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Material Type: Notes; Class: Numerical Soln Diff Eqn; Subject: Mathematics; University: University of California - Davis; Term: Fall 2009;
Typology: Study notes
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Immersed boundary method is useful for simulating fluid structure problems. It uses two types of grids: Eulerian grid for the fluid, and Lagrangian grid for the structure which moves the structure.
Given a closed curve ( or a surface in 3D ) moving in the normal direction with a given velocity. Track the location of this curve as it evolves. How to solve this numerically? E.g., use Lagrangian grid
Discretize s and represent the curve (x(s, t), y(s, t)) by discrete points xj (t) = (xj (t), yj (t)).
Lagrangian methods: Put discrete points on the curve and move then at the local velocity. Assume F > 0 constant. The velocity at a point: v = F n is given. Compute the (outside) normals
ni,j =
−(yj+1 − yj− 1 ) (xj+1 − xj− 1 )
((yj+1 − yj− 1 )^2 + (xj+1 − xj− 1 )^2 )^1 /^2
dxj dt
= vj = F nj. Ex: Suppose F = const.
Figure 3.1: Self intersecting.
Figure 3.2: Topological changing.
They may not be a physical solution.
Level set methods represent the curves/surfaces as the zero level set of some (smooth) function φ.
φ(x, y) = 0 in 2D, or φ(x, y, z) = 0 in 3D.
What does this mean about the level set? What is the velocity depends on the curvature? It does in many applications. Assume F = −bκ where κ is the mean curvature, consider
φt − bκ |∇φ| = 0 , κ = ∇ · n = ∇ ·
∇φ |∇φ|
and thus
φt = b∇ ·
∇φ |∇φ|
|∇φ|.
If ∇φ ≈ 1, then φt ≈ b∇ · (∇φ),
a diffusion equation. We need b > 0 for a well-posed problems.
Figure 3.3: Driven by mean curvature flow, the shape approaches a circle of decreasing radius.
Motion by mean curvature smooth the curve. This suggest how to look for a viscosity solution to
φt + F |∇φ| = 0
with F ǫ^ = F 0 − ǫκ.
φǫt + F 0 |∇φǫ^ | = ǫ∇ ·
∇φǫ |∇φǫ^ |
|∇φǫ^ |.
For ǫ > 0, the solution is smooth and we’re interested in φ = limǫ→ 0 φǫ. How to solve φt + F |∇φ| = 0 for φ(x, t) numerically? Try using centered differences to compute ∇φ.
(φx)i,j =
φi+1,j − φi− 1 ,j 2 h
One will get numerical oscillation near phase where the derivatives are large.
To figure out what to do, we look at 1D Hamilton-Jacobi equation
φt + H(φx) = 0.
Take the x-derivative, φtx + (H(φx))x = 0.
Let u = φx, we have ut + (H(u))x = 0.
The derivative of φ satisfies a hyperbolic conservation law. We should use conservative schemes:
un j +1− unj ∆t
j+ 12 −^ Fj− (^12) h
where Fj− 12 is an approximation to H at point xj− 12. Recall that consistent, conservative, monotone method
converges to the viscosity solution (right solution). (Godunov or Lax-Friedrich are monotone methods.) Let’s consider numerical flux function of the form
Fj− 12 = f (ui− 1 , uj ).
For example,
fLF (uL, uR) =
H(uL) +^ H(uR)^ −
diffusion ︷ ︸︸ ︷ h ∆t (uR − uL)
uL + uR 2
h 2∆t
(uR − uL)
How to use this to solve the Hamilton-Jacobi equation? Recall that φt + H(φx) = 0,
(D+φ)j =
φj+1 − φj h
, (D−φ)j =
φj − φj− 1 h
If F > 0,
(φx)^2 i,j = max
max(D−i,jx , 0)^2 , min(D+i,jx , 0)^2
If F < 0, swap D−i,jx and Di,j+x :
(φx)^2 i,j = max
min(D−i,jx , 0)^2 , max(D+i,jx , 0)^2
For level set method,
φn i,j+1 = φni,j − ∆t
max(Fi,j , 0)∇+^ + min(Fi,j , 0)∇−
where
∇+^ =
max
max(Di,j−x , 0)^2 , min(D+i,jx , 0)^2
max(D−i,jy , 0)^2 , min(D+i,jy , 0)^2
Stable provided
∆t ≤
h |H 1 | + |H 2 |
F φx |∇φ|
F φy |∇φ|
just like the constraint for the donor cell upwind.
How to choose φ? We want φ = 0 on the initial surface, φ < 0 inside and φ > 0 outside. φ should be somewhat smooth. A common choice is signed distance function. Let Γ be the zero level set (initial surface). The distance d is defined by
d(x) = min y∈Γ
|x − y|
that is, the distance to φ = 0 level set. Clearly d = 0 only on the surface. Define
φ(x) =
d(x) x outside −d(x) x inside
We can show |∇φ| = 1 for the signed distance function.
Figure 3.5: Level set in 1D: the interfaces is the two points.
Figure 3.6: Inside the circle centered at x with radius d(x), there are no points of Γ. All points on the line segment connecting x and xc have closest point xc.
The gradient direction is (^) |xx−−xxcc |. The magnitude of the gradient is the absolute value of the directional
derivative in the gradient direction. Therefore |∇d| = 1 at all points where ∇d is defined. Having |∇φ| = 1 simplifies some formulas:
n̂ = ∇φ |∇φ|
= ∇φ , κ = ∇ · ̂n = ∆φ.
The accuracy degrades if the level sets become too distorted. Some ways of maintaining (approximating) the signed distance function:
How to compute the signed distance function? Consider a curve moving in the normal direction at speed F = 1. Let T (x) be the time or arrival of the curve. This is the distance function.
Figure 3.7: Solving |∇φ| = 1 on Γ. Compute arrival times at neighbor points and add the smallest arrival time to the accepted values. Fast Marching Methods.
