High-order Solution Transfer between Curved Triangular Meshes

Cite paper

This is work with Per-Olof Persson.

This originally appeared in my thesis and is now split out
into a standalone paper.

This repository is laid out in a manner described in
Good Enough Practices in Scientific Computing.

The content itself has been uploaded to the arXiv and was submitted to
the journal CAMCoS on October 12, 2018. The paper has been accepted and
was published on January 8, 2025.

Abstract

The problem of solution transfer between meshes arises frequently in
computational physics, e.g. in Lagrangian methods where remeshing
occurs. The interpolation process must be conservative, i.e. it
must conserve physical properties, such as mass. We extend previous
works — which described the solution transfer process for straight sided
unstructured meshes — by considering high-order isoparametric meshes
with curved elements. To facilitate solution transfer, we numerically
integrate the product of shape functions via Green's theorem along the
boundary of the intersection of two curved elements. We perform a numerical
experiment and confirm the expected accuracy by transferring test fields
across two families of meshes.

Citation

To cite this paper:

@article{Hermes2025,
  doi = {10.2140/camcos.2025.20.1},
  url = {http://dx.doi.org/10.2140/camcos.2025.20.1},
  year = {2025},
  month = {Jan},
  publisher = {Mathematical Sciences Publishers},
  volume = {20},
  number = {1},
  pages = {1--27},
  author = {Hermes, Danny and Persson, Per-Olof},
  title = {High-order solution transfer between curved triangular meshes},
  journal = {Communications in Applied Mathematics and Computational Science},
  ISSN = {1559-3940}
}

Installation

The code used to build the manuscript, generate images and verify
computations is written in Python. To run the code, Python 3.7
should be installed, along with nox:

python -m pip install --upgrade 'nox == 2024.4.15'

Once installed, the various build jobs can be listed. For example:

$ nox --list-sessions
...
Sessions defined in .../noxfile.py:

* build_tex
* blacken -> Run black code formatter.

sessions marked with * are selected, sessions marked with - are skipped.

To run nox -s build_tex (i.e. to build the PDF), pdflatex and
bibtex are required.

Images

The images/ in this repository are from my thesis, in particular:

• images/preliminaries/:
  • inverted_element.pdf
  • main_figure01.pdf
  • main_figure26.pdf
• images/bezier-intersection/:
  • bbox_check.pdf
  • locate_in_triangle.pdf
  • main_figure21.pdf
  • main_figure22.pdf
  • main_figure23.pdf
  • main_figure24.pdf
  • subdivide_curve.pdf
  • subdivision_linearized.pdf
  • subdivision_process.pdf
• images/solution-transfer/:
  • main_figure00.pdf
  • main_figure02.pdf
  • main_figure11.pdf
  • main_figure13.pdf
  • main_figure14.pdf
  • main_figure15.pdf
  • main_figure16.pdf
  • main_figure25.pdf
  • main_figure30.pdf

Scripts to generate these images can be found there.