recent advances on large time–step schemes for mean ... · recent advances on large time–step...
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Recent advances on large time–step
schemes for Mean Curvature Motion
Elisabetta Carlini, Maurizio Falcone and Roberto Ferretti
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• ”Level set” techniques and MCM (in codimension one)
• Analytical setting and Soner-Touzi representation formula
• Discretization and convergence analysis
• Numerical results (part 1a)
• Adaptive time–stepping
• Numerical results (part 1b)
• MCM in codimension two - The Ambrosio–Soner approach
• Post–processing of the solution
• Numerical results (part 2)
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Interfaces evolution and ”level set” techniques
• Geometric problem: to follow the evolution of a manifold (inter-
face) which moves in the direction of the normal vector with a speed
dependent (for example) on position and curvature.
• Motivations: Combustion, porous media, semiconductors,...
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Interfaces evolution and ”level set” techniques
• Geometric problem: to follow the evolution of a manifold (inter-
face) which moves in the direction of the normal vector with a speed
dependent (for example) on position and curvature.
• Motivations: Combustion, porous media, semiconductors,...
• Classical approach: Let the manifold evolve using a parametrization
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Interfaces evolution and ”level set” techniques
• Geometric problem: to follow the evolution of a manifold (inter-
face) which moves in the direction of the normal vector with a speed
dependent (for example) on position and curvature.
• Motivations: Combustion, porous media, semiconductors,...
• Classical approach: Let the manifold evolve using a parametrization
• ”Level set” approach: Consider the manifold as a level surface of
the solution of a suitable evolutive PDE
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Model problem: Motion by Mean Curvature
The most typical case of application of level set techniques to inter-
face propagation is the degenerate parabolic problem: vt(x, t) = |Dv| div
(Dv(x,t)|Dv(x,t)|
)v(x,0) = v0(x)
(1)
in the unknown v = v(x, t), with x ∈ RN , t ≥ 0 and with v0 a continu-
ous function such that
Γ0 = {x ∈ RN |v0(x) = 0}
(initial interface position)
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Analytical questions (1)
Equation (1) is:
• nonlinear (it is in the form vt + H(u, Dv, D2v) = 0)
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Analytical questions (1)
Equation (1) is:
• nonlinear (it is in the form vt + H(u, Dv, D2v) = 0)
• degenerate (not uniformly parabolic)
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Analytical questions (1)
Equation (1) is:
• nonlinear (it is in the form vt + H(u, Dv, D2v) = 0)
• degenerate (not uniformly parabolic)
• indefinite at points where Dv = 0 (Dv/|Dv| makes no sense)
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Analytical questions (1)
Equation (1) is:
• nonlinear (it is in the form vt + H(u, Dv, D2v) = 0)
• degenerate (not uniformly parabolic)
• indefinite at points where Dv = 0 (Dv/|Dv| makes no sense)
Such problems have been solved from an analytical point of view by
introducing the notion of viscosity solution (Evans-Spruck,Chen-Giga-
Goto,1991).
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Analytical questions (2)
On the basis of the theory of viscosity solutions, it is possible to define
a generalized evolution of Γ0 by Mean Curvature as
Γt = {x ∈ RN |v(x, t) = 0} ∀t ≥ 0.
• It is possible to prove that such definition matches the classical one
as far as the latter is well–defined, but can also handle generation of
singularities and topology changes of the interface (in this situations,
MCM cannot be defined in a classical sense).
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Analytical questions (2)
On the basis of the theory of viscosity solutions, it is possible to define
a generalized evolution of Γ0 by Mean Curvature as
Γt = {x ∈ RN |v(x, t) = 0} ∀t ≥ 0.
• It is possible to prove that such definition matches the classical one
as far as the latter is well–defined, but can also handle generation of
singularities and topology changes of the interface (in this situations,
MCM cannot be defined in a classical sense).
• Note that Γt does not depend on v0 (i.e. on how the initial position
of the interface is represented)
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Soner–Touzi formula (1)
Recently Soner and Touzi (2002) have proposed a stocastic represen-
tation formula for solutions of a large class of geometric second–order
Hamilton–Jacobi equations, including (1); in our case (and provided
Dv does not vanish - it can be generalized to situations in which
Dv = 0 at some point) it reads
v(x, t) = E{v0(y(x, t, t))} (2)
where the curves y(x, t, s) play the role of generalized characteristics.
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Soner–Touzi formula (2)
The function y(x, t, s) appearing in (2) solves the stochastic initial
value problem (written in R2 for simplicity):{dy(x, t, s) = σ(y, t, s)dW (s)y(x, t,0) = x,
(3)
where dW is the differential of a standard Wiener process, and
σ(y, t, s) =
√2
|Dv(y, t− s)|
(vx2(y, t− s)−vx1(y, t− s)
)(such matrix projects the diffusion on the space orthogonal to the
gradient of the solution v)
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Discretization (1)
Writing Soner–Touzi representation formula on a single time step
tk → tk+1 one obtains:
v(x, tk+1) = E{v(y(x, tk+1,∆t), tk)} (4)
where y(x, tk+1,∆t) is a single step (backwards from the point (x, tk+1))
along the generalized characteristic: dy(x, tk+1, s) =√
2|Dv(y,tk+1−s)|
(vx2(y, tk+1 − s)−vx1(y, tk+1 − s)
)dW (s)
y(x, tk+1,0) = x.
(5)
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Discretization (2)
In order to set up (4), (5) in a fully discrete form:
• The computation of v(·, tk) is replaced by a numerical reconstruction
I[vk](·) (Lagrange, ENO, WENO,...)
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Discretization (2)
In order to set up (4), (5) in a fully discrete form:
• The computation of v(·, tk) is replaced by a numerical reconstruction
I[vk](·) (Lagrange, ENO, WENO,...)
• Partial derivatives vxi(y, tk+1) are replaced by finite differences com-
puted at time tk (a further error term is introduced to avoid an implicit
scheme)
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Discretization (2)
In order to set up (4), (5) in a fully discrete form:
• The computation of v(·, tk) is replaced by a numerical reconstruction
I[vk](·) (Lagrange, ENO, WENO,...)
• Partial derivatives vxi(y, tk+1) are replaced by finite differences com-
puted at time tk (a further error term is introduced to avoid an implicit
scheme)
• An approximation of the expectation E{v(y(x, tk+1,∆t), tk)} is com-
puted by weak convergence scheme for SDEs
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Discretization (3)
Given the stochastic Cauchy problem (which we assume as scalar for
simplicity) {dy(x, t) = a(t, y(x, t)) + σ(t, y(x, t))dW (t)y(x, t0) = x,
the simplest choice in order to approximate E{h(y(x,∆t))} is the weak
stochastic Euler method:{y1(∆W ) = x + a(t0, x)∆t + σ(t0, x)∆W
E{h(y(x,∆t))} = 12
(h(y1(
√∆t)) + h(y1(−
√∆t))
)+ O(∆t2)
In this case, the expectation is approximated with the same order of
consistency as in the deterministic case (provided a(·, ·), σ(·, ·), h(·) are
smooth enough)
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Discretization (4)
We finally obtain the fully discrete scheme (in R2):
vn+1ij =
1
2
(I[vn](xij + σn
ij
√∆t) + I[vn](xij − σn
ij
√∆t)
)in which the derivatives in σn
ij have been replaced with finite differ-
ences:
σnij =
√2√
Dn1,ij
2 + Dn2,ij
2
(Dn
2,ij−Dn
1,ij
)
and, for example:
Dn1,ij =
vni+1,j − vn
i−1,j
2∆x, Dn
2,ij =vni,j+1 − vn
i,j−1
2∆x
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Consistency
Assume the scheme approximating the SDE to converge with order
1, the interpolation I[v](·) to be of degree r and the approximation
of derivatives to converge with order q. Then, the consistency error
has the form:
L∆x,∆t(x, t) = O
(∆t1/2 +
∆xr+1
∆t+
∆xq
∆t1/2
).
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Consistency
Assume the scheme approximating the SDE to converge with order
1, the interpolation I[v](·) to be of degree r and the approximation
of derivatives to converge with order q. Then, the consistency error
has the form:
L∆x,∆t(x, t) = O
(∆t1/2 +
∆xr+1
∆t+
∆xq
∆t1/2
).
• It is possible to determine a relationship of the form ∆x = ∆tα
which makes the scheme consistent.
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Consistency
Assume the scheme approximating the SDE to converge with order
1, the interpolation I[v](·) to be of degree r and the approximation
of derivatives to converge with order q. Then, the consistency error
has the form:
L∆x,∆t(x, t) = O
(∆t1/2 +
∆xr+1
∆t+
∆xq
∆t1/2
).
• It is possible to determine a relationship of the form ∆x = ∆tα
which makes the scheme consistent.
• Suitable conditions ensure consistency of the scheme where Dv = 0.
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Monotonicity
Consistency along with monotonicity would imply convergence by a
general convergence result (Barles-Souganidis, 1991).
Unfortunately, even perturbing the scheme with an artificial viscosity
term, the relationship ∆t/∆x giving monotonicity is opposite to the
relationship giving consistency.
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Numerical tests (1)
Evolution of a circle
We take as initial condition the function
v0(x) = max{(1− (x21 + x2
2))10,0}.
on the domain [−2,2]2. In this case the exact solution can be explicitly
computed as
v(x, t) = max{(1− (x21 + x2
2)− 2t)10,0}.
The following table reports numerical errors computed at t = 0.16,
with a scheme based on a first–order reconstruction.
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Numerical tests (2)
Evolution of a circle - error table
∆x ∆t ‖ · ‖∞ ‖ · ‖1 L∞ − order L1 − order
0.04 0.08 6.44 · 10−2 6.37 · 10−4
0.02 0.053 2.40 · 10−2 4.54 · 10−4 1.4 0.50.01 0.032 1.05 · 10−2 1.49 · 10−4 1.2 1.60.005 0.02 8.36 · 10−3 4.55 · 10−5 0.3 1.7
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Numerical tests (3)
Evolution of a square
We take as an initial condition the function
v0(x) = 1− ‖Qx‖1
with Q a rotation matrix (in order to avoid alignment between the
level curve and the grid).
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Numerical tests (4)
Fattening
We use an initial condition given by
v0(x) = x41 − x2
1 + x22 + 1
which has a saddle point at critical level v = 1. In this situation, it
is well–known that the level curve at the critical level may develop an
interior. This phenomenon is known as fattening and in the following
figures it has been illustrated by plotting two level curves, one slightly
above and one slightly below the critical level.
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Numerical tests (5)
Three–dimensional torus
We consider now the evolution of a toroidal surface in three space
dimensions. Is is known that there exists a critical ratio between the
two radii of the torus. When above this ratio, the torus evolves to-
wards a circle and extinguishes as such, when below the torus changes
topology evolving towards a sphere and collapsing in a point.
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Adaptive time–stepping
• Large time–step schemes do not allow in general to efficiently detect
small scales (e.g. corners) when the level–set function is nonsmooth
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Adaptive time–stepping
• Large time–step schemes do not allow in general to efficiently detect
small scales (e.g. corners) when the level–set function is nonsmooth
• Reducing time step in order to resolve small structures leads to a
drop in efficiency as soon as the solution is smoothed out
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Adaptive time–stepping
• Large time–step schemes do not allow in general to efficiently detect
small scales (e.g. corners) when the level–set function is nonsmooth
• Reducing time step in order to resolve small structures leads to a
drop in efficiency as soon as the solution is smoothed out
• Resolution of the scheme is related to both space and time step,
but our first attempt is only towards time–step adaptivity
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• The numerical domain of depen-
dence is made of two regions of re-
construction which are 2√
2∆t apart
• This is precisely what causes the
smaller scales to be underresolved
• In some sense, the problem is
accuracy rather than stability
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-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
10 time steps, ∆t = 0.01
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
1 time step, ∆t = 0.1
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• Avoiding any such ”hole” in the numerical domain of dependence
would require the parabolic CFL condition ∆t ∼ ∆x2
• Asymptotically, ”holes” are filled at a given time T under the weaker
condition:
∆t = o(T2/3∆x2/3
)
• The behaviour of the scheme at fixed time step improves for large
times (or in the limit case of computing the regime solution)
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We follow a strategy based on the estimation of the local (time–)
truncation error as used for ODEs:
• Time–step derefinement is performed assuming the solution is smooth
and the error introduced on a single time step is estimated by
ε∆x,∆t(xj, tn) ∼ C1∆t3/2 + C2∆xr + C3∆t1/2∆xq
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We follow a strategy based on the estimation of the local (time–)
truncation error as used for ODEs:
• Time–step derefinement is performed assuming the solution is smooth
and the error introduced on a single time step is estimated by
ε∆x,∆t(xj, tn) ∼ C1∆t3/2 + C2∆xr + C3∆t1/2∆xq
• Time–step refinement is performed assuming the solution is only
Lipschitz continuous and the error introduced on a single time step is
ε∆x,∆t(xj, tn) ∼ C4∆t1/2 + C5∆x
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We follow a strategy based on the estimation of the local (time–)
truncation error as used for ODEs:
• Time–step derefinement is performed assuming the solution is smooth
and the error introduced on a single time step is estimated by
ε∆x,∆t(xj, tn) ∼ C1∆t3/2 +C2∆xr + C3∆t1/2∆xq
• Time–step refinement is performed assuming the solution is only
Lipschitz continuous and the error introduced on a single time step is
ε∆x,∆t(xj, tn) ∼ C4∆t1/2 +C5∆x
• We assume space discretization error is negligible
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We compute the (n + 1)–th time step with 1 step ∆t and 2 steps
∆t/2 and estimate the local truncation error by:
ε∆t ∼v∆t − v∆t/2
1− 1√2
in smooth conditions (→ derefinement), and by
ε∆t ∼v∆t − v∆t/2√
2− 1
for Lipschitz solutions (→ refinement).
• The L∞ norm of this estimate is computed on a suitable set S (a
”narrow band” around the interface)
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The local error estimate is compared with a refinement and a dere-
finement threshold:(1−
1√2
)τD∆x <
∥∥∥∥∥v∆t − v∆t/2
|Dv|
∥∥∥∥∥L∞(S)
< (√
2− 1)τR∆x
• The renormalization by |Dv| allows the adaptation to be independent
of the function v0 (i.e. ”geometric”)
• The thresholds are set proportional to ∆x in order to take into
account the parabolic CFL condition in nonsmooth conditions
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Numerical tests
L∞ errors in a neighbourhood of the level curve for the shrinking
circle:
∆x Adaptive ∆t Fixed ∆t
1.428 · 10−1 1.581 · 10−3 1.066 · 10−3
7.070 · 10−2 3.872 · 10−4 2.587 · 10−4
3.517 · 10−2 1.039 · 10−4 6.725 · 10−5
In this situation the fixed step scheme performs slightly better then
the adaptive scheme with the same number of steps. However, we
expect adaptivity to become crucial in nonsmooth situations, as in
the evolution of a Lipschitz level set function.
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-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
Fixed time step, t ∈ [0,0.3]
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
Adaptive time step, t ∈ [0,0.3]
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0
0.05
0.1
0.15
0.2
0 2 4 6 8 10
Adapted time step, truncation error and thresholds, t ∈ [0,0.3]
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The case of codimension 2 (curves in R3)
In this case the framework is similar, but:
• We use the Ambrosio-Soner approach: the curve is interpreted as
zero level curve of a non–negative function (e.g. using the squared
distance from the manifold Γ0 as an initial condition)
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The case of codimension 2 (curves in R3)
In this case the framework is similar, but:
• We use the Ambrosio-Soner approach: the curve is interpreted as
zero level curve of a non–negative function (e.g. using the squared
distance from the manifold Γ0 as an initial condition)
• The Soner-Touzi formula has a generalized form (in particular, there
appears a minimization on all the directions orthogonal to Dv)
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The evolution equation describing the codimension–k MCM reads:vt = Fk(D2v, Dv) = inf
ν[trace(D2vP ν)]
v(x,0) = v0(x) with v0 = 0 on Γ0
with |ν| = 1, ν · Dv(x) = 0 and P ν an orthogonal projection. The
related Soner–Touzi representation formula is
u(x, t) = infµ(·)
(E {u(yµ(x, t, t),0)})
where µ(·) plays the role of a ”control”:dyµ(x, t, s) =√
2µ(t− s)dW (s) s ∈ (0, t]
yµ(x, t,0) = x
with |µ(·)| = 1, and µ(t− s) ·Dv(yµ(x, t, s)) = 0.
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-1
-0.5
0
0.5
1-1 -0.5 0 0.5 1
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The fully discrete scheme has the same structure as for the codimension–
1 case, unless for a minimization:
vn+1j = min
ν∈S2
{1
2I[vn](xj+
√2∆tν)+
1
2I[vn](xj−
√2∆tν)+
1
ε(Dj[v
n]·ν)2}
where:
• I[v](·) is a numerical reconstruction
• Dj[v] is a centered–difference approximation of the gradient
• The constraint ν ·Dv = 0 has been treated by penalization
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Known problems in this approach:
• Following a curve made of stationary points is an inherently ill–
conditioned operation
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Known problems in this approach:
• Following a curve made of stationary points is an inherently ill–
conditioned operation
• The ε–sublevel set gets thicker and thicker because of the parabolic
behaviour of the equation
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Known problems in this approach:
• Following a curve made of stationary points is an inherently ill–
conditioned operation
• The ε–sublevel set gets thicker and thicker because of the parabolic
behaviour of the equation
• It is analytically known that the fattening phenomenon cannot be
avoided (Bellettini, Novaga, Paolini ’98).
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Post–processing of the solution (1)
Rather than follow the zero–level set, we move on Γt from a point ξj
to the next point ξj+1 by constrained minimization:
vn(ξj+1) = min|ξ−ξj|=∆ξ
vn(ξ)
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Post–processing of the solution (1)
Rather than follow the zero–level set, we move on Γt from a point ξj
to the next point ξj+1 by constrained minimization:
vn(ξj+1) = min|ξ−ξj|=∆ξ
vn(ξ)
• The admissible displacement ξj+1 − ξj is required to satisfy the
restriction (ξj+1 − ξj) · (ξj − ξj−1) ≥ 0 in order to avoid changing
direction on Γt
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Post–processing of the solution (1)
Rather than follow the zero–level set, we move on Γt from a point ξj
to the next point ξj+1 by constrained minimization:
vn(ξj+1) = min|ξ−ξj|=∆ξ
vn(ξ)
• The admissible displacement ξj+1 − ξj is required to satisfy the
restriction (ξj+1 − ξj) · (ξj − ξj−1) ≥ 0 in order to avoid changing
direction on Γt
• The step ∆ξ is of the order of ∆x
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Post–processing of the solution (2)
The Ambrosio–Soner theory gives no analytical way to single out the
”physical” solution when fattening occurs
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Post–processing of the solution (2)
The Ambrosio–Soner theory gives no analytical way to single out the
”physical” solution when fattening occurs
• As a general rule to check numerical schemes, double points should
be ”cut” keeping acute angles connected
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Post–processing of the solution (2)
The Ambrosio–Soner theory gives no analytical way to single out the
”physical” solution when fattening occurs
• As a general rule to check numerical schemes, double points should
be ”cut” keeping acute angles connected
• Although in our case fattening corresponds to an ”almost constant”
region for the solution, the post–processing phase tends to select the
physically relevant evolution of the curve
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Numerical tests
Evolution of two linked circles
In this test we follow the evolution of two linked circles in R3. In this
case by tracking the ε–level set we are unable to resolve the ambiguity
arising after the double point generation, due to fattening. On the
contrary, post–processing singles out the meaningful solution
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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−1
0
1
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
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Evolution of a helical curve
We consider the evolution of a helix in three dimensions, with periodic
conditions. The Mean Curvature Flow preserves the shape of the
curve, but causes its straightening towards the axis. The figures also
compare the ε–level set with the post–processed curve
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00.2
0.40.6
0.81 0
0.20.4
0.60.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.2
0.40.6
0.81 0
0.20.4
0.60.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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00.2
0.40.6
0.81 0
0.20.4
0.60.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.2
0.40.6
0.81 0
0.20.4
0.60.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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00.2
0.40.6
0.81 0
0.20.4
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