## Current collection of multiphysics modeling notes (ece1254)

October 27, 2014 ece1254

I’ve now posted
v.1 of my current notes collection for “Modeling of Multiphysics Systems” (ECE1254H1F), a course I’m now taking taught by Prof. Piero Triverio. This includes the following individual lecture or personal notes, which may or may not have been posted as separate blog entries:

October 27, 2014 Struts and Joints, Node branch formulation

October 24, 2014 Conjugate gradient method

October 22, 2014 Nonlinear equations

October 22, 2014 Conjugate gradient methods

October 21, 2014 Nonlinear systems

October 21, 2014 Sparse factorization and iterative methods

October 15, 2014 Illustrating the LU algorthm with pivots by example

October 14, 2014 Modified Nodal Analysis

October 09, 2014 Numerical LU example where pivoting is required

October 09, 2014 Numeric LU factorization example

October 02, 2014 Singular Value Decomposition

October 01, 2014 Matrix norm, singular decomposition, and conditioning number

September 30, 2014 Numerical error and conditioning.

September 26, 2014 Solving large systems

September 24, 2014 Nodal Analysis

September 23, 2014 Assembling system equations automatically

September 23, 2014 Analogies to circuit systems

## 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.

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.

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.

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.

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