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DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massively Coupled 3D Mi

DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massively Coupled 3D Microelectronic Structures Indranil Chowdhury, Vikram Jandhyala Dipanjan Gope* ACE Research Department of Electrical Engineering University of Washington

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DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massively Coupled 3D Mi

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  1. DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massively Coupled 3D Microelectronic Structures Indranil Chowdhury, Vikram Jandhyala Dipanjan Gope* ACE Research Department of Electrical Engineering University of Washington Design and Technology Solutions INTEL Corporation Supported by: NSF, SRC and DARPA

  2. Class of Problems Magnetostatic Problems Electrostatic Problems DiMES: FAST DIRECT SOLVER ALGORITHM Electric Field Integral Equations Magnetic Field Integral Equations PMCHW: Multi-Region Dielectric Problems

  3. Outline • Focus Application: Accurate Charge Distribution • - Circuit Parasitic Estimation • - MEMS Charge Distribution • Motivation behind Fast Direct Solution • - Large Number of RHS Vectors • - Re-simulation Advantages • DiMES: Fast Direct FMM based Solver • - Sparsification of MoM Using FMM • - Sparse 1.3 Solution • Numerical Results

  4. Deep Sub-Micron and Nano Fabrication Technology - Gate delay reduces Overall chip size does not decrease - More functionalities added to the same chip Switching Speed: Function of Interconnect Parsitics • = 0.0103pF • = 0.0103pF • = 0.0153pF • = 0.0153pF • = 0.0069pF • = 0.0069pF • = 0.0153pF • = 0.0069pF • = 0.0103pF • = 0.0153pF • = 0.0103pF • = 0.0069pF • = 0.0153pF • = 0.0103pF • = 0.0069pF • = 0.0153pF • = 0.0103pF • = 0.0069pF • Spacing between traces reduced • Spacing between traces reduced • Spacing between traces reduced • Spacing between traces reduced • Spacing between traces reduced • Spacing between traces reduced • Aspect Ratio (H/W) Increases • Aspect Ratio (H/W) Increases • Aspect Ratio (H/W) Increases • Aspect Ratio (H/W) Increases • Aspect Ratio (H/W) Increases • Aspect Ratio (H/W) Increases Size Size Size 250nm 250nm 250nm 70nm 70nm 70nm ITRS Data ITRS Data ITRS Data Spacing Spacing Spacing 340nm 340nm 340nm 100nm 100nm 100nm H/W H/W H/W 1.8:1 1.8:1 1.8:1 2.7:1 2.7:1 2.7:1 Courtesy: VLSI Systems WPI web-course Increasing Interconnect Parasitics

  5. MEMS: Electrical Force Computations MEMS Electrical Force Computation Requires Accurate Simulation of Charge Distribution • Approximate Solutions: Inaccurate Charge Distribution • Inaccurate Charge Distribution: Inaccurate Force Computation

  6. Solution Scheme Solution Scheme Analytic Numerical Inexpensive but Inaccurate Accurate for 3D Arbitrary Shaped Objects Accurate Prediction of Charge Distribution • Method of Moments (MoM) • Well-Conditioned System • Smaller Sized Matrix • Dense Matrix

  7. Surface is Discretized into Patches (Basis Functions) Pulse Method of Moments • Basis Functions Interact through the Green’s Function • Generates a Dense Method of Moments Matrix

  8. Courtesy: Ansoft Corporation Practical problems: N ~ 1 million Fast Solvers: Significance N = Number of basis functions; (50,000) p = Number of iterations per RHS; r = Number of RHS • Fast Iterative Methods: Mature Field • - Fast Multipole Method (FastCap) [Nabors and White 1992] • - Pre-Corrected FFT Method [Phillips and White 1997] • QR Based Method (IES3) [Kapur and Long 1997] • QR Based Method (PILOT) [Gope and Jandhyala 2003] • O(N)-O(NlogN) Matrix Vector Products • Why Look Any Further?

  9. Outline • Focus Application: Accurate Charge Distribution • - Circuit Parasitic Estimation • - MEMS Charge Distribution • Motivation behind Fast Direct Solution • - Large Number of RHS Vectors • - Re-simulation Advantages • DiMES: Direct Multipole Expansion Solver • - Sparsification of MoM Using FMM • - Sparse 1.3 Solution • Numerical Results

  10. α=2; β=1 Fast Setup and Solve p1=2xp ILL-Conditioned Problem α=2; β=2 Fast LU Setup α=3; β=2 Direct / LU Fast Direct Fast Iterative Fast Iterative Fast Iterative Fast Iterative Fast Direct Fast Direct • ILL-Conditioned Problems (More Prominent for EFIE) • Large Number of Excitations / Large Number of RHS Vectors Motivn 1: Large Number of RHS Vectors Direct Setup + Solve Fast Iterative Setup + Solve Setup Solve Solve Setup N=10,000 p=90

  11. Motivn 2: Fast Updates in Re-simulation Critical Transition: Analysis to Solution 1. Schur Complement 2. SMW-Updates = B + A Ax+By=z1; Cx+Dy=z2; (A+BD-1C)x=z1-BD-1z2 D A U V M C Repeated Simulation: Update vs. Re-Solve

  12. Advances In Fast Direct Solvers NOT Comparable To Advances In Fast Iterative Solvers Existing Literature • Michielssen, Boag and Chew (1996) • - Reduced Source Field Representation • Canning and Rogovin (1999) • - SMW Method • - LUSIFER • Hackbusch (2000) • - H-Matrices • Gope and Jandhyala (2001) • - Compressed LU Method • Yan, Sarin and Shi (2004) • - Inexact Factorization • Forced Matrix Structure Unsuitable for Arbitrary 3D Shapes • Fillins: Chief Cost Factor / Neglected

  13. Outline • Focus Application: Accurate Charge Distribution • - Circuit Parasitic Estimation • - MEMS Charge Distribution • Motivation behind Fast Direct Solution • - Large Number of RHS Vectors • - Re-simulation Advantages • DiMES: Direct Multipole Expansion Solver • - Sparsification of MoM Using FMM • - Sparse 1.3 Solution • Numerical Results

  14. Number of degrees of freedom << Number of basis functions (Well-separated groups) = = Translation: Same Size Translation Fast Multipole Basics 1D Geometry MoM Matrix

  15. Multilevel Multipole Operators Q – Q2M – M2M – M2L – L2L – L2P – P M2L Finest - 1 Level M2M L2L L2L M2M M2L Finest Level M2L Q2M L2P Down Tree Up Tree Across Tree

  16. = = Q2P = = Reconstruct with Multipoles Problems in Single Matrix Formation M2L M2Ms Q2M L2Ls L2P Fast Matrix Vector Products Fast Multipole Iterative Method Does Not Inherently Lend Itself to Fast Direct Solution

  17. Step 1: Increase LHS Size Step 2: Use Multipole Expansions Step 1 On Its Own Will NOT Expedite; Step 1 is ONLY Required To Achieve Step2 Modified LHS Z q V Are We Simply Increasing the Size of the Matrix to Make it Sparse? No = • Size of the Matrix Increases • Non-Zero Entries = O(No) • Non-Zero Entries NOT No2 q ML Nn Multipole Expansions ML-1 LL Local Expansions LL-1

  18. Q2P q Q2M ML L2P ML-1 M2M LL LL-1 M2L L2L Modified Set of Equations LHS • 1st Set of Equations: Formation of V • - Contribution from q via Q2P (Finest Level) • - Contribution from L via L2P (Finest Level) • 2nd Set of Equations: Formation of M • - Contribution from q via Q2M (Finest Level) • - Contribution from M (From Level Below) via M2M • 3rd Set of Equations: Formation of L • - Contribution from M via M2L (Same Level) • - Contribution from L (From Level Above) via L2L

  19. = 4 Level Sparse Matrix Set 1 Set 2 Set 3 • Total Number of Non-zero Entries is O(N)

  20. Optimization: Number of Levels • Increase Levels: More Sparsity • Increase Levels: Larger Size of the Matrix Dry Run: Pre-Estimation of Number of Levels • Re-Order The Unknowns Based on Geometry • Dry-Run Cost is a Function of Fillin-Factor (w)

  21. Outline • Focus Application: Accurate Charge Distribution • - Circuit Parasitic Estimation • - MEMS Charge Distribution • Motivation behind Fast Direct Solution • - Large Number of RHS Vectors • - Re-simulation Advantages • DiMES: Direct Multipole Expansion Solver • - Sparsification of MoM Using FMM • - Sparse 1.3 Solution • Numerical Results

  22. Hughes Test Chip ic_hrl_tc1 Validation Example Multipole Order (p): 2 1.5GB RAM and 1.6GHz Processor Speed Capacitance Matrix Norm Difference < 1e-3

  23. Hughes Test Chip ic_hrl_tc1 Time and Memory α=3 Memory Time: LU Setup β=2 α=1.8 β=1.2 β=2 Time: LU Solve β=1.2

  24. Substrate Coupling Problem 2500 Metal Contacts; 6500 Charge Basis Functions

  25. Comparison with FastCap Cutoff Point: 360 RHS Vectors Below Cutoff: Fast Iterative Solver Above Cutoff: Fast Direct Solver

  26. Highlight Conclusions and Future Work • Conclusions: • First of Its Kind Multilevel Multipole-based Direct Solver • Matrix Structure is Not Forced: • - Valid for Arbitrary 3D Structures • Fillins are Not Neglected • - Guaranteed High Accuracy • Future Work: • Reduce Setup Time • - Increasing N will Increase Cut-off Point More than Linearly

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