josephcslater / mousai

Harmonic balance solvers

Mousai

Python-based Harmonic Balance solvers and relevant tools.

Please see the manual for usage and installation, and a nice overview from my Dayton Engineering Sciences Symposium 2017 presentation. You can play around with it by selecting the launch binder button above.

Mousai can solve sets of first-order and second-order ordinary differential equations written in state-space form (solved for acceleration for second-order form) subject to a harmonic excitation. All you need to provide is the name of a Python function which may simply be a wrapper to an external code. The function must accept:

1. The states (displacements and derivatives for second order forms)
2. Time (interpreted as time within the period)
3. Fundamental frequency of the harmonic representation

The function must return the state derivatives (acceleration for second order form). It is expected that in most cases this will simply be a python wrapper function to call an external finite element, electro-magnetic, molecular dynamic, computational fluid dynamic, or other simulation code.

No traditional numerical integration is performed in harmonic balance, so issues of stability, reliability, etc. are separate concerns. An inability to determine a solution is reported if failure occurs. A good initial guess is always helpful, sometimes essential, and may simply have the form of a nearby solution. Sweeping through frequencies or increasing amplitudes of excitation is the best way to ensure swift convergence.

Quasi-linear models can (not yet implemented) obtain good low amplitude solutions that can be used as starting guesses for solutions (appropriate scaling may also help).

The manual is still under development, but the Tutorial provides two examples of solutions to the Duffing Oscillator, one in first order form, the other in second order form.

The Reference section of the manual illustrates supporting code for building out more time-refined solutions, finding velocities from displacements, and specifics on function calls.

Please see the manual for usage and installation.

Please see release notes for updates, fixes, and plans.

If you use this for any work, please cite it as: Slater, Joseph C., "Mousai: An Open Source Harmonic Balance Solver for Nonlinear Systems," 13th ASME Dayton Engineering Sciences Symposium, Oct. 30, 2017, Dayton, OH.

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mousai's Issues

lack of initial guess causes fail: try with fewer points and fewer harmonics to initiate.

def duff_osc_ss(x, params):
    omega = params['omega']
    t = params['cur_time']
    xdot = np.array([[x[1]],[-x[0]-.1*x[0]**3-.1*x[1]+1*sin(omega*t)]])
    return xdot


t, x, e, amps, phases = ms.hb_freq(duff_osc_ss, num_variables = 2, omega = 0.1,
                                   eqform='first_order', num_harmonics=5, num_time_steps=51,             
                                   mask_constant=True, f_rtol = 1e-8)#, f_rtol=1e-8)
time, xc = ms.time_history(t, x)
plt.plot(time, xc.T, label='5 harmonics')

Results in failed start.

num_harmonics =1 works fine. This would be a good fall-back call should convergence fail.

frequency method higher harmonics causes wrong solutions

single harmonic works. With higher, it breaks down. Unknown cause at this time.

Should creation of frequency function be done with decorator?

https://www.reddit.com/r/Python/comments/azbdvr/python_decorators_made_simple_part_1/

Improve help to solver

Provide details and explanation for structure of function to be called.

Incorporate fftw as an option.

https://hgomersall.github.io/pyFFTW/pyfftw/builders/builders.html

FFTW is much faster, but license restricted. The code should allow the user to optionally call fftw, but not rely on it.

Further, speedups can be obtained with SciPy as seen at https://blog.mide.com/matlab-vs-python-speed-for-vibration-analysis-free-download

Use pytorch

https://pytorch.org/blog/the-torch.fft-module-accelerated-fast-fourier-transforms-with-autograd-in-pyTorch/

Need to raise exception when failure to find solution

Need to raise exception and create working demo

Pseudo Arc-length continuation

I've started adding an implementation of pseudo-arc length continuation to the code. I've checked it to be working for the SDOF Duffing oscillator. You can have a look at the code in the branch "Nidish" in my fork of the project (https://github.com/Nidish96/mousai/tree/Nidish).
I'll raise a pull request once I'm done qualifying the rest. I'm raising this issue here for any suggestions you may give me. I'm having the following things on my agenda:

• Implement Jacobian through finite-differences for both Frequency domain as well as Time domain user supplied routines
• Qualify the code for MDOF systems
• Switch to a better solver
• Create a time-marching-based Monodromy stability estimation method

[DOC] Second or First order Derivative

In https://josephcslater.github.io/mousai/tutorial/demos/Theory_and_Examples.html
Do you use \ddot{z} or \dot{z} ?

VS

Needs automated initial guess

Automated initial guess of proper size (of zeros).

Quality automated guess by linearization

Calculate Jacobian and use eigen solution for guess?

Initial guess is a challenge. Not feasible to be very accurate for hard nonlinearities.

Make default form of derivatives state space

Form of derivative function should be state space by default. Defaulting fails.

Change hb_so name, deprecate it.