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A modular approach for topology optimization

Project description



Modular topology optimization framework with semi-automatic derivatives. The two main types Module and Signal are used to implement a problem formulation to be optimized. The Module implements functionality (and design sensitivity calculations) and Signal carries data of both the variables and their derivatives.

Sensitivity calculations are done based on backpropagation. The final value of interest is seeded with sensitivity value $\frac{\textup{d}f}{\textup{d}f}=1$. Then the modules are executed in reverse order, each applying the chain rule. As example for a Module which maps $x\rightarrow y$, only the following operation needs to be implemented:

$$ \frac{\textup{d}f}{\textup{d}x} = \frac{\textup{d}f}{\textup{d}y}\frac{\textup{d}y}{\textup{d}x}\text{.} $$

In this way, the order of modules can easily be adapted without having to worry about sensitivities, as these are automatically calculated.

Quick start installation

  1. Make sure you have Python running in some kind of virtual environment (e.g. conda, miniconda, venv)
  2. Install the pymoto Python package (and its dependencies)
    • Option A (conda): If you are working with Conda, install by conda install -c aatmdelissen pymoto
    • Option B (pip): Type pip install pymoto into your console to install
  3. Download one of the examples found in the repository's example folder (here)
  4. Run the example by typing python in the console

A local installation for development in pyMOTO can be done by first downloading the entire git repo, and then calling pip install -e . in the pyMOTO folder (of course from within your virtual environment).


  • NumPy - Dense linear algebra and solvers
  • SciPy - Sparse linear algebra and solvers
  • SymPy - Symbolic differentiation for MathGeneral module
  • Matplotlib - Plotting and visualisation
  • (optional) SAO - Sequential approximated optimizers
  • (optional) opt_einsum - Optimized function for EinSum module

For fast linear solvers for sparse matrices:

  • (optional) pypardiso - Uses the Intel OneAPI PARDISO solver (recommended)
  • (optional) scikit-umfpack - Fast LU linear solver based on UMFPACK
  • (optional) scikit-sparse - Fast Cholesky solver based on CHOLMOD
  • (optional) cvxopt - Another fast Cholesky solver based on CHOLMOD

Note on linear solvers for sparse matrices: Scipy implements a version of LU which is quite slow. To increase the speed of the optimization, pypardiso is recommended as it contains a very robust and flexible solver for symmetric and asymmetric matrices. An alternative is scikit-umfpack which provides a fast LU factorization. For symmetric matrices a Cholesky factorization is recommended (not provided with Scipy), which can be used by either installing scikit-sparse or cvxopt.


pyMOTO is available under te MIT License.

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