[Hacktoberfest Athens] Fin with two holes: a Gmsh heat-conduction tutorial (Team A) #5

Fin with two holes: a Gmsh heat-conduction tutorial, Team A (no AI)

This issue is a hands-on example prepared for the Hacktoberfest Meetup Athens x FEAScript, 2026.

Add a new browser tutorial to the FEAScript website: 2D steady-state heat conduction in a
rectangular fin with two circular holes, meshed with Gmsh and solved with FEAScript.
This team works without AI assistants: use the existing tutorials, the Gmsh docs and each other.

Start from the existing rhomboid tutorial:
tutorials/heat-conduction-2d-rhom-fin-gmsh.html (+ rhom.geo, rhom_quad.msh)

The geometry

• Rectangle 4 m × 2 m, corners (0, 0) and (4, 2)
• Two circular holes of radius 0.3 m, centred at (1.3, 1) and (2.7, 1)
• The holes act as cooling channels: heat leaves the fin through their surface by convection

The physics

Steady heat conduction, k = 1 (default), no heat source.

FEAScript boundary key = Gmsh physical group tag − 1.

Making the mesh with Gmsh

Write fin-two-holes.geo:

1. Outer rectangle: 4 points and 4 lines (bottom, right, top, left), lc = 0.4
2. Each hole: a centre point plus 4 points on the circle, joined by 4 Circle arcs
   (e.g. Circle(11) = {11, 10, 12}; means start, centre, end); use a finer size lch = 0.2
3. Surface: one Curve Loop per boundary, then Plane Surface(1) = {outer, hole1, hole2};
4. Physical groups: exactly as in the table. Both holes (all 8 arcs) in one group. The surface gets the last tag (6)
5. Elements: FEAScript needs quadrilateral elements:

   Recombine Surface{1};
   Mesh.RecombinationAlgorithm = 1;
   Mesh.SubdivisionAlgorithm = 1; // all-quad mesh
   Mesh 2;
   

6. Export as MSH 4.1 ASCII:
   gmsh fin-two-holes.geo -2 -format msh41 -o fin_two_holes_quad.msh
7. Check the size: the second number after $Nodes in the .msh is the node count.
   Aim for about 3,000 nodes or fewer.
8. Upload the mesh to a Gist: create a Gist at https://gist.github.com with your
   fin_two_holes_quad.msh (public or secret), open it, click Raw and copy the URL.
   Remove the commit hash so the link keeps working after edits:
   https://gist.githubusercontent.com/<user>/<gist-id>/raw/fin_two_holes_quad.msh

Starter skeleton for fin-two-holes.geo (complete the TODOs):

lc  = 0.4; // Mesh size at the outer boundary
lch = 0.2; // Finer mesh size around the holes
r   = 0.3; // Hole radius

// Outer rectangle
Point(1) = {0, 0, 0, lc};
Point(2) = {4, 0, 0, lc};
Point(3) = {4, 2, 0, lc};
Point(4) = {0, 2, 0, lc};
Line(1) = {1, 2}; // bottom
Line(2) = {2, 3}; // right
Line(3) = {3, 4}; // top
Line(4) = {4, 1}; // left

// Hole 1: centre (1.3, 1), four arcs
Point(10) = {1.3,     1,     0, lch}; // centre
Point(11) = {1.3 + r, 1,     0, lch};
Point(12) = {1.3,     1 + r, 0, lch};
Point(13) = {1.3 - r, 1,     0, lch};
Point(14) = {1.3,     1 - r, 0, lch};
Circle(11) = {11, 10, 12}; // start, centre, end
Circle(12) = {12, 10, 13};
Circle(13) = {13, 10, 14};
Circle(14) = {14, 10, 11};

// TODO: Hole 2, centre (2.7, 1), same pattern (use points 20-24 and circles 21-24)

// TODO: one Curve Loop per boundary (outer, hole 1, hole 2)
// TODO: Plane Surface(1) = {outer loop, hole 1 loop, hole 2 loop};

// TODO: physical groups as in the table (tags 1-5 with no gaps, surface = 6)
// Syntax example: Physical Line("bottom", 1) = {1};

Recombine Surface{1};
Mesh.RecombinationAlgorithm = 1;
Mesh.SubdivisionAlgorithm = 1;
Mesh 2;

The FEAScript part

Start from heat-conduction-2d-rhom-fin-gmsh.html and complete the TODOs:

// Load the mesh from your Gist (raw URL without commit hash)
const response = await fetch("https://gist.githubusercontent.com/<user>/<gist-id>/raw/fin_two_holes_quad.msh");
if (!response.ok) throw new Error(`Failed to load mesh: ${response.status}`);
const meshFile = new File([await response.text()], "fin_two_holes_quad.msh");

model.setModelConfig("heatConductionScript");
model.setMeshConfig({
  parsedMesh: await importGmshMesh(meshFile),
  meshDimension: "2D",
  elementOrder: /* TODO: must match how you generated the mesh */,
});

// TODO: add one boundary condition per row of the physics table.
// Remember: FEAScript key = Gmsh physical group tag − 1.
// Hint: the left edge needs no line at all.

model.setSolverMethod("lusolve"); // frontal currently fails on this mesh
const result = model.solve();
plotInterpolatedSolution(model, result, "contour", "resultsCanvas");

Note: the rhomboid tutorial uses quadratic elements, a local mesh file, the deprecated constantTemp
and symmetry. This tutorial uses linear elements, a Gist mesh, constantTemperature, and leaves
insulated edges without a condition.

Hint: Performing sanity check. solve() returns { solutionVector, nodesCoordinates }.

// Find min and max of the temperature
const minT = Math.min(...result.solutionVector);
const maxT = Math.max(...result.solutionVector);

Files to add (in tutorials/)

• fin-two-holes.geo
• heat-conduction-2d-fin-two-holes-gmsh.html

The .msh file is not committed: it lives in your Gist and the page fetches it from there.

Required features

• Page loads the mesh from your Gist raw URL with fetch and importGmshMesh
• Linear elements and lusolve
• Contour with plotInterpolatedSolution, with both holes masked (empty)
• Console prints min/max temperature

Definition of done

• Page works when opened by double-click (as file://): the mesh loads from the Gist
• The Gist raw URL has no commit hash and opens the mesh text in the browser
• All temperatures lie within [20, 200] (sanity check)
• Every team member has at least one commit
• PR includes a screenshot of the mesh (from Gmsh) and of the result
• PR description includes the link to your mesh Gist
• PR description contains Closes #<this issue number>

Bonus goals

• Compare insulated holes (remove the holes' condition) vs convective holes: how does the max/min T change?
• Investigate mesh refinement: Generate a finer quadrilateral mesh and compare the minimum/maximum temperatures.

Where and how to contribute

• Repository: FEAScript-website
• Workflow:
  1. One team member forks FEAScript-website and adds the others as collaborators on the fork
  2. Everyone clones the fork and works on the branch fin-two-holes
  3. Split the work: one person on the .geo/.msh and the Gist, one on the HTML/JS, one on the
     text and screenshots
  4. Commit small steps with clear messages (e.g. "Add two-hole geometry", "Add tutorial page")
  5. Push and open a pull request to FEAScript/FEAScript-website main