Your first simulation

This page walks through a complete simulation: steady-state heat conduction in a 2D fin, solved in the browser and plotted with Plotly. The same steps apply to every model.

The problem

A $4 \times 2$ fin is heated at its bottom and right sides, cooled by convection at its top side, and symmetric about its left side. The temperature $T$ satisfies $\nabla^2 T = 0$ (heat conduction with $k = 1$ and no heat source), with:

Side	Boundary key	Condition
Bottom	"0"	Constant temperature, $T = 200$
Left	"1"	Symmetry (zero heat flux)
Top	"2"	Convection, $h = 1$, $T_\infty = 20$
Right	"3"	Constant temperature, $T = 200$

The code

Save the following as an .html file and open it through a local web server (for example npx serve in the same folder):

<!DOCTYPE html>
<html>
  <head>
    <meta charset="utf-8" />
    <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjs/11.12.0/math.min.js"></script>
    <script src="https://cdnjs.cloudflare.com/ajax/libs/plotly.js/2.35.3/plotly.min.js"></script>
  </head>
  <body>
    <div id="resultsCanvas"></div>

    <script type="module">
      import {
        FEAScriptModel,
        plotSolution,
      } from "https://cdn.jsdelivr.net/npm/feascript@0.2.0/dist/feascript.esm.js";

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

      // 2. Define the mesh: 8 × 4 quadratic quadrilaterals on 0 ≤ x ≤ 4, 0 ≤ y ≤ 2
      model.setMeshConfig({
        meshDimension: "2D",
        elementOrder: "quadratic",
        numElementsX: 8,
        numElementsY: 4,
        maxX: 4,
        maxY: 2,
      });

      // 3. Apply the boundary conditions
      model.addBoundaryCondition("0", ["constantTemp", 200]); // Bottom
      model.addBoundaryCondition("1", ["symmetry"]); // Left
      model.addBoundaryCondition("2", ["convection", 1, 20]); // Top
      model.addBoundaryCondition("3", ["constantTemp", 200]); // Right

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

      // 5. Plot the results
      plotSolution(model, result, "contour", "resultsCanvas");
    </script>
  </body>
</html>

Step by step

1. Create the model and select the physics. FEAScriptModel holds the whole configuration of the simulation. setModelConfig takes the name of the model, here heat conduction.
2. Define the mesh. setMeshConfig generates a structured mesh from the size of the domain and the number of elements (see Mesh), or uses a mesh imported from Gmsh.
3. Apply the boundary conditions. The sides of a structured 2D mesh have the keys "0" (bottom), "1" (left), "2" (top) and "3" (right). See Boundary conditions for the condition types of every model.
4. Solve. setSolverMethod is optional (the default is "lusolve"). solve() returns the solutionVector and the nodesCoordinates of the mesh (see Solvers).
5. Plot the results. plotSolution draws the result in the resultsCanvas element.

The result

result.solutionVector holds the temperature at every node, and result.nodesCoordinates the coordinates of the nodes, in the same order:

const { solutionVector, nodesCoordinates } = result;
const { nodesXCoordinates, nodesYCoordinates } = nodesCoordinates;

for (let i = 0; i < solutionVector.length; i++) {
  console.log(nodesXCoordinates[i], nodesYCoordinates[i], solutionVector[i]);
}

The same simulation in Node.js

import * as math from "mathjs";
global.math = math;

import { FEAScriptModel } from "feascript";

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

// 2. Define the mesh
model.setMeshConfig({
  meshDimension: "2D",
  elementOrder: "quadratic",
  numElementsX: 8,
  numElementsY: 4,
  maxX: 4,
  maxY: 2,
});

// 3. Apply the boundary conditions
model.addBoundaryCondition("0", ["constantTemp", 200]); // Bottom
model.addBoundaryCondition("1", ["symmetry"]); // Left
model.addBoundaryCondition("2", ["convection", 1, 20]); // Top
model.addBoundaryCondition("3", ["constantTemp", 200]); // Right

// 4. Solve
const { solutionVector, nodesCoordinates } = model.solve();

// 5. Print the results
console.log(`Number of nodes: ${nodesCoordinates.nodesXCoordinates.length}`);
console.log("Temperature:", solutionVector);

Run it with node fin.js (with "type": "module" in your package.json, and feascript@0.2.0 and mathjs installed, see Installation).

Next steps

• Read about the physical models and their equations.
• Import an unstructured mesh with importGmshQuadTri.
• Run the simulation in a web worker, or on the GPU with the "jacobi-gpu" solver.
• Explore the examples of FEAScript-core 0.2.0.