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Mesh Generation for Implicit Geometries
PerOlof Persson ([email protected])
Department of Mathematics
Massachusetts Institute of Technology
Ph.D. Thesis Defense, December 8, 2004
Advisors: Alan Edelman and Gilbert Strang

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Topics
1. Introduction
2. The New Mesh Generator
3. Mesh Size Functions
4. Applications
5. Conclusions

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Mesh Generation
Given a geometry, determine node points and element connectivity
Resolve the geometry and high element qualities, but few elements
Applications: Numerical solution of PDEs (FEM, FVM, BEM),
interpolation, computer graphics, visualization
Popular algorithms: Delaunay refinement, Advancing front, Octree

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Geometry Representations
Explicit Geometry
Parameterized boundaries
(x, y) = (x(s), y(s))
Implicit Geometry
Boundaries given by zero level set
(x, y) = 0
(x, y) < 0
(x, y) > 0

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Discretized Implicit Geometries
Discretize implicit function on background grid
Cartesian, Octree, or unstructured
Obtain (x) for general x by interpolation
Cartesian Quadtree/Octree Unstructured

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Explicit vs Implicit
Most CAD programs represent geometries explicitly (NURBS and Brep)
Most mesh generators require explicit expressions for boundaries
(Delaunay refinement, Advancing front)
Increasing interest for implicit geometries:
Easy to implement: Set operations, offsets, blendings, etc
Extend naturally to higher dimensions
Image based geometry representations (MRI/CT scans)
Level set methods for evolving interfaces

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Topics
1. Introduction
2. The New Mesh Generator
3. Mesh Size Functions
4. Applications
5. Conclusions

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The New Mesh Generator
1. Start with any topologically correct initial mesh, for example random node
distribution and Delaunay triangulation
2. Move nodes to find force equilibrium in edges
Project boundary nodes using
Update element connectivities

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Internal Forces
F1
F2
F3
F4
F5
F6
For each interior node:
i
Fi = 0
Repulsive forces depending on edge
length and equilibrium length 0:
F =
k(0 ) if < 0,
0 if 0.
Get expanding mesh by choosing 0
larger than desired length h

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Reactions at Boundaries
F1
F2
F3
F4
R
For each boundary node:
i
Fi + R= 0
Reaction force R:
Orthogonal to boundary
Keeps node along boundary

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MATLAB Demo

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Topics
1. Introduction
2. The New Mesh Generator
3. Mesh Size Functions
4. Applications
5. Conclusions

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Mesh Size Functions
Function h(x) specifying desired mesh element size
Many mesh generators need a priori mesh size functions
Our forcebased meshing Advancing front and Paving methods
Discretize mesh size function h(x) on a coarse background grid

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Mesh Size Functions
Based on several factors:
Curvature of geometry boundary
Local feature size of geometry
Numerical error estimates (adaptive solvers)
Any userspecified size constraints
Also: h(x) g to limit ratio G = g + 1 of neighboring element sizes

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Mesh Gradation
For a given size function h0(x), find gradient limited h(x) satisfying
h(x) g
h(x) h0(x)
h(x) as large as possible
Example: A pointsource in Rn
h0(x) =
hpnt, x = x0
, otherwise h(x) = hpnt + gx x0
1 0.5 0 0.5 10
0.5
1
h0
(x)
1 0.5 0 0.5 10
0.5
1h(x)

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Mesh Gradation
Example: Any shape, with distance function d(x), in Rn
h0(x) =
hshape, d(x) = 0
, otherwise h(x) = h
shape+ gd(x)
Example: Any function h0(x) in Rn
h(x) = miny
(h0(y) + gx y)
Too expensive and inflexible for practical use

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The Gradient Limiting Equation
The optimal gradient limited h(x) is the steadystate solution to
h
t+ h = min(h, g),
h(t = 0,x) = h0(x).
Nonlinear hyperbolic PDE (a HamiltonJacobi equation)
Theorem: For g constant, and convex domain, the steadystate solution is
h(x) = miny
(h0(y) + gx y)
Proof. Use the HopfLax theorem (see paper).
The optimal minimumoverpointsources solution!

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Gradient Limiting in 1D
Dashed = Original function, Solid = Gradient limited function
Very different from smoothing!
Max Gradient g = 4 Max Gradient g = 2
Max Gradient g = 1 Max Gradient g = 0.5

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Mesh Size Functions 2D Examples
Feature size computed by hlfs(x) = (x) + MA(x)
Med. Axis & Feature Size Mesh Size Function h(x) Mesh Based on h(x)

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Mesh Size Functions 3D Examples
Mesh Size Function h(x) Mesh Based on h(x)

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Mesh Size Functions 3D Examples

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Mesh Size Functions 3D Examples

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Topics
1. Introduction
2. The New Mesh Generator
3. Mesh Size Functions
4. Applications
5. Conclusions

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Moving Meshes
Iterative formulation wellsuited for geometries with moving boundaries
Mesh from previous time step good initial condition, a new mesh obtained
by a few additional iterations
Even better if smooth embedding of boundary velocity move all nodes
Density control if area changes

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Level Sets and Finite Elements
Level Set Methods superior for interface propagation:
Numerical stability and entropy satisfying solutions
Topology changes easily handled, in particular in three dimensions
Unstructured meshes better for the physical problems:
Better handling of boundary conditions
Easy mesh grading and adaptation
Proposal: Combine Level Sets and Finite Elements
Use level sets on background grid for interface
Mesh using our iterations, and solve physical problem using FEM
Interpolate boundary velocity to background grid

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Structural Vibration Control
Consider eigenvalue problem
u = (x)u, x
u = 0, x .
with
(x) =
1 for x / S
2 for x S.
Solve the optimization
minS
1 or 2 subject to S = K.
\S, 1
S, 2

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Structural Vibration Control
Level set formulation by Osher and Santosa:
Calculate descent direction = v(x) using solution i, ui
Find Lagrange multiplier for area constraint using Newton Represent interface implicitly, propagate using level set method
Use finite elements on unstructured mesh for eigenvalue problem
More accurate interface condition
Arbitrary geometries
Graded meshes

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Structural Design Improvement
Linear elastostatics, minimize compliance
g u ds =
A(u) (u) dx subject to = K.
Boundary variation methods by Murat and Simon, Pironneau,
Homogenization method by Bendsoe and Kikuchi
Level set approaches by Sethian and Wiegmann (Immersed interfacemethod) and Allaire et al (Ersatz material, Youngs modulus )

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Stress Driven Rearrangement Instabilities
Epitaxial growth of InAs on a GaAs substrate
Stress from misfit in lattice parameters
Quasistatic interface evolution, descent direction for elastic energy andsurface energy
+ F(x) = 0, with F(x) = (x) (x)
Electron micrograph of defectfree InAs quantum dots

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Stress Driven Rearrangement Instabilities
Initial Configuration
Final Configuration, = 0.20
S D R

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Stress Driven Rearrangement Instabilities
Initial Configuration
Final Configuration, = 0.10
S D i R I bili i

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Stress Driven Rearrangement Instabilities
Initial Configuration
Final Configuration, = 0.05
Fl id D i ith M i B d i

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Fluid Dynamics with Moving Boundaries
Solve the incompressible NavierStokes equations
u
t 2u + u u + P = f
u = 0
Spatial discretization using FEM and P2P1 elements
Backward Euler for viscous term
Lagrangian node movement for convective term
Improve poor mesh elements using new force equilibrium
Interpolate velocity field (better: L2projections, ALE, SpaceTime)
Discrete form of Chorins projection method for incompressibility
Fluid Dynamics with Moving Boundaries

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Fluid Dynamics with Moving Boundaries
Lagrangian Node Movement Mesh Improvement
Fluid Dynamics with Moving Boundaries

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Fluid Dynamics with Moving Boundaries
Lagrangian Node Movement Mesh Improvement
Fluid Dynamics with Moving Boundaries

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Fluid Dynamics with Moving Boundaries
Lagrangian Node Movement Mesh Improvement

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MATLAB Demo
Meshing Images

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Meshing Images
Images are implicit representations of objects
Segment image to isolate objects from background
Active contours (snakes), available in standard imaging software
Level set based methods [Chan/Vese]
Apply Gaussian smoothing on binary images (masks)
Mesh the domain:
Find distance function using approximate projections and FMM
Find size function from curvature, feature size, and gradient limiting
Mesh using our forcebased implicit mesh generator
Extends to three dimensions (MRI/CT scans)
Meshing Images

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Meshing Images
Original Image
Meshing Images

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Meshing Images
Segmented Image Mask
Meshing Images

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Meshing Images
Mask after Gaussian Smoothing
Meshing Images

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Meshing Images
Distance Function from Approximate Projections and FMM
Meshing Images

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Meshing Images
Medial Axis from Shocks in Distance Functions
Meshing Images

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Meshing Images
Curvature, Feature Size, Gradient Limiting Size Function
Meshing Images

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g g
Triangular Mesh
Meshing Images  Lake Superior

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g g p
Original Image
Meshing Images  Lake Superior

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g g p
Triangular Mesh
Meshing Images  Medical Applications

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g g pp
Volume Mesh of Iliac Bone
Topics

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1. Introduction
2. The New Mesh Generator
3. Mesh Size Functions
4. Applications
5. Conclusions
Conclusions

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New iterative mesh generator for implicit geometries
Simple algorithm, high element qualities, works in any dimension
Automatic computation of size functions, PDEbased gradient limiting
Combine FEM and level set methods for moving interface problems in
shape optimization, structural instabilities, fluid dynamics, etc
Mesh generation for images (in any dimension)
Future work: Spacetime meshes, sliver removal, performance and
robustness improvements, quadrilateral/hexahedral meshes, anisotropic
gradient limiting, more applications, . . .