Skip to content

Latest commit

 

History

History
77 lines (47 loc) · 3.25 KB

README.md

File metadata and controls

77 lines (47 loc) · 3.25 KB

Lattice (Oscillators on String) Simulator

Numerically simulated program that emulates the famous experiment conducted in 1953 by Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou. The program will solve a dual system of ODEs and find displacements of oscillators in a 1D lattice with user-specified parameters, display an animation of the oscillators' displacements through time and finally perform a Fourier analysis.

## Video Demonstration (Non-Linear Case)

How to run the program

  • Download or clone repository
  • Run view.py
  • In the Chrome browser, navigate to http://127.0.0.1:3000/
  • Fill in the HTML form and submit
  • Wait for data to be processed, then watch the Matplotlib-generated animated solution

Simulation Program Structure

HTML GUI

Two HTML forms

Form 1

  • Has every input apart from initial conditions, x(t0) for all i.
  • Hides submit button until every input field contains a number.

Form 2

  • First asks what shape the string will take initially. Options are limited to sine, half-sine or parabola (since x0 and xL must be zero).
  • If user inputs "sine" or "half-sine", ask for amplitude and number of wavelengths or half-wavelengths, resp. Check no. of wavelengths is an integer.
  • If user inputs "parabola", ask for stationary point height.
    • Extra feature: Add another option: "polynomial":
      • Allow user to chooose polynomial degree and coefficients
        • i.e. if degree is "4", ask user for five coefficients (in order)
      • allow user to click button to randomize coefficients before/after choosing polynomial degree
      • keep button shaded and unclickable until polynomial satisfies the constraints of x0 = xL = 0
        • or, if polynomial doesn't fit constraints, translate it until it does without changing (challenging)

Python integration

  • Take data from user and call main "simulate()" function with paramaters
  • Represent processed data and solution on new HTML page to show user

Solve System of ODEs

Simultaneous Equations

  • dx_i/dt for every oscillator i, i.e. N simultaneous equations

Perform RungeKutta 4th-order method on d^2x/dt^2 to determine discretised velocities

  • Create array of arrays, where each array holds all velocity values for one of N oscillators

Repeat process for dx/dt for displacements

  • Instead of using a derivative function, we'll have to index the velocity arrrays to find dx_i/dt at each time-step

Plot x(t, i)

  • X-axis will be the distance along string (i-value)
  • To show the progression through time, we will either animate the graph or plot M graphs in one figure, where M is the number of time-steps between t_0 and t_final
    • Matplotlib has an animation class

Analyse the change in the first few Fourier Series coefficients over time in the non-linear case

Additional Features - Ideas

  • If user inputs any parameters that may cause instability, kill the program and print error or prevent user from submitting the Parameters form in this case.

  • Create a range of preset polynomials that satisfy the boundary conditions for the user to choose from when setting initial conditions. Create a button that when clicked displays a subsection of polynomial graphs that can be selected and simulated.