Mesh Generation and Size Functions using Implicit...
Transcript of Mesh Generation and Size Functions using Implicit...
Mesh Generation and Size Functionsusing Implicit Geometries and PDEs
Per-Olof Persson ([email protected])
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)