Introduction to Simulation Lecture 24

1
Introduction to Simulation - Lecture 24
Model-Order Reduction
Jacob White
Thanks to Luca Daniel, Guillaum Lassauxx, Jing
Li, Mark Reichelt, Deepak Ramaswamy, Michal
Rewienski, Mary Tolikas, Karen Veroy and Karen
Willcox
2
MOR Outline

• Need For Model Reduction
• Circuits, MEMS, Jet Engines
• Steady-State Case (linear and nonlinear)
• Dynamic Linear Case
• Eigenmodes and Rational Functions
• Projection Framework, Krylov, TBR
• Nonlinear Case
• Projection Framework

3
Todays Outline

• Need For Model Reduction
• Circuits, MEMS, Optics, Jet Engines
• Simple Example Problem
• Heat Conducting bar example
• Steady-State Case (linear and nonlinear)
• Dynamic Linear Case
• Truncating Eigenmodes
• Rational Function Fitting

4
Signal Wire in an Integrated Circuit
Application Example
Signal Wire
Wire has resistance
Wire and ground plane form a capacitor
Logic Gate
Logic Gate
Ground Plane
Reduced-Order Model

• Assess wiring impact on IC performance
• Wire Model must preserve terminal behavior

5
Jet Engine Design
Application Example

• Generate Low-order models directly from
  Navier-Stokes Equation based physical simulators.
• Reduced model must preserve instabilities.

Operating Conditions
Sensing
Reduced-Order Inlet Model
Actuator Model
Flow Disturbances
Actuation
Flow features of Interest
6
Micromechanical Resonators in a Wireless
transceiver
Application Examples
RF Front end with micromachined resonators for
the oscillator

• What is system performance (noise, distortion,
  etc).
• Will poly-substrate separation (changes Q)
  matter?
• How tight must manufacturing tolerances be?

7
Micromechanical Resonators in a Wireless
transceiver (cont)
Application Examples
Need to simulate ENTIRE system with dynamically
accurate macromodels for all the components
8
Common features
Application Examples

• Devices have Well-defined Inputs and Outputs
• Signal Transmission on a Wire
• Left and right end Voltages and Currents
• Jet Engine Nozzle Design
• Nozzle in and out flow, Nozzle size
• Microresonator
• Comb finger voltages and currents
• Dynamics is important
• Internal state must be somehow represented

9
Traditional Approach to Generating Models
Application Examples
6 months
Interconnect Expert
6 months
MEMS Expert
Lucent
6 months
Resonator Expert
10
The Numerical Macromodeling or Model Reduction
Paradigm
Generate a Reduced-Order Model Directly from 3-D
Geometry and Physics
Automatic
Cheap to evaluate model which captures input
(u)/output(y) behavior
Complicated Geometry, Coupled Electrostatics,
Fluids, Elastics
11
Whats Needed
The Numerical Macromodeling or Model Reduction
Paradigm

• Fast Solvers for complicated 3-D geometries
• (Fast enough to solve ENTIRE devices)
• for fluids, electrostatics, mechanics,
• Approaches for coupled domain analysis
• Multilevel-Newton methods
• Automatic extraction of reduced order models

12
Heat Conducting Bar
Demonstration Example
Lamp
Input of Interest
Output of Interest
13
Heat Conducting Bar
Demonstration Example
Basic Equations

• Temperature Differential Equation
• Spatial Discretization (except at end)

14
Heat Conducting Bar
Demonstration Example
Input-Output Discrete Equations
15
Heat Conducting Bar
Demonstration Example
State-Space Description
Given the right scaling
16
Heat Conducting Bar
Demonstration Example
Temperature Dependent Thermal Conductivity

• Temperature Differential Equation
• Simple Spatial Discretization (not at ends)

17
Heat Conducting Bar
Demonstration Example
Nonlinear State-Space Description
18
Linear example
No Dynamics (Steady-State) Case

• Original System - Single Input/Output
• Reduced System
• Satisfies Reduced Model Criteria
• Cheap to evaluate
• Exactly reproduces I/O Behavior

19
Nonlinear Example
No Dynamics Case

• Original System - Single Input/Output
• Reduced System
• Is g(u) a reduced-order model?
• Depends how we represent g!

20
Nonlinear Example
No Dynamics Case
Representation of Reduced Model

• Could use an interpolated table of data
• Table is a reduced order model
• Cheap to evaluate
• Accurate if enough points used

21
Model Construction Time
No Dynamics Case

• Linear Case, one solve, one inner product
• Solve
• Form
• Nonlinear Case (if an interpolated table is used)
• Solve
• Form
• Nonlinear Reduction adds a representation problem
  to model reduction
• Table is a reduced order model
• Cheap to evaluate
• Accurate if enough points used

22
State-Space Description
Dynamic Linear case

• Original Dynamical System - Single Input/Output
• Reduced Dynamical System
• q ltlt N, but input/output behavior preserved

23
Reminder about Eigenanalysis
24
Reminder about Eigenanalysis Cont.
Decoupled Equations
Output Equation
25
Reminder about Eigenanalysis Cont.
Solving Decoupled Equations
Assuming Zero Initial Conditions
Output Equation
26
Reduced models via mode truncation
Dynamic Linear Case
Output Equation
27
Reduced models via mode Truncation
Dynamic Linear Case
Why?

• Certain modes are not affected by the input
• Certain modes do not affect the output
• Keep least negative evals (slowest modes)
• Look at response to a constant input

28
Reduced models via mode truncation
Dynamic Linear Case
Heat Conducting bar Results
N100
q1
q3
q10
Exact
Keep qth slowest modes
29
An Aside on Transfer Functions Laplace
Transform
Rewrite the ODE in transformed variables
? Transfer Function
30
An Aside on Transfer Functions Meaning of H(s)
For Stable Systems, H(jw) is the frequency
response
?Sinusoid
Sinusoid with shifted phase and amplitude
31
An Aside on Transfer Functions EigenAnalysis
Transfer Function
Apply Eigendecomposition
32
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Original System Transfer Function
Reduced Model Transfer Function
Model Reduction Find a low order rational
function matching H(s)
33
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Degrees of Freedom
Reduced Model Dynamical System
Reduced Model Transfer Function
coefficients
coefficients
34
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Variable Changes Do not change transfer functions
Reduced Model Transfer Function
Similarity (x Sw) Transformed Transfer Function
Many Dynamical Systems have the same transfer
function!!
35
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Rational Function Fitting by point matching

• Can match 2q points
• cross multiplying generates a linear system

For i 1 to 2q
36
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Point Matching Matrix can be ill-conditioned

• Columns contain progressively higher powers of
  the test frequencies
• Must orthogonalize columns during construction

37
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Importance of Fitting at low frequency
Correct Steady State behavior requires accurate
match at low frequencies
38
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Taylor Series Expansion and Moments
Original System Transfer Function Moments
Moments
39
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Moment Matching for accurate low frequency
behavior
Reduced Model Matches Original Systems Moments
Cross-Multiplying and Matching Terms
40
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Explicit Moment Matching Problem
System of equations extremely ill-conditioned
Columns become linearly dependent for large q!
41
Rational Transfer Function Representation
An Aside on Transfer Functions Continued
Dynamic Linear Case
Problems with explicit fitting methods

• Linear Systems for fitting ill-conditioned
• Need specialized algorithms which avoid explicit
  fitting matrix construction
• Rational function must be converted to
  state-space
• Needed by most simulation tools
• Requires root finding procedure, very sensitive
  to parameter variation

42
Summary

• Need For Model Reduction
• Circuits, MEMS, Optics, Jet Engines
• Simple Example Problem
• Heat Conducting bar example
• Steady-State Case (linear and nonlinear)
• Dynamic Linear Case
• Truncating Eigenmodes
• Loss correct steady state values
• Select modes to delete
• Rational Function Fitting
• Generates ill-conditioned matrices