Heat Conduction in a 2D Rhomboid Fin Tutorial (Gmsh Mesh)

This example demonstrates how to import a Gmsh-generated mesh (.msh file format) and solve a heat conduction problem.

<body>
  <!-- ...body region... -->
  <script type="module">
    import {
      FEAScriptModel,
      importGmshMesh,
      plotInterpolatedSolution,
      printVersion,
    } from "https://core.feascript.com/dist/feascript.esm.js";

    window.addEventListener("DOMContentLoaded", async () => {
      console.log("FEAScript version:", printVersion);

      // 1. Create the model and select the physics
      const model = new FEAScriptModel();
      model.setModelConfig("heatConductionScript");

      // 2. Load the Gmsh mesh (.msh format version 4.1)
      const response = await fetch("./rhom_quad_quadratic.msh");
      if (!response.ok) {
        throw new Error(`Failed to load the mesh file: ${response.status} ${response.statusText}`);
      }
      const meshFile = new File([await response.text()], "rhom_quad_quadratic.msh");
      const mesh = await importGmshMesh(meshFile);
      model.setMeshConfig({ parsedMesh: mesh });

      // 3. Apply the boundary conditions (key = Gmsh physical group tag - 1)
      model.addBoundaryCondition("0", ["constantTemperature", 200]); // Bottom
      model.addBoundaryCondition("1", ["symmetry"]);                 // Left
      model.addBoundaryCondition("2", ["convection", 1, 20]);        // Top
      model.addBoundaryCondition("3", ["constantTemperature", 200]); // Right

      // 4. Solve
      model.setSolverMethod("lusolve");
      const result = model.solve();

      // 5. Plot the results
      plotInterpolatedSolution(model, result, "contour", "resultsCanvas");
    });
  </script>
  <!-- ...rest of body region... -->
</body>

For details on the Gmsh workflow, physical groups, and boundary condition mapping, please refer to the rectangular fin tutorial.

Below is the rhom.geo file used in this tutorial. It defines a rhomboid domain with a 4 m base and a 2 m height, whose top edge is shifted by 1 m, with a physical line for each boundary edge:

// 2D rhomboid: 4 m (base width) x 2 m (height), top edge shifted by dx
// Generate the mesh with: gmsh rhom.geo -2 -format msh41 -o rhom_quad_quadratic.msh

// 1. Mesh size and shape
lc = 0.7; // Characteristic length
dx = 1.0; // Horizontal shift of the top edge (sets the rhomboid angle)

// 2. Points (x, y, z, mesh size)
Point(1) = {0,      0, 0, lc}; // Bottom left
Point(2) = {4,      0, 0, lc}; // Bottom right
Point(3) = {4 + dx, 2, 0, lc}; // Top right (shifted)
Point(4) = {dx,     2, 0, lc}; // Top left (shifted)

// 3. Lines and surface
Line(1) = {1, 2};            // Bottom
Line(2) = {2, 3};            // Right
Line(3) = {3, 4};            // Top
Line(4) = {4, 1};            // Left
Line Loop(1) = {1, 2, 3, 4}; // Closed boundary
Plane Surface(1) = {1};      // Domain

// 4. Physical groups, tagged bottom, left, top, right (the side order of FEAScript structured meshes)
Physical Line("bottom", 1) = {1};    // FEAScript boundary "0"
Physical Line("left", 2)   = {4};    // FEAScript boundary "1"
Physical Line("top", 3)    = {3};    // FEAScript boundary "2"
Physical Line("right", 4)  = {2};    // FEAScript boundary "3"
Physical Surface("domain", 5) = {1}; // Domain

// 5. Mesh
Recombine Surface{1};  // Quadrilaterals instead of triangles
Mesh.ElementOrder = 2; // Quadratic elements
Mesh 2;

To generate a mesh file from this .geo script, you would run: gmsh rhom.geo -2 -format msh41 -o rhom_quad_quadratic.msh in your terminal.

Below is a visualization of the quadrilateral mesh generated with Gmsh. The mesh consists of 24 elements.

Quadrilateral mesh (rhom_quad_quadratic.msh) generated using the rhom.geo script with Gmsh

Below is the 2D contour plot of the computed temperature distribution. This plot is generated in real time using FEAScript.