heatConduction2DFinGmsh

FEAScript vs Elmer and scikit-fem benchmark: 2D fin on Gmsh meshes

This folder compares FEAScript with the Elmer finite element solver and the scikit-fem finite element library on unstructured meshes made with Gmsh. The problem is the 2D fin of heatConduction2DFin, which has an exact solution. All the codes read the same meshes, so this tests what structured meshes do not: reading a mesh file, triangles, distorted quadrilaterals, and elements with no regular numbering.

Contents

• Results
• The problem
• What is measured
• How to run the benchmark
• License

Results

The report of each run is in reports/<machine>/<FEAScript version>/. The results of each code are in <code>/results/<machine>/<its version>/, and those of FEAScript also by runtime, in feascript/results/<machine>/<FEAScript version>/<runtime>/<its version>/:

Machine	FEAScript version	Results
AMD Ryzen 5 9600X	0.3.1	reports/amd-ryzen-5-9600x/0.3.1/
AMD Ryzen 5 9600X	0.4.0	reports/amd-ryzen-5-9600x/0.4.0/
Intel N100	0.3.1	reports/intel-n100/0.3.1/
Intel N100	0.4.0	reports/intel-n100/0.4.0/

Each report has a README with the plots of that run, a table of all the numbers (results_table.md) and the details of the run (meta.json): the versions of all the software, the machine and the date. The 0.4.0 results were measured before the release, on commit fb17f89 (AMD Ryzen 5 9600X) and commit 367d095 (Intel N100) of the release candidate, whose solver code is the same.

About the comparison: Elmer runs here on one core, as a single MPI process, on small 2D problems, so that the codes are compared on the same terms. See About the comparison.

The problem

• Domain: 4 × 2 rectangle, thermal conductivity k = 1
• Constant temperature: T = 200 at the bottom and right edges
• Symmetry: zero heat flux on the left edge
• Convection: h = 1 and T∞ = 20 on the top edge
• Exact solution: the Fourier series of heatConduction2DFin, evaluated to better than 1e-6

What is measured

gmsh/fin.geo describes the geometry, and the benchmark runs Gmsh on it for each mesh:

• Triangles: linear (3-node) and quadratic (6-node), with target element sizes 0.2, 0.1, 0.05 and 0.025
• Quadrilaterals: linear (4-node) and quadratic (9-node), with target element sizes 0.4, 0.2, 0.1 and 0.05. Gmsh splits every triangle into three quadrilaterals, which halves the element size and gives distorted elements

On each mesh, the benchmark runs:

• FEAScript: the frontal and lusolve direct solvers, reading the .msh file with importGmshMesh
• Elmer: the UMFPACK direct solver, on the mesh converted with ElmerGrid
• scikit-fem: its default direct solver (SuperLU from SciPy), reading the .msh file with meshio, once with the Gauss rules of FEAScript and once with a much finer rule

FEAScript also runs in Chromium and Firefox on the three finest quadratic meshes of each shape, with frontal and lusolve: see Browsers.

The compared quantities are the RMS and largest error against the exact solution, the largest difference from FEAScript at the same nodes, the assembly and solve time, the end-to-end time and the peak memory, measured as in heatConduction2DFin. Reading the mesh is part of the end-to-end time, not of the assembly and solve time.

On a triangle with straight sides, the Jacobian is constant, so Gauss quadrature integrates the stiffness matrix exactly. On a distorted quadrilateral it does not: the integrand is no longer a polynomial, and the result depends on the rule. FEAScript uses 2 × 2 points on linear quadrilaterals and 3 × 3 on quadratic ones, which is why the main scikit-fem run uses the same rules, and the run with a finer rule shows what they cost in accuracy.

How to run the benchmark

The benchmark needs what heatConduction2DFin needs, plus Gmsh on the PATH and the Python package meshio:

~/.venvs/feascript-benchmark/bin/pip install meshio
~/.venvs/feascript-benchmark/bin/python benchmark.py   # writes <code>/results/ and reports/
python3 make_report.py <machine> <version>             # writes the plots, results_table.md and README.md

It takes the same environment variables as heatConduction2DFin. ONLY runs some meshes, e.g. ONLY=tri-linear-h0.2, and NO_BROWSERS=1 and ONLY_BROWSERS=1 leave out or run only the browsers.

The files in this folder:

• benchmark.py: generates the meshes, runs all the codes, compares them and writes the results
• make_report.py: draws the plots and writes the results table and README of a run
• gmsh/fin.geo: the geometry and mesh settings
• feascript/run_feascript.mjs: runs one FEAScript case
• elmer/case.sif.in: the Elmer input template
• scikit-fem/run_skfem.py: runs one scikit-fem case
• gmsh/meshes/ and elmer/meshes/: the meshes, generated by the benchmark and not committed
• feascript/case.mjs: the FEAScript model, shared by Node.js and the browsers
• feascript/results/<machine>/<FEAScript version>/<runtime>/<its version>/: the results of FEAScript in Node.js, Chromium and Firefox: results.json, and in Node.js nodal results (not committed)
• elmer/results/<machine>/<its version>/ and scikit-fem/results/<machine>/<its version>/: the latest results of Elmer and scikit-fem: results.json, nodal results (not committed) and the Elmer .sif file
• reports/<machine>/<FEAScript version>/: the report of one run: README, plots, results table, meta.json, results.json and browsers.json

License

This benchmark is released under the MIT license. Note that the core library of FEAScript is distributed under the terms of the MIT license.