1 Introduction to Model Order Reduction Thanks to Jacob White, Kin Sou, Deepak Ramaswamy, Michal...

1

Introduction to Model Order Reduction

Thanks to Jacob White, Kin Sou, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

I.2.a – Assembling Models from

MNA Modified Nodal Analysis

Luca Daniel

2

Power Distribution for a VLSI Circuit

Cache ALU Decoder+3.3

v

Power Supply

Main power wires

• Select topology and metal widths & lengths so that a) Voltage across every function block > 3 volts b) Minimize the area used for the metal wires

3

Heat Conducting BarDemonstration Example

endT0 0T

Output of Interest

lamp power u t

Lamp Input of Interest

Select the shape (e.g. thickness) so that a) The temperature does not get too high b) Minimize the metal used.

4

Load Bearing Space Frame

Attachment to the ground

Joint

Beam

Vehicle

Cargo

Droop

Select topology and Strut widths and lengths so that a) Droop is small enough b) Minimize the metal used.

5

Assembling Systems from MNA

• Formulating Equations– Circuit Example– Heat Conducting Bar Example– Struts and Joints Example

• Modified Nodal Analysis Stamping Procedure– Nodal Analysis (NA)– Modified Nodal Analysis (MNA)

• From MNA to State Space Models– e.g. circuits– e.g. struts and joints

6

Given the topology and metal widths & lengths determine

a) the voltage across the ALU, Cache and Decoder

b) the temperature distribution in the engine block

c) the droop of the space frame under load.

First Step - Analysis Tools

Droop

Cache ALU Decoder+3.

3 v

Lamp

7

Cache ALU Decoder+3.3 v

Modeling VLSI circuit Power Distribution

• Power supply provide current at a certain voltage.• Functional blocks draw current. • The wire resistance generates losses.

8

Modeling the Circuit

Supply becomes

A Voltage Source

sV V +

+ Power supply

Physical Symbol

+ Voltage current

Current element

Constitutive Equation

IsV

V

9


Functional blocks become

Current Sources

+ -

sI IALU

Physical Symbol

Circuit Element

Constitutive Equation

VI

sI

10


Metal lines become

Resistors

+ -0IR V

Length

R resistivityArea

Physical Symbol Circuit model Constitutive Equation(Ohm’s Law)

IV

Material PropertyDesign

Parameters

11

Modeling VLSI Power Distribution

Cache ALU Decoder

+-

IC IDIALU

• Power Supply voltage source• Functional Blocks current sources• Wires become resistors

Result is a schematic

Putting it all together

12

Formulating Equations from Schematics

Circuit Example

Step 1: Identifying Unknowns

Assign each node a voltage, with one node as 0

01

2

34

1si2si 3si

13


Circuit Example

Assign each element a current

01

2

3

4

1i

3i

5i

4i

2i

1si2si 3si


14


Circuit Example

Sum of currents = 0 (Kirchoff’s current law)

01

2

3

4

1i2i

3i

5i

4i

1 5 4 0i i i

1si0211 iiis

2si 3si

2 3 2 5 0s si i i i

033 sii4 1 2 3 0s si i i i

Step 2: Conservation Laws

15


Circuit Example

01

2

3

4

Use Constitutive Equations to relate branch currents to node voltages

1R 2R

3R4R

5R

3 3 3 4R i V V

2 2 1 2R i V V

5 5 20R i V

1 1 10R i V

4 4 4 0R i V

Step 3: Constitutive Equations

16





17

Heat Conducting BarDemonstration Example

endT0 0T

Output of Interest

lamp power u t

Lamp Input of Interest

18

Conservation Laws and Constitutive Equations

Heat Flow1-D Example

(0)T (1)T

Unit Length Rod

Near End Temperature

Far End Temperature

Question: What is the temperature distribution along the bar

(0)T(1)T

x

T

Incoming Heat

19


Heat FlowDiscrete Representation

(0)T(1)T

1) Cut the bar into short sections

1T 2T NT1NT

2) Assign each cut a temperature

20


Heat FlowConstitutive Relation

iT

Heat Flow through one section

1iT

x

1,i ih

11, heat flow i i

i i

T Th

x

1iTiT

xR

thermal

1

1,i ih

21


Heat FlowConservation Law

1, , 1i i i sih hh x

Heat in from left

Heat out from right

Incoming heat per unit length

Net Heat Flow into Control Volume = 0

~

~

1iT iT 1iT 1,i ih , 1i ih

x

“control volume”Incoming Heat ( )sh

22


Heat FlowCircuit Analogy

+-

+-

1

R x

ssi xh (0)sv T (1)sv T

Temperature analogous to VoltageHeat Flow analogous to Current

1T NT

~

23





24

Oscillations in a Space Frame Application

Problems

• What is the oscillation amplitude?

Ground

Bolts

Struts

Load

Example Simplified for Illustration

Application Problems

Simplified Structure

Oscillations in a Space Frame

Application Problems Modeling with Struts, Joints and

Point Masses

Oscillations in a Space Frame

Point Mass

Strut

• Replace cargo with point mass.

Constructing the Model• Replace Metal Beams with Struts.

1:20

27

Strut Example To Demonstrate Sign convention

Two Struts Aligned with the X axis

Conservation Law

1 2At node 1: 0x xf f

2At node 2: - 0x Lf f

1f

1 1, 0x y 2 2, 0x y

2f Lf

28



Constitutive Equations

1 22 0 1 2

1 2

x

x xf L x x

x x

11 0 1

1

0 0

0x

xf L x

x

1f

1 1, 0x y 2 2, 0x y

2f Lf

),( *** yxr

),( yxr

rrLrr

rrf

*0*

**

29


Two Struts Aligned with the X axisReduced (Nodal) Equations

1 1 20 1 0 1 2

1 1

2

2

0

x

x x xL x L x

x x x

f

x

1 20 1 2

1 2

2

0

x

L

x xL x x f

x x

f

021 xx ff

02 Lx ff

30



Solution of Nodal Equations

1 0

10x L

2 1 0

10x x L

1f

1 1, 0x y 2 2, 0x y

2f Lf

direction) x positivein (force ˆ10 e.g. 1efL

31



Notice the signs of the forces

2 10 (force in positive x direction)xf

1 10 (force in negative x direction)xf

1f

1 1, 0x y 2 2, 0x y

2f Lf

32


Struts Example

C

DA B

Assign each joint an X,Y position, with one joint as zero.

0,0

Y

X

hinged1,0

Step 1: Identifying Unknowns 11, yx

22 , yx

33


Struts Example

C

DA B

Assign each strut an X and Y force component.

loadf


*,

*, , yAxA ff *

,*, , yBxB ff

*,

*, , yCxC ff

*,

*, , yDxD ff

34


Struts Example

loadf

C

DA B

0,0 1,0

Force Equilibrium Sum of X-directed forces at a joint = 0 Sum of Y-directed forces at a joint = 0

Step 2: Conservation Laws

*,

*, , yAxA ff *

,*, , yBxB ff

*,

*, , yCxC ff

*,

*, , yDxD ff

0

0*,

*,

*,

*,

*,

*,

yCyByA

xCxBxA

fff

fff

0

0

,*

,*,

,*

,*,

yloadyDyC

xloadxDxC

fff

fff


Struts Example

loadf

C

DA B

1,00

12

Use Constitutive Equations to relate strut forces to joint positions.

Step 3: Constitutive Equations

11, yx 22 , yx

AAA

yA

AAA

xA

LLL

yf

LLL

xf

0,1*

,

0,1*

,

CC

CyC

CCC

xC

LLL

yyf

LLL

xxf

0,12*

,

0,12*

,

DDD

yD

DDD

xD

LLL

yf

LLL

xf

0,2*

,

0,2*

,

BBB

yB

BBB

xB

LLL

yf

LLL

xf

0,1*

,

0,1*

,

36


Comparing Conservation Laws

1iT iT 1iT 1,i ih , 1i ih

x

Incoming Heat ( )sh

iV1iV 1iV

si

BiAi

BRAR

0 sBA iii

A B

Lf

~

0** LBA fff

0~

,11, xhhh siiii

37

Summary of key points

Two Types of UnknownsCircuit - Node voltages, element currentsStruts - Joint positions, strut forcesBar – Node Temperatures, heat flows

Two Types of Equations Conservation/Balance Laws

Circuit - Sum of Currents at each node = 0Struts - Sum of Forces at each joint = 0

Bar - Sum of heat flows into control volume = 0

Constitutive EquationCircuit – current-voltage relationship

Struts - force-displacement relationship Bar - temperature drop-heat flow relationship

38


• Formulating Equations– Heat Conducting Bar Example– Circuit Example– Struts and Joints Example



39

0 2 3 2 1 22 5

1 1( ) 0s si i V V V

R R

1 2 4 4 34 3

1 1( ) 0s si i V V V

R R

1) Number the nodes with one node as 0.2) Write a conservation law at each node. except (0) in terms of the node voltages !

1V 2V

3V4V

1si 2si3si

1R 2R

3R4R

5R1 1 1 2

1 2

1 1( ) 0si V V V

R R

Nodal Formulation Generating MatricesCircuit Example

40

1

2

3

4

sIG

v

v

v

v

One row per node, one column per node.

For each resistorR1n 2n


4R4i

2si

1R

1i2R

2i

3R3i

1si3si

0

5R5i

1

1

R 2

1

R

2

1

R

2

1

R

2

1

R 5

1

R

3

1

R 3

1

R

3

1

R

3

1

R 4

1

R

2si

1si

3si

2si

3si

1si

1V2V

3V

4V

41

Nodal Matrix Generation Algorithm


RnnGnnG

1)1,1()1,1(

RnnGnnG

1)2,1()2,1(

RnnGnnG

1)1,2()1,2(

42

X

X

X

X

X

X

X

X

X

1

2

3

4

5

6

7

8

9

2 x 2 block

X =

Sparse MatricesApplications

Space Frame

Space FrameX

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

XX

X X

X

X

X X

X

Unknowns : Joint positionsEquations : forces = 0

X

Nodal Matrix

X

X

X

X

X

X

X

XX

X

X

43

N

IVGN sn

2

2

j LJ G u F

J

(Struts and Joints)

(Resistor Networks)

Nodal Formulation

Generating Matrices

44

1 2 3 4 1m m

2m

1m 2m 3m 2m

( 1) ( 1)m m Unknowns : Node VoltagesEquations : currents = 0


Resistor Grid

45

Nodal Formulation

Matrix non-zero locations for 100 x 10 Resistor Grid


Resistor Grid

46

Nodal FormulationSparse MatricesApplications

Temperature in a cube

Temperature known on surface, determine interior temperature

1 2

1m 2m

2 1m 2 2m CircuitModel

47





48

Nodal Formulation Voltage Source

Can form Node-Branch Constitutive

Equation with Voltage Sources

1R

1i2R

2i

3R3i4R

4i0

1 2

3

4

5R5i

+

6isV

5

Problem Element

2

1

R

2

1

R

2

1

R

2

1

R 5

1

R

3

1

R 3

1

R

3

1

R

3

1

R 4

1

R

2si

1si

3si

2si

3si

1si

4

3

2

1

v

v

v

v1

1

R 1R

Vs

5R

Vs

49

Rigid rod

Nodal Formulation Rigid Rod

Problem Element

),( *** yxr

),( yxr rrL

rr

rrf

*0*

**

fixed2*2* )( Lyyxx

constitute equation

The constitute equation does not contain forces!

50





51

State-Space Models State-Space Models

• Linear system of ordinary differential equations Linear system of ordinary differential equations (ABCDE form) (ABCDE form)

State Input

Output

)()()(

)()(

tDutCxty

tButAxdt

dxE

52

State-Space Model Example:State-Space Model Example:Interconnect Segment Interconnect Segment

• Step 1: Identify internal state variablesStep 1: Identify internal state variables– Example : MNA uses node voltages & inductor current Example : MNA uses node voltages & inductor current

1v 3v

LI

2v

53


• Step 2: Identify inputs & outputs Step 2: Identify inputs & outputs – Example : For Z-parameter representation, choose port Example : For Z-parameter representation, choose port

currents inputs and port voltage outputs currents inputs and port voltage outputs

in1I

in2I

out2vout

1v

1v 3v

LI

2v

1out1 vv

3out2 vv

54


• Step 3: Write state-space & I/O equations Step 3: Write state-space & I/O equations – Example : KCL + inductor equation Example : KCL + inductor equation

01211 inI

Rvv

dtdv

C

1out1 vv

3out2 vv

012 LIR

vv 023 in

L IIdtdv

C

32 vvdtdI

L L

in1I

in2I

out2vout

1v LI

55


• Step 4: Identify state variables & matrices Step 4: Identify state variables & matrices

L

C

C

E0

00

10

00

01

B

LI

v

v

v

x3

2

1

11

1

111

11

RR

RR

A

0100

0001C

00

00D

2

1

in

in

I

Iu

out

out

v

vy

2

1

56

State-Space Model:State-Space Model:circuits more in generalcircuits more in general

:)(

:)(

)( t

ti

txc

L

)(tu

)(ty

LARGE!LARGE!

)()(

)()(

txcty

tButAxdt

dxE

T

KCL/KVL

57





58

Application Problems

A 2x2 Example

Define v as velocity (du/dt) to yield a 2x2 System

Constitutive Equations

2

2m

d uf M

dt

Conservation Law

0s mf f sf

mf0

0 0

cs c

y y EAf E A u

y y

0y y u

Struts, Joints and point mass example

0

00

0 11 0

cdv EA

M vdt ydu u

dt

1:39

59

Summary MNA formulations




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1 Introduction to Model Order Reduction Thanks to Jacob White, Kin Sou, Deepak Ramaswamy, Michal...

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