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<!DOCTYPE HTML>
<head>
<title>Surface Evolver Documentation - Constraints</title>
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<body>
<h1 class="center"><a href="http://www.susqu.edu/brakke/evolver/evolver.htm"
class="comic">
Surface Evolver</a> Documentation</h1>
<a href="evolver.htm#doc-top">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
<a id="general-constraints"></a>
<h1>Constraints and Boundaries</h1>
<a id="global-constraints"></a>
<a id="pointwise-constraints"></a>
<p>The usual mode of operation of the Surface Evolver is to minimize energy
subject to constraints. There are two broad categories of constraints:
</p>
<ul>
<li>Pointwise constraints:
<ul>
<li><a HREF="elements.htm#fixed-vertex">Fixed vertices</a> </li>
<li><a HREF="#level-set-constraints">Level set constraints</a> </li>
<li><a HREF="#one-sided-constraints">One-sided constraints</a> </li>
<li><a HREF="#parametric-boundaries">Parametric "boundary" curves and surfaces</a>
</li>
</ul>
</li>
<li>Global constraints:
<ul>
<li><a HREF="elements.htm#target-volume">Volume constraints</a> </li>
<li><a HREF="#named-quantity-constraints">Named quantity constraints</a>
</li>
</ul>
</ul>
<p>
<hr><a id="level-set-constraints"></a>
<h2>Level set constraints</h2>
A level-set constraint is a restriction of vertices to lie on the zero
level-set of a function.
The formula may include any expressions whose values
are known to the Evolver, given the particular vertex.
Most commonly one just uses the coordinates (x,y,z) of the
vertex, but one can use variables, quantity values,
or vertex extra attributes. Using a vertex extra
attribute is a good way to customize one formula to
individual vertices. For example, if there were a
vertex extra attribute called zfix, one could force
vertices to individual z values with one constraint
with the formula z = zfix, after of course assigning
proper values to zfix for each vertex.
A level set constraint may have several roles:
<p>
<ul>
<li>Vertices may be required to lie on a constraint (equality constraint)
or on one side (<a HREF="#one-sided-constraints">inequality constraint</a>).
A constraint may be declared
<a href="datafile.htm#constraint-decl">GLOBAL</a>,
in which case it applies to all vertices.
See <a HREF="mound.htm">mound.fe</a> for an example. </li>
<li>A constraint may have an
<a href="datafile.htm#constraint-decl">energy vectorfield</a>
associated with it that is integrated
over edges lying in the constraint to give an
<a href="energies.htm#constraint-energy-integrals">energy</a>.
This is useful for
specifying wall contact angles and for calculating gravitational energy.
Integrals are not evaluated over edges that are FIXED. See
<a href="mound.htm">mound.fe</a> for
an example.
In the <a href="model.htm#string-model">string model</a>,
the energy integrand is a single component evaluated on vertices
on the constraint.</li>
<li> A constraint may have a
<a href="datafile.htm#constraint-decl">content vectorfield</a>
associated with it that is integrated
over edges lying in the constraint to give a <a href="elements.htm#body-volume">
volume</a> contribution to a body
whose boundary facets contain the edges. This is useful for getting correct
volumes for bodies without completely surrounding them with otherwise useless
facets. It is important to understand how the content is added to the body
in order to get the signs right. The integral is evaluated along the positive
direction of the edge. If the edge is positively oriented on a facet, and
the facet is positively oriented on a body, then the integral is added
to the body. This may wind up giving the opposite sign to the integrand
from what you think may be natural. Integrals are not evaluated over edges
that are FIXED. See tankex.fe for an example.
In the <a href="model.htm#string-model">string model</a>,
the content integrand is a single component evaluated on vertices
on the constraint.</li>
<li>
There is a constraint attribute <code>content_rank</code> such that if a vertex (string model) or
an edge (soapfilm model) is on multiple boundaries with content
integrals (say where walls meet) then if content ranks are present,
the content integral with the least rank will contribute to the
content on the negative side body and the highest rank content
will contribute to the content of the positive side body.
<li><a id="convex-constraint"></a>A constraint may be declared
<a href="datafile.htm#constraint-decl">CONVEX</a>,
in which case edges in the
constraint have an energy associated with them that is proportional to
the area between the straight edge and the curved wall. This energy (referred
to as "<a href="energies.htm#gap-energy">gap energy</a>")
is meant to compensate for the tendency for flat
facets meeting a curved wall to minimize their area by lengthening some
edges on the wall and shortening others, with the net effect of increasing
the net gap between the edges and the wall. See tankex.fe for an example.
</li>
</ul>
<p>Level set constraints are declared in the top section of the <a HREF="datafile.htm#constraint-decl">datafile</a>.
They may be applied to <a HREF="elements.htm#vertex-constraints">vertices</a>,
<a HREF="elements.htm#edge-constraints">edges</a>,
or <a HREF="elements.htm#facet-constraints">facets</a>.
Constraints are usually applied to vertices and edges, as in
<a href="mound.htm">mound.fe</a>. Remember that you need to apply a constraint
to an edge to get that constraint to apply to vertices created on that edge
by refining. Sometimes one applies constraints to facets, usually to get
the facet to conform to a predetermined shape. Be sure that the constraints
applied to a vertex are linearly independent at the vertex.
<p>
Constraints are usually applied in the datafile
<a href="datafile.htm#vertices-section">vertices</a>,
<a href="datafile.htm#edges-section">edges</a>, and
<a href="datafile.htm#faces-section">faces</a> sections, but they
may also be set or removed with the <a href="commands.htm#set">set</a>
or <a href="commands.htm#unset">unset</a> commands. Example:
<pre> set vertex[4] constraint 4
unset edge constraint 1 where id < 10
</pre>
It does not hurt to unset an element that isn't on the constraint.
When a vertex is set to a constraint, the vertex coordinates are
immediately projected to the constraint. Setting an edge on a
constraint does not set its vertices. Likewise for facets.
<p>
Several element attributes are useful in connection with level-set constraints:
<ul>
<li> <a href="elements.htm#on_constraint">on_constraint</a>
- Boolean attribute that tells whether the
element has been put on a constraint.
<li> <a href="elements.htm#hit_constraint">hit_constraint</a>
- Boolean vertex attribute that tells whether
the constraint exactly satisfies the constraint. Meant for
one-sided constraints; always 1 for exact constraints.
<li> <a href="elements.htm#value_of_constraint">value_of_constraint</a>
- Vertex attribute that is the value of the constraint formula at the vertex.
<li> <a href="elements.htm#v_constraint_list">v_constraint_list</a>,-
<a href="elements.htm#e_constraint_list">e_constraint_list</a>,-
<a href="elements.htm#f_constraint_list">f_constraint_list</a>--
Vector attribute whose first element is the number of constraints the
element is on, followed by a list of the constraint numbers (using the
internal numbers for named constraints).
<li><a href="elements.htm#constraint-normal">constraint normal</a>
- unit vector perpendicular to the level set at a vertex.
</ul>
<a id="constraint-fixed"></a>
Whether a particular level-set constraint is an equality constraint or
a one-sided constraint can be queried at runtime by the expression
<pre> is_constraint[number].fixed
or is_constraint[name].fixed
</pre>
"number" may be an expression; "name" is the unquoted name of the
constraint, if it has one. This has value 1 if the constraint is an
equality constraint, else the value is zero. Example of listing all
the equality constraints that vertices are on:
<pre> foreach vertex vv do
{ for ( inx := 1 ; inx < vv.v_constraint_list[1] ; inx++ )
if is_constraint[vv.v_constraint_list[i+1]].fixed then
printf "Vertex %d on equality constraint %d\n",
vv.id,vv.v_constraint_list[i+1];
};
</pre>
<a id="one_sided_lagrange"></a> <a id="nonnegative"> </a>
<a id="nonpositive"></a>
<hr><a id="one-sided-constraints"></a>
<h2>One-sided constraints</h2>
If a <a HREF="#level-set-constraints">level set constraint</a> is declared
<code>NONNEGATIVE</code> or <code>NONPOSITIVE</code> in the <a HREF="datafile.htm#constraint-decl">datafile</a>,
the vertices subject to the constraint must stay in that part of the domain
of the level set function. It is usually unwise to give edge integrals
to edges on one-sided constraints, or to declare them <a HREF="energies.htm#gap-energy">CONVEX</a>.
Whether a vertex exactly satisfies the constraint may be queried with the
vertex <a HREF="elements.htm#hit_constraint">hit_constraint</a> attribute.
The 'g' iteration step will check for a vertex wanting to leave a one-sided
constraint it has hit, but hessian commands do not; therefore it is wise
to intersperse 'g' with hessian or hessian_seek when there are one-sided
constraints involved.
<p>
Example: Suppose one wanted to keep a bubble inside a spherical tank
of radius 5. Then one would define the constraint in the datafile
<pre>constraint 1 nonpositive
formula: x^2 + y^2 + z^2 = 25
</pre>
For purposes of evaluating nonnegativity or nonpositivity, all terms are
shifted to the left side of the formula. One would then apply this
constraint to all vertices, edges, and facets of the bubble surface.
<p>
If you define the real-valued vertex extra attribute one_sided_lagrange, the
Lagrange multipliers for vertices hitting one-sided constraints will
be recorded. one_sided_lagrange may be defined as an array. If a vertex
hits more constraints than the size of one_sided_lagrange, then the
first ones that fit will be recorded.
<p>
The type of a constraint can be queried at runtime as expressions <pre>
is_constraint[number].nonnegative
is_constraint[name].nonnegative
is_constraint[number].nonpositive
is_constraint[name].nonpositive
is_constraint[number].fixed
is_constraint[name].fixed </pre>
which have value 1 if the constraint is of the give type, else the value is 0.
"number" may be an expression; "name" is the unquoted name of the
constraint, if it has one.
Example:
<pre>
print is_constraint[floorcon].nonnegative
</pre>
<hr><a id="boundary"></a>
<a id="boundaries"></a>
<a id="parametric-boundaries"></a>
<h2>Parametric "boundary" curves and surfaces</h2>
Vertex locations may be given in terms of parameters on a parameterized
curve or surface. Such curves or surfaces are called "boundaries"
in Evolver terminology, since they are usually used as boundary curves
of surfaces, for example a soap film on a wire loop could have the wire
implemented as a boundary. Vertices, edges, and facets may be deemed to
lie in a boundary. For a vertex, this means that the fundamental parameters
of the vertex are the parameters of the boundary, and its coordinates are
calculated from these. Vertices on boundaries may move during iteration,
unless declared fixed. See <a href="catenoid.htm">cat.fe</a> for an example. <p>
<p>
Boundaries are defined in the top section of the
<a href="datafile.htm#boundary-decl">datafile</a>. Vertices on boundaries
are listed in the datafile with their parameter values instead of their
coordinates, with "<code>boundary </code><i>n</i>" appended to each such
<a href="datafile.htm#vertices-section">vertex definition</a>.
Edges and faces on boundaries are defined as usual, but with
"<code>boundary </code><i>n</i>" appended to each definition.
So the datafile has lines like these:
<pre>
boundary 1 parameters 1
x1: cos(p1)
x2: sin(p1)
x3: 0.75
...
Vertices
1 0.0 boundary 1
2 pi/3 boundary 1
...
Edges
1 1 2 boundary 1
...
</pre>
<p>Putting an edge on a boundary means that vertices created on that edge
will be on the boundary. An edge on a boundary must have at least one
endpoint on the boundary, for use in extrapolating the boundary parameters
of any created vertices. Extrapolating instead of interpolating
midpoint parameters solves the problem of wrap-arounds on a boundary such as
a circle or cylinder. However if you do want interpolation, you can use
the keyword
<code><a href="datafile.htm#interp_bdry-decl">INTERP_BDRY_PARAM</a></code>
in the top of the datafile, or use the
toggle command <a href="toggle.htm#interp_bdry_param">interp_bdry_param</a>.
<a id="extra_boundary"></a><a id="extra_boundary_param"></a>
Interpolation requires that both endpoints of an edge be on the same
boundary, which cannot happen where edges on different boundaries meet.
To handle that case, it is possible to add extra boundary information to
a vertex by declaring two particular vertex extra attributes,
<code>extra_boundary</code> and <code>extra_boundary_param</code>:
<pre>
interp_bdry_param
define vertex attribute extra_boundary integer
define vertex attribute extra_boundary_param real[1]
</pre>
Then declare attribute values on key vertices, for example
<pre>
vertices
1 0.00 boundary 1 fixed extra_boundary 2 extra_boundary_param 2*pi
</pre>
If the extra_boundary attribute is not set on a vertex when wanted,
Evolver will silently fall back on interpolation.
<p> Putting a face on a boundary means that all edges and vertices created
from refining the face will be on the boundary. In this case, the boundary
should have two parameters (or whatever the dimension of the surface is).
This is good for getting a surface to conform to a known parametric
shape.
<p>
Edges on boundaries have energy and content integrals like level-set
constraints edges, but they are internally implemented as.
<a href="quants.htm#named-quantities">named quantities</a>.
</p>
<p>
Whether an element is on a particular boundary can be queried with
the <a href="elements.htm#on_boundary">on_boundary</a> Boolean attribute.
Elements can be removed from boundaries with the
<a href="commands.htm#unset">unset</a> command, and they can be
<a href="commands.htm#set">set</a> on boundaries. A typical use
of <code>unset</code> is to define an initial surface using a 2-parameter
boundary, refine a couple of times, then unset. Examples:
<pre> list vertex where on_boundary 2
unset vertex boundary 1 where on_boundary 1
unset edge boundary 1
unset facet boundary 1
set vertex[10] boundary 1
</pre>
It does not hurt to unset an element not on the boundary.
<p>
<a id="__v_boundary"></a><a id="__e_boundary"></a>
<a id="__f_boundary"></a>
The number of the boundary that an element is on is in the read-only
attribute __v_boundary, __e_boundary, or __f_boundary. These are
0 if the element is not on a boundary. These attributes are not
present if there are no boundaries defined. Note that named
boundaries are internally assigned numbers, which are what
show up here.
<p>
Vertex parameters can be accessed in expressions as the attribute
p1 (and p2,... for further parameters).
Vertex parameters can be changed with the <a href="commands.htm#set">set</a>
command. Example:
<pre> print vertex[5].p1
set vertex p1 p1+.1 where id < 4
vertex[2].p1 := 3
</pre>
It is not an error to access the parameters of a vertex not on a boundary
as long as some vertex is on a boundary (so that space is allocated in
the vertex structure for parameters).
<p>A general guideline is to use constraints for two-dimensional walls
and boundaries for one-dimensional wires. If you are using a boundary wire,
you can probably declare the vertices and edges on the boundary to be FIXED.
Then the boundary becomes just a guide for refining the boundary edges.
</p>
<p>NOTE: A vertex on a boundary cannot also have constraints.
<hr><a id="named-quantity-constraints"></a>
<h2>Named quantity constraints</h2>
See <a href="quants.htm#named-quantities">fixed named quantities</a>.
<hr><a HREF="evolver.htm#doc-top">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
<p>
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