Mesh Generation and Size Functionsusing Implicit Geometries and PDEs

Per-Olof Persson (persson@mit.edu)

Department of Mathematics, MIT

http://www.mit.edu/∼persson

13th International Meshing Roundtable, September 20, 2004

Mesh Generation for Implicit Geometries

Geometry Specification

• Geometry given by φ(x) ≤ 0, with

discretized φ (Cartesian, octree, un-

structured)

• Natural representation for

– Smooth objects

– Moving meshes (level sets)

– Medical data (MRI)

– Geophysical data

Unstructured Meshing

• Smoothing-based approach with lo-

cal connectivity updates

• Project boundary nodes using ∇φ

• Needs a priori element size function!

• Persson/Strang, SIREV 2004:2

Initial Guess Final Configuration

Mesh Size Functions

Signed Distance Function

• Compute d(x) from φ(x) s.t.

d = 0 ⇔ φ = 0 and |∇d| = 1.

• O(n log n) time: Fast marching

method [Sethian]

Curvature Adaptation

• Compute curvature from φ by

κ = ∇ · (∇φ/|∇φ|)

• Size function with grid correction:

hcurv =∣∣∣1+κijkφijk

Kκijk

∣∣∣ + gφijk

Feature Size

• Solve hyperbolic PDE numerically:∂h∂t

+∇d · ∇h = 0, h(t0) = d

• Alternatives: Second arrival, MAT

Mesh Gradation

• Solve gradient limiting PDE:∂h∂t

+ |∇h| = min(|∇h|, g)

h(x)|t=0 = h0(x)

• Analytical solution for Rn:

h(x) = miny(h0(y) + g|x− y|)

• Persson, Proc. of 13th IMR

Mesh Size Function h(x) Mesh Based on h(x)