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Unformatted text preview: 11/15/2010 Level Sets
Numerical Considerations Lagrangian Formulation
r c
• Consider the curve given by ø(x,y)=0.
r
V ( x, y )
• Suppose that the velocity is known for every point on the curve.
• To move the points on the curve with this velocity, we needr to solve
dc r
= V ( x, y )
dt • For every point (x,y) on the curve. Math Modeling II, Fall 2010 (Dr. Xin Li) 1 11/15/2010 Eulerian Formulation
• Now, for each t, there is an equation for the r
c (t ) φ (t , x, y ) = 0
curve : .
• Differentiate with respect to t gives:
r
• (1)
φt + V ⋅ ∇φ = 0
r
r
V = (u, v)
• Recall that if , then V ⋅ ∇φ = uφ x + vφ y
• We refer to equation (1) as the level set equation • We need to solve (1) in the following sense:
– We will use the initial curve for t=0
– Increase the time step by Δt
φ (t + Δt , x, y )
– Solve for the value of on a grid in the xy‐plane
– Even though V may be given only for points on the curve, we will assume that it is also given (or could be extended) as a vector field for the whole xy‐
plane (or for a region). Math Modeling II, Fall 2010 (Dr. Xin Li) 2 11/15/2010 Upwind Differencing
• We need to evolve ø forward in time and move the curve (interface).
• Assume a grid in the xy‐plane is given ‐ in digital image case, we have a very natural grid of the pixels
• At time step tn, let øn= ø(tn)
• Update ø in time = finding values of ø at every grid point after some time increment Spatial Discretization
• First order accurate method for the time discretization is the forward Euler method given by
n +1
n
φ •
•
•
• −φ
+ u nφ xn = 0
Δt How about the spatial derivatives of ø?
Great care must be exercised!
We will consider one‐dimensional spatial case
Can be applied to higher in a dimension‐by‐
dimension manner Math Modeling II, Fall 2010 (Dr. Xin Li) 3 11/15/2010 Start Upwinding
• At a specific spatial point x i
φi n +1 − φi n
Δt + ui (φ x ) in = 0
n ui > 0
• If , the values of ø are moving from left to xi
right, so we need to look to the left of to determine what value of ø will land on the point at the end of a time step. Upwinding
φx
uin > 0
• So, D‐ø should be used to approximate when and D+ø should be used otherwise
• Since D‐ and D+ are first order accurate, the combination of the forward time and upwind differencing is a consistent finite difference approximation to the partial differential equation (1)
• According to Lax‐Richtmyer equivalence theorem, a finite difference approximation to a linear differential equation is convergent if it is consistent
and stable. Math Modeling II, Fall 2010 (Dr. Xin Li) 4 11/15/2010 Stability
• The stability condition is Δt max{ u v
+
Δx Δy }= α < 1 • Instead of upwinding, the spatial derivatives could be approximated with more accurate methods.
• But we stay with upwind method
• The first order upwind can be improved by using more accurate approximation Homework
• Write done the steps needed for the implementation of the level sets method for solving equation (1) (do not need to implement in Matlab yet)
Step 0: Ste up the grid in xy‐plane with uniform step‐length h; select time step k
Step 1: Initialization of the level set function ø(0,x,y)
at every grid point (x,y). If only a curve is given, we use signed distance function to initialize ø(0,x,y)=+or‐distance(<x,y>,C)
Step 2: … Math Modeling II, Fall 2010 (Dr. Xin Li) 5 ...
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 Fall '10
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 Math, Sets

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