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<a href="http://www.susqu.edu/brakke/evolver/evolver.htm" class="comic">
Surface Evolver</a> Documentation</h1>
<a href="evolver.htm#doctop">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
<hr><a id="gradientdescentiteration"> </a>
<h2>Gradient Descent Iteration</h2>
Basic surface evolution is done by gradient descent iteration.
Relevant topics are:
<ul>
<li><a href="#iterationstep">Iteration step</a>
<li><a href="#scalefactor">Scale factor</a>
<li><a href="#optimizingscale">Optimizing scale factor</a>
<li><a href="#conjugategradient">Conjugate gradient</a>
<li><a href="#diffusion">Diffusion</a>
<li><a href="#stability">Stability</a>
</ul>
<hr>
<a id="iterationstep"></a><h2>Gradient descent iteration step</h2>
Each <a href="single.htm#g">g</a> command iteration does the following:
<ul>
<li> Calculates the force vector at each vertex as the gradient
of the total <a href="energies.htm">energy</a>.
<li> Calculates the gradients at each vertex of constrained
<a href="elements.htm#bodyvolume">body
volumes</a> and
<a href="quants.htm#fixedquantity">named quantities</a>.
Also calculates the multiple of
gradient motion needed to restore target values of the constraints.
<li> Projects the forces to be tangent to
<a href="constrnt.htm#levelsetconstraints">level set constraints</a>,
volume constraints (pressure is calculated here),
and fixed quantity constraints.
<li> If any <a href="model.htm#mobility">mobility</a> options are in effect,
the proper force to velocity conversion is done.
<li> If the <a href="toggle.htm#normal_motion">normal_motion</a> toggle is
on, then velocities are projected to the surface normal.
<li> Does the
<a href="#conjugategradient">conjugate gradient</a>
projection, if conjugate gradient is in effect.
<li> The current vertex positions are saved.
<li> If the optimizing mode is on, then trial motions are made to
find the optimal scale factor.
<li> The scale factor is multiplied by the value of the
<a href="syntax.htm#scale_scale">scale_scale</a> internal variable.
(Useful only if you want to muck around with modifying the optimal
scale factor for some strange reason.)
<li> Each vertex is moved by the scale factor times the velocity,
plus the volume and fixed quantity restoring motions. If
<a href="toggle.htm#runge_kutta">runge_kutta</a>
is in effect, then a fourthorder RungeKutta step is done instead of
a simple Euler step.
<li> All level set constraints are enforced.
Vertices
violating an inequality or equality constraint are
projected to the constraint (Newton's method). Several
projection steps may be needed, until the violation
is less that a certain tolerance or a certain number
of steps are done (which generates a warning message).
The default constraint tolerance is 1e12, but it can
be set with the
<a href="datafile.htm#constraint_tolerancedecl">CONSTRAINT_TOLERANCE</a>
option in the datafile,
or setting the
<a href="syntax.htm#constraint_tolerance">constraint_tolerance</a> variable.
<li> If <a href="toggle.htm#jiggle">jiggling</a> is on, the
surface is randomly perturbed.
<li> If <a href="toggle.htm#autopop">autopop</a> or
<a href="toggle.htm#autochop">autochop</a> are on, then the
appropriate edge deletion or division is done.
<li> New energies, volumes, and quantities are calculated.
<li> If <a href="toggle.htm#check_increase">check_increase</a> is
on and the surface energy has increased, then the vertices are
restored to their old values.
<li> If <a href="toggle.htm#estimate">estimating</a> is on, then
a linear estimate of the energy change is printed. This is calculated
from the velocities, gradients, and scale factor.
<li> The number of iterations left, the new area, energy, and scale factor
are printed.
<li> If a graphics display is active and set to
<a href="toggle.htm#autodisplay">autodisplay</a>, then graphics
are redrawn.
</ul>
<hr>
<a id="scalefactor"></a><h2>Scale factor</h2>
Once a direction of motion is found, the direction must be
multiplied by a scale factor to compute the actual motion.
If one interprets the direction of motion as a velocity
(as in motion by mean curvature), then the scale factor becomes
a time step. The scale factor may be fixed with the
<a href="single.htm#m">m</a> command, or it may be in
<a href="#optimizingscale">optimizing</a> mode. The default
is to start in optimizing mode.
<hr>
<a id="optimizingscale"></a><h2>Optimizing scale</h2>
In using gradient descent to seek a minimum energy, one finds
a direction of motion and does a line search along that direction
to find the minimum energy. Evolver will do that in optimizing
scale mode. The line search consists of halving or doubling
the current <a href="#scalefactor">scale factor</a> until an
energy minimum is bracketed; then quadratic interpolation is
used to estimate the optimum scale. Optimizing scale is the
default; it also may be turned on with the <a href="single.htm#m">m</a>
command or the <a href="commands.htm#optimizing">optimizing</a> command.
<p>
For safety, there is an upper bound to the scale; it defaults to 1
but may be changed with the <a href="commands.htm#optimizing">optimizing</a>
command. There is also a lower bound; if Evolver gets a scale
below 1e12 of the scale bound when attempting to find a minimum,
it gives up and just uses scale 0. Scale 0 is not a null operation
since it still projects to constraints, if they are not exactly
satisfied.
<p>
In general, a good scale factor depends on the type of energy being
minimized and the level of refinement. However, for minimizing area,
when the triangulation is wellbehaved and
<a href="model.htm#areanormalization">area normalization</a> is off,
the best scale factor is usually around 0.2, independent of refinement.
In optimizing
mode, a scale factor getting small, say below 0.01, indicates
triangulation problems. Too large a fixed scale factor will show up as
total energy increasing. If you have motion by area normalization ON
use a small scale factor, like 0.001, until you get a
feel for what works.
<p>
If <a href="toggle.htm#check_increase">check_increase</a> is toggled
on, then the motion is not done if it would increase energy. But
be aware that energy sometimes may have to increase in order to
satisfy constraints.
<hr>
<a id="conjugategradient"></a><h2>Conjugate gradient</h2>
"Conjugate gradient" is a method of accelerating gradient descent.
In ordinary gradient descent, one uses the gradient of energy to
find the steepest downhill direction, then moves along that line
to the minimum energy in that direction. Hence successive steps
are at right angles. However, this can be very inefficient, as
you can spend a lot of time zigzagging across an energy "valley"
without making much progress "downstream". With conjugate gradient,
the search direction is chosen to be in a "conjugate" direction to
the previous direction. For a mathematical explanation, see any
decent book in numerical analysis, such as <a href="biblio.htm#refP">[P]</a>.
In practice, the conjugate gradient method remembers a cumulative
"history vector", which it combines with the ordinary gradient to
figure out the conjugate gradient direction. The upshot is that
conjugate gradient can converge much faster than ordinary gradient
descent.
<p>
Conjugate gradient can be toggled with the <a href="single.htm#U">U</a>
command, or with the <a href="toggle.htm#conj_grad">conj_grad</a> toggle.
It should always be used with <a href="#optimizingscale">optimizing scale</a>.
<p>
Notes: The conjugate gradient method is designed for quadratic energy
functions. As long as the energy function is nearly quadratic, as it
should be near an energy minimum, conjugate gradient works well.
Otherwise, it may misbehave, either by taking too big steps or by
getting stalled. Both effects are due to the history vector being
misleading. To prevent too big steps, one should iterate without
conjugate gradient for a few steps whenever significant changes are
made to the surface (refining, changing a constraint, etc.).
On the other hand, if it looks like conjugate gradient is converging,
it may have simply become confused by its own history. See the
<a href="catenoid.htm">catenoid</a> example for a case in point.
A danger signal is the scale factor going to zero. If you are suspicious,
toggle conjugate gradient off and on ("<code>U 2</code>" does nicely)
to erase the history vector and start over.
<hr>
<a id="diffusion"></a><h2>Diffusion</h2>
<a id="diffusion_coeff"></a>
<a id="edge_diffusion"></a>
<a id="facet_diffusion"></a>
The Evolver can simulate the reallife phenomenon
of gas diffusion between neighboring bubbles. This
diffusion is driven by the pressure difference across
a surface. This is invoked by the keyword DIFFUSION
in the first part of the datafile, followed by the
value of the diffusion constant. The amount diffused
across a facet during an iteration is calculated as
scale*diffusion_constant*facet_area*pressure_difference.
The scale factor is included as the time step of
an iteration. The amount is added to or subtracted
from the prescribed volumes of the bodies on either
side of the facet. The diffusion constant can be
accessible at runtime through the readwrite
internal variable <code>diffusion_coeff</code>.
<p> If you want finer control over the rate of diffusion across
various surfaces, you can define the edge_diffusion edge attribute
in the string model or the facet_diffusion facet attribute in
the soapfilm model and give individual values for edges or facets
as you desire. If the attribute is defined, then its value is
used instead of the global diffusion constant.
<hr>
<a id="stability"></a><h2>Stability</h2>
The timestep of an iteration should not be so large as to amplify
perturbations of the surface. Short wavelength perturbations
are most prone to amplification. This section contains a sketch of
the stability characteristics of the various mobility modes, enough
to let the user relate the maximum timestep to the minimum facet or
edge size. Two examples are discussed: a zigzag string and a
nearly flat surface with equilateral triangulation. Effective
area is not included, as it is an insignificant correction for nearly
flat surfaces. The general moral of this section is that the
maximum time step in iteration is limited by the length of the
shortest edge or the area of the smallest facet, except in one case.
<h3>Zigzag string</h3>
Let the amplitude of the perturbation about the midline be Y and
the edge length L. Then the force on a vertex is F = 4Y/L
for small Y. Let the timestep (the Evolver scale factor) be
\Delta t. Let V be the vertex velocity. Then the critical
timestep for amplification of the perturbation is given by
V\Delta t = 2Y, or \Delta t = 2Y/V.
<p>
Vertex mobility. Here V = F, so \Delta t = L/2.
<p>
Area normalization. Here the vertex star has length 2L,
so V = F/L and \Delta t = L^2/2.
<p>
Approximate curvature. It turns out that the zigzag is
an eigenvector of the mobility matrix M for the largest eigenvalue 3/L,
so V = 3F/L and \Delta t = L^2/6. This is a major disadvantage
of approximate curvature. If perturbation instability is the
limitation on the timestep, it will take three times as many iterations
as with area normalization to do the same evolution.
<h3> Perturbed sheet with equilateral triangulation.</h3>
Consider a plane surface triangulated with equilateral triangles
of area A.
The perturbation consists of a tiling of hexagonal dimples with their centers
of height Y above their peripheries. The force at a central vertex turns
out to be sqrt(3)Y and the force at a peripheral vertex sqrt(3)Y/2.
<p>
Vertex mobility. The critical time step is given by
<br>
(\sqrt(3)Y + sqrt(3)Y/2)\Delta t = 2Y,
<br>
so \Delta t = 4/3*sqrt(3). Note that this is independent of the triangle
size. This is consistent with experience in evolving with optimizing
scale factor, where the optimum time step is in the range 0.2  0.3
indenpendent of triangle size. This is a definite advantage of this
version of mobility, since different parts of the surface can have
different size triangulations and one size time step can work for all.
<p>
Area normalization. The star area of each vertex is 6A, so
the velocities become sqrt(3)Y/2A and sqrt(3)Y/4A, and the
critical time step \Delta t = 4/6*sqrt(3)A. Hence on a surface
with varying size triangles, the timestep is limited by the area of
the smallest triangles.
<p>
Approximate area. This force turns out to be an eigenvector
of the mobility matrix with eigenvalue 2/A. Hence the velocity
is four times that of the area normalization, and the critical
time step four times shorter.
<hr>
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