Damped Newton’s method

Final notes for ECE1254, Modelling of Multiphysics Systems

December 27, 2014 ece1254 No comments , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

Capture

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 (http://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

ECE1254H Modeling of Multiphysics Systems. Lecture 12: Struts and Joints, Node branch formulation. Taught by Prof. Piero Triverio

October 27, 2014 ece1254 No comments , , , ,

[Click here for a PDF of this post with nicer formatting]

Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

Struts and Joints, Node branch formulation

Let’s consider the simple strut system of fig. 1 again.

lecture12Fig1

fig. 1. Simple strut system

 

Our unknowns are

  1. Forces at each of the points we have a force with two components
    \begin{equation}\label{eqn:multiphysicsL12:20}
    \Bf_A = \lr{ f_{A,x}, f_{A,y} }
    \end{equation}We construct a total force vector\begin{equation}\label{eqn:multiphysicsL12:40}
    \Bf =
    \begin{bmatrix}
    f_{A,x} \\
    f_{A,y} \\
    f_{B,x} \\
    f_{B,y} \\
    \vdots
    \end{bmatrix}
    \end{equation}
  2. Positions of the joints\begin{equation}\label{eqn:multiphysicsL12:60}
    \Br =
    \begin{bmatrix}
    x_1 \\
    y_1 \\
    y_1 \\
    y_2 \\
    \vdots
    \end{bmatrix}
    \end{equation}

Our given variables are

  1. The load force \( \Bf_L \).
  2. The joint positions at rest.
  3. parameter of struts.

Conservation laws

The conservation laws are

\begin{equation}\label{eqn:multiphysicsL12:80}
\Bf_A + \Bf_B + \Bf_C = 0
\end{equation}
\begin{equation}\label{eqn:multiphysicsL12:100}
-\Bf_C + \Bf_D + \Bf_L = 0
\end{equation}

which we can put in matrix form

\begin{equation}\label{eqn:multiphysicsL12:120}
\begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 & -1 & 0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
f_{A,x} \\
f_{A,y} \\
f_{B,x} \\
f_{B,y} \\
f_{C,x} \\
f_{C,y} \\
f_{D,x} \\
f_{D,y}
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
-f_{L,x} \\
-f_{L,y}
\end{bmatrix}
\end{equation}

Here the block matrix is called the incidence matrix \( \BA \), and we write

\begin{equation}\label{eqn:multiphysicsL12:160}
A \Bf = \Bf_L.
\end{equation}

Constitutive laws

Given a pair of nodes as in fig. 2.

lecture12Fig2

fig. 2. Strut spanning nodes

 

each component has an equation relating the reaction forces of that strut based on the positions

\begin{equation}\label{eqn:multiphysicsL12:180}
f_{c,x} = S_x \lr{ x_1 – x_2, y_1 – y_2, p_c }
\end{equation}
\begin{equation}\label{eqn:multiphysicsL12:200}
f_{c,y} = S_y \lr{ x_1 – x_2, y_1 – y_2, p_c },
\end{equation}

where \( p_c \) represent the parameters of the system. We write

\begin{equation}\label{eqn:multiphysicsL12:220}
\Bf =
\begin{bmatrix}
f_{A,x} \\
f_{A,y} \\
f_{B,x} \\
f_{B,y} \\
\vdots
\end{bmatrix}
=
\begin{bmatrix}
S_x \lr{ x_1 – x_2, y_1 – y_2, p_c } \\
S_y \lr{ x_1 – x_2, y_1 – y_2, p_c } \\
\vdots
\end{bmatrix},
\end{equation}

or

\begin{equation}\label{eqn:multiphysicsL12:260}
\Bf = S(\Br)
\end{equation}

Putting the pieces together

The node branch formulation is

\begin{equation}\label{eqn:multiphysicsL12:280}
\begin{aligned}
A \Bf – \Bf_L &= 0 \\
\Bf – S(\Br) &= 0
\end{aligned}
\end{equation}

We’ll want to approximate this system using the Jacobian methods discussed, and can expect the cost of that Jacobian calculation to potentially be expensive. To move to the nodal formulation we eliminate forces (the equivalent of currents in this system)

\begin{equation}\label{eqn:multiphysicsL12:320}
A S(\Br) – \Bf_L = 0
\end{equation}

We cannot use this nodal formulation when we have struts that are so stiff that the positions of some of the nodes are fixed, but can work around that as before by introducing an additional unknown for each component of such a strut.

Damped Newton’s method

We want to be able to deal with the oscillation that we can have in examples like that of fig. 3.

lecture12Fig3

fig. 3. Oscillatory Newton’s iteration

 

Large steps can be dangerous. We want to modify Newton’s method as follows

Our algorithm is

  • Guess \( \Bx^0, k = 0 \).
  • REPEAT
  • Compute \( F \) and \( J_F \) at \( \Bx^k \)
  • Solve linear system \( J_F(\Bx^k) \Delta \Bx^k = – F(\Bx^k) \)
  • \( \Bx^{k+1} = \Bx^k + \alpha^k \Delta \Bx^k \)
  • \( k = k + 1 \)
  • UNTIL converged

with \( \alpha^k = 1 \) we have standard Newton’s method. We want to pick \( \alpha^k \) so that we minimize

Continuation parameters

Newton’s method converges given a close initial guess. We can generate a sequence of problems where the previous problem generates a good initial guess for the next problem.

An example is a heat conducting bar, with a final heat distribution. We can start the numeric iteration with \( T = 0 \), and gradually increase the temperatures until we achieve the final desired heat distribution.

Suppose that we want to solve

\begin{equation}\label{eqn:multiphysicsL12:340}
F(\Bx) = 0.
\end{equation}

We modify this problem by introducing

\begin{equation}\label{eqn:multiphysicsL12:360}
\tilde{F}(\Bx(\lambda), \lambda) = 0,
\end{equation}

where

  • \( \tilde{F}(\Bx(0), 0) = 0 \) is easy to solve
  • \( \tilde{F}(\Bx(1), 1) = 0 \) is equivalent to \( F(\Bx) = 0 \).
  • (more on slides)

The source load stepping algorithm is

  • Solve \(\tilde{F}(\Bx(0), 0) = 0 \), with \( \Bx(\lambda_{\text{prev}} = \Bx(0) \)
  • (more on slides)

This can still have problems, for example, when the parameterization is multivalued as in fig. 4.

lecture12Fig4

fig. 4. Multivalued parameterization

 

We can attempt to adjust \( \lambda \) so that we move along the parameterization curve.