Recent developments for MOR in the electronics industry
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1 Recent developments for MOR in the electronics industry Wil Schilders Reduced Order Models in Computational Science and Engineering Aachen, January 30-31, 2014
2 Full proposal oc for a new COST action EU-MORNET: European Model Reduction Network Proponent Wil Schilders
3 Applications of Model Reduction Multibody systems Medical and dental modeling Control and power Tsunami risk analysis Automotive Social networ ks PAGE 3
4 What is Model Reduction? Systems and control, Lyapunov, Truncated Balanced Realization Scientific computing, numerical MOR, Krylov methods, linear algebra, tensors Pade-via-Lanczos and PRIMA Passivity preserving Structure preserving Linear and nonlinear problems Parameterized methods Tensor analysis Large-scale Lyapunov systems Balanced realizations Observability and controllability Gramians Port-Hamiltonian systems Hankel singular values Projection methods MOR at operator level Mathematical modeling, behavioral modeling, models via data Karhunen-Loeve expansions Neural networks Vector fitting Behavioral models PAGE 4
5 Why these topics together? Scientific computing, numerical MOR, Krylov methods, linear algebra, tensors Systems and control, Lyapunov, Truncated Balanced Realization Remarkably, they share the same common phenomenology! Mathematical modeling, behavioral modeling, models via data PAGE 5
6 More information Kick-off meeting May/June 2014 Management Committee currently being formed After kick-off, addditional partners can join the network Basis for new European projects EID: European Industrial Doctorate (1 univ, 1 ind) ITN/ETN W. H. A. Schilders, J. M. L. Maubach and S. Lungten PAGE 6
7 MOR in the electronics industry Methods like PVL, PRIMA were developed by people associated with electronics companies The electronics industry has been extremely stimulating for progress in model order reduction, as designers need very fast simulations of extremely large structures PAGE 7
8 Motivation for the work Accurate Simulations require a high level of detail Huge models whose simulations are computationally very expensive! Models inner states may be too detailed Solution: find a smaller yet equivalent model Model Order Reduction But EM effects are becoming more relevant! Scaling reduction Integration Increase Higher Frequency UvA, PAGE 8
9 The impossible made possible.. A complete package layout used for full wave signal integrity analysis 8 metallization layers and 40,000 devices The FD solver used 27 million mesh nodes and 5.3 million tetrahedrons The transient solver model of the full package had 640 million mesh cells and 3.7 billion of unknowns 9
10 Avoiding brute force The behaviour of this MOS transistor can be simulated by solving a system of 3 partial differential equations discrete system for at least unknowns Insight? Even worse: an electronic circuit consists of MOS devices Solution: compact device model (made by physicists/engineers based on many device simulations, ~50 unk) Can we construct such compact models in an automated way? 10
11 Another example Integrated circuits need 3-D structure for wiring 10 years ago 1-2 layers of metal, no influence on circuit performance Present situation: 8-10 layers of metal Delay of signals, parasitic effects due to high frequencies 3-D solution of Maxwell equations leads to millions of extra unknowns Gradual Can we construct a compact model for the interconnect in an automated way? circuit 11
12 Model Order Reduction is about capturing dominant features 12
13 Example: MOR for transmission line (RC) Original Reduced Netlist size (Mb) # ports # internal nodes # resistors # capacitors
14 Many challenges in the electronics industry Extremely large resistor networks are used to describe substrate propagation, power transistors and more Designers want to know the sensitivity of their designs with respect to many (!) parameters The positioning of analog components on a chip is vital for the performance Challenges in linear & nonlinear MOR PAGE 14
15 Outline Low-order models at work in the electronics industry: MOR for purely algebraic systems MOR for nonlinear systems MOR for DAEs Conclusions 15 15
16 Outline Low-order models at work in the electronics industry: MOR for purely algebraic systems MOR for nonlinear systems MOR for DAEs Conclusions 16 16
17 Challenge: Large resistor networks Obtained from extraction programs to model substrate and interconnect Networks are typically extremely large, up to millions of resistors and thousands of inputs/outputs Network typically contains: Resistors Internal nodes ( state variables ) External nodes (connection to outside world, often to diodes) Model Order Reduction needed to drastically reduce number of internal nodes and resistors 17
18 Reduction of resistor networks 18
19 Deleting all internal nodes: not an option 19
20 Simple network 274 external nodes (pads, in red) 5384 internal nodes 8007 resistors/branches Can we reduce this network by deleting internal nodes and resistors, still guaranteeing accurate approximations to the path resistances between external nodes? NOTE: there are strongly connected sets of nodes (independent subsets), so the problem is to reduce each of these strongly connected components individually 20
21 Two-Connected Components can delete all internal nodes here 21
22 Other examples January 2010 PAGE 22
23 Solution: Block Bordered Diagonal form Algorithm was developed to put each of the strongly connected components into BBD form (see figure) Internal nodes can be deleted in the diagonal blocks, keeping only the external nodes and crucial internal nodes The reduced BBD matrix allows extremely fast calculation of path resistances (work of Duff et al) 23
24 Reduction results See also thesis Roxana Ionutiu (2011) 24
25 Outline Low-order models at work in the electronics industry: MOR for purely algebraic systems MOR for nonlinear systems MOR for DAEs Conclusions 25 25
26 Challenge: Fast and accurate simulation of oscillators ( nonlinear MOR) RC Cal SD- ADC Test Pins Xtal CLK DAC MIXER PA + Matching Digital Core Modem and control Tripler LNA + Matching IF Amp VCO + PLL PLL Filter 26
27 Classification of circuits Engineer Mathematician Analysis by Jaijeet Roychowdhury 27
28 Nonlinear modeling of perturbed oscillators 1. PSS of oscillator: d/dt [q(x PSS )]+ j(x PSS )=0, T=T OSC 2. u 1 (t)= d/dt(x PSS )(t) ( right Floquet eigenfunction) 3. v 1 (t) solves a linearized adjoint system ( left Floquet eigenfunction) 4. Perturbed oscillator: d/dt [q(x)]+ j(x)= b(t) has solution x(t)=x PSS (t+ (t))+x n (t), [ (t) phase noise] 5. (t) satisfies a non-linear scalar differential equation d/dt ( )(t) = v 1 (t+ (t)).b(t), (0)=0 28
29 Example- three stage ring oscillator 153kHz ring oscillator Unlocked : i inj = 6 * 10-5 *sin(1.04ω 0 * t) Locked osc: i inj =6 * 10-5 * sin(1.03ω 0 * t) 29
30 MOR: Fast and accurate modeling of VCO pulling VCO pulling due to PA and other blocks/oscillators needs to be analyzed before production Full system simulation is CPU intensive or infeasible Behavioural model order reduction gives fast and accurate insight in pulling/locking and coupling Full mathematical theory of locking/unlocking mechanisms and conditions lacking! 30
31 Outline Low-order models at work in the electronics industry: MOR for purely algebraic systems MOR for nonlinear systems MOR for DAEs Conclusions 31 31
32 General idea of Index-aware Model Order Reduction (IMOR) PAGE 32
33 Differential algebraic systems Why did we develop IMOR? PAGE 33
34 Model of a generator PAGE 34
35 Model of a generator PAGE 35
36 PRIMA reduced order generator model January 2010 PAGE 36
37 Index-aware MOR needed PRIMA may run into problems for higher index systems Besides, we feel that it is always good to mimic the structure and properties of the original problem Mimetic methods are gaining popularity, but have been developed for a long time: Exponentially fitted schemes for singularly perturbed and stiff differential equations Modified ICCG method for iterative solution of linear systems MOR for port-hamiltonian systems As the basis for our IMOR method, we use a method developed in the 1990 s 11 October 2011 PAGE 37
38 März decoupling procedure January 2010 PAGE 38
39 März decoupling procedure January 2010 PAGE 39
40 Modification of decoupling procedure January 2010 PAGE 40
41 Modified index-1 system January 2010 PAGE 41
42 IMOR-1 method descriptor form January 2010 PAGE 42
43 IMOR-1 method reduced order form January 2010 PAGE 43
44 Modified index-2 system January 2010 PAGE 44
45 Modified index-2 system January 2010 PAGE 45
46 IMOR-2 method descriptor form January 2010 PAGE 46
47 IMOR-2 method reduced order form January 2010 PAGE 47
48 Why a new method IIMOR? The IMOR method leads to algebraic systems that are explicit in the algebraic variables This is due to the way the decoupling method is described/constructed Not attractive in practice: if we start with a large resistor network (purely algebraic), IMOR would need the inverse of the system matrix Question: can we develop a projection method that leads to implicit algebraic systems? so that we can use the methods we developed for the reduction of purely algebraic systems January 2010 PAGE 48
49 The Implicit IMOR method PAGE 49
50 Delaying the inversion in the decoupling January 2010 PAGE 50
51 Implicit index-1 decoupled system January 2010 PAGE 51
52 Descriptor form January 2010 PAGE 52
53 Construction of bases for projector January 2010 PAGE 53
54 Numerical results for IIMOR method PAGE 54
55 January 2010 PAGE 55
56 January 2010 PAGE 56
57 January 2010 PAGE 57
58 January 2010 PAGE 58
59 January 2010 PAGE 59
60 January 2010 PAGE 60
61 January 2010 PAGE 61
62 January 2010 PAGE 62
63 January 2010 PAGE 63
64 January 2010 PAGE 64
65 January 2010 PAGE 65
66 January 2010 PAGE 66
67 January 2010 PAGE 67
68 Conclusions PAGE 68
69 Needed (future work): Use the methods we developed for purely algebraic systems also in this IIMOR context (cf paper by Schilders, Marcotte, Shontz in COMPEL, 2012) January 2010 PAGE 69
70 References January 2010 PAGE 70
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