Mesh generators

const mesh = generateTriangleMesh(geometry, { size, minAngle, elementOrder, maxNodes });
const mesh = generateQuadrilateralMesh(geometry, { size, divisions, elementOrder });
model.setMesh(mesh);

The two generators behind setGeometry and setMeshConfig, exported for a mesh that is inspected, plotted or reused before it is solved on. Both take a geometry, in the form of setGeometry ({ rectangle } or { shape, holes, points }), and return a mesh object, the same as importGmshMesh, for model.setMesh(mesh). The boundaries of the mesh are keyed by the names of the geometry.

Both throw a FEAScriptError with the code "INVALID_INPUT" for a polygon that is not simple (its edges cross), a hole outside the shape or inside another hole, a point outside the domain, a path that does not end at its start, or a setting out of range, listing every problem found.

generateTriangleMesh(geometry, options)

Meshes a polygon, with or without holes, with triangles: a constrained Delaunay triangulation of the input points, refined by Ruppert's algorithm until every element has angles of at least minAngle and an area of at most that of the equilateral triangle of side size. See Algorithm and software.

Option	Type	Description
size	number or function	The length of an element side, as a number, or as a function (x, y) of the element center for a mesh size that varies. Without it, only the angles are refined
minAngle	number	The smallest angle of the elements, in degrees (default 20). The refinement always stops below 20.7 degrees, and usually up to about 33; above that, it may reach maxNodes
elementOrder	string	"linear" (3-node triangles, default) or "quadratic" (6-node triangles)
maxNodes	number	The limit on the number of vertices (the corner nodes of the elements) before the refinement stops with a "MESH_TOO_FINE" error (default 100000)

Example

A rectangular plate with a square hole, quadratic elements of side 0.15 at most:

import { FEAScriptModel, generateTriangleMesh } from "feascript";

const mesh = generateTriangleMesh(
  {
    rectangle: [2, 1],
    holes: [
      {
        name: "hole",
        points: [
          [0.8, 0.4],
          [1.2, 0.4],
          [1.2, 0.6],
          [0.8, 0.6],
        ],
      },
    ],
  },
  { size: 0.15, minAngle: 25, elementOrder: "quadratic" },
);

const model = new FEAScriptModel();
model.setModelConfig("heatConduction", { coefficientFunctions: { thermalConductivity: 1 } });
model.setMesh(mesh);
model.addBoundaryCondition("bottom", { type: "constantTemperature", temperature: 100 });
model.addBoundaryCondition("right", { type: "symmetry" });
model.addBoundaryCondition("top", { type: "constantTemperature", temperature: 0 });
model.addBoundaryCondition("left", { type: "symmetry" });
model.addBoundaryCondition("hole", { type: "convection", h: 1, ambientTemperature: 20 });
const { solutionVector, nodesCoordinates } = model.solve();

The function works the same in the browser and in Node.js; it reads no files.

generateQuadrilateralMesh(geometry, options)

Meshes a rectangle, or any convex shape of exactly four points without holes or embedded points, with a grid of quadrilateral elements mapped onto the shape by its corners (a bilinear map). The rows of nodes run from the bottom side (the first edge of the shape) to the top side, and the columns from the left side to the right side; the nodes are numbered column by column, as in Node numbering.

Option	Type	Description
size	number	The length of an element side: each side of the shape is divided into round(length / size) elements, the longer of two opposite sides deciding
divisions	number[]	[nx, ny], the number of elements along the bottom and the left side, instead of size
elementOrder	string	"linear" (4-node elements, default) or "quadratic" (9-node elements)

A shape with more than four points, a non-convex one, holes, embedded points, or a size function throw "INVALID_INPUT"; triangles cover those cases.

import { generateQuadrilateralMesh } from "feascript";

const mesh = generateQuadrilateralMesh(
  { rectangle: [4, 2] },
  { elementOrder: "quadratic", divisions: [8, 4] },
);
console.log(mesh.nodesXCoordinates.length); // 153 nodes

Algorithm and software

The triangle generator works in two stages:

1. Constrained Delaunay triangulation of the input points. The Delaunay triangulation is computed with Delaunator (ISC license), and the edges of the shape and of the holes are then forced into it with Constrainautor (ISC license), by flipping the edges that cross them. Triangles outside the shape or inside a hole are removed.
2. Delaunay refinement, Ruppert's algorithm [1]: a boundary edge that has a vertex inside its diametral circle is split at its midpoint, and a triangle that is too skinny or too large gets a new vertex. The new vertex is the off-center of the triangle [2], which lies closer to its short edge than the circumcenter and gives fewer vertices; a vertex that would fall inside the diametral circle of a boundary edge splits that edge instead, as in Triangle [3]. Where two edges of the input meet at a small angle, they are split at the same distances from the corner, in powers of two, and the skinny triangles between them are accepted, so that the refinement ends [4]. The vertices are inserted in passes, and the triangulation is rebuilt after each pass.

The orientation and in-circle tests use the exact, adaptive-precision predicates of Shewchuk [5], through robust-predicates (Unlicense), so a point exactly on an edge or on a circle is decided correctly. The same predicates check the geometry (edges that cross, holes and points inside the shape) for both generators. The libraries are listed with their licenses in the NOTICE file of FEAScript-core.

The quality guarantees are those of the algorithm: the refinement always ends for a minAngle below 20.7 degrees, and usually up to about 33, apart from the triangles at the small input angles, which keep their angles.

The quadrilateral generator is the transfinite (bilinear) map of a grid onto the four corners, the mapped mesh of the finite element textbooks [6]; it needs no external software.

References

1. J. Ruppert, "A Delaunay refinement algorithm for quality 2-dimensional mesh generation," Journal of Algorithms, vol. 18, no. 3, pp. 548–585, 1995.
2. A. Üngör, "Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations," in LATIN 2004: Theoretical Informatics, Lecture Notes in Computer Science, vol. 2976, pp. 152–161, 2004.
3. J. R. Shewchuk, "Delaunay refinement algorithms for triangular mesh generation," Computational Geometry: Theory and Applications, vol. 22, no. 1–3, pp. 21–74, 2002.
4. G. L. Miller, S. E. Pav and N. J. Walkington, "When and why Ruppert's algorithm works," in Proceedings of the 12th International Meshing Roundtable, pp. 91–102, 2003.
5. J. R. Shewchuk, "Adaptive precision floating-point arithmetic and fast robust geometric predicates," Discrete & Computational Geometry, vol. 18, no. 3, pp. 305–363, 1997.
6. W. J. Gordon and C. A. Hall, "Construction of curvilinear co-ordinate systems and applications to mesh generation," International Journal for Numerical Methods in Engineering, vol. 7, no. 4, pp. 461–477, 1973.