level set methods - win.tue.nl€¦ · fast marching method update “downwind” level set methods...

Level Set methodsSandra Allaart-Bruin

[email protected]

Level Set methods – p.1/24

Overview

Introduction

Boundary Value Formulation

Initial Value Formulation

Numerical schemes

Fast Marching Methods

Narrow Band Level Set


Introduction

moving interface problem

Local propertiesGlobal properties

Independent properties


Introduction

moving interface problemSpeed function � ��

Local propertiesGlobal properties



Introduction

moving interface problemSpeed function � � � � � � �Local properties

Global properties



Introduction

moving interface problemSpeed function � � � � � � �Local propertiesGlobal properties



Formulations





Speed function � �

Arrival Time

interface is given by



Speed function � �

Arrival Time

interface is given by

��



distance � rate · time

� �

��

More dimensions:

|∇ | � �

with � �

on�


Example Boundary Value Formulation

�

is circular, � �

:



Level Set Function

��

Interface is given by

��

.



level set function

��

By the chain rule:

� ∇

��

· � ′ � � � � �

is speed in outward normal direction, then′ · , ∇

|∇ |

|∇ |

given



level set function

��

By the chain rule:

� ∇

��

· � ′ � � � � �

is speed in outward normal direction, then

� ′

� � �

· � � , � � ∇

�

|∇

�

|

|∇ |

given



level set function

��

By the chain rule:

� ∇

��

· � ′ � � � � �

is speed in outward normal direction, then

� ′

� � �

· � � , � � ∇

�

|∇

�

|

� |∇ | � �

given

��


Example Initial Value Formulation


Summary

Boundary Value Formulation ( � �

)

|∇ | � �Front =

� � � �

= {

��

|��

}


|∇ |

Front = = { | }


Summary

Boundary Value Formulation ( � �

)

|∇ | � �Front =

� � � �

= {

��

|��

}


� |∇ | � �

Front =� � � �

= {

��

|

��

}


Numerical Scheme for BVP

|∇ |

��

front:

� � {� |

��

}

−

where − −− Backward Scheme

and − Forward Scheme.


Numerical Scheme for BVP

|∇ |

��

front:

� � {� |

��

}

� � ��

− ��

where − �� − ��

−

�� Backward Scheme

and

� �� − �� Forward Scheme.


Numerical Scheme for IVP

� |∇ | � �

front:

� � {� |

��

}

− ∇ ∇−

where

∇ −

∇− −


Numerical Scheme for IVP

� |∇ | � �

front:

� � {� |

��

}

� � �� − � � � �� ∇

� � � ��

∇−

�

where

∇

� � � � � � � − ��

� �

∇− � � � � � � � �� − ��

��


CFL condition

Time Step restriction

� ��

≤ �


Notes

Extensions to higher order schemes

Schemes for non-convex speed function

These Numerical Schemes costs a lot ofcomputations


Fast Marching Method

Beginning of Fast Marching MethodLevel Set methods – p.16/24


Update “downwind”Level Set methods – p.16/24


A

B

D

C

Compute new possible valuesLevel Set methods – p.16/24


A

B

D

C

Choose smallest dark pink sphere (A)Level Set methods – p.16/24


A

B

D

C

Freeze value at A, update neighboring downwindpoints



Initialisation: Tag points as Known, Trial andFar .



Begin Loop: Let A be the Trial point with thesmallest T value




Add the point A to Known ; remove it fromTrial




Add the point A to Known; remove it fromTrial

Tag as Trial all neighbors of A that are notKnown . If neighbor is in Far , remove andadd to the set Trial .



Recompute the values of T at all Trialneighbors of A according to

� �� − �� −� ��

�

� �� − � � �� −� � � ��

�

� � � − � � �� −

� � � ��

� ��

��



Begin Loop: Let A be the Trialpoint with thesmallest T value

Add the point A to Known ; remove it fromTrial

Tag as Trial all neighbors of A that are notKnown . If neighbor is in Far , remove andadd to the set Trial .

Recompute the values of T at all Trialneighbors of A according to . . .

Return to top of loop.Level Set methods – p.17/24


Tag Alive points in narrow band

Build Land Mines to indicate near edge

Initialize Far Away points outside the narrowband with large positive (negative) values ifvalues are outside (inside) the front itself

Solve level set equation until Land Mine hit

Rebuild and loop


Advantages of Narrow Band Level Set

Speed

instead of in 3D

TimestepCFL condition on narrow band instead ofentire domain

≤



Speed

� � � �

instead of

� � �

in 3D


≤



Speed

� � � �

instead of

� � �

in 3D

Timestep

CFL condition on narrow band instead ofentire domain

≤



Speed

� � � �

instead of

� � �

in 3D


� �� ≤ �


Next lecture

Applications


End


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