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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– Doesn’t change too fast– For all p,q lfs(p) <= lfs(q) + k*dist(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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• LFS of the Sample Domain

Example of Modifying Local Feature Size

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• Modify the Geometric Domain to Capture a New Feature

Example of Modifying Local Feature Size

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• LFS of the Modified Domain

Example of Modifying Local Feature Size

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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 aren’t 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 didn’t 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– Don’t 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