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Merge pull request #342 from robbievanleeuwen/joss
JOSS Paper
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‎.github/workflows/draft-pdf.yml‎

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name: Draft JOSS paper
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on: [push]
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jobs:
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paper:
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runs-on: ubuntu-latest
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name: Paper Draft
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steps:
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- name: Checkout
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uses: actions/checkout@v3
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- name: Build draft PDF
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uses: openjournals/openjournals-draft-action@master
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with:
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journal: joss
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# This should be the path to the paper within your repo.
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paper-path: paper/paper.md
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- name: Upload
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uses: actions/upload-artifact@v1
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with:
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name: paper
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# This is the output path where Pandoc will write the compiled
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# PDF. Note, this should be the same directory as the input
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# paper.md
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path: paper/paper.pdf

‎.pre-commit-config.yaml‎

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name: Check for added large files
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entry: check-added-large-files
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language: system
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args: ["--maxkb=750"]
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- id: check-toml
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name: Check Toml
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entry: check-toml

‎README.md‎

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[![Codecov](https://codecov.io/gh/robbievanleeuwen/section-properties/branch/master/graph/badge.svg)][codecov]
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[![pre-commit](https://img.shields.io/badge/pre--commit-enabled-brightgreen?logo=pre-commit&logoColor=white)][pre-commit]
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[![Black](https://img.shields.io/badge/code%20style-black-000000.svg)][black]
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[![DOI](https://zenodo.org/badge/104638241.svg)][zenodo]
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[pypi_]: https://pypi.org/project/sectionproperties/
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[status]: https://pypi.org/project/sectionproperties/
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[codecov]: https://app.codecov.io/gh/robbievanleeuwen/section-properties
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[pre-commit]: https://github.com/pre-commit/pre-commit
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[black]: https://github.com/psf/black
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[zenodo]: https://zenodo.org/badge/latestdoi/104638241
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`sectionproperties` is a python package for the analysis of arbitrary cross-sections
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using the finite element method. `sectionproperties` can be used to determine

‎docs/user_guide/theory.rst‎

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@@ -320,9 +320,10 @@ describe the resulting modified stiffness matrix, and solution and load vector:
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0 \\
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\end{bmatrix}
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where :math:`\textbf{C}` is a row vector of ones and :math:`\lambda` may be though of as
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a force acting to enforce the constraints, which should be relatively small when
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compared to the values in the force vector and can be omitted from the solution vector.
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where :math:`\textbf{C}` is the assembly of :math:`\int_{\Omega} \textbf{N} \, d \Omega`
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and :math:`\lambda` may be thought of as a force acting to enforce the constraints, which
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should be relatively small when compared to the values in the force vector and can be
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omitted from the solution vector.
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Calculation of Cross-Section Properties
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---------------------------------------
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following expressions:
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.. math::
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\textbf{k}^e &= \sum_{i=1}^3 w_i \textbf{B}_i^{\rm T} \textbf{B}_i J_e \\
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\textbf{k}^e &= \sum_{i=1}^6 w_i \textbf{B}_i^{\rm T} \textbf{B}_i J_e \\
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\textbf{f}^e &= \sum_{i=1}^6 w_i \textbf{B}_i^{\rm T}
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\begin{bmatrix}
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\textbf{N}_i \textbf{y}_e \\

‎paper/example/paper_example.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "219c61c6-eef9-43bf-bcb9-49d2de2dd879",
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"metadata": {},
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"outputs": [],
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"source": [
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"from sectionproperties.analysis import Section\n",
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"from sectionproperties.pre.library import bulb_section"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "a1f98f26-78d8-4861-b0ba-66096e32e5cd",
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"metadata": {},
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"outputs": [],
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"source": [
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"geom = bulb_section(d=200, b=50, t=12, r=10, n_r=8)\n",
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"geom.create_mesh(mesh_sizes=5)\n",
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"sec = Section(geometry=geom)\n",
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"sec.plot_mesh(materials=False)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "2bad83b0-d055-4fef-9b32-70dc8308e48c",
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"metadata": {},
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"outputs": [],
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"source": [
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"sec.calculate_geometric_properties()\n",
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"sec.calculate_warping_properties()\n",
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"post = sec.calculate_stress(mzz=1e6)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0b62f94f-1fa9-4659-a354-a6281628f973",
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"metadata": {},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"\n",
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"\n",
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"sec.plot_centroids(alpha=0.2)\n",
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"ax = sec.plot_centroids(alpha=0.2, nrows=1, ncols=2, render=False)\n",
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"fig = ax.get_figure()\n",
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"post.plot_stress(\n",
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" stress=\"mzz_zxy\",\n",
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" cmap=\"viridis\",\n",
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" normalize=False,\n",
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" alpha=0.2,\n",
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" title=\"Torsion Stress\",\n",
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" ax=fig.axes[1],\n",
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" fmt=\"{x:.3f}\",\n",
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")\n",
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"fig.savefig(\"../figures/example.png\", dpi=300)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.9.17"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

‎paper/figures/example.png‎

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‎paper/paper.bib‎

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@book{pilkey,
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author = {Pilkey, W. D.},
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address = {New York},
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booktitle = {Analysis and Design of Elastic Beams: Computational Methods},
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doi = {10.1002/9780470172667},
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isbn = {9780471381525},
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language = {eng},
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publisher = {Wiley},
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title = {Analysis and Design of Elastic Beams: Computational Methods},
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year = {2002},
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}
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@book{larson,
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author = {Larson, M. G. and Bengzon, F.},
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address = {Netherlands},
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booktitle = {The Finite Element Method},
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doi = {10.1007/978-3-642-33287-6},
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edition = {1},
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isbn = {9783642332869},
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language = {eng},
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publisher = {Springer Berlin, Heidelberg},
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series = {Texts in Computational Science and Engineering},
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title = {The Finite Element Method: Theory, Implementation, and Applications},
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volume = {10},
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year = {2013},
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}
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@software{shapely,
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author = {Gillies, Sean and van der Wel, Casper and Van den Bossche, Joris and Taves, Mike W. and Arnott, Joshua and Ward, Brendan C. and {others}},
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doi = {10.5281/zenodo.5597138},
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license = {BSD-3-Clause},
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month = oct,
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title = {Shapely},
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url = {https://github.com/shapely/shapely},
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version = {2.0.2},
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year = {2023}
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}
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@misc{triangle,
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author = {Rufat, D.},
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title = {Triangle},
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year = {2023},
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publisher = {GitHub},
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journal = {GitHub repository},
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url = {https://github.com/drufat/triangle}
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}
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@article{shewchuck,
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author = {Shewchuk, J. R.},
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doi = {10.1016/S0925-7721(01)00047-5},
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journal = {Computational Geometry},
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language = {eng},
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number = {1},
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pages = {21-74},
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publisher = {Elsevier B.V},
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title = {Delaunay refinement algorithms for triangular mesh generation},
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volume = {22},
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year = {2002},
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}

‎paper/paper.md‎

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---
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title: "sectionproperties: A Python package for the analysis of arbitrary cross-sections using the finite element method"
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tags:
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- python
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- computational mechanics
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- finite element method
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- cross-section
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- stress analysis
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- engineering
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authors:
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- given-names: Robbie
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non-dropping-particle: van
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surname: Leeuwen
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orcid: 0009-0004-8056-3977
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affiliation: 1
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- name: Connor Ferster
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orcid: 0009-0005-0861-2428
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affiliation: 2
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affiliations:
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- name: Independent Researcher, Australia
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index: 1
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- name: Independent Researcher, Canada # update
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index: 2
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date: 13 October 2023
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bibliography: paper.bib
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---
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# Summary
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Properties of plane cross-sections are often required in engineering research, analysis,
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and design. For example, cross-sectional properties are used to determine the
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displacements, natural frequencies, and stresses within beams under complex loading.
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`sectionproperties` is a Python package for the analysis of _arbitrary_ cross-sections
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using the finite element method. `sectionproperties` can be used to determine
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geometric and warping properties, as well as visualising cross-sectional stresses
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resulting from combinations of applied loads. `sectionproperties` aims to provide a
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pre-processor, analysis engine, and post-processor, in a single open source and
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accessible package, that can be used by researchers, practising engineers, and students.
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# Statement of Need
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Obtaining the geometric properties of simple shapes is a classical engineering problem
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with well-defined analytical solutions. However, obtaining warping properties, e.g. for
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torsion and shear analyses, involves solving partial differential equations [@pilkey].
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While some analytical solutions exist for a small subset of geometries, the method for
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obtaining these results is not able to be generalised to shapes commonly used in
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engineering pratice. Further, the analysis of arbitrary composite geometries, in which a
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cross-section could consist of any shape with any number of internal holes, and be made
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from any number of materials, complicates both geometric and warping computations.
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To the best of our knowledge, there is no open source software available for the
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computation of both geometric and warping propreties for composite, arbitary
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cross-sections. While there are several commercial solutions available, e.g.
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[`RSECTION 1`](https://www.dlubal.com/en/products/cross-section-properties-software/rsection),
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[`ShapeDesigner SaaS`](http://mechatools.com/en/shapedesigner.html), or
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[`CADRE Profiler`](https://www.cadreanalytic.com/profiler.htm), none of these are
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open source or provide an application programming interface (API) that would enable
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these programs to be used for research. As a result, `sectionproperties` supports both
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engineering practice and research, by implementing an open source solution to the
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complex modelling problem that is arbitrary composite geometric and warping analyses.
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# Implementation
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`sectionproperties` harnesses the power of Shapely [@shapely] to streamline geometry
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generation, and triangle [@triangle] (a python port of Triangle [@shewchuck]) to produce
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a triangular mesh of six-noded quadratic elements. The finite element method is used to
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solve for the geometric and warping properties, the latter involving the solution of
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partial differential equations and boundary value problems [@pilkey]. For example, the
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Saint-Venant torsion constant ($J$) is obtained by solving for the warping function,
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$\omega$ [@pilkey]:
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$$
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\nabla^2 \omega = 0
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$$
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subject to the boundary condition:
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$$
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\frac{\partial \omega}{\partial x} n_x + \frac{\partial \omega}{\partial y} n_y = y n_x - x n_y
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$$
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Using the finite element method, this problem is reduced to a set of linear equations of
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the form:
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$$
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\textbf{K} \boldsymbol{\omega} = \textbf{F}
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$$
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where the stiffness matrix and load vector at the element level are defined as:
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$$
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\textbf{k}^e = \sum_{i=1}^6 w_i \textbf{B}_i^{\rm T} \textbf{B}_i J_i
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$$
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$$
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\textbf{f}^e = \sum_{i=1}^6 w_i \textbf{B}_i^{\rm T}
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\begin{bmatrix}
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\textbf{N}_i \textbf{y}_e \\
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-\textbf{N}_i \textbf{x}_e \\
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\end{bmatrix} J_i
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$$
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In the above, $\textbf{N}$ and $\textbf{B}$ are the shape functions and their
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derivatives, and $w_i$ and $J_i$ are the weights and Jacobians of the current
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integration point. The boundary conditions neccesitate the inversion of a nearly
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singular global stiffness matrix. As such, the Lagrangian multiplier method is used to
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solve the set of linear equations of the form $\textbf{K} \textbf{u} = \textbf{F}$ by
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introducing an extra constraint on the solution vector, whereby the mean value is equal
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to zero [@larson].
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$$
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\begin{bmatrix}
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\textbf{K} & \textbf{C}^{\rm{T}} \\
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\textbf{C} & 0 \\
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\end{bmatrix}
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\begin{bmatrix}
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\textbf{u} \\
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\lambda \\
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\end{bmatrix} =
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\begin{bmatrix}
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\textbf{F} \\
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0 \\
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\end{bmatrix}
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$$
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where $\textbf{C}$ is the assembly of $\sum_{i} w_i \textbf{N}_i J_i$, and $\lambda$
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may be though of as a relatively small force acting to enforce the constraints. Once the
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warping function has been evaluated, the Saint-Venant torsion constant can be calculated
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as follows:
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$$
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J = I_{xx} + I_{yy} - \boldsymbol{\omega}^{\rm T} \textbf{K} \boldsymbol{\omega}
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$$
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The calculation of plastic properties is meshless, and is conducted using an iterative
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method to enforce plastic equilibrium, yielding the plastic centroids. A full
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description of the theoretical background underpinning `sectionproperties` can
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be found in the
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[documentation](https://sectionproperties.readthedocs.io/en/stable/user_guide/theory.html).
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An example of some of the visualisation generated by `sectionproperties` can be seen in
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\autoref{fig:example} below.
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![Plot of the centroids and torsion stress distribution for a bulb-section modelled in `sectionproperties`.\label{fig:example}](figures/example.png)
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# Software Development
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The `sectionproperties` package is available on [GitHub](https://github.com/robbievanleeuwen/section-properties),
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where the source code, issue tracker, CI workflow, and discussion board can be found.
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Pre-commit hooks are used to ensure code quality and style is consistent across all
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contributions. There is an extensive testing and validation suite used to ensure that
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the output produced by `sectionproperties` is verified and repeatable, including a set
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of benchmarking tests. `sectionproperties` has an actively maintained and complete
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[documentation](https://sectionproperties.readthedocs.io), including
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[installation instructions](https://sectionproperties.readthedocs.io/en/stable/installation.html),
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a detailed [user guide](https://sectionproperties.readthedocs.io/en/stable/user_guide.html),
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a list of [examples](https://sectionproperties.readthedocs.io/en/stable/examples.html),
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and an [API reference](https://sectionproperties.readthedocs.io/en/stable/api.html).
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# Conclusion
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In this paper we have described `sectionproperties`, a Python package that calculates
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the section properties of arbitrary sections. It is our hope that this project is used
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by researchers and practising engineers to improve their experimental and analysis
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workflows.
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# Acknowledgements
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We acknowledge the contributions from all the
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[contributors](https://github.com/robbievanleeuwen/section-properties/graphs/contributors)
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to `sectionproperties`.
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# References

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