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

This page demonstrates solving the 2D heat conduction fin problem using a Gmsh-generated mesh. Gmsh External Link Icon is a powerful mesh generation tool that can create complex geometries and meshes for finite element analysis. For the mathematical formulation and theory, see the basic tutorial.

Gmsh File Import

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 FEAScript library
    import { FEAScriptModel, importGmshQuadTri, plotSolution, printVersion } from "https://core.feascript.com/dist/feascript.esm.js";
  
    window.addEventListener("DOMContentLoaded", async () => {
      // Print FEAScript version in the console
      printVersion();
  
      // Fetch the mesh file
      const response = await fetch("./rect_quad_unstruct.msh"); // .msh version 4.1 is currently supported
      if (!response.ok) {
        throw new Error(`Failed to load mesh file: ${response.status} ${response.statusText}`);
      }
      const meshContent = await response.text();
  
      // Create a File object with the actual content
      const meshFile = new File([meshContent], "rect_quad_unstruct.msh");
  
      // Create and configure model
      const model = new FEAScriptModel();
      model.setSolverConfig("heatConductionScript");
  
      // Parse the mesh file
      const result = await importGmshQuadTri(meshFile);
  
      // Define mesh configuration with the parsed result
      model.setMeshConfig({
        parsedMesh: result,
        meshDimension: "2D",
        elementOrder: "quadratic",
      });
  
      // Apply boundary conditions using Gmsh physical group tags
      model.addBoundaryCondition("0", ["constantTemp", 200]); // bottom boundary
      model.addBoundaryCondition("1", ["constantTemp", 200]); // right boundary
      model.addBoundaryCondition("2", ["convection", 1, 20]); // top boundary
      model.addBoundaryCondition("3", ["symmetry"]); // left boundary
  
      // Solve
      model.setSolverMethod("lusolve");
      const { solutionVector, nodesCoordinates } = model.solve();
  
      // Plot results
      plotSolution(
        solutionVector,
        nodesCoordinates,
        model.solverConfig,
        model.meshConfig.meshDimension,
        "contour",
        "resultsCanvas",
        "unstructured" // Important: specify unstructured mesh type for Gmsh meshes
      );
    });
  </script>
  <!-- ... rest of body region... -->
</body>

Important notes about the Gmsh workflow:

You can create your own Gmsh files by writing .geo scripts or using Gmsh's GUI. For this example, we used a simple rectangular domain defined in a .geo file with specific physical groups for each boundary.

Example Gmsh .geo File

Below is the rect.geo file used in this tutorial. It defines a 4 m × 2 m rectangular domain with physical lines for each boundary edge: 

// 2D Rectangle: 4m (width) x 2m (height) with physical lines for boundary labeling

lc = 0.7; // Characteristic length (mesh density)

// 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, 2, 0, lc}; // Top right
Point(4) = {0, 2, 0, lc}; // Top left

// Lines
Line(1) = {1, 2}; // Bottom boundary
Line(2) = {2, 3}; // Right boundary
Line(3) = {3, 4}; // Top boundary
Line(4) = {4, 1}; // Left boundary

// Line Loop and Surface
Line Loop(1) = {1, 2, 3, 4};
Plane Surface(1) = {1};

// Physical Lines
Physical Line("bottom") = {1};
Physical Line("right")  = {2};
Physical Line("top")    = {3};
Physical Line("left")   = {4};

// Physical Surface (optional, for FEM domains)
Physical Surface("domain") = {1};

// Generate 2D mesh
Recombine Surface{1}; // Turn triangle elements into quadrilaterals
Mesh.ElementOrder = 2; // Set quadratic elements 
Mesh 2;

Note how the physical line tags in the .geo file correspond to the boundary conditions in our FEAScript code:

To generate a mesh file from this .geo script, you would run: gmsh rect.geo -2 in your terminal, which creates a rect.msh file that can be imported into FEAScript.

Generated Mesh

Below is a visualization of the quadrilateral mesh generated with Gmsh. The mesh consists of 30 elements.

Quadrilateral mesh generated by Gmsh

Quadrilateral mesh (rect_quad_unstruct.msh) generated using the rect.geo script with Gmsh

The unstructured mesh approach would be particularly advantageous for complex geometries, where the mapped mesh approach could not be applied effectively.

Results

Below is the 2D contour plot of the computed temperature distribution. This plot is generated in real time using FEAScript. Please note that solutions computed on unstructured meshes like this may exhibit small numerical differences compared to solutions on structured orthogonal meshes. This occurs because derivative calculations in non-orthogonal elements are inherently less precise due to the Jacobian transformation process. These small differences are expected and acceptable for most engineering applications, but may be noticeable in regions with steep temperature gradients.