The Eikonal Equation

Transcription

1 The Eikonal Equation Numerical efficiency versus computational compleity Shu-Ren Hysing III Institute of Applied Mathematics LSIII University of Dortmund

2 Level set - methodology By embedding an interface in a higher dimensional function simplifies the treatment of interface type problems Consider a circular interface, Γ The zero contour represents the interface

3 Level set - methodology The equation governing the evolution of the level set function φ reads φ t + u φ 0 with initial condition φγ,t00 This transport equation can be efficiently solved with standard solution tools!

4 Level set - properties A smoothness constraint on the level set field relaes the solver requirements The natural choice is to restrict the level set function to a signed distance function φ F At any given point the magnitude of the level set function will thus represent the shortest distance to the interface

5 Level set - problems The distance function property is only preserved if the velocity fulfils u φ 0 φ t + u φ 0 Stretching and folding is unwanted: Poor evaluation of geometrical quantities Possible solver failure

6 Level set - solutions A stretched and folded level set function can be periodically reinitialized in order to maintain the distance function property

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8 Reinitialization - methods Brute force redistancing Algebraic Newton approach Fast marching method Fast sweeping method PDE based redistancing

9 Brute force algorithm Approimate the interface curve with M line segments Calculate the minimum distance to all segments for each point of interest Algorithmic compleity ON M

10 Algebraic Newton approach Algebraic Newton approach Solve for each grid point 0, where Ψ is a given approimate distance field Algorithmic compleity is ON 0 0 ψ ψ L 0

11 Newton system Newton system Typical Newton iteration k+1 k -δ k [J k ] -1 L k New distance is given by φ 0-0 T yy y y y y y y y y y L J y y y L + + ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ,

12 Fast marching method Update grid points in order of increasing distance with the aid of a difference formula The solution will thus correspond to an upwind solution Algorithmic compleity ON logn

13 Fast sweeping method Update grid points in predetermined characteristic directions to try to capture the propagation of information Uses the same difference formula as the fast marching method Algorithmic compleity potentially ON

14 Difference update Difference update Given some known distance values a difference approimation to the Eikonal equation is constructed as 0 1 2, 1, Qb b Qb a Qa a Q b a v v v v T T T T i i i T T φ φ φ φ φ PP PP P P 1 P , 2, i i i i o i o i i o i o b a b a i i i i φ φ φ P

15 Quadrilateral treatment Difference update requires simplees subdivide each quadrilateral into virtual triangles Upwinding constraint minimized some acute angles will always be generated

16 PDE based redistancing By calculating the stationary limit of the following PDE, we can also reinitialize a given approimate distance function φ 0 φ t * + u φ * S φ 0, u S φ 0 φ φ * * where Sφ 0 is an appropriately chosen smoothed sign function

17 Reinitialization comments Brute force method both accuracy and speed dependent upon interface approimation Algebraic Newton method smoothness properties of the approimate distance function determines accuracy and convergence rate Difference update first derivatives of the distance function are needed for second order Fast sweeping method sweeping order unclear for unstructured grids

18 Performance? What is the performance of the different algorithms for redistancing a non-trivial interface problem? Which algorithm gives the most accurate results and which algorithm is quickest: Accuracy versus CPU time

19 Test case Represent a typical level set simulation The test case should contain smooth regions and shocks An eact solution is available φ y, min y

20 Mesh cases Unstructured and structured grids Number of grid points: *10 6 Cartesian Perturbed Unstructured

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22 Results - time comparison

23 Results - time comparison

24 Results - time comparison

25 Results fast sweeping

26 Results - accuracy

27 Results - accuracy

28 Results - accuracy

29 Results - accuracy

30 Results - accuracy

31 Results - accuracy

32 Conclusions Brute force algorithm most robust results but scales very poorly Algebraic Newton dependent on approimate distance function Fast sweeping method Additional sweeps too costly Unstructured sweeping order unclear

33 Conclusions The fast marching method scales very well with increasing grid density really deserves the epithet fast, redistancing 2.4*10 6 grid points in ~13 s Second order difference update only marginally more costly than first order somewhat unstable for highly perturbed grids The fast marching method is therefore our preferred algorithm!