Level Set methods Sandra Allaart-Bruin sbruin@win.tue.nl Level Set methods p.1/24
Overview Introduction Level Set methods p.2/24
Overview Introduction Boundary Value Formulation Level Set methods p.2/24
Overview Introduction Boundary Value Formulation Initial Value Formulation Level Set methods p.2/24
Overview Introduction Boundary Value Formulation Initial Value Formulation Numerical schemes Level Set methods p.2/24
Overview Introduction Boundary Value Formulation Initial Value Formulation Numerical schemes Fast Marching Methods Level Set methods p.2/24
Overview Introduction Boundary Value Formulation Initial Value Formulation Numerical schemes Fast Marching Methods Narrow Band Level Set Level Set methods p.2/24
Introduction moving interface problem Level Set methods p.3/24
Introduction moving interface problem Speed function Level Set methods p.3/24
Introduction moving interface problem Speed function Local properties Level Set methods p.3/24
Introduction moving interface problem Speed function Local properties Global properties Level Set methods p.3/24
Introduction moving interface problem Speed function Local properties Global properties Independent properties Level Set methods p.3/24
Formulations Boundary Value Formulation Initial Value Formulation Level Set methods p.4/24
Boundary Value Formulation Speed function Level Set methods p.5/24
Boundary Value Formulation Speed function Arrival Time interface is given by Level Set methods p.5/24
Boundary Value Formulation distance rate time More dimensions: with on Level Set methods p.6/24
Example Boundary Value Formulation is circular, : Level Set methods p.7/24
Initial Value Formulation Level Set Function Interface is given by. Level Set methods p.8/24
Initial Value Formulation level set function By the chain rule: Level Set methods p.9/24
Initial Value Formulation level set function By the chain rule: is speed in outward normal direction, then, Level Set methods p.9/24
Initial Value Formulation level set function By the chain rule: is speed in outward normal direction, then, given Level Set methods p.9/24
Example Initial Value Formulation Level Set methods p.10/24
Summary Boundary Value Formulation ( ) Front = = { } Level Set methods p.11/24
Summary Boundary Value Formulation ( ) Front = = { Initial Value Formulation } Front = = { } Level Set methods p.11/24
Numerical Scheme for BVP front: { } Level Set methods p.12/24
Level Set methods p.12/24 Numerical Scheme for BVP } { front: Backward Scheme Forward Scheme. where and
Numerical Scheme for IVP front: { } Level Set methods p.13/24
Level Set methods p.13/24 Numerical Scheme for IVP } { front: where
CFL condition Time Step restriction Level Set methods p.14/24
Notes Extensions to higher order schemes Level Set methods p.15/24
Notes Extensions to higher order schemes Schemes for non-convex speed function Level Set methods p.15/24
Notes Extensions to higher order schemes Schemes for non-convex speed function These Numerical Schemes costs a lot of computations Level Set methods p.15/24
Fast Marching Method Beginning of Fast Marching Method Level Set methods p.16/24
Fast Marching Method Update downwind Level Set methods p.16/24
Fast Marching Method B C D A Compute new possible values Level Set methods p.16/24
Fast Marching Method B C D A Choose smallest dark pink sphere (A) Level Set methods p.16/24
Fast Marching Method B C D A Freeze value at A, update neighboring downwind points Level Set methods p.16/24
Fast Marching Methods Initialisation: Tag points as Known, Trial and Far. Level Set methods p.17/24
Fast Marching Methods Begin Loop: Let A be the Trial point with the smallest T value Level Set methods p.17/24
Fast Marching Methods Begin Loop: Let A be the Trial point with the smallest T value Add the point A to Known ; remove it from Trial Level Set methods p.17/24
Fast Marching Methods Begin Loop: Let A be the Trial point with the smallest T value Add the point A to Known; remove it from Trial Tag as Trial all neighbors of A that are not Known. If neighbor is in Far, remove and add to the set Trial. Level Set methods p.17/24
Level Set methods p.17/24 Fast Marching Methods Recompute the values of T at all Trial neighbors of A according to
Fast Marching Methods Begin Loop: Let A be the Trialpoint with the smallest T value Add the point A to Known ; remove it from Trial Tag as Trial all neighbors of A that are not Known. If neighbor is in Far, remove and add to the set Trial. Recompute the values of T at all Trial neighbors of A according to... Return to top of loop. Level Set methods p.17/24
Narrow Band Level Set Level Set methods p.18/24
Narrow Band Level Set Level Set methods p.18/24
Narrow Band Level Set Level Set methods p.18/24
Narrow Band Level Set Tag Alive points in narrow band Level Set methods p.19/24
Narrow Band Level Set Tag Alive points in narrow band Build Land Mines to indicate near edge Level Set methods p.19/24
Narrow Band Level Set Tag Alive points in narrow band Build Land Mines to indicate near edge Initialize Far Away points outside the narrow band with large positive (negative) values if values are outside (inside) the front itself Level Set methods p.19/24
Narrow Band Level Set Tag Alive points in narrow band Build Land Mines to indicate near edge Initialize Far Away points outside the narrow band with large positive (negative) values if values are outside (inside) the front itself Solve level set equation until Land Mine hit Level Set methods p.19/24
Narrow Band Level Set Tag Alive points in narrow band Build Land Mines to indicate near edge Initialize Far Away points outside the narrow band with large positive (negative) values if values are outside (inside) the front itself Solve level set equation until Land Mine hit Rebuild and loop Level Set methods p.19/24
Narrow Band Level Set Level Set methods p.20/24
Narrow Band Level Set Level Set methods p.21/24
Advantages of Narrow Band Level Set Speed Level Set methods p.22/24
Advantages of Narrow Band Level Set Speed instead of in 3D Level Set methods p.22/24
Advantages of Narrow Band Level Set Speed Timestep instead of in 3D Level Set methods p.22/24
Advantages of Narrow Band Level Set Speed instead of in 3D Timestep CFL condition on narrow band instead of entire domain Level Set methods p.22/24
Next lecture Applications Level Set methods p.23/24
End Level Set methods p.24/24