Lamps of Aladdin1 Moving Mesh Adaptation Techniques Todd Phillips Gary Miller Mark Olah

Lamps of Aladdin2 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?

Lamps of Aladdin3 The Basic Setup A Function Exists

Lamps of Aladdin4 The Basic Setup A Mesh Also Exists

Lamps of Aladdin5 The Basic Setup We Approximate the Function with the Mesh

Lamps of Aladdin6 The Basic Setup We can adapt the Mesh to Better Approximate the Function

Lamps of Aladdin7 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.

Lamps of Aladdin8 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

Lamps of Aladdin9 Example of Modifying Local Feature Size A Sample Geometric Domain

Lamps of Aladdin10 LFS of the Sample Domain Example of Modifying Local Feature Size

Lamps of Aladdin11 Modify the Geometric Domain to Capture a New Feature Example of Modifying Local Feature Size

Lamps of Aladdin12 LFS of the Modified Domain Example of Modifying Local Feature Size

Lamps of Aladdin13 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.

Lamps of Aladdin14 An Example in One Dimension: Function

Lamps of Aladdin15 An Example in One Dimension: Function and Uniform Mesh

Lamps of Aladdin16 An Example in One Dimension: Function, Mesh, and Error

Lamps of Aladdin17 An Example in One Dimension: Function and Adaptive Mesh

Lamps of Aladdin18 An Example in One Dimension: Adaptive Mesh LFS

Lamps of Aladdin19 Will this work? Original Feature Size must be Small Enough Actual Function must be Differentiable

Lamps of Aladdin20 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

Lamps of Aladdin21 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

Lamps of Aladdin22 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)

Lamps of Aladdin23 Some 2-D Examples

Lamps of Aladdin24 Some 2-D Examples

Lamps of Aladdin25 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