this case will be shown as the second example inSection 5.3.Finally, I will continue to perform simulation problems for a woven pattern based on anetwork-based model.2.3 Network-Based Simulation ConceptNetwork-based simulation is an approach based on the information of the whole network. Ithas a concept of network calculation on nodes and corresponding connections of the entirenetwork. Each node has detailed information between adjacent nodes and itself. For IC inter-connect simulations, there are netlist files describing the coordinates of components, the valuesof capacitance and resistance, and the properties of materials. The list of a net provides detaileddata including the values of resistors, properties of materials and the information of adjacentnodes (e.g., the net-list format of SPICE). The net-based simulation can be composed of severalnets. Those nets may be coupled. By the net-based approach, each net is solved individuallythen the nets are assembled from the layout database to form the entire network. Using the212.4. Summarynet-based approach, the total calculation time does not increase with the entire model size,but rather with the size of each net, and the number of nets. Some earlier works introduced a“Model Reduction” concept (for relatively coarse computational grids) to minimize the timeof simulation in temperature prediction of interconnects [38–41] , but they lost the details ofinformation of temperature distribution for the whole system.As described in Section 2.2, I adopt a modified RC network with thermal module from theelectrical module [37, 42, 43]. Using this modified RC network, the net-based approach candemonstrate resistor-level accuracy of the full-chip temperature estimation.2.4 SummaryIn the chapter, I introduced “Network-Based Simulation” for temperature estimation of IC in-terconnects and textile fabrics along with underlying theories. I discussed an analogy betweenthermal and electrical problems. Based on those, I proposed a RC network model from theconcept of electromigration in circuits. With the RC network, I can simulate large networkswith full-chip calculations without losing the detailed information on each component and theaccuracy of temperature distribution. Moreover, this method can be conveniently adapted totextile materials. The program implementation will be described in the next chapter.22Chapter 3Therminator3DThis chapter introduces the Therminator3D network-based calculation method. The first partbegins with background and input formats of this program. The symbolic network analysissolution is shown in the second part. The third part demonstrates the iterative and multiphysicsevaluation procedure. Finally, a summary of the Therminator3D is presented in the end of thechapter.3.1 The Analytical Solution of Networked ElementsIn this section, the analytical solutions of the one-dimensional heat equation are summarizedfor fibre segment resistors and vias that represent the contact of overlapping fibres. A typicalinterconnect network with vias is shown in Figure 3.1. The network formed by such resistorsand vias is often treated in metal interconnect electromigration reliability (EMR) problems[34]. An equivalent multilevel Manhattan interconnect model of the net-based interconnectswith vias is shown in Figure 3.2. The interconnect columns (1-4) and rows (5-8) are embeddedin dielectric and divided into rectangular segments (“resistors”).3.1.1 Resistor-Capacitor NetworkThe electric current in each resistor can be found by DC analysis of the interconnect RC net-work, which is described as equivalent pi-networks that consist of resistors and current sourcesproportional to interconnect parasitic capacitance as shown in Figure 3.3. This equivalent net-work is commonly used for electromigration analysis [6]. The network nodes correspond to233.1. The Analytical Solution of Networked ElementsFigure 3.1: A net-based interconnect model with vias shown with an ‘X’.Figure 3.2: A two-layer interconnect grid with vias showing approximate segmentation toconnected resistors.243.1. The Analytical Solution of Networked Elementsthe resistor endpoints. The system of analytical boundary value equations (2.14) and (2.15) canbe evaluated over the same RC network by assigning the appropriate values to the circuit sym-bols in Figure 3.3. From equation (2.14) and (2.15) , for interconnect resistors, the equivalentquantities are:α = GlongξslashBigsinh(ξ L), (3.1)ψ = 2Φl (cosh(ξ L)−1)slashbig(ξ sinh(ξ L)) (3.2)β = α (cosh(ξ L)−1). (3.3)For vias:α = GlongslashBigL, (3.4)ψ = ΦlL. (3.5)Connected resistors have common temperatures at their junctions and heat flux along theirlength is conserved. Analytical trajectories of equation (2.14) have been previously applied tostudy thermal scaling of interconnect architectures [44].3.1.2 The Format of Input FilesThe two input files for Therminator3D are the netlist file and the coupling file. The formatsof these two files are shown in Table 3.1. The netlist file is formed by the resistor and nodenumbers, the coordinates, length, and thermal resistance values. Every line in the coupling fileconsists “CAP”, the resistor number , the “coupled to” resistor number, capacitance value, andthe value of lateral thermal conductance Glat.253.1. The Analytical Solution of Networked ElementsFigure 3.3: EMR methodology diagrams. Typical EMR methodology, each of the resistorsthat form a net is represented by the equivalent pi network. In the EMR methodology network(a) is used for via resistors and network (b) for other resistors, but with conductance β= 0 inelectrical analysis and β > 0 in thermal analysis.The coupling file presents the heat conductance to the neighbouring dielectric, substrate,and resistors. The pair-wise Glat through the dielectric between each resistor and its neigh-bouring resistors (analogous to their coupling capacitance) can be extracted from the layout bya parasitic capacitance extractor and summed to form the total Glat for that resistor, as shownin Figure 3.4. The equations for Kin−plane and Kcross−plane are [45]:Kin−plane = 1(1−r)(kvP+kf (1−P))−1 + Prkv + (1−P)rk f(3.6)Kcross−plane = (1−r)( Pkv+ Pkf)−1 +(kv−kf )Pr +kf r (3.7)where P is the porosity of the dielectric/matrix, kv is the thermal conductivity of the inclu-sion medium, kf is the thermal conductivity of the host medium, r is the ratio of the parallelcomponent area to the total area.Although a simple area model is used in this study, the more sophisticated coupling modelsfound in most capacitance extractors could potentially be used for more accurate results ifsupplied with appropriate thermal conductivity parameters.For example, in application to woven materials (Figure 3.5), the simple area model is not263.1. The Analytical Solution of Networked Elementsappropriate to describe the Glat to represent the phenomenon. Currently, there are no experi-mental data to achieve the formulization of Glat as related to the woven structures and couplingareas, and the matrix between the fibres. In this research, I initially estimate Glat by the simplearea model in Figure 3.5 then calibrate the values of Glat with the result of ABAQUS’s sim-ulation. The Glat for the twist part with angular section coupled to wire 3, Glattwist, is around0.15 of Kcross−planew/2h. Then the result of Therminator3D simulation is closest to the resultof ABAQUS simulation. That is because the angular section enhances the coupling effects toadjacent wires as the distant between wires changes.Table 3.1: Therminator3D input file format and typical values.netlist file( resistor node0 nodeL R # xlowerle ftylowerle ftxupperrightyupperrightLRlong)(current source node0 nodeL Irms). . . ...RES27 25 26 83.046936 # 40 -20 40 -10 10.000000 3433.110212RES28 26 27 83.046936 # 40 -10 40 0 10.000000 3433.110212. . . . . .AMP1 0 23 0.000000RESAMP1 0 23 3.000000 # 35 -45 45 -35 1.000000 0.0001. . . . . . . . .RES4-27 4 27 3.000000 # 35 -5 45 5 1.000000 0.001......coupling file(CAP RESISTOR {RESISTOR|0(SUBSTRATE)} c # Glat......CAP RES27 0 12.666667 # 0.00000040086361458CAP RES28 0 6.333333 # 0.00000020043180686CAP RES28 RES4 9.500000 # 0.00000030064771072CAP RES28 RES5 9.500000 # 0.00000030064771072......273.2. Symbolic Network Analysis SolutionFigure 3.4: Typical Lateral thermal conductance model in IC Interconnects. Lateral thermalconductance Glat between each resistor and its assumed uniform background temperature T re fis the weighted average of temperatures to neighbouring resistors and the substrate, using sim-ple area models of conductance. Anisotropic thermal conductance is indicated by Kin−plane andKcross−plane values. [46]3.2 Symbolic Network Analysis SolutionThe sparse N x N system of linear algebraic equationsAz = b, (3.8)can be decomposed to lower and upper triangular matrices L and U with order O(N3)operations by the Crout algorithm. The system can then be evaluated for z by two consecutiveback-substitution operations (O(N) if null operations are not performed):A = LU (3.9)Az = (LU)z = L(Uz) = b (3.10)283.2. Symbolic Network Analysis SolutionFigure 3.5: Lateral thermal conductance for woven case (shown by the capacitor symbols) (a)Lateral thermal conductance Glat between each resistor in woven case. (b) Equivalent layoutin T3D. The value of multi-directional coupling Glat1 is calibrated by the results of ABAQUSFEA.293.2. Symbolic Network Analysis SolutionLy = b (3.11)Uz = y (3.12)The elements li j and ui j of L and Ucan be found by equations (3.13), (3.14), and (3.15) overthe elements ai j of A [47, 48].li j = 0, for i < j,ui j = 1 for 1 ≤ i ≤ N,and ui j = 0,fori > j,where 1 ≤ i, j ≤ N (3.13)lim = aim−m−1∑µ=1liµuµm, i = m,m+1,....,N (3.14)um j = 1lmmparenleftBiggam j −m−1∑µ=1lmµuµ jparenrightBigg, j = m+1,m+2,...,N. (3.15)Therminator3D, a C language program, applies dynamic storage allocation and data structuresto store the symbolic structure of the solution. Each resistor in the net-list file is given a datastructure containing all numerical data and results for the problem to be solved. All the resistorstructures are referenced through ‘hash’ tables. The problem matrix A is represented by alinked list of O(N) non-null elements of A. Figure 3.6 shows the distribution of elements inA. The cross marks represent the non-zero values. The distribution of Figure 3.6 is the least-sparse case with all vias between columns and rows. The vector b consists of links to heatsource elements and boundary conditions. Matrix A is decomposed to the L and U linked listsby the Crout factorization formulas. Vectors z and y are constructed as linked lists pointingto components of L, U, and b. After the entire system of linked lists has been created, it isstraightforward to evaluate the specific physical problem (mechanical, thermal, electrical, etc).For thermal problems, the temperature trajectories T(x) are determined after the z vector (the303.2. Symbolic Network Analysis Solutiontemperature of each node) has been found. Average and maximum values are stored in the datastructure for convenience. The O(N) evaluation can be iterated for nonlinear problems (forexample when klong is temperature-dependent) or for multiphysics problems.Figure 3.6: The distribution of problem matrix A (the 20 columns and 20 rows case with viabetween each row and column).Consider the Joule self-heat generated by an electric current flowing through a resistiveconductor. In an IC, multiple active devices (e.g. transistors) are connected electrically bya metal interconnect network, or “net” of uniform-width resistor segments connected at theirendpoints, or “nodes” that may be “extracted” to a file. Any system of 1-D boundary valueequations, such as the steady-state heat conduction, can be solved over the entire net by con-necting the individual solutions in each resistor segment. As mentioned earlier, Therminator3Dapplies the Crout algorithm for LU decomposition and back-substitution to the system andrecords only the non-null operations, thus forming a generic symbolic solution for the entirenet. This symbolic solution is then evaluated with electrical parameters to solve the electricalproblem first, and then evaluated again with thermal parameters to solve the resulting Jouleheat problem. The resistance of each resistor can increase with the increasing temperature.The final resistance and temperature is therefore found iteratively. Also, thermal conductance313.3. Iterative and Multiphysics Evaluationthrough the insulator between adjacent resistors on different nets may be extracted in the sameway as capacitive coupling for IC interconnect. By composing the ambient, reference temper-ature for each resistor from temperatures of resistors to which it couples through the dielectric,it is possible to iteratively couple separate nets’ temperature trajectories. To form the symbolicsolution requires O(N3) operations, but for a sparse system such as an interconnect networkthere are only O(N) non-null operations (where N is the number of nodes). Thus, the electri-cal and thermal evaluations require only O(N) operations. Therminator3D requires a databasethat contains network information, such as the nodal relationships, coupling between resistors,input sources, and boundary conditions. A netlist file extracted from IC layout with a standardCAD tool would be the most common source of this information.3.3 Iterative and Multiphysics EvaluationThe analytical solution procedure involves the coupling of thermal and electrical calculationuntil the convergence of temperature solution for steady-state is reached. After the symbolicsolution is done, the L and U matrices in linked-list forms pass to the analytical solution loop.For the electrical problem, the pi equivalent network values α, β, and Ψ are consideredfrom electrical circuit values for each resistor as described in Section 3.1.1. In IC interconnects,the electrical conductivity of dielectric is almost zero, so β = 0. The value of non-zero ai j isthe sum of the α field of linked resistors. The elements of y matrix and z matrix are calculatedby the back-substitution.For the thermal problem, the pi equivalent network values α, β, and Ψ are consideredfrom thermal and electrical current values for each resistor using (3.1), (3.2), and (3.3) and theevaluation procedure is repeated. The z array (nodal temperature in the thermal problem) isthen found. The temperature of resistors are updated and coupled to form the T re f (uniform323.3. Iterative and Multiphysics Evaluationthermal reference):Glatnm(Tnm −T re fnm ) =N∑v=0Mv∑µ=1Glatnmvµ (Tnm −Tvµ )(n negationslash= v), (3.16)T re fnm =N∑v=0Mv∑µ=1Glatnmvµ TvµGlatnm , (3.17)whereT = 1LLintegraldisplayx=0T(x)dx f or each resistor. (3.18)The temperature at each junction between resistors is stored. The temperature of eachresistor is updated from equation (3.18) and the values of Irms and T re f .Since T re f is also updated, the temperatures of nodes of the entire net have to be recalcu-lated through electrical and thermal solutions until convergence is reached. Figure 3.7 showsthe flow chart of Therminator3D. There are several published reports that present differentschemes of thermo-electrical coupling calculation [49–51]. These reports calculate the Jouleheating first, then pass and couple the electrical results to an additional thermal calculationpackage, then return the calculations of thermal problem for post-processing. The procedureis repeated until convergence is reached. This type of thermo-electrical coupling calculationcould lose some consistency of data due to interpretations between different packages. Ther-minator3D program uses one package to calculate both electrical and thermal problems usingthe same symbolic solution procedure. Hence, the Therminator3D analytical approach may bemore suitable for large systems by making only minor modifications from the EMR verifica-tion methodology [34, 52]. The modified RC-network model describes all possible propertiesof a resistor or component. Additional coupling data represent that the behaviour of thermalcoupling could also represent other mechanical coupling properties. This gives Therminator3Da potential to solve large networks of multi-physics problems with high speed and accuracy.333.3. Iterative and Multiphysics EvaluationFigure 3.7: The flowchart of Therminator3D thermo-electrical simulation343.3. Iterative and Multiphysics Evaluation3.3.1 On Convergence of Non-linear Problems in Therminator3DIn Therminator3D, as described in Section 3.2, the symbolic solution of Az = b is reformedusing the following steps:1. Construct the matrices L and U.2. Solve Ly = b for y using forward substitution.3. Solve Uz = y for z using back substitution.If the matrix A is not constant, the system becomes non-linear and to find the solution theprogram use an iterative method. There are several iterative methods available in the literature.Examples include:- Fixed-point method,- Newton-Raphson method,- Quasi-Newton method,- Dynamical relaxation method, and- Continuation or Arc-length methodsThe algorithm used in Therminator3D for iterations is “Fixed-point method”. A generalfixed-point problem is defined as the follows [53]:De finitionSolve the non−linear fixed point system o f−→zk =−→g (−−→zk−1) (3.19)given one initial value −→P0 and generating a sequence −→Pk which converges to the final solution−→P where−→G(−→Pk) =−→P (3.20)For the case of this research, the sequence for solving the non-linear system A(z)z = b usingthe fixed-point method is as follows (here I assume the load vector b is constant):353.3. Iterative and Multiphysics Evaluation1. Azk−1zk = b2. zk = A−1zk−1b3. zk = g(zk−1) where k = 1,2,. . . ...(iteration number)The convergence criteria of fixed-point algorithm is proven to be |dg/dz|< 1 [53]. Figure3.8 shows this criterion in one dimensional case.Figure 3.8: Fixed point iteration for a general function g(x) for the four cases of interest.(Cited from [54])Let us consider a sample non-linear heat problem shown in Figure 3.9Assume T3 =0 , thus 1R1 0−1R21R2 T1T2= QQ (3.21)363.3. Iterative and Multiphysics EvaluationFigure 3.9: Sample circuit 1 for Therminator3D convergence verificationAlso assume ∆RR0 = α0∆T where R0 is a constant resistance at initial temperature T0 , and α0 isa temperature-dependent coefficient.R1 = R0[1+α0(T1−T0)] (3.22)R2 = R0[1+α0(T2−T0)] (3.23)Using (3.22) and (3.23) in (3.21), I have: 1R0[1+α0(T1−T0)] 0−1R0[1+α0(T2−T0)]1R0[1+α0(T2−T0)] T1T2= QQ (3.24)Hence, the temperature solution will follow:T1 = QR0[1+α0(T1−T0)] (3.25)T2 = QR0[1+α0(T2−T0)] (3.26)373.3. Iterative and Multiphysics EvaluationOr in a general form, for both temperature solutions:T = QR0[1+α0(T −T0)] (3.27)Using the fixed-point iteration method to solve equation (3.27) results inT = g(T) = QR0 [1+α0(T −T0)] (3.28)For convergence, I should havevextendsinglevextendsinglevextendsingle∂g∂Tvextendsinglevextendsinglevextendsingle< 1 => |QR0α0|< 1. For example, using values ofthe netlist in Table 3.1, assumed α0 ∼ 0.33% for copper, its thermal resistant is 3433.110 (K/W)for a 10 µm x 10 µm x 40 µm (WxDxL) brick resistor, and its electrical resistant is 83.047 Ω.To ensure the above convergence, Irms should be smaller than 3.26 x10−2 Amp for a range ofQ < 1.063x10−3.As a second example, to evaluate Therminator3D in a small case with Glat, consider a sys-tem containing two infinitely-long, coupled nets, each with one resistor (Figure 3.10), coupledby the Glat1,2 and couplings Glat1,subst and Glat2,subst (all positive) to the substrate. Also assume tem-perature of substrate, Tsubst = 0. From equation (3.16), the overall lateral thermal conductanceof a given resistor is the sum of all conductances to neighbouring resistors and the substrate:Glatnm =N∑v=0Mv∑µ=1Glatnmvµ (n negationslash= v), (3.29)For the infinitely-long nets, note T ∞i = Ti = Ti.From equation (3.17), I haveT re f1 = Glat1,2T2Glat1,2 +Glat1,subst= Glat1,2Glat1 T2 (3.30)383.3. Iterative and Multiphysics EvaluationFigure 3.10: Sample circuit 2 for Therminator3D convergence verification with infinitely-longresistors and substrate.T re f2 = Glat1,2T1Glat1,2 +Glat2,subst= Glat1,2Glat2 T1 (3.31)The temperature of each wire is given by equation (2.13), and thusT1 = Φl1Glat1+T re f1= Φl1Glat1+ Glat1,2Glat1 T2= Φl1Glat1+ Glat1,2Glat1parenleftbiggΦl2Glat2 +Tre f2parenrightbigg= Φl1Glat1+ Glat1,2Glat1Φl1Glat2 +Glat1,2Glat1Glat1,2Glat2 T1= g(T1) (3.32)Although for temperature-independent case, equation (3.32) can be solved in closed-form,for the general case solved by Therminator3D a fixed-point iteration scheme is used (e.g., when393.4. SummaryR and thus Φ is a function of T ). The convergence condition of a one-dimensional fixed-pointiteration scheme, 0 <vextendsinglevextendsinglevextendsingledg(x)dxvextendsinglevextendsinglevextendsingle< 1, yieldsdg(T1)dT1 =parenleftBigGlat1,2parenrightBig2Glat1 Glat2 < 1 (3.33)Thus, it is proven that fixed-point iteration of the simple system described will always converge.However, coupling of more segments, with more complex formula for T given above, does notchange the argument given here. So the general case must also converge. Also, the nearerthe derivative is to unity, the more rapid the convergence. Thus, when coupling between netsbecomes weaker relative to coupling to the substrate, say because the nets are moved furtherapart, more iterations will be required for convergence.3.4 SummaryTherminator3D, a network-based integrated simulation tool, was introduced in this chapter.Therminator3D applies network topology to the solution of fully-coupled electric and thermalproblems. This program derives a modified RC model from EMR and has the ability to sim-ulate multi-physics problems. It can also deal with textile materials by changing the physicalproperties of columns and vias to represent the fibre yarns and conditions. The matrix betweenyarns is analogous to the dielectric between wires in Therminator3D. The porosity of dielectriccan be interpreted to the properties of matrix by coupling effects. Those interpretations showthat Therminator3D can simulate textile materials with a network-based topology. Chapter 5will demonstrate results of several examples of temperature distributions in IC interconnectsand textile materials.40Chapter 4Finite Element AnalysisThis chapter describes details of the finite element analysis (FEA) used in this work and in-troduces the well-known FEA software - ABAQUS. The first part of the chapter discusses thebasic simulation model using FEA. The second section describes the concept of mesh meth-ods. Different mesh methods applied on shapes of objects can affect the accuracy of simulationresults. The third section shows that the FEA software, ABAQUS, is capable of conducting thethermo-electrical stimulations and can be successfully used for comparison with Thermina-tor3D. The last part of this chapter is a summary of the FEA using in this work.4.1 Basic Physical Model of Finite Element AnalysisThe finite element method (FEM) is a numerical procedure for solving physical problems bya series of ordinary and partial differential equations. This method was originally developedto solve stress and strain mechanical analysis. Today, it has numerous of applications such asheat transfer analysis, thermo-electrical analysis, fluid mechanical analysis, etc. The classicalFEA approach includes a series of equations to represent the continuity of physical behaviours,transferring the solution domain to a finite element mesh (such as the interpolation of shapefunctions), assembling equations of elements, applying the boundary conditions, and comput-ing the solutions of system equations. There are numerous articles and books on FEA. TheFEM handbook by Kardestuncer (1987) has described most of these applications [55]. For theapplication of this research, I focus on the heat transfer and coupled electro-thermal problemsof IC interconnects and textile materials. Before solving these problems, the FEM has to form414.1. Basic Physical Model of Finite Element Analysisa CAD model to represent the layout pattern. There are two common (often interchangeable)ways, unit-cell and Representative Volume Element (RVE) modeling, used for this purpose inthe literature.Unit-cell is a concept from the material crystal structure. In crystallography, crystal struc-ture is a unique arrangement of atoms in a crystalline solid or liquid with symmetrical pattern.The symmetrical pattern in a given lattice is called a “Unit-cell”. Unit cells can be stacked toform a model which represent the meso or macro-level problem that I want to solve. Figure 4.1(a) shows a typical unit cell that emphasizes the opening between fibres of a non-crimp fabrics[56].Representative Volume Element (RVE) is an effective volume contained a set of microstruc-ture elements. It has to be smaller than the original sample. W.J. Drugan and J. R. Willis [57]has a definition of RVE – “the smallest material volume element of the composite for whichthe usual spatially constant “overall modules” macroscopic constitutive representative is a suf-ficiently accurate model to represent mean constitutive response”. They pointed out the im-portance meaning of “mean constitutive response”. It expresses that the RVE has to be smallenough to represent the mean of surrounding materials. Figure 4.1(b) shows a typical RVE.The spaces beside the wire/yarn could be filled with a matrix material. The sum of all proper-ties of materials in the square would be replaced by the effective volume of each mechanicalproperty. It is very difficult to get a good representative cell. This concept has extracted manyarticles which propose their research in finding the best fitted RVE that is suitable to theirspecific application [58–60].4.1.1 Mesh MethodTo successfully implement a FEA, the meshing methodology is important to locate joint points(nodes) and continuous elements over the modelled object. There are several meshing meth-ods such as lines for one-dimensional models, triangles and quadrilaterals for two-dimensionalmodels, and tetrahedral, triangular prisms and hexahedra for three-dimensional models. Fig-424.2. Simulation with ABAQUSFigure 4.1: Typical models of Finite Element Method (a) stitched unit cell of a NCF, (b) aclosed-form RVE of the same fabric.ure 4.2 exhibits samples of these mesh methods and typical elements shapes in them. FEAaccuracy can be improved when more elements and correct mesh methods are used. However,the simulation will then spend much more computing resources, as more elements and meshesmean larger numbers of equations to be solved.4.2 Simulation with ABAQUSABAQUS is a well-known commercial FEA software originally developed by HKS, USA, andnow is under SIMULIA of Dassault Systemes. It has a full capability of performing coupledthermo-electrical analysis as described in its documentation. In ABAQUS, while coupling,electrical conductivity can be temperature-dependent, and the internal heat can also be a func-tion of electrical current. Theories of thermo-electrical applications in ABAQUS are such thatthe solver of electrical problem is based on Ohm’s law equation for the flow of the electricalcurrent, then it derives the amount of thermal energy generated by electrical current (Jouleheating); the solver of thermal part is essentially based on heat conduction; however, it couldbe extended to heat convection and radiation from its library [61].The coupled thermo-electrical problem is an unsymmetrical problem. It is impossible to434.2. Simulation with ABAQUSFigure 4.2: Typical finite element shapes and mesh points in one through three dimensions.444.2. Simulation with ABAQUScalculate thermal and electrical evaluations simultaneously. It has to first reach the convergenceof either the thermal or the electrical solution, then couples the solution of the first solver to theother solver, reaches the convergence of the new problem and couples back to the first solver,and so on, until both solvers are convergent and the system equilibrum is reached. There aretwo types of coupled thermo-electrical analysis in ABAQUS - exact and approximate Newton’smethods.The exact implementation has a non-symmetric Jacobian matrix to represent the coupledequations: KVV KV TKTV KT T∆V∆T=RVRT (4.1)where ∆V and ∆T are the respective corrections to the incremental electrical potential andtemperature. Ki j are submatrices of the fully coupled Jacobian matrix, and RV and RT are theelectrical and thermal residual vectors, respectively.The coupled thermo-electrical analysis will be quadratically convergent when the solutionestimate is within the radius of convergence of the algorithm.For problems with weak coupling between thermal and electrical solutions, ABAQUS usesthe approximate implementation. The KV T and KTV are assumed to be relatively small to thecomponents KVV and KT T . Equation 4.1 is turned to equation 4.2. The rate of convergence isnot quadratic any more and depends strongly on the magnitude of the neglected coupling effect.This approximate method generally needs more iterations to achieve equilibrium, compared tothe exact implementation of Newton’s method. KVV 00 KT T∆V∆T=RVRT (4.2)In the implementations of this research, I use the exact implementation of Newton’s methodto execute the coupling thermo-electrical simulation.454.2. Simulation with ABAQUSThe electrical and thermal conductivity value can be temperature dependent in particularapplications. Applying the temperature dependence makes the problem complicated and in-creases the cost of calculations significantly. There are numerous articles using the results ofABAQUS simulation to compare them to experiments. To know the capability and accuracy ofcoupled thermal- electrical stimulation of ABAQUS, Wang and Hilali (1995) ran a comparisonwith an experiment in automotive electrical fuse [62]. They reported that the finite elementanalysis results are in some agreement with experimental results. The differences between thesimulations in ABAQUS and experiments may be due to the type of model chosen, overes-timated infrared temperature measurement, etc. Zhang, Zavaliangos, and Groza (2003) alsoreported that their electrical-thermal prediction using ABAQUS mostly match the experimen-tally observed values [63]. There are other papers that have used the coupled thermo-electricalsimulation of ABAQUS in different areas such as in the powder compact/die/punch assemblyduring the spark plasma sintering process [27], or in blanking of electrically heated engineeringmaterials [26].For textile material analysis, ABAQUS/CAE has predicted the thermal transport behaviourof woven ceramic matrix composites with unit cell FE modeling [56], and tension and strain onmeso-scale with representative volume element [64]. In ABAQUS, to draw a layout of a textilepattern is always a challenge. The difficulties are not only modelling in shapes of fabrics butalso planning designated contacts and boundary conditions. These reported woven models forsimulations on mechanical properties in ABAQUS or other FEA tools chose Unit-Cell or RVEand expand the successful results to larger networks by uniform behaviour assumptions.ABAQUS has capabilities of solving highly non-linear problems in various fields. Thecoupled electrical-thermal problem may not be as difficult as other coupling mechanical ap-plications such as fluid-solid interactions. However, time and memory consumption are verylarge while applying the simulations to a larger networks even for thermo-electrical problems.464.3. SummaryFigure 4.3: Comparison between the finite element and experimental results in a coupledthermo-electrical problem (cited from Wang and Hilali [62])4.3 SummaryIn summary, thermo-electrical simulation using ABAQUS is a convenient and efficient tool fordesign, specially when there are no or limited experimental data. The more the elements ormeshes included in a model, the more the accurate results would be in a simulation. The mainreason that ABAQUS was chosen in this study to compare with Therminator3D program isthat there are several published papers comparing their experimental results with simulationsof ABAQUS based on finite element method [26][27][56][62][63][64]. Reports show that theresults from ABAQUS have been reasonably accurate. I use ABAQUS to build the layouts ofIC interconnects and textile composite models, run the simulations of thermal and/or electricalproblems, and post-process the results. The next chapter will illustrate several examples ofTherminator3D and ABAQUS and compare the results using the two programs. The mainpoint of interest is to evaluate how Therminator3D (network-based approach) compared toABAQUS (finite element method) under computation time and CPU memory criteria.47Chapter 5Applications and ComparisonsIn this chapter, the results of thermo-electrical simulations from Therminator3D will be com-pared with those of ABAQUS in different examples. For all simulations, I have used an iden-tical MacBook machine, which has MAC OSX 10.5 OS, 4G RAM, Intel Core 2 Duo 2.4 GHz.The compiler for Therminator3D is gcc version 4.3. ABAQUS is installed in windows XP prosystem VMFUSE virtual machine in the same MacBook. While performing the simulations ofABAQUS, I disable unnecessary running programs in MACOSX system and allow ABAQUSsoftware in the VMFUSE could take as many resources of the MACOSX system as possible.In the first section, I verify the solutions of Therminator3D by basic layout examples. Thefollowing sections in this chapter present results of thermal simulations of Manhattan layout ofIC interconnects with/without current, non-crimp fabric composites with temperature bound-ary conditions and with/without a body heat flux (current), woven fabric composites and 20by 20 large network results. Finally, a large-scale model with multiple boundary conditions isdemonstrated. The comparison of computation cost for each example case is shown in the endof the example. A summary of performance of Therminator3D is given in the last part of thechapter.5.1 Basic Layout Examples: Verification of Therminator3DIt is first necessary to verify the accuracy of solution of Therminator3D in basic problems be-fore I continue my simulations in more complex problems. Let us start with a single wire(with dielectric) simulation which has a current of 0.001 A in the wire. The size of wire is485.1. Basic Layout Examples: Verification of Therminator3D10µm x 10µm x 400µm (width x depth x length). The layout is shown in Figure 5.1. Re-sults of this thermo-electrical simulation are shown in Figure 5.2. Materials of the wire anddielectric are given in Table 5.1. The results of the exact calculation from Equations (2.14)and (2.15), Therminator3D, and ABAQUS are well matched. For example, on node (x=22.5µm), the temperature calculated using ABAQUS was 10.674200 oC, the exact calculation fromExcel resulted was a temperature of 10.673000oC, and for Therminator3D the temperature was10.672037oC. The percentage of difference of exact calculation and Therminator3D simulationis 0.01%. The total number of nodes and elements in Therminator3D were 11 and 12, respec-tively. I minimized the meshes of ABAQUS to 10 elements which formed 44 nodes to make itclose to Therminator3D. The percentage of difference of exact calculation and ABAQUS sim-ulation is 0.0112%. The percentage of difference of Therminator3D and ABAQUS simulationis 0.02%.The next basic example is a layout of three wires with the same size. The system washeated in the middle wire by an electrical current (0.001 A). The layout is shown in Figure5.3 and the comparison of results is included in Figure 5.4. There is no exact calculation forthis case because of the complexity of coupling behaviour among the three wires. However, Ican compare the results between ABAQUS and Therminator3D. For the node position x=22.5µm on the middle wire, the temperature calculated using ABAQUS was 9.107710 oC, and forTherminator3D the temperature was 9.087000 oC. The percentage of difference is 0.0227 %,which is slightly higher than the previous example with one-wire.Results of these two simple layouts suggest that Therminator3D can perform thermo-electrical simulations as accurate as ABAQUS. Next, I move forward to more complicatedsimulation cases in IC interconnects, textile materials and finally large-scale networks. I willcompare results from the two programs with respect to their simulation time and the CPU/mem-ory used.495.1. Basic Layout Examples: Verification of Therminator3DFigure 5.1: The layout of one wire simulation.Figure 5.2: The temperature distribution of one wire simulation.505.1. Basic Layout Examples: Verification of Therminator3DFigure 5.3: The layout of three-wire simulationFigure 5.4: The temperature distribution of three-wire simulation.515.2. IC Interconnects NetworkTable 5.1: Material properties used in simulations adjusted by narrow-line effects [46] [65]Material ThermalConductivity(W/m-K)Wire (Row and Column) Cu 710Vias Cu 710Dielectric SiO2 1.4Table 5.2: Material properties used in simulations.Material ThermalConductivity(W/m-K)Wire (Row and Column) Bulk-Cu 401Vias Bulk-Cu 401Dielectric SiO2 1.45.2 IC Interconnects NetworkThe IC interconnect modelling is a fundamental application of Therminator3D. The rapid andaccurate simulations on IC interconnects by Therminator3D was originally demonstrated byLabun and Jagjitkumar [17]. In this section, I present two examples with Manhattan lay-outs. They are Joule heating simulations with and without a dielectric, and a fixed temperatureboundary condition on one end of a wire.A four-column by four-row, two-layer grid (Figure 5.5) was rendered as an RC network forTherminator3D, and written as a coupling file and a netlist file. Temperature and thermal fluxboundary conditions were achieved using equivalent current sources and resistors at the end ofthe corresponding rows and columns (Figure 5.6). As shown in Figure 5.7, both electrical andthermal conductivities of wires are considered to be temperature dependent. The electrical andthermal conductivities of dielectric (matrix) are considered to temperature-independent. Eachrow and column was divided into 12 identical resistors. Each resistor coupled to those opposite525.2. IC Interconnects Networkto it by the same Glat value. The values of Glat were calculated and calibrated by ABAQUSsimulations. Each via was also represented by a resistor. For ABAQUS without dielectric,each resistor was modeled as a single, three-dimensional rectangular element (a “brick”). Thecoarsest mesh of a brick is consistent with an accurate FEA solution in order to minimizeCPU time and memory. Dielectric fill required a more complex mesh in the finite elementmethod. To simulate scenarios without vias in the presence of dielectric, the material type ofvia was simply changed to dielectric. In ABAQUS, I used a “tie” type of constraint without anyspace tolerance. The materials of wire and vias were copper and the dielectric was SiO2 withproperties shown in Table 5.1. The dimension of each interconnect is 10µmx10µmx120µm(WxDxL).The selected mesh size in Figure 5.8 was to make a fair comparison between two simula-tions by ABAQUS and Therminator3D. More meshes in the ABAQUS’s model will increasethe accuracy of simulations. However, it will also increase the consumption of computationalresources. In Therminator3D, the inside temperature distribution of a resistor is calculated byequation (2.14), which is based on an analytical solution as opposed to the approximation usedin FEA.Two examples were executed: a passive thermal problem with vias and a Joule heatingproblem without vias.The temperature distribution in the passive network was calculated by Therminator3D (Fig-ure 5.8 (a) and (b)). The ABAQUS model with an automatic incrementation found the temper-ature contours shown in Figure 5.8 (c) and (d). Figure 5.9 shows the consistency of temperaturesolution along column 3, obtained by both Therminator3D and ABAQUS. The CPU time forthese cases (Table 5.3) is the average of three simulation runs; note that due to the variation ofactual user- and system- time from one run to another, the total time and memory on the samemachine is somewhat variable for each result.ABAQUS meshes the dielectric, and it significantly increases the number of nodes N, theCPU time, and memory requirements (Table 5.3). The network of Therminator3D does not535.2. IC Interconnects Networkrequire more nodes to include dielectric heat conduction, but instead couples through the di-electric by updating T re f on each resistor while it iterates the evaluations. Therminator3D’sCPU time is dominated by the O(N3) symbolic solution. Iterative evaluations require O(N)operations and incur negligible incremental CPU time.The next example is to remove all vias and apply the electrical current (0.01 A) on column3. The temperatures of endpoints of all interconnects are set to 0 oC. The resulting temper-ature distribution calculated by Therminator3D for this network is shown in Figure 5.10 (a)and (b). A similar model was established in ABAQUS with automatic incrementation and thetemperature contours were plotted in Figure 5.10 (c) and (d). Figures 5.10 to 5.12 show theconsistency of the temperature distribution, obtained by Therminator3D and ABAQUS. Over-head dominates ABAQUS CPU time for small number of elements, but for larger problems(such as the cases with dielectric) CPU time scales with the number of iterations.Figure 5.5: Four-by-four, two-layer grid with vias showing approximate segmentation intoconnected resistors.Temperature at each end and on substrate is set to T = 0 oC, except T =100 oC is applied to one end of interconnect 3. Note that here both electrical and thermalconductivities of wires are assumed to be temperature dependent (i.e., non-linear analysis).The porosity of SiO2 is considered to be 0 %.545.2. IC Interconnects NetworkFigure 5.6: Boundary Conditions achieved by equivalent circuit source and resistors (a)Temperature boundary condition circuit schematic (b) Heat flux boundary condition circuitschematic.Figure 5.7: Temperature-dependent thermal and electrical conductivities of copper.555.2. IC Interconnects NetworkFigure 5.8: Interconnect temperature distribution for the network shown in Figure 5.5 com-puted by Therminator3D with (a) and without (b) dielectric, and by ABAQUS with (c) andwithout (d) dielectric (T=100oC applied to the end of column 3 at (x,y)=(80,60)); note: thedielectric material mesh around the wires is not shown in figures (c) and (d).565.2. IC Interconnects NetworkTable 5.3: Comparison of CPU time and memory required by Therminator3D and ABAQUSfor Joule heat analysis with temperature-dependent electrical and thermal conductivities.Dielectric Number ofElementsCPUTime(sec)Number ofIterationsMinimumMemoryRequiredT 3DFig.5.8(a)yes 96 + 8 (vias)(104 nodes) 0.063 12 2.6 MBABAQUSFig.5.8(c)741(2212 nodes)12.0 7 24 MBT 3DFig.5.8(b)no 96 + 8 (vias)(104 nodes) 0.054 11 2.5 MBABAQUSFig.5.8(d)104(480 nodes)1.1 7 18 MBT 3DFig.5.10(a)yes 96(96 nodes) 0.053 13 2.5 MBABAQUSFig.5.10(c)741(2212 nodes)19.8 13 24 MBT 3DFig.5.10(b)no 96(96 nodes) 0.049 13 2.5 MBABAQUSFig.5.10(d)96(416 nodes)1.1 9 18 MB575.3. Non-Crimp Fabric NetworkFigure 5.9: Comparison of column 3 temperature distribution in Figure 5.5, calculated byTherminator3D and ABAQUS.5.3 Non-Crimp Fabric NetworkIn this section, I simulate non-crimp fabric network model which may be an interpreted ICinterconnects model. The overall layout of non-crimp fabric is made by layers with stitchesthat fix the fabric shape. I can conveniently translate the whole non-crimp fabric networkinto an interconnect network. The yarn of NCF materials is analogous to the interconnect ofIC circuits. The vias, which can be assigned different values in Therminator3D, express thecontact behaviour between yarns in NCF. The dielectric in IC circuits is analogous to the resinmaterial in NCF networks. Overlapping yarns are assumed to be in perfect contact without agap. The resin material is analogous to the 0% porosity SiO2.A dry biaxial NCF structure made of copper is considered. In Therminator3D, only lon-gitudinal and overlap contact thermal conduction is assumed (i.e., Glat =0 for all rows andcolumns of the network). The latter assumption was to simplify the problem for comparisonpurposes with other FE models. The size and computational cost of Therminator3D analysis585.3. Non-Crimp Fabric NetworkFigure 5.10: Interconnect temperature distribution for the network shown in Figure 5.5with vias removed, computed by Therminator3D with (a) and without (b) dielectric, and byABAQUS with (c) (partial cut-view) and without (d) dielectric (Irms= 0.01 Amp through col-umn 3). Notice that in the ABAQUS model, the dielectric material should be actually meshed.595.3. Non-Crimp Fabric NetworkFigure 5.11: Comparison of column 3 temperatures shown in Figure 5.5 with vias removed,calculated by Therminator3D and ABAQUS.Figure 5.12: Comparison of column 4 temperature distribution in Figure 5.5 with vias removed,calculated by Therminator3D and ABAQUS.605.3. Non-Crimp Fabric Networkwas not significantly affected by the neglect of lateral thermal conduction [17]; note that theinclusion of Glat would result in an unsymmetrical system matrices based on the two differentorders of differential equations appearing in Equation 2.11. It is also assumed that there areno Joule self-heating and no heat loss into the environment (adiabatic condition). Vias, withrelatively small dimensions compared to the interconnect (Glat = 0), represent a perfect thermalcontact between overlapping yarns. Nodal temperature (i.e., essential) and thermal flux (i.e.,free) types of boundary conditions were achieved using equivalent current sources and resistorsat the end of corresponding rows/columns.The first test problem is shown in Figure 5.13, which consists of a four-by-four, two-layergrid. The thermal conductivity was assumed to be constant. Values of material properties usedare given in Table 5.2. Each yarn is 10 µm x 10 µm x 100 µm. The resulting temperaturedistribution in the network was calculated by Therminator3D and is shown in Figure 5.14(a).A similar model was established in the ABAQUS finite element package with automatic incre-mentation and the temperature contours were obtained (Figure 5.14(b)). Figure 5.15(a) showsthe temperature variation along column 7, obtained by T3D and ABAQUS. Next, the procedurewas repeated by allowing a variation of conductivity with temperature (similar to the examplein Figure 5.5). Comparison of results for the new nonlinear case is shown in Figure 5.15(b).Table 5.4 also includes results for an unbalanced NCF where the thermal conductivity value ofrows is doubled (for columns, it was unchanged).Table 5.4 compares the CPU time required for these two examples using T3D and ABAQUSin an Apple MacBook. Each value shown is the average of three repeats of a computer experi-ment; note that due to the variation of actual user- and system- time from one repeat to another,the total time in the same machine can be non-repeatable. Values in parentheses refer to theCPU times normalized by the number of temperature nodes ( DOF). The temperature trajectoryfor the temperature-independent thermal resistance case is found in a single T3D evaluation.As compared to ABAQUS results, the convergence of the self-consistent temperature calcula-tion for the temperature dependent case was faster in T3D due to the linear time required for615.3. Non-Crimp Fabric NetworkFigure 5.13: An NCF represented by a two-layer interconnect grid with vias showing approx-imate segmentation into connected resistors. All endpoint temperatures are set to T = 0 oCexcept T = 100 oC is applied to one end of interconnect/yarn 7. To mimic a dry fabric, the zeroporosity of dielectric can be used in Therminator3D.the evaluation of the symbolic solution. For example, the computation time was 0.056 sec-ond using T3D and 1.53 second using ABAQUS. The estimation time of T3D is 29 times fasterthan that of ABAQUS (80 elements). From Figure 5.15(a), I notice that in ABAQUS I require alarge number of elements (640 elements) to achieve a temperature distribution almost identicalto that of T3D with 80 elements (this would suggest that the convergence rate of solution dur-ing mesh refinements would be faster in T3D). For example in temperature independent case,the temperature calculated using Therminator3D was 36.1 oC, ABAQUS (80 elements) was37.6 oC, and ABAQUS (640 elements) was 36.3 oC on the point at coordinate x=60 µm andy=0 µm. The temperature distribution calculated using T3D was close to that using ABAQUS(640 elements).Finally, a fourth example was chosen to demonstrate the application of T3D for thermo-electrical analysis of fabrics (e.g., for E-textiles where electronic components are embeddedinto a fabric structure). The NCF geometry of four-by-four, two-layer grid was consideredand an electric current (0.001 Amp) was applied to interconnect 7 (Figure 5.16). Each wireis 10 µm x 10 µm x 180 µm. In this case, the Joule heat creates a body heat flux through-625.3. Non-Crimp Fabric NetworkFigure 5.14: Interconnect temperature distribution for the network shown in Figure 5.13, com-puted by Therminator3D (a) by ABAQUS (b) (T=100oC applied to the end of column 7 at(x,y)=(80,60)).635.3. Non-Crimp Fabric NetworkFigure 5.15: The row 3 temperature trajectory for the network in Figure 5.13, computed byT3D and ABAQUS with thermal resistance independent (a) and dependent (b) of temperature(results are for the balanced fabric case).645.4. Woven Fabric Composite NetworkTable 5.4: Comparison of Therminator3D and ABAQUS CPU times for the heat analysis of a4x4 metal non-crimp structure.T3D ABAQUS ABAQUSNo. of Elements 80 80 640CPU Time (sec) forTemperature IndependentBalanced NCF Example0.055(6.88E-04)1.636.44E-033.03(2.00E-03)CPU Time (sec) forTemperature DependentBalanced NCF Example0.056(7.00E-04)1.53(6.05E-03)2.93(1.94E-03)CPU Time (sec) forTemperature DependentUnbalanced NCF Example0.057(7.13E-04)1.66(6.56E-03)2.96(1.96E-03)out column 7. The result of the network calculation is shown in Figure 5.17 and the resultof ABAQUS simulation is shown in Figure 5.18. Self-consistent thermo-electrical calculationwith temperature-dependent thermal and electrical resistance required only 0.096 sec CPUtime. Figure 5.19 shows the temperature comparison on all 8 wires. The temperature distribu-tion lines of ABAQUS and Therminator3D along all 8 wires are reasonably matched.5.4 Woven Fabric Composite NetworkWoven fabric composite materials are composed of interlacing yarns with good mechanicalproperties, specially for impact applications. The exact shape of woven fibres is rather difficultto reproduce using CAD tools. Descriptions of the exact contact behaviours between fibresare also challenging. In Therminator3D, the shape of yarns can be represented by switchingthe coordinates of nodes in the NCF case and replacing new coupling data for the thermaland electrical capacitances. I have made a comparison of the two simulation (ABAQUS andTherminator3D) results in this section to show the potential of Therminater3D for thermal andelectrical evaluations of woven materials.The layout of the fabric considered is four by four dry plain weave wires (here, I also referto the wires as yarns) with partial vias shown in Figure 5.20. The dimension of each yarn655.4. Woven Fabric Composite NetworkFigure 5.16: A two-layer interconnect grid layout with vias and Joule heat generated on inter-connect 7.665.4. Woven Fabric Composite Networka0a1 a2a3 a4a5 a6a7 a8a9 a10 a10a11 a12 a13a14 a15 a16a17 a18 a19a20 a21 a22a23 a24 a25a26 a27 a28a29 a30 a31a32a33 a34a35 a36a37 a38a39 a40a41 a42 a42a43 a44 a45a46a47a48a49a50a51 a52a53a54a55a56a57a58a59a60 a61 a62 a63 a64a65 a66 a67 a68 a69 a70 a71 a72 a73 a74Figure 5.17: T3D implementation of NCF layout in Figure 5.16 .(Avg: 75%)TEMP+0.000e+00+4.493e−01+8.986e−01+1.348e+00+1.797e+00+2.247e+00+2.696e+00+3.145e+00+3.594e+00+4.044e+00+4.493e+00+4.942e+00+5.392e+00Step: Step−1Increment      1: Step Time =    1.000Primary Var: TEMPDeformed Var: not set   Deformation Scale Factor: not setODB: a−microm−matrix−vias−sur.odb    Abaqus/Standard Version 6.8−1    Thu Jul 15 20:10:00 Pacific Daylight Time 2010XYZFigure 5.18: ABAQUS implementation of NCF layout in Figure 5.16 .675.4. Woven Fabric Composite Network(a) Wire1 (b) Wire2(c) Wire3 (d) Wire4(e) Wire5 (f) Wire6(g) Wire7 (h) Wire8Figure 5.19: The comparison of all 8 wires between ABAQUS and Therminator3D with viasand heated wire 7685.4. Woven Fabric Composite Networkis 10µmx10µmx180µm (WxDxL). The material properties used are listed in Table 5.2. Acurrent (0.001 AMP) was applied to yarn 5. The resulting temperature distributions of T3Dand ABAQUS are shown in Figure 5.21 and 5.22, respectively.To further evaluate differences between two simulations by ABAQUS and Therminator3D,I listed the temperature profiles of all 8 yarns in Figure 5.23. As a second test case, I ap-plied the same current to yarn 7. Those temperature distributions on all 8 yarns have beenshown in Figure 5.24. In the woven fabric case, since there is no simple method to computethe exact coupling data, I used a calibration with the results of ABAQUS to estimate the Glatparameters (Figure 3.5). The Therminator3D results of all yarns are in a good agreement withthe simulation of ABAQUS, except for yarn 7 in the first example (Figure 5.23) and for yarn5 in the second example (Figure 5.24). The reason those yarns do not match well the tem-perature distribution lines of ABAQUS is the complexity of coupling behaviours estimated inTherminator3D. For example, the Therminator3D temperature of yarn 7 is colder than that ofABAQUS in the case of heated yarn 5 (Figure 5.23). In fact, in Therminator3D, there is nocoupling data for yarns 5 and 7 to see each other. In future work, it is worth continuing toderive more comprehensive coupling formulas in order to obtain more accurate results fromTherminator3D. In Table 5.5, I list the data of CPU time and memory required by Thermi-nator3D and ABAQUS for the performed woven fabric simulations. The results clearly showthat Therminator3D has higher speed and memory saving capabilities. The CPU time for thesimulation of Therminator3D is 0.06 sec. The CPU time for the simulation of ABAQUS is 8.1sec.695.4. Woven Fabric Composite NetworkFigure 5.20: A woven fabric composites represented by a two-layer interconnect grid with viasshowing approximate segmentation into connected resistors. The “X” indicate the yarn havevias presented and woven. All endpoint temperatures are set to T = 0 oC.Table 5.5: Comparison of CPU time and memory required by Therminator3D and ABAQUS onwoven fabric materials. Cases for Joule heat analysis with temperature-independent electricaland thermal conductivities. A current of 0.001 AMP was applied to yarn 5Dielectric No. ofElementsCPUTime(sec)MinimumMemoryRequiredT3DFig. 5.21 yes96 + 8 (Vias) +2(Sky)(106 nodes)0.06 2.5 MBABAQUSFig. 5.2212893(3751 nodes)8.1 31 MB705.5. Large-Scale Network Simulationa75a76 a77a78 a79a80 a81a82 a83a84 a85a86a87 a88a89 a90a91 a92a93 a94a95 a96 a96a97 a98 a99a100 a101 a102a103 a104 a105a106 a107 a108a109 a110 a111 a112a113 a114 a115 a116 a117a118 a119 a120a121 a122 a123 a124a125 a126 a127 a128 a128a129 a130 a131a132a133a134a135a136a137 a138a139 a140a141 a142a143 a144a145 a146 a147 a148 a149 a150a151 a152 a153 a154a155 a156 a157 a158 a159Figure 5.21: Yarn temperature distribution for the network shown in Figure 5.20, computed byTherminator3D.5.5 Large-Scale Network SimulationThis research expects that Therminator3D has high capability to simulate large-scale networks.In this section, I use two examples to verify this. The first example shown in Figure 5.25 isa 20 (columns) by 20 (rows) network with vias. The dimension of each wire is 10x10x400µm (WxDxL). The material properties are given in Table 5.2. The boundary condition is 0oCtemperature on substrate and a current of 0.001 AMP is applied to wire 3. This network has1680 nodes including vias for Therminator3D and 45538 nodes for ABAQUS. The CPU timeof simulation of Therminator3D was found to be 181.19 sec (including 179 seconds to form thesymbolic solution and only 2 seconds for performing the numerical solution). The CPU time ofsimulation of ABAQUS was 391.70 sec. The symbolic solution in Therminator3D took mostof the CPU time before entering iteration loops of numerical solution. The number of iterationsin Therminate3D was 35 iterations (in ABAQUS, it was 2 iterations). However, the speed of715.5. Large-Scale Network Simulation(Avg: 75%)TEMP+0.000e+00+1.312e+00+2.625e+00+3.937e+00+5.250e+00+6.562e+00+7.875e+00+9.187e+00+1.050e+01+1.181e+01+1.312e+01+1.444e+01+1.575e+01Step: Step−1Increment      1: Step Time =    1.000Primary Var: TEMPDeformed Var: not set   Deformation Scale Factor: not setODB: wonven−001amp.odb    Abaqus/Standard Version 6.8−1    Fri Jul 16 09:40:55 Pacific Daylight Time 2010XYZFigure 5.22: Yarn temperature distribution for the network shown in Figure 5.20, computed byABAQUS.iterations of Therminator3D was very fast, each iteration only took 0.0571 sec, compared toABAQUS which took 185.85 sec.The second example shown in Figure 5.26 is a 20 (columns) by 20 (rows) network withoutvias and a temperature of 0oC is applied to the ends of all wires and the substrate. It has1680 nodes without vias for Therminator3D and 45538 nodes for ABAQUS. The CPU time ofTherminator3D is 181.18 sec. The CPU time of ABAQUS is 325.40 seconds. When comparedto the previous example, the CPU time of Therminator3D is not much affected by the presenceor absence of vias. It also shows that Therminator3D’s iterations only require O(N) additionaloperations and do not correlate with CPU time.Figure 5.27 shows the simulation results of Therminator3D program and Figure 5.28 showsthe simulation results of ABAQUS program. From those two figures, the temperature distribu-tions of T3D and ABAQUS are consistent. Table 5.6 shows that Therminator3D could reachthe purpose of fast simulation on large scale networks. In these large-scale simulations, I found725.5. Large-Scale Network Simulation(a) Yarn1 (b) Yarn2(c) Yarn3 (d) Yarn4(e) Yarn5 (f) Yarn6(g) Yarn7 (h) Yarn8Figure 5.23: The comparison of all 8 yarns between ABAQUS and Therminator3D on wovencase without vias and with heated yarn 5. (the dimension is changed to 10x10x180 (WxDxL)µm).735.5. Large-Scale Network Simulation(a) Yarn1 (b) Yarn2(c) Yarn3 (d) Yarn4(e) Yarn5 (f) Yarn6(g) Yarn7 (h) Yarn8Figure 5.24: The comparison of all 8 yarns between ABAQUS and Therminator3D on wovencase without vias and with heated yarn 7.745.5. Large-Scale Network Simulationthat the order of memory of the worst case (which has vias connected with columns and rows)to form the L and U matrices is near to N3 . The reason such a large memory is requiredwhile the size of networks increased, is the solver of the Crout algorithm to form the linked-listhashtables of L and U matrices. However, the iteration time of numerical solution is hardlyaffected by the scale of network.Finally, I include an additional simulation case of 18 (columns) by 18 (rows) with multi-boundary condition as shown in Figure 5.29. The dimensions of each wire is 10x10x400 µm(WxDxL) and its material properties are the same as those of previous large-scale networks.It is very straight forward to apply boundary conditions in Therminator3D. I use a set of cur-rents, current resistors and one resistor to create a heat source and apply any node designatedto the netlist file as shown in Figure 5.6. This shows the potential of Therminator3D programto demonstrate complicated boundary conditions on large-scale networks. The simulation timefor the latter example was 195.3 seconds and the required CPU for iteration became 1.9 sec-onds.Table 5.6: Comparison of CPU time required by Therminator3D and ABAQUS on Large Scalenetwork. Cases for Joule heat analysis with temperature-independent electrical and thermalconductivities.Dielectric No. ofElementsCPUTime(sec)IterationsT3DFig. 5.27(a) yes1680 + 400 (Vias)(1680 nodes)181.19(179 secondsfor symbolicsolution, 2seconds fornumericalsolution)35ABAQUSFig. 5.28(a)52263(45538nodes)391.70 2T3DFig. 5.27(b) yes1680(1680 nodes)181.18 35ABAQUSFig. 5.28(b)52263(45538 nodes)325.40 2755.5. Large-Scale Network SimulationFigure 5.25: The layout of a 20 (columns) by 20 (rows) large scale network with vias (X) with aheated wire (red line). The temperature of substrate is 0oC. The porosity of SiO2 is consideredto be 0 %.765.5. Large-Scale Network SimulationFigure 5.26: The layout of 20 (columns) by 20 (rows) large scale network without vias andwith a heated wire. The boundary condition of temperature of 0oC is applied to both ends ofall wires and the substrate. The porosity of SiO2 is considered to be 0 %.775.5. Large-Scale Network Simulationa160 a161a160 a162a160 a160 a162 a161a160 a163a160 a160 a163 a161a160 a164a160 a160 a164 a161a160 a165a160 a160 a165 a161a160 a166a161a160a160a161a160a162a160 a160a162 a161a160a163a160 a160a163 a161a160a164a160 a160a164 a161a160a165a160 a160a160a160 a167a161a162a162a167a161a163a163a167a161a164a160a160 a167a161a162a162a167a161a163a163a167a161a164a168 a169 a170 a171 a172a173 a174 a175 a176 a177a178 a179 a180 a181 a182a183 a184a183 a185a183 a183 a185 a184a183 a186a183 a183 a186 a184a183 a187a183 a183 a187 a184a183 a188a183 a183a183a184a183a185a183 a183a185 a184a183a186a183 a183a186 a184a183a187a183 a183a187 a184a183a188a183 a183a183a184a185a183a185 a184a186a183a186 a184a187a183a183a184a185a183a185 a184a186a183a186 a184a187a183a189 a190 a191 a192 a193a194 a195 a196 a197 a198a199 a200 a201 a202 a203Figure 5.27: The simulation results of Therminator3D for a 20x20 net. (a) Layout as Figure5.25. (b) Layout as Figure 5.26.785.5. Large-Scale Network Simulation(a)(Avg: 75%)TEMP+0.000e+00+2.284e−01+4.567e−01+6.851e−01+9.135e−01+1.142e+00+1.370e+00+1.599e+00+1.827e+00+2.055e+00+2.284e+00+2.512e+00+2.740e+00Step: Step−1Increment      1: Step Time =    1.000Primary Var: TEMPDeformed Var: not set   Deformation Scale Factor: not setODB: a−nm−20−20−new−a−test1.odb    Abaqus/Standard Version 6.8−1    Sat Jul 10 16:53:31 Pacific Daylight Time 2010XYZ(b)(Avg: 75%)TEMP+0.000e+00+2.277e+00+4.555e+00+6.832e+00+9.110e+00+1.139e+01+1.366e+01+1.594e+01+1.822e+01+2.050e+01+2.277e+01+2.505e+01+2.733e+01Step: Step−1Increment      1: Step Time =    1.000Primary Var: TEMPDeformed Var: not set   Deformation Scale Factor: not setODB: a−nm−20−20−new−b−test1.odb    Abaqus/Standard Version 6.8−1    Sat Jul 10 10:15:32 Pacific Daylight Time 2010XYZFigure 5.28: The simulation results of ABAQUS for a 20x20 net. (a) Layout as Figure 5.25.(b) Layout as Figure 5.26.795.6. Evaluation of Effective Thermal Conductivitya204 a205 a206a207 a208 a208 a209 a210 a211a212 a213 a213 a214 a215 a216a217 a218 a218 a219 a220 a221a222 a223 a223 a224 a225 a226a227 a228 a228 a229 a230 a231a232 a233 a233 a234 a235 a236a237 a238 a238 a239 a240 a241a242 a243 a243a244a245 a246a247 a248a249 a250a251 a252a253 a254a255 a0a1 a2a3 a4a5 a6a7a8 a9a10 a11a12 a13a14 a15a16 a17a18 a19a20 a21a22 a23a24 a25a26 a27 a28 a29 a30 a31 a32 a33 a34 a35a36 a37 a38 a39 a40Figure 5.29: The multiple boundary condition simulation results by Therminator3D for a 18x18net.5.6 Evaluation of Effective Thermal ConductivityTherminator3D program is also a tool which could rapidly perform sensitivity analysis oneffective thermal conductivities of textile networks. Let us consider a three-factor and three-level design of experiment (DOE) to evaluate the effect of significant factors on the effectivethermal conductivity of a typical woven fabric. Figure 5.30 is the cause-effect diagram toindicate these factors. Detailed steps of the conducted DOE are shown in Appendix D.Similar parametric studies could be used for other mechanical properties of textiles. I couldalso use Therminator3D to find optimal textile materials on an specific design objective suchas maximization of effective thermal conductivity. For maximizing the thermal conductivityof a given textile material, varying the degree of via contacts should be considered as a firstattempt, without a need to change the fibre/matrix materials. As a result, different optimumproducts could be manufactured with minor changes in fabrication process. In fact, in my DOE805.7. Summarystudy, the vias effect (see Figure C.2 in Appendices D ) proved to be the highest among thestudy variables (width of wires, vias, and porosity of matrix).Effective Thermal Conductivity Vias Width of Wire  Porosity of matrix Connection type Thermal conductivity Volume of main wire  Percentage of porosity Thermal conductivity y Thermal conductivity Figure 5.30: Cause-effect diagram used in the DOE study.5.7 SummaryIn summary, the symbolic solution procedure and the incorporation of analytical heat solutionover each interconnect segment allows the net-based approach in Therminator3D to efficientlyaddress the equivalent resistance circuits that incorporate the thermal transport of intercon-nects. From examples performed in this chapter, the speed and accuracy of Therminator3Dare very satisfactory. The examples also showed that Therminator3D can solve large scalesimulations in a short time and the consumed CPU time is not significantly affected by the iter-ations required for nonlinear problems. It is also able to deal with several boundary conditionsproblems. The analytical model and network structure mean Therminator3D uses fewer nodes,and thus less memory than FEA, while offering a competitive accuracy. In FEA, the dielectricmaterial is actually included in the CAD model and meshed, whereas in Therminator3D noadditional elements are used for the dielectric (matrix). Future studies are needed to use layout815.7. Summaryextractors, such as Magic, to extract netlists and more accurate coupling input files, speciallyfor woven networks.82Chapter 6ConclusionsIn this chapter, I summarize the main conclusions of the conducted research and present somefuture work directions for further development and improvement of the work.6.1 Summary of ContributionsThe new network-based temperature simulation program, Therminator3D, was successfullydeveloped. This program employs a modified RC network which is developed from electromi-gration reliability verification concept. Performance of the proposed network-based simulationhas been verified through several test cases. Therminator3D appeared to be a fast and accuratethermo-electrical simulation tool throughout this thesis. In the case of large network simula-tions, the memory need was large during symbolic solution procedure because of forming LUmatrices by Crout algorithm. However, after the LU matrices are formed, the CPU-time usageof iterations to reach the thermal solution was found to be much less than traditional FEA tool.The main characteristics of Therminator3D can be summarized as follows.1. Network-based approach in Therminator3D efficiently addresses the equivalent resis-tance circuits that incorporate the thermal transport of interconnects, and fibre yarns inthe case of textiles.2. Therminator3D can solve large interconnect/textile composite simulations in a short timeand the consumed time is not significantly affected by the iterations required for nonlin-ear problems.836.2. Future Work3. The analytical models and network structures mean that Therminator3D uses fewer nodesand thus less memory than FEA. In FEA, the dielectric/matrix material is actually in-cluded in the CAD model and meshed, whereas in Therminator3D no additional elementsare used for the dielectric/matrix.4. Therminator3D can be used as a fast and reliable simulation tool in optimal deign of ICinterconnect/textile networks at micro/meso scales.There are assumptions applied to the applications of this study:1. This research only calculated heat conduction behaviour. Heat convection and heat radi-ation are not considered in this study.2. This research only simulated steady-state problems.3. The temperature in each conductor varies only in the lengthwise direction.4. The property of via represents the degree of contact between column and row. Theapplications in this research assumed that those contacts are perfectly connected withcolumns and rows, and without Glat.6.2 Future WorkThe work represented in this thesis was a first step to improve the simulation of large scale net-work models using Therminator3D. I only illustrated the thermal and electrical problems on ICinterconnects and textile materials. However, there is a good potential to develop several othermechanical simulations on material behaviours, such as strain-stress response which is nor-mally obtained by FEA, and multi-physic problems which often require higher computationalresources.To achieve these future goals, there are four phases to follow:846.2. Future Work1. Develop more advanced coupling formulas to include layout variables such as the angleof yarn, distance, effective properties, etc. Also consider coupling effects between morewires in complex layouts such as woven fabrics. Since the simple area model is not per-fectly working in the woven structures, formulation of the coupling value with differentcurvilinear shapes should be calibrated by experiments and/or FEA results. The otheroption to improve the accuracy of coupling data is using a field solver to calculate thecoupling data.2. Modify an open-source layout extractor, such as Magic, to extract netlist input files withdetailed thermal coupling data. In this research, netlist and coupling data were the keyinput information. Using a friendly and precise layout tool will improve accuracy ofsimulations and avoid unexpected convergence errors.3. Modify the Therminator3D code to increase its speed even further. Therminator3Ddemonstrated a fast and accurate temperature estimation on large scale IC interconnectsand textile composite networks. However, for the large scale problem, the Crout algo-rithm in symbolic solution increased the consumption of memory in forming LU matri-ces dramatically. One can employ a different decomposition algorithm, such as SuiteS-parseQR Decomposition [66], to store the information of large scale networks whichwould decrease the convergence order from O(N3) to O(N∼1.7).4. Simulate different mechanical properties in textile composite materials and IC fields.There are very complicated and ultra-large structures which could be considered for me-chanical stress analysis coupled with an interaction of the electrical behaviour of wires/-yarns.85Bibliography[1] “2007 itrs roadmap for semiconductors,interconnect table intc1, 2,http://www.itrs.net/links/2007itrs/home2007.htm,” 2007.[2] J. Black, “Electromigration;a brief survey and some recent results,” Electron Devices,IEEE Transactions on, vol. 16, no. 4, pp. 338 – 347, Apr 1969.[3] S. Sankaran, S. Arai, R. Augur, M. Beck, G. Biery, T. Bolom, G. Bonilla, O. Bravo,K. Chanda, M. Chae, F. Chen, L. Clevenger, S. Cohen, A. Cowley, P. Davis, J. De-marest, J. Doyle, C. Dimitrakopoulos, L. Economikos, D. Edelstein, M. Farooq, R. Fil-ippi, J. Fitzsimmons, N. Fuller, S. M. Gates, S. E. Greco, A. Grill, S. Grunow, R. Han-non, K. Ida, D. Jung, E. Kaltalioglu, M. Kelling, T. Ko, K. Kumar, C. Labelle, H. Landis,M. Lane, W. 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Potluri and V. Thammandra, “Influence of uniaxial and biaxial tension on meso-scalegeometry and strain fields in a woven composite,” Composite Structures, vol. 77, no. 3,pp. 405 – 418, 2007.[65] W. Steinhoegl, G. Schindler, G. Steinlesberger, M. Traving, and M. Engelhardt, “Scalinglaws for the resistivity increase of sub-100 nm interconnects,” in Simulation of Semi-conductor Processes and Devices, 2003. SISPAD 2003. International Conference on, 3-52003, pp. 27 – 30.[66] T. A. Davis, “Users guide for SuiteSparseQR, a multifrontal multithreaded sparse QRfactorization package,” 2008.94Chapter 6. Bibliography[67] C.-S. Liao, A. Labun, and A. S. Milani, “A rapid method to compute the temperaturedistribution in non-crimp fabric composites,” in 17th International Conference on Com-posite Materials, Edinburgh, UK, 2009.95Appendix AThe Input Parameters of Weave ProgramThere is a pre-processor program, called “Weave”, that was used instead of layout extractiontools. It created netlist file and coupling file of an interconnect layout for the examples Idemonstrated in this thesis. The following technology parameters used in this program areshown as samples:##### Technology parameters########### Vertical distance between rows and substrate in nmA=10# Row thickness in nmB=10# Vertical distance between rows and columns in nmC=10# Column thickness in nmD=10# zero-porosity isotropic dielectric coefficient of dielectricE=3.8# electric potential on voltage sources (V) or current on current sources# (A), depending on preceding (v) or (i) flagF=1# bulk thermal conductivity of interconnect W/nm-K# True value: G=7.1E-796Appendix A. The Input Parameters of Weave ProgramG=7.1E-7# Number of rows per netK=2# Number of columns per netL=2# Number of nets (1 or 2)M=1# Number of resistors between vias on a row in net 1N=4# number of resistors between vias on a row in net 2n=4# % porosity of dielectric (0 <= O <= 100)O=0# Number of resistors between vias on a column in net 1P=4# Number of resistors between vias on a column in net 2p=4# Temperature coefficient of resistance (%/K)#Q=0.33Q=0# bulk electrical resistivity of interconnect 0hm-nm# [From Steinhoegl, et al, 2003, Fig. 6, for T=300K]R=22# Resistance of vias in ohmsS=2.2# zero-porosity thermal conductivity of dielectric W/nm-K# True value: T=1.4E-997Appendix A. The Input Parameters of Weave ProgramT=1.4E-9# Row pitch in nm (for rows on the same net!)U=40# Column pitch in nm (for columns on the same net!)V=40# Row width in nmW=10# Column width in nmX=10# Factor for thermal & electrical source resistances (i.e. source resistance = Y * via resistance)Y=1000# Multiplier for longitudinal thermal resistance on row terminating resistorsZ=1.0E398Appendix BModified Netlist and Coupling Data forSimulationsThe netlist and coupling data of different simulation cases can be modified to apply differentboundary conditions. Below an example is shown:Original format of a netlist file: (for a four columns by four rows case )Resistor node0 nodeL R # xlowerle ftylowerle ftxupperrightyupperrightLRlongCurrent source node0 nodeL IrmsRES4 36 3.300000 # 135 90 150 90 10.000000 211267.605634AMP1 0 37 0.000000RESAMP1 0 37 30000000.000000 # 25 -35 35 -25 0.001 0.0001. . . . . . . . .RES2-39 2 39 300000000000.000000 # 25 -5 35 5 10.000000 192061.459667How to apply the boundary conditions on the netlist file:1. Keep the original values of Rlong from the weave program, the simulation will keep the“Adiabatic” condition .2. To set a heat flux condition, I input the value of current Irms into the AMP set.3. To make the temperature zero degree on one specific point, I simply apply a very smallRlong (large Glong) on that point and connect it to the substrate.99Appendix B. Modified Netlist and Coupling Data for Simulations4. To make a specific temperature BC, I use a set of regular resistor, current sources andvery large current resistors. The large current resistor is to keep the current in the set(e.g., along a given wire). The format of this set is shown below for a 20-column by20-row case with a fixed temperature boundary condition applied on node 1228.AMP41 0 1681 1.000000RESAMP40 0 1681 3000000000.000000 # 405 405 415 415 10.000000 192061459.667093RES1681 1228 1681 2.200000 # 405 415 415 425 10.000000 140845.070423AMP42 0 1228 -1.000000RESAMP40 0 1228 3000000000.000000 # 415 415 425 425 10.000000 192061459.667093Woven material application :For woven material applications, I simply change the coupling data with actual location ofthe woven resistor.Example of an original coupling data with the interconnects coupled to the ground substrateis shown below (the example is a four-column by four-row case):(cap resistor {resistor|0(substrate)} c # GlatCAP RES23 0 6750 # 0.000000067581395396CAP RES24 0 6750 # 0.000000067581395396CAP RES25 0 750 # 0.000000007509043933CAP RES26 0 750 # 0.000000007509043933CAP RES27 0 750 # 0.000000007509043933CAP RES28 0 500 # 0.000000005006029289. . . . . . ..The modified coupling data for the woven material coupled to the ground and an upper levelsubstrate (RES97 is the upper level substrate)is as follows (the following information is for thewoven case in Figure 5.20)CAP RES1 0 646.6666667 # 0.00000000647446455100Appendix B. Modified Netlist and Coupling Data for SimulationsCAP RES2 0 2833.333333 # 0.00000002836749930CAP RES3 0 2833.333333 # 0.00000002836749930CAP RES4 0 480 # 0.00000000480578812CAP RES5 0 480 # 0.00000000480578812. . . . . . ..CAP RES1 RES97 2833.333333 # 0.00000002836749930CAP RES2 RES97 646.6666667 # 0.00000000647446455CAP RES3 RES97 646.6666667 # 0.00000000647446455CAP RES4 RES97 2833.333333 # 0.00000002836749930CAP RES5 RES97 2833.333333 # 0.00000002836749930CAP RES6 RES97 480 # 0.00000000480578812CAP RES7 RES97 480 # 0.00000000480578812. . . . . . ..101Appendix CUsing DOE in the Evaluation of EffectiveThermal Conductivity of the Fabric withT3DThe effective thermal conductivity of a bar with an uniform cross-section can be assessed by:Ke f f = qAx∆TKe f f is the effective thermal conductivity, q is heat flux , x is the length, and ∆T is the differenceof temperature,Previous studies show that Ke f f increases as the ratio of yarn width to gap (pitch) spacingincreases, and also increases as the yarn or resin materials conductivity increases [67].In my DOE study using Therminator3D, the model simulated was two-column by two-rowmodel as shown in Figure C.1. All parameters are the same as those shown in Table 5.2. The listof chosen factors and their levels is shown in Table C.1. The ANOVA anaysis was performedin Minitab. From the effects plots, shown in Figure C.2 and C.3, I could easily identify themost effective factors by the slopes of the plots. In this study, the present of vias and the widthof yarn (represents the inverse of gap between two parallel wires) seem to be key factors tomodify the effective thermal conductivity of the fabric (Table C.2 and Figure C.2). The viasfactor contributes 95% to the effective thermal conductivity variation while the width of yarn(wire) has 3.95% contribution. The porosity has only 0.00378941% contribution. Figure C.3shows that there are no interaction effects between the parameters (The interaction between102Appendix C. Using DOE in the Evaluation of Effective Thermal Conductivity of the Fabric with T3Dvias and the yarn width is as low as 0.23%). The ANOVA model has a p-value < 1.34E-07. Itmeans that the model has more than 99.9999% confidence.Figure C.1: The layout used in the DOE study.103Appendix C. Using DOE in the Evaluation of Effective Thermal Conductivity of the Fabric with T3DTable C.1: Study factors and their levels.Experiment No. Porosity Width Vias Ke f f1 0 10 yes 5.591924392 0 10 no 3.453290903 90 10 yes 5.591631824 90 10 no 3.437952265 0 15 yes 6.159309386 0 15 no 3.814176907 90 15 yes 6.159082768 90 15 no 3.77329034Table C.2: ANOVA results and effect estimates from minitab.Parameters Effect Estimate Sum of Squares Percent ContributionWidth (A) 0.457765004 SSA 0.419097597 3.94576573%Porosity (B) -0.014186096 SSB 0.000402491 0.00378941%Vias (C) 2.255809488 SSC 10.17735289 95.81885115%AB -0.006370492 SSAB 8.12E-05 0.00076417%AC 0.109652961 SSAC 0.024047544 0.22640544%BC 0.013926506 SSBC 0.000387895 0.00365200%ABC 0.006403465 SSABC 8.20E-05 0.00077211%Sserror 4.62E-14SST 10.62145159104Appendix C. Using DOE in the Evaluation of Effective Thermal Conductivity of the Fabric with T3DFigure C.2: Main effects plot for Ke f f .105Appendix C. Using DOE in the Evaluation of Effective Thermal Conductivity of the Fabric with T3DFigure C.3: Interaction effects plot for Ke f f .106

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