I’ve now finished my first grad course, Modelling of Multiphysics Systems, taught by Prof Piero Triverio.

I’ve posted notes for lectures and other material as I was taking the course, but now have an aggregated set of notes for the whole course posted.

This is now updated with all my notes from the lectures, solved problems, additional notes on auxillary topics I wanted to explore (like SVD), plus the notes from the Harmonic Balance report that Mike and I will be presenting in January.

This version of my notes also includes all the matlab figures regenerating using http://www.mathworks.com/matlabcentral/fileexchange/23629-export-fig, which allows a save-as pdf, which rescales much better than Matlab saveas() png’s when embedded in latex. I’m not sure if that’s the best way to include Matlab figures in latex, but they are at least not fuzzy looking now.

All in all, I’m pretty pleased with my notes for this course. They are a lot more readable than any of the ones I’ve done for the physics undergrad courses I was taking (https://peeterjoot.com/writing/). While there was quite a lot covered in this course, the material really only requires an introductory circuits course and some basic math (linear algebra and intro calculus), so is pretty accessible.

This was a fun course. I recall, back in ancient times when I was a first year student, being unsatisfied with all the ad-hoc strategies we used to solve circuits problems. This finally answers the questions of how to tackle things more systematically.

Here’s the contents outline for these notes:

Preface

Lecture notes

1 nodal analysis

1.1 In slides

1.2 Mechanical structures example

1.3 Assembling system equations automatically. Node/branch method

1.4 Nodal Analysis

1.5 Modified nodal analysis (MNA)

2 solving large systems

2.1 Gaussian elimination

2.2 LU decomposition

2.3 Problems

3 numerical errors and conditioning

3.1 Strict diagonal dominance

3.2 Exploring uniqueness and existence

3.3 Perturbation and norms

3.4 Matrix norm

4 singular value decomposition, and conditioning number

4.1 Singular value decomposition

4.2 Conditioning number

5 sparse factorization

5.1 Fill ins

5.2 Markowitz product

5.3 Markowitz reordering

5.4 Graph representation

6 gradient methods

6.1 Summary of factorization costs

6.2 Iterative methods

6.3 Gradient method

6.4 Recap: Summary of Gradient method

6.5 Conjugate gradient method

6.6 Full Algorithm

6.7 Order analysis

6.8 Conjugate gradient convergence

6.9 Gershgorin circle theorem

6.10 Preconditioning

6.11 Symmetric preconditioning

6.12 Preconditioned conjugate gradient

6.13 Problems

7 solution of nonlinear systems

7.1 Nonlinear systems

7.2 Richardson and Linear Convergence

7.3 Newton’s method

7.4 Solution of N nonlinear equations in N unknowns

7.5 Multivariable Newton’s iteration

7.6 Automatic assembly of equations for nonlinear system

7.7 Damped Newton’s method

7.8 Continuation parameters

7.9 Singular Jacobians

7.10 Struts and Joints, Node branch formulation

7.11 Problems

8 time dependent systems

8.1 Assembling equations automatically for dynamical systems

8.2 Numerical solution of differential equations

8.3 Forward Euler method

8.4 Backward Euler method

8.5 Trapezoidal rule (TR)

8.6 Nonlinear differential equations

8.7 Analysis, accuracy and stability (Dt ! 0)

8.8 Residual for LMS methods

8.9 Global error estimate

8.10 Stability

8.11 Stability (continued)

8.12 Problems

9 model order reduction

9.1 Model order reduction

9.2 Moment matching

9.3 Model order reduction (cont).

9.4 Moment matching

9.5 Truncated Balanced Realization (1000 ft overview)

9.6 Problems

Final report

10 harmonic balance

10.1 Abstract

10.2 Introduction

10.2.1 Modifications to the netlist syntax

10.3 Background

10.3.1 Discrete Fourier Transform

10.3.2 Harmonic Balance equations

10.3.3 Frequency domain representation of MNA equations

10.3.4 Example. RC circuit with a diode.

10.3.5 Jacobian

10.3.6 Newton’s method solution

10.3.7 Alternative handling of the non-linear currents and Jacobians

10.4 Results

10.4.1 Low pass filter

10.4.2 Half wave rectifier

10.4.3 AC to DC conversion

10.4.4 Bridge rectifier

10.4.5 Cpu time and error vs N

10.4.6 Taylor series non-linearities

10.4.7 Stiff systems

10.5 Conclusion

10.6 Appendices

10.6.1 Discrete Fourier Transform inversion

Appendices

a singular value decomposition

b basic theorems and definitions

c norton equivalents

d stability of discretized linear differential equations

e laplace transform refresher

f discrete fourier transform

g harmonic balance, rough notes

g.1 Block matrix form, with physical parameter ordering

g.2 Block matrix form, with frequency ordering

g.3 Representing the linear sources

g.4 Representing non-linear sources

g.5 Newton’s method

g.6 A matrix formulation of Harmonic Balance non-linear currents

h matlab notebooks

i mathematica notebooks

Index

Bibliography