Hierarchical Interpolation with Projection, Evaluation, and Derivation

The purpose of this code is to define arbitrary interpolation trees with some utilities (evaluation, projection, and derivation) that can be used, for instance, in multi-material topology optimization algorithms ¹², as well as in other applications using projected gradient descent.

This code is a major upgrade of ³, based itself on ⁴.

Quickstart

Start by running the example files in the folder "examples". If your Matlab version supports it, you can run the LiveScripts (in "examples/notebooks"), or traditional .m files (in "examples/mfiles"), in alphabetical order.

NB: The code has been successfully tested on Matlab R2023a.

Purpose

Let $a\in \mathbb{R}^d$, and $x \in \mathcal{D}$ a convex polytope in 1,2 or 3 dimensions. The polytope has $n$ vertices and looks like one in the following figure :

This repository contains a general framework to define a mapping $m : \mathcal{D} \times \mathbb{R}^d \rightarrow \mathbb{R}^d$, which define an interpolation between $n$ functions $f_i : a \rightarrow f_i(a)$ on $\mathcal{D}$.

$$ m(a,x) = \sum_{i}^n \omega_i(x) f_i(a),$$

each of the $f_i$ being associated to a vertex of $\mathcal{D}$, and each of the weights $\omega_i$ being a generalized barycentric coordinate of $\mathcal{D}$ ⁴. This code can also handle penalizations $P_i$ :

$$ m(a,x) = \sum_{i}^n P_i(\omega_i(x)) f_i(a),$$

and extend the interpolation domain recursively by defining a rooted tree of interpolations :

$$ m(a,x) = \sum_{i}^n P_i(\omega_i(x)) m_i(a,x)$$

An illustration of such a hierarchical interpolation domain is given in the following figure.

The routines computing the evaluation of the interpolation, the derivatives, and the projection of exterior $x$ onto $\mathcal{D}$ are provided, making it suitable for a projected gradient descent in optimization.

Contents

• examples
  • mfiles
    • a_SimpleDomains.m : examples of simple interpolation domains
    • b_SimpleScalarInterp.m : interpolation of scalar functions on a simple domain.
    • c_SimpleVectorInterp.m : interpolation of vector functions on a simple domain.
    • d_HierarchicalScalarInterp.m : interpolation of scalar functions on hierarchized domains.
    • e_HierarchicalVectorInterp.m : interpolation of vector functions on hierarchized domains.
    • f_VertexFunctionOperations.m : some playaround with the functions
  • notebooks (same as in mfiles, but .mlx format)
• src

  • class (main code is here, definition of the classes)

    • Domain.m
    • Interpolation.m
    • Penalization.m
    • ShapeFunction.m
    • VertexFunction.m

  • others (auxiliary functions for computation)

    • mult.m (function to simplify pagewise multiplications)
    • t.m (just a shorthand for pagetranspose)

  • vizualization (auxiliary functions for vizualization)

    • getTransformedPlan.m
    • legend2color.m
    • plotcolor2D.m
    • rot3D.m
    • rotPoint.m
• LICENSE
• README.md

Citation

Please use the following citation reference if you use the code:

T. Cherrière. tcherrie/Hiped: Hierarchical Interpolation with Projection, Evaluation, and Derivation (v1.0.0), May 2024. Zenodo. https://doi.org/10.5281/zenodo.10718117

Bibtex entry:

@software{tcherrie_2024_10718117,
author       = {Cherrière, Théodore},
title        = {tcherrie/Hiped: Hierarchical Interpolation with Projection, Evaluation and Derivation},
month        = may,
year         = 2024,
publisher    = {Zenodo},
version      = {v1.0.0},
doi          = {10.5281/zenodo.10718117},
url          = {https://doi.org/10.5281/zenodo.10718117}
}

License

Copyright (C) 2024 Théodore CHERRIERE (theodore.cherriere@ricam.oeaw.ac.at)

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.

References

Footnotes

1. Cherrière, T., Laurent, L., Hlioui, S. et al., (2022). "Multi-material topology optimization using Wachspress interpolations for designing a 3-phase electrical machine stator." Structural and Multidisciplinary Optimization, 65(352). https://doi.org/10.1007/s00158-022-03460-1 ↩

2. Cherrière, T. (2024) "Toward topology optimization of hybrid-excited electrical machines using recursive material interpolation", SCEE 2024, Darmstadt, Germany. ↩

3. Cherrière, T. and Laurent, L. (2023), Wachspress2D3D, https://doi.org/10.5281/zenodo.7701776 ↩

4. M. Floater, A. Gillette, and N. Sukumar, (2014). “Gradient bounds for Wachspress coordinates on polytopes,” SIAM J. Numer. Anal., vol. 52, no. 1, pp. 515–532, doi: 10.1137/130925712 ↩ ↩²