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Example Unsteady Formation Flight

This is Ptera Software: a fast, easy-to-use, and open-source package for analyzing flapping-wing flight.


In late 2018, I became curious about biological flight. To sate this curiosity, I wanted to computationally simulate some flapping-wing fliers. I quickly realized I had two options:

  1. Spend thousands of dollars on a closed-source CFD program, which would take hours to solve a simple case.
  2. Try to learn someone else's open-source, unsteady solver written in a language I didn't know, or using a framework that is overly complicated for my use case.

Neither of these seemed like the right choice.

Thankfully, my friend, Peter Sharpe, had just released his own open-source aerodynamics solver: AeroSandbox. With his support, I have used AeroSandbox as a jumping-off point to develop a solver package capable of unsteady simulations.

Through the combined efforts of Peter Sharpe, Suhas Kodali, and me, Ptera Software was born. It is an easy-to-use, open-source, and actively-maintained UVLM package capable of analyzing flapping-wing flight. Moreover, it's written in Python, is well documented, tested, and validated.

With your help, I hope we will increase the open-source community's interest and understanding of biological flight.


  1. Various Aerodynamic Simulation Methods
    • Steady simulations can be run with a standard horseshoe vortex-lattice method (VLM) or a ring VLM.
    • Unsteady simulations use a ring unsteady VLM (UVLM) solver.
    • Unsteady simulations support both fixed and free wakes.
    • Unsteady simulations implement vortex aging to reduce numerical instabilities.
  2. Customizable Aircraft Geometry
    • Aircraft can be defined as a collection of one or more wings of any dimensions and positions.
    • Wings can be defined as a collection of two or more wing cross sections of any dimensions and positions.
    • Wing cross sections can be specified to match the mean camber line of an airfoil.
    • The package comes with a massive database of airfoil to chose from.
    • Wings are automatically discretized into panels with customizable sizes and spacings.
  3. Customizable Aircraft Motion
    • The relative motion of wings and wing cross sections can be defined using any time-dependent functions of sweep, pitch, and heave angles.
  4. Customizable Operating Points
    • Parameters such as the free-stream velocity, density, angle of attack, angle of sideslip, etc. can be changed by the user.
  5. High-Speed Simulations
    • Using Just-In-Time compilation, Ptera Software can solve many unsteady flapping-wing simulations in less than a minute!
    • Steady simulations take only seconds!
  6. Simulations of Formation Flight
    • Since v2.0.0, Ptera Software has supported simulations with more than one airplane.
    • This feature can be used to analyze the aerodynamics of flapping-wing formation flight!
  7. Features for Flapping-Wing Vehicle Design
    • Ptera Software is focused on developing features to facilitate designing flapping-wing vehicles.
    • For example, use the functions in the trim module to automatically search for a trim operating point for steady and unsteady simulations of aircraft.

Installation and Use

First things first, you will need a copy of Python 3.8. Python 3.9 is not yet supported due to a dependency issue in VTK. Download Python 3.8 from the official Python website. At this time, I do not recommend using a version from the Anaconda distribution as it could introduce compatibility issues with PyPI.

There are two ways to use Ptera Software. The first is by downloading GitHub release, which will provide you your own copy of the source code, in which you can get a feel for how it works (this can also be accomplished by forking the main branch). The second is by importing the Ptera Software package using PyPI, which will allow you to call Ptera Software's functions in your own scripts. If you are new to this tool, I recommend first downloading a release, as this will give you access to the "examples" directory.

Next, make sure you have an IDE in which you can run Ptera Software. I recommend using the Community Edition of PyCharm, which is free, powerful, and well documented. If you've never set up a Python project before, follow this guide to set up a new project in PyCharm. If you'll be downloading a release, follow that tutorial's "Open an existing project guide." Otherwise, follow the "Create a new project guide."

Downloading A Release

To download a release, navigate to the releases page and download the latest zipped directory. Extract the contents, and set up a python project as described in the PyCharm tutorial.

Then, open a command prompt window in your project's directory and enter:

pip install -r requirements.txt

via the command prompt in your fork's directory. You may also want to run:

pip install -r requirements_dev.txt

if you plan on making significant changes to the software.

Finally, open the "examples" folder, which contains several heavily commented scripts that demonstrate different features and simulations. Read through each example, and then run them to admire their pretty output!

Importing As A Package

If you wish to use this package as a dependency in your own project, simply run:

pip install pterasoftware

via the command prompt in your project's directory. Then, in a script that you'd like to use features from Ptera Software, add:

import pterasoftware as ps

If you haven't previously downloaded Ptera Software's source code, you can also learn about the available functions by reading their docstrings, which should be fetched automatically by many IDEs. Otherwise, you can return to the GitHub and read through the docstrings there.

I am hoping to implement a web-based documentation guide soon! If you'd like to contribute to this, feel free to open a feature request issue and start a conversation!

What If I'm Having Trouble Getting Set Up?

Not to worry! I am working on a video that walks through getting Ptera Software up and running. It will include every step, from downloading Python for the first time to setting up your IDE to running the software. In the meantime, feel free to open an issue for guidance.

Example Code

The following code snippet is all that is needed (after running pip install pterasoftware) to run the steady horseshoe solver on an airplane with custom geometry.

import pterasoftware as ps

airplane = ps.geometry.Airplane(

operating_point = ps.operating_point.OperatingPoint()

problem = ps.problems.SteadyProblem(
    airplanes=[airplane], operating_point=operating_point

solver = (

ps.output.draw(solver=solver, scalar_type="lift", show_streamlines=True)

Example Output

This package currently supports three different solvers, a steady horseshoe vortex lattice method (VLM), a steady ring VLM, and an unsteady ring VLM (UVLM). Here are examples of the output you can expect to receive from each of them.

Steady Horseshoe VLM

Example Steady Horseshoe VLM Output

Steady Ring VLM

Example Steady Ring VLM Output

Unsteady Ring VLM

Example Unsteady Ring VLM Animation Output

Example Unsteady Ring VLM Force Coefficient Output

Example Unsteady Ring VLM Moment Coefficient Output


Here are the requirements necessary to run Ptera Software:

  • matplotlib >= 3.5.3, < 3.6.0
  • numpy >= 1.23.5, < 1.24.0
  • pyvista >= 0.37.0, < 1.0.0
  • scipy >= 1.10.0, < 2.0.0
  • numba >= 0.56.4, < 1.0.0
  • cmocean >= 2.0.0, < 3.0.0
  • tqdm >= 4.64.1, < 5.0.0
  • webp >= 0.1.4, < 1.0.0
  • PySide2 >=, <

Additionally, these packages are useful for continued development of the software:

  • codecov >= 2.1.12, < 3.0.0
  • black >= 22.12.0, < 23.0.0
  • pre-commit >= 2.21.0, < 3.0.0
  • build >= 0.9.0, < 1.0.0
  • twine >= 4.0.2, < 5.0.0
  • PyInstaller >= 5.7.0, < 6.0.0
  • setuptools >= 65.5.3, < 66.0.0
  • wheel >= 0.38.4, < 0.39.0


Since the release of version 1.0.0, Ptera Software is now validated against experimental flapping-wing data! See the "validation" directory to run the test case and read a report on the software's accuracy.

How to Contribute

As I said before, the primary goal of this project is to increase the open-source community's understanding and appreciation for unsteady aerodynamics in general and flapping-wing flight in particular. This will only happen through your participation. Feel free to request features, report bugs or security issues, and provide suggestions. No comment is too big or small!

Here is a list of changes I would like to make in the coming releases. If you want to contribute and don't know where to start, this is for you!


  • We should make sure that all the integration tests compare output against expected results. This means getting rid of all the "test_method_does_not_throw" tests.
  • We should maintain the repository's testing coverage to be at least 80%.

Style and Documentation

  • Maintain the repository's A CodeFactor Rating.
  • We should fill in any of the "Properly document this..." TODO statements.
  • We should ensure that all files be at least 30% comment lines.
  • We should continue to ensure that all source code is formatted using Black.


  • We should create a setup tutorial video and add it to the documentation. This should be geared toward a user who doesn't have Python, an IDE, or Ptera Software installed on their computer yet.
  • We should implement a leading-edge model to account for flow separation. See "Modified Unsteady Vortex-Lattice Method to Study Flapping Wings in Hover Flight." by Bruno Roccia, Sergio Preidikman, Julio Massa, and Dean Mook for details.
  • We should create a command-line interface or GUI.
  • We should try to implement aeroelastic effects in Ptera Software's solvers.
  • Flapping wing controls is both fascinating and complicated. We should try to create a workflow in Ptera Software for controls systems identification for flapping-wing vehicles.


Here is a list of all the people and packages that helped me created Ptera Software in no particular order. Specific citations can be found in the source code's docstrings where applicable.

  • Suhas Kodali
  • Peter Sharpe
  • Ramesh Agarwal
  • Joseph Katz
  • Allen Plotkin
  • Austin Stover
  • AeroSandbox
  • Black
  • Codecov
  • Travis CI
  • NumPy
  • SciPy
  • PyVista
  • MatPlotLib
  • Numba
  • Pre-Commit
  • SetupTools
  • GitIgnore
  • PyPI
  • Wheel
  • Twine
  • SemVer
  • GitFlow
  • Cmocean
  • Tqdm
  • WebP
  • Build


To the best of my ability, I am following SemVer conventions in naming my releases. I am also using the GitFlow method of branching for this project's development. This means that nightly builds will be available on the develop branch. The latest stable releases can be found on the master branch.