Project Description
This project is aimed at solving a simple one dimensional wave equation, which we are specifying here as a vibrating string fixed at both ends. We intend to simulate this system varying certain key parameters which influence the oscillations.
Scope and Complexity
Description
One of the most simple techniques to visualise a simulation which involves solving a PDE with certain boundary conditions is to save the string profile for certain time intervals and plot them against appropriate axes. A fundamental conditions required to initialise a vibration, is to have a source function at one end which generates the wave. After this at every time interval the old and new profiles of the string is updated through an iteration. The output we intend to achieve is to visualise the wave pattern with varying frequencies and modes.
One primary objective is to visualise standing waves.
Requirements
- Theoretical Skills
A basic understanding of certain elementary topics in mechanics such as tension and displacements in elastic strings is required. Additionally choice of material needed to run the computational simulation is needed. (For instance,if no boundary conditions are specified on the ultimate strength of the material,the compiler will evaluate results which might not be physically possible. The string can always snap off if the force magnitude is higher).Knowledge of Elliptic PDE's (and its solution interpretation) is an added bonus.
- Computational skills
The program can be solved in any computational platform (such as MATLAB, Python(preferred), C++). Basic knowledge of iterators,pointers, arrays, vectors (in some languages), function (and function calling) is required to solve the problem.
- Team Interaction
In class discussion and using social platforms such as Slack etc.
Possible Improvements
This section mainly talks about what we consider as out of scope for the project
We are evaluating the problem assuming the wave to be at steady state and slowly progressing with time.An additional manipulation could have been introducing a friction coefficient and calculating the time rate of decay of the wave.Another interesting application could have been visualising the transient state of the wave profile and the time it takes to transform from a non periodic wave profile to a periodic one.
Project Workflow
- Intialize input from user with function 'get_user_input()'
- Receive user input from the command line, (or a text file), by prompting the user with questions about the geometry of the problem and initial conditions.
- Output a dictionary called inputs that contains the string length, pluck position, pluck displacement, yield strength, and time scale.
- Solve Physics
- Setup Finite Difference function
- Given user input, sets up solution matrix
- Build first time step of matrix
- Initially we load the initial condition into u_1 (solution array at one time level back)
- Update u with special function that utilised u_1.
- Solve matrix and update time step
- Loop over temporal intervals
- Loop over spatial intervals
- Apply finite difference formula
- Insert boundary conditions after each time step
- Switch variables before next time step
- Setup Finite Difference function
- Write and View Output
- Output file of results at all spatial and temporal points
- Create a dictionary that contains the node number, displacement, and time step
- Output table in text file with function 'output_results_file'.
- Output file of plots at desired timesteps
- Ask for what timesteps the user requests a plot of the results by prompting the user with questions in the command line.
- Create and output plots
- Output file of results at all spatial and temporal points