Posted: 09/10/21

Due: 09/24/21, 11:55PM (first draft, comments will be returned, revision due on 10/1)

Consider a dynamical system

a generalization to multiple degrees of freedom of the damped oscillator equation

In (1), $𝒙\left(t\right)$ are the time-depenent coordinates of the system, $𝒇\left(t\right)$ the forces acting on the system, and $𝑴,𝑫,𝑲$ are the mass, drag, stiffness matrices, respectively.

It is often the case that $m\gg 1$, and reduced description is sought by linear combination of $n\ll m$ basis vectors

Choose $𝑩\in {ℝ}^{m\times n}$ to have orthonormal columns, and project (1) onto $C\left(𝑩\right)$ by multiplication with the projector $𝑷=𝑩{𝑩}^T$

Since $N\left(𝑩\right)=\left\{𝟎\right\}$, deduce $𝒛=𝟎$, hence

Introduce notations

for the reduced mass, drag, stiffness matrices, with $\stackrel{\sim }{𝑴},$$\stackrel{\sim }{𝑫}$,$\stackrel{\sim }{𝑲}\in {ℝ}^{n\times n}$ of smaller size. The reduced coordinates and forces are

This project explores model reduction for the simple case of a linear array of coupled oscillators (Fig. 1), modeled as unit point masses with $x_i\left(t\right)$ the displacement from equilibrium. The equation of motion for point mass $i$ is

with end conditions $x_0\left(t\right)=x_{m+1}\left(t\right)=0$.

Figure 1. Linear array of oscillators, with $x_i\left(t\right)$ denoting displacement from equlibrium position of point mass $i$.

1Track 1 & 2 common problems

Consider $g(x_{i+1},x_i)=a\cdot (x_{i+1}-x_i)$, $v({\dot{x}}_{i+1},{\dot{x}}_i)=p\cdot$$({\dot{x}}_{i+1}-{\dot{x}}_i)$, $f_i\left(t\right)=0$, and initial conditions

1. Write out formulas for matrices $𝑴,𝑫,𝑲$. Write code to compute these matrices.

2. Coarse-graining seeks to replace groups of $J$ unit point masses with a single point mass of mass $J$ by arithmetic averaging

   ${\displaystyle z_j\left(t\right)=\frac{1}{J}\sum _{i=(j-1)J+1}^{jJ}{x}_i\left(t\right),j=1,2,\dots ,\frac{m}{J}=n.}$

   Assume $mmodJ=0$, and determine the matrix $𝑪$

   ${\displaystyle 𝒛=𝑪𝒙.}$

   Write code to compute $𝑪$.

3. Using the SVD of $𝑪=𝑼𝚺{𝑽}^T$, the pseudo-inverse is ${𝑪}^{+}=𝑽{𝚺}^{+}{𝑼}^T$, and reduced coordinates can be defined as

   ${\displaystyle \stackrel{\sim }{𝒙}=𝑽𝒚.}$

   Find the reduced matrices $\stackrel{\sim }{𝑴},$$\stackrel{\sim }{𝑫}$,$\stackrel{\sim }{𝑲}\in {ℝ}^{n\times n}$, and write code to compute them.

4. For $𝑫=𝟎$, (i.e., $p=0$), the analytical solution to this problem is

   ${\displaystyle x_i\left(t\right)=\cos (\omega t)\sin \left(\frac{ik\pi }{m+1}\right).}$

   Find $\omega$, and plot the error

   ${\displaystyle e\left(t\right)=||\stackrel{\sim }{𝒙}\left(t\right)-𝒙\left(t\right)||,}$

   as a function of time $t$. Experiment with different choices of $m,n$.

2Track 2 additional problems

Benner, Gugercin & Wilcox [1] provide an overview into the topic of projective model reduction, in which $𝑴,𝑫,𝑲$ are now longer constant, but depend on some parameter $p$.

1. From among the papers citing [1], choose three close to your own interests, and present a synopsis of how parametric model reduction is used in applications. Use a bibliographic database (i.e., Web of Science) to find citations of [1].

2. Redo problem (4) above for $p>0$.

Bibliography