Implicit Meshing

Mesh generation based on implicit functions is a major focus of MyMesh. It offers a way to generate surface and volume meshes of objects defined by mathematical functions.

What is an implicit function?

In three dimensions, implicit functions represent surfaces with the form:

\[f(x,y,z) = 0\]

For example, a sphere defined by \(x^2+y^2+z^2=r^2\) can be represented implicitly by the function \(f(x,y,z) = x^2+y^2+z^2 - r^2 = 0\). Any point satisfying the condition \(f(x,y,z) = 0\) is considered to be on the surface, while points where \(f(x,y,z) < 0\) are considered to be inside the surface and points where \(f(x,y,z) > 0\) are considered to be outside the surface. This convention is assumed by default throughout the implicit meshing functions of MyMesh (others may adopt the opposite convention elsewhere).

Defining implicit functions

Implicit functions being defined for use with MyMesh should generally be defined as functions of three variables (x, y, z) and accept vectorizable inputs (if x, y, z are scalars, func(x, y, z) should return a scalar, if x, y, z are numpy arrays, func(x, y, z) should return an array). For example, the “surface of genus 2” example from wikipedia can be defined as follows:

def example_func(x,y,z):
    f = 2*y*(y**2 - 3*x**2)*(1-z**2) + (x**2 + y**2)**2 - (9*z**2 - 1)*(1-z**2)
    return f

Several built-in implicit function are available in mymesh.implicit, including a sphere, box, and torus. For example, the implicit function of a unit sphere (center = (0,0,0), radius=1) can be obtained from: func = implicit.sphere([0,0,0], 1)

More complex geometries can be obtained by combining multiple functions using union, intersection, and difference operations (see Implicit CSG for more details).

Meshing Implicit Functions

Surface Meshing

m = implicit.SurfaceMesh(example_func, [-2,2,-2,2,-2,2], .05)

Volume Meshing

m = implicit.VoxelMesh(example_func, [-2,2,-2,2,-2,2], .1)

m = implicit.TetMesh(example_func, [-2,2,-2,2,-2,2], .05)