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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<HEAD><TITLE>Surface Evolver Documentation - Energies</title></head>
<BODY>
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<center>
<h1><a href="http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.htm">
Surface Evolver</a> Documentation</h1>
</center>
<a href="evolver.htm#doc top">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
<a name="energies"></a>
<a name="energy"></a>
<h1>Energies</h1>
The Evolver usually works by minimizing the total energy of the
surface, subject to constraints.
This energy can have several components:
<ul>
<li> <a href="#surface tension">Surface tension</a>
<li> <a href="#gravity energy">Gravitational potential energy</a>
<li> <a href="#constraint energy integrals">Constraint energy integrals</a>
<li> <a href="#named quantity energy">Named quantity energies</a>
<li> <a href="#gap energy">Convex constraint gap energy</a>
<li> <a href="#pressure energy">Prescribed pressure energy</a>
<li> <a href="#compressibility energy">Compressibility energy</a>
<li> <a href="#crystalline energy">Crystalline energy</a>
</ul>
<hr>
<a name="surface tension"></a><h2>Surface tension energy</h2>
Soap films and interfaces
between different fluids have an energy content proportional
to their area. Hence they shrink to minimize energy. The
energy per unit area can also be regarded as a surface
tension, or force per unit length. Each facet has a surface
tension, which is 1 unless the datafile specifies otherwise
(see <a href="datafile.htm#faces section">TENSION</a> attribute
for faces).
Different facets may have different surface tensions.
Facet tensions may be changed interactively with the
<a href="commands.htm#set"><tt>set facet tension ...</tt></a> command.
The contribution to the total energy is the sum
of all the facet areas times their respective surface
tensions. The surface tension of a facet may also be specified
as depending on the <a href="elements.htm#facet phase">phases</a> of the bodies it separates.
In the <a href="model.htm#string model">string model</a>, the
tension resides on edges instead of facets.
<p> Example datafile: <a href="cube.htm">cube.fe</a>
<hr>
<a name="gravity energy"></a><h2>Gravitational potential energy</h2>
If a body has
a <a href="elements.htm#body density">density</a>,
then that body contributes its gravitational
energy to the total. The acceleration of gravity
<a href="syntax.htm#G value">G</a> is
under user control with the <a href="single.htm#G">G</a> command.
Letting \rho be the body density,
the energy is defined as
<pre>
E = \int\int\int_{body} G \rho z dV
</pre>
but is calculated using the Divergence Theorem as
<pre>
E = \int\int_{body surface} G\rho {z^2\over 2} \vec k \cdot \vec{dS}.
</pre>
This integral is done over each facet that bounds a body.
If a facet bounds two bodies of different density, then
the appropriate difference in density is used. Facets
lying in the z = 0 plane make no contribution, and may be
omitted if they are otherwise unneeded. Facets lying
in constraints may be omitted if their contributions to
the gravitational energy are contained in
<a href="#constraint energy integrals">constraint energy integrals</a>.
In the <a href="model.htm#string model">string model</a>,
all this happens in one lower dimension.
<p> Example datafile: <a href="mound.htm">mound.fe</a>
<hr>
<a name="constraint energy integrals"></a><h2>Constraint energy integrals</h2>
An edge on a <a href="constrnt.htm#level set constraints">
level-set constraint</a>
may have an energy given by integrating a vectorfield
<b>F</b> over the oriented edge:
<pre>
E = \int_{edge} <b>F . dl</b>.
</pre>
The integrand is defined in the <a href="datafile.htm#constraint decl">
constraint declaration</a> in the datafile.
The integral uses the innate orientation of the edge,
but if the orientation attribute of the edge is negative,
the value is negated.
This is useful for prescribed contact angles on walls
(in place of wall facets with equivalent tension)
and for gravitational potential energy that would
otherwise require facets in the constraint.
The <a href="mound.htm">mound example</a> illustrates this.
<hr>
<a name="named quantity energy"></a><h2>Named quantity energies</h2>
There are a large number of <a href="quants.htm">named methods</a>
for calculating various quantities, which all follow the same syntax.
These may be used as energy by defining an
<a href="quants.htm#energy quantity">energy-type</a>
<a href="quants.htm#named quantities">named quantity</a> in the
<a href="datafile.htm#named quantity decl">datafile</a>.
<p> Example datafile: <a href="ringblob.htm">ringblob.fe</a>
<hr>
<a name="gap energy"></a><h2>Convex constraint gap energy</h2>
Consider a soap film spanning
a circular cylinder. The Evolver must approximate
this surface with a collection of facets. The straight
edges of these facets cannot conform to the curved wall,
and hence the computed area of the surface leaves out
the gaps between the outer edges and the wall. The
Evolver will naturally try to minimize area by moving
the outer vertices around so the gaps increase, ultimately
resulting in a surface collapsed to a line. This is not
good. Therefore there is provision for a "gap energy"
to discourage this. A
<a href="constrnt.htm#level set constraints">level-set constraint</a>
may be declared CONVEX
in the <a href="datafile.htm#constraint decl">datafile</a>.
For an edge on such a constraint, an
energy is calculated as
<pre> E = <i>k</i>\left\Vert \vec S \times \vec Q \right\Vert / 6
</pre>
where \vec S is the edge vector and \vec Q is the
projection of the edge on the tangent plane
of the constraint at the tail vertex of the edge.
The constant <i>k</i> is a global constant called the "gap
constant". A gap constant of 1 gives the best
approximation to the actual area of the gap. A larger
value minimizes gaps and gets vertices nicely spread
out along a constraint. You can set the
value of <i>k</i> in the
<a href="datafile.htm#gap constant decl">datafile</a> or
with the <a href="single.htm#k">k</a> command.
<p>
The gap energy falls off quadratically as the surface
is refined. That is, refining once reduces the gap
energy by a factor of four. You can see if this energy
has a significant effect on the surface by changing the
gap constant.
<p> Note: gap energy is effective only in the
<a href="model.htm#linear model">linear model</a>.
<p> Example datafile: tankex.fe
<hr>
<a name="pressure energy"></a><h2>Prescribed pressure energy</h2>
Each body with a prescribed
pressure P contributes energy E = PV.
where V is the actual volume of the body.
This can be used to generate surfaces of prescribed
mean curvature, since mean curvature is proportional
to pressure. Pressure can be prescribed in the
<a href="datafile.htm#bodies section">bodies section</a> of the
datafile, and can be changed with the <a href="single.htm#b">b</a>
command, or by assigning a value to the <a href="elements.htm#body pressure">
pressure attribute</a> of a body.
<hr>
<a name="compressibility energy"></a><h2>Compressibility energy</h2>
If the ideal gas mode is in effect (set by the
<a href="toggle.htm#ambient_pressure">PRESSURE</a> keyword
in the datafile), then each body contributes an energy
<pre>
E = P*V_0*ln(V/V_0) </pre>
where P is the ambient pressure, V_0 is the target volume of the body,
and V is the actual volume. To account for work done against the
ambeint pressure, each body also makes a negative contribution of
<pre>
E = -P*V. </pre>
The ambient pressure can be set in the datafile or with the
<a href="single.htm#p">p</a> command. This energy is calculated
only for bodies given a target volume.
<hr>
<a name="crystalline energy"></a><h2>Crystalline energy</h2>
The Evolver can model energies of crystalline
surfaces. These energies are proportional to the
area of a facet, but they also depend on the direction
of the normal. The energy is given by the largest dot
product of the surface normal with a set of vectors known
as the Wulff vectors. Surface area can be regarded as
a crystalline integrand whose Wulff vectors are the unit sphere.
See the datafile section on Wulff vectors
for more. A surface has either crystalline energy
or surface tension, not both. Use is not recommended since
nonsmoothness makes Evolver work poorly.
<p> Example datafile: crystal.fe
<hr>
<a href="evolver.htm#doc top">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
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