Introduction to Model Order Reduction

Introduction to Model Order Reduction I.2.a – Assembling Models from MNA Modified Nodal Analysis Luca Daniel Thanks to Jacob White, Kin Sou, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

Power Distribution for a VLSI Circuit Cache Decoder ALU + 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

Heat Conducting Bar Demonstration Example 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. Output of Interest

Droop Joint Beam Cargo Attachment to the ground Vehicle Load Bearing Space Frame 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

Droop Decoder Cache ALU 3.3 v + First Step - Analysis Tools Lamp • Given the topology and metal widths & lengths determine • the voltage across the ALU, Cache and Decoder • the temperature distribution in the engine block • the droop of the space frame under load.

Modeling VLSI circuit Power Distribution Cache ALU Decoder + 3.3 v • Power supply provide current at a certain voltage. • Functional blocks draw current. • The wire resistance generates losses.

+ + Voltage Supply becomes Modeling the Circuit A Voltage Source Power supply + current Constitutive Equation Physical Symbol Current element

Functional blocks become Modeling the Circuit Current Sources + - ALU Constitutive Equation Physical Symbol Circuit Element

Metal lines become Modeling the Circuit Resistors + - Physical Symbol Circuit model Constitutive Equation (Ohm’s Law) Material Property Design Parameters

ID • Power Supply voltage source • Functional Blocks current sources IC + - IALU Modeling VLSI Power Distribution Putting it all together Cache ALU Decoder • Wires become resistors Result is a schematic

Circuit Example Formulating Equations from Schematics Step 1: Identifying Unknowns 1 0 2 3 4 Assign each node a voltage, with one node as 0

Circuit Example Formulating Equations from Schematics Step 1: Identifying Unknowns 1 0 2 3 4 Assign each element a current

Circuit Example Formulating Equations from Schematics Step 2: Conservation Laws 1 0 2 3 4 Sum of currents = 0 (Kirchoff’s current law)

Use Constitutive Equations to relate branch currents to node voltages Circuit Example Formulating Equations from Schematics Step 3: Constitutive Equations 1 0 2 3 4

Heat Conducting Bar Demonstration Example Lamp Input of Interest Output of Interest

Conservation Laws and Constitutive Equations Heat Flow 1-D Example Incoming Heat Far End Temperature Near End Temperature Unit Length Rod Question: What is the temperature distribution along the bar T x

2) Assign each cut a temperature Conservation Laws and Constitutive Equations Heat Flow Discrete Representation 1) Cut the bar into short sections

Conservation Laws and Constitutive Equations Heat Flow Constitutive Relation Heat Flow through one section

“control volume” Conservation Laws and Constitutive Equations Heat Flow Conservation Law Net Heat Flow into Control Volume = 0 ~ ~ Incoming heat per unit length Heat in from left Heat out from right

+ - + - Conservation Laws and Constitutive Equations Heat Flow Circuit Analogy Temperature analogous to Voltage Heat Flow analogous to Current ~

Oscillations in a Space Frame Application Problems • What is the oscillation amplitude?

Ground Oscillations in a Space Frame Application Problems Simplified Structure Bolts Struts Load Example Simplified for Illustration

Point Mass Strut Oscillations in a Space Frame Application Problems Modeling with Struts, Joints and Point Masses Constructing the Model • Replace Metal Beams with Struts. • Replace cargo with point mass. 1:20

Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Conservation Law

Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Constitutive Equations

Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Reduced (Nodal) Equations

Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Solution of Nodal Equations

Strut Example To Demonstrate Sign convention Two Struts Aligned with the X axis Notice the signs of the forces

Y X Struts Example Formulating Equations from Schematics Step 1: Identifying Unknowns C D B A hinged Assign each joint an X,Y position, with one joint as zero.

Struts Example Formulating Equations from Schematics Step 1: Identifying Unknowns C D B A Assign each strut an X and Y force component.

Struts Example Formulating Equations from Schematics Step 2: Conservation Laws C D B A Force Equilibrium Sum of X-directed forces at a joint = 0 Sum of Y-directed forces at a joint = 0

Struts Example Formulating Equations from Schematics Step 3: Constitutive Equations 1 2 C D B A 0 Use Constitutive Equations to relate strut forces to joint positions.

Formulating Equations from Schematics Comparing Conservation Laws ~ B A

Summary of key points Two Types of Unknowns Circuit - Node voltages, element currents Struts - Joint positions, strut forces Bar – Node Temperatures, heat flows Two Types of Equations Conservation/Balance Laws Circuit - Sum of Currents at each node = 0 Struts - Sum of Forces at each joint = 0 Bar - Sum of heat flows into control volume = 0 Constitutive Equation Circuit – current-voltage relationship Struts - force-displacement relationship Bar - temperature drop-heat flow relationship

Assembling Systems from MNA • Formulating Equations • Heat Conducting Bar Example • Circuit 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

Circuit Example Generating Matrices Nodal Formulation 0 • ) Number the nodes with one node as 0. • ) Write a conservation law at each node.except (0) in terms of the node voltages !

Circuit Example One row per node, one column per node. For each resistor Generating Matrices Nodal Formulation 0

Circuit Example Generating Matrices Nodal Formulation Nodal Matrix Generation Algorithm

5 7 9 3 4 6 8 2 1 Applications Sparse Matrices Space Frame Nodal Matrix Space Frame X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Unknowns : Joint positions Equations : forces = 0

Generating Matrices Nodal Formulation (Resistor Networks) (Struts and Joints)

Applications Sparse Matrices Resistor Grid Unknowns : Node Voltages Equations : currents = 0

Applications Sparse Matrices Nodal Formulation Resistor Grid Matrix non-zero locations for 100 x 10 Resistor Grid

Applications Sparse Matrices Nodal Formulation Temperature in a cube Temperature known on surface, determine interior temperature Circuit Model

+ Problem Element Nodal Formulation Voltage Source Can form Node-Branch Constitutive Equation with Voltage Sources 5 1 2 3 0 4

Problem Element Nodal Formulation Rigid Rod Rigid rod constitute equation The constitute equation does not contain forces!

Introduction to Model Order Reduction