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

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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

Power Distribution for a VLSI Circuit

Cache ALU Decoder+3.3

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

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.

Load Bearing Space Frame

Attachment to the ground

Vehicle

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

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

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

Cache ALU Decoder+3.

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.

Modeling the Circuit

Supply becomes

A Voltage Source

sV V +

+ Power supply

Physical Symbol

+ Voltage current

Current element

Constitutive Equation

Functional blocks become

Current Sources

sI IALU

Physical Symbol

Circuit Element

Constitutive Equation

Metal lines become

Resistors

+ -0IR V

Length

R resistivityArea

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

Material PropertyDesign

Parameters

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

Formulating Equations from Schematics

Circuit Example

Step 1: Identifying Unknowns

Assign each node a voltage, with one node as 0

1si2si 3si

Circuit Example

Assign each element a current

1si2si 3si

Circuit Example

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

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

Circuit Example

Use Constitutive Equations to relate branch currents to node voltages

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

Heat Conducting BarDemonstration Example

endT0 0T

Output of Interest

lamp power u t

Lamp Input of Interest

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

Incoming Heat

Heat FlowDiscrete Representation

(0)T(1)T

1) Cut the bar into short sections

1T 2T NT1NT

2) Assign each cut a temperature

Heat FlowConstitutive Relation

Heat Flow through one section

1,i ih

11, heat flow i i

thermal

1,i ih

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

“control volume”Incoming Heat ( )sh

Heat FlowCircuit Analogy

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

Temperature analogous to VoltageHeat Flow analogous to Current

Oscillations in a Space Frame Application

Problems

• What is the oscillation amplitude?

Ground

Struts

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

• Replace cargo with point mass.

Constructing the Model• Replace Metal Beams with Struts.

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

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

Constitutive Equations

1 22 0 1 2

x xf L x x

11 0 1

xf L x

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

),( *** yxr

),( yxr

Two Struts Aligned with the X axisReduced (Nodal) Equations

1 1 20 1 0 1 2

x x xL x L x

1 20 1 2

x xL x x f

021 xx ff

02 Lx ff

Solution of Nodal Equations

10x x L

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

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

Notice the signs of the forces

2 10 (force in positive x direction)xf

1 10 (force in negative x direction)xf

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

Struts Example

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

hinged1,0

Step 1: Identifying Unknowns 11, yx

22 , yx

Struts Example

Assign each strut an X and Y force component.

*, , yAxA ff *

,*, , yBxB ff

*, , yCxC ff

*, , yDxD ff

Struts Example

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

yCyByA

xCxBxA

yloadyDyC

xloadxDxC

Struts Example

Use Constitutive Equations to relate strut forces to joint positions.

Step 3: Constitutive Equations

11, yx 22 , yx

Comparing Conservation Laws

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

Incoming Heat ( )sh

iV1iV 1iV

0 sBA iii

0** LBA fff

,11, xhhh siiii

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

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

0 2 3 2 1 22 5

1 1( ) 0s si i V V V

1 2 4 4 34 3

1 1( ) 0s si i V V V

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 !

1si 2si3si

5R1 1 1 2

1 1( ) 0si V V V

Nodal Formulation Generating MatricesCircuit Example

One row per node, one column per node.

For each resistorR1n 2n

1si3si

Nodal Matrix Generation Algorithm

RnnGnnG

1)1,1()1,1(

RnnGnnG

1)2,1()2,1(

RnnGnnG

1)1,2()1,2(

2 x 2 block

Sparse MatricesApplications

Space Frame

Space FrameX

Unknowns : Joint positionsEquations : forces = 0

Nodal Matrix

IVGN sn

j LJ G u F

(Struts and Joints)

(Resistor Networks)

Nodal Formulation

Generating Matrices

1 2 3 4 1m m

1m 2m 3m 2m

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

Resistor Grid

Nodal Formulation

Matrix non-zero locations for 100 x 10 Resistor Grid

Resistor Grid

Nodal FormulationSparse MatricesApplications

Temperature in a cube

Temperature known on surface, determine interior temperature

2 1m 2 2m CircuitModel

Nodal Formulation Voltage Source

Can form Node-Branch Constitutive

Equation with Voltage Sources

3R3i4R

Problem Element

Rigid rod

Nodal Formulation Rigid Rod

Problem Element

),( *** yxr

),( yxr rrL

fixed2*2* )( Lyyxx

constitute equation

The constitute equation does not contain forces!

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

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

• 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

out2vout

1out1 vv

3out2 vv

• 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

1out1 vv

3out2 vv

012 LIR

vv 023 in

L IIdtdv

32 vvdtdI

out2vout

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

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

LARGE!LARGE!

tButAxdt

KCL/KVL

Application Problems

A 2x2 Example

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

Constitutive Equations

d uf M

Conservation Law

0s mf f sf

y y EAf E A u

0y y u

Struts, Joints and point mass example

0 11 0

cdv EA

M vdt ydu u

Summary MNA formulations

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

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