Posted: 09/27/21

Due: 10/11/21, 11:55PM (first draft, comments will be returned, revision due on 10/18)

This is a continuation of Project 1, but instead of coarse-graining of nearest neighbors, a reduced description is sought in the eigenbasis of the stiffness matrix. Consider the non-dissipative dynamical system

In (1), $𝒙\left(t\right)$ are the time-dependent coordinates of the unit point mass system, $𝒇\left(t\right)$ the forces acting on the system, and $𝑲$ is the stiffness matrix.

Solutions are sought for $𝒇=0$ of the form

leading to the eigenproblem

As in Project 1, consider a linear force-displacement dependence $g(x_{i+1},x_i)=a\cdot (x_{i+1}-x_i)$, leading to $𝑲$ symmetric.

1Track 1 & 2 common problems

1. For $m=100$, find the eigenvalues $\lambda$ of $𝑲$ through the Julia eigvals() function, and plot the set $\{(k,{\lambda }_k),k=1,\dots ,m\}$.

2. Superimpose in one plot the five eigenvectors of $𝑲$ that correspond to the largest eigenvalues and the five eigenvectors that correspond to the smallest eigenvalues. Use the Julia eigenvec() function. Each eigenvector is to be plotted as a line graph of $\{(i,z_{ik}),i=1,\dots ,m\}$ with $z_{ik}$ the $i^{\mathrm{th}}$ component of the $k^{\mathrm{th}}$ eigenvector. Comment on what you observe.

3. Carry out model reduction by choosing a basis set formed from $n=10$ eigenvectors. In matrix form the eigenproblem is stated as $𝑲𝒁=𝒁𝚲$. Assume that eigenvectors are ordered $\left|{\lambda }_1\right|⩾\left|{\lambda }_2\right|⩾\cdots ⩾\left|{\lambda }_m\right|$. Let ${𝒁}_n\in {ℝ}^{m\times n}$ denote the first $n$ columns of $𝒁$. Introduce the model reduction

   ${\displaystyle 𝒙=𝑩𝒚\Rightarrow \stackrel{\sim }{𝑲}={𝑩}^T𝑲𝑩,𝑩={𝒁}_n.}$

   Plot the first five eigenvectors of $\stackrel{\sim }{𝑲}$. How do these compare with the the eigenvectors of $𝑲$?

2Track 2 additional problems

1. Compare the eigenmode model reduction with coarse graining from Project 1. Which is more accurate? Carefully consider which theoretical concepts from linear algebra can be used to assess accuracy of one model versus another.

2. Read [1]. Write a short summary of the approach to reduced model construction based on greedy algorithms.

Bibliography