Moving Mesh Adaptation Techniques

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Moving Mesh Adaptation Techniques

• Todd Phillips
• Gary Miller
• Mark Olah

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Introduction

• The Meshing Problem
• Discretize a Spatial Domain
• Minimize Size (Number of Triangles)
• Maximize Quality (Shape of Triangles)
• Mesh Adaptation
• Introduce More Elements
• Base Decisions on the Function being Interpolated
• How do we make such a decision?

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The Basic Setup

• A Function Exists

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The Basic Setup

• A Mesh Also Exists

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The Basic Setup

• We Approximate the Function with the Mesh

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The Basic Setup

• We can adapt the Mesh to Better Approximate the
  Function

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Background Local Feature Size

• What is Local Feature Size?
• Roughly, the Size of Triangles We Should Be Using
• The Smallest Distance to Two Geometric Domain
  Features
• Local Feature Size Should be k-Lipschitz
• Doesnt change too fast
• For all p,q lfs(p) lt lfs(q) kdist(p,q)
• This is like having derivative bounded by k.

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Optimal Meshing and Adaptive Meshing

• Optimal Meshing
• Consider the LFS of the Input Domain
• Output a Mesh
• Triangle Size is within a constant of LFS
• Triangle Aspect Ratio is bounded
• Note the Impossabilty of this if LFS is not
  Lipschitz
• Adaptive Meshing
• LFS Accounts for Other Features Provided by
  Some Oracle
• Introduce Geometric Features to Accommodate
• Perturb the Original Mesh to obtain a new Optimal
  Mesh

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Example of Modifying Local Feature Size

• A Sample Geometric Domain

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Example of Modifying Local Feature Size

• LFS of the Sample Domain

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Example of Modifying Local Feature Size

• Modify the Geometric Domain to Capture a New
  Feature

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Example of Modifying Local Feature Size

• LFS of the Modified Domain

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How can we Define an Oracle?

• Function-Angle Based Refinement
• Some Function is Approximated by the Mesh
• This Embeds the Mesh as a surface in one-higher
  dimension
• Observe the angle between faces of this surface
• Introduce Features Where This Angle is Small
• Insert Circumcenters, Split Edges, etc.

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An Example in One Dimension Function
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An Example in One Dimension Function and Uniform
Mesh
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An Example in One Dimension Function, Mesh, and
Error
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An Example in One Dimension Function and
Adaptive Mesh
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An Example in One Dimension Adaptive Mesh LFS
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Will this work?

• Original Feature Size must be Small Enough

• Actual Function must be Differentiable

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Practicalities

• Real Mesh Adaptation Requires Coarsening As Well
• Sometimes Functional Features Move or Disappear
• Thrashing is a dirty word
• Real Solutions arent always differentiable
• Give up after a Minimum Feature Size

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Why is this so great for Moving Meshes?

• Unlike the One Dimensional Movie
• Fixed Mesh ( Grid points didnt move left/right )
• Constantly Chasing the Feature
• Mesh moves with the Velocity Function
• Hence, Mesh moves with the Velocity features
• After features are captured, very little Mesh
  Adaptation

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How does this compare to other methods?

• Sensitivity Analysis and a posteriori Error
  Estimates
• Double the Mesh Size Everywhere, compare
  solutions
• Require multiple meshes of different sizing be
  maintained
• Require multiple solves at every iteration
• Dont take advantage of Moving Mesh
• (Persistance of Features)

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Some 2-D Examples
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Some 2-D Examples
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Conclusion

• Summary
• Use an oracle based on Function-Angles
• Introduce new geometric features
• Use existing geometrically driven meshing
  algorithms
• Future
• Tighter Coupling of Oracle with Meshing
  Algorithms
• Adaptive Time Refinement