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Index.
Named quantities and methods
This is the systematic scheme of calculating global quantities
such as area, volume, and surface integrals that replaces the
original ad hoc scheme in the Evolver. Briefly,
methods are built-in
functions, and
named quantities
are combinations of
instances of methods.
See the ringblob datafile
for an example.
The original ad hoc calculations are still the default
where they exist, but all new quantities are being added
in the named quantity scheme. Some new features will work
only with named quantities. To convert everything to named
quantities, start Evolver with the -q
option or use the
convert_to_quantities command.
This has not been made the default since named quantities can be
slower than the originals.
The sample datafiles qcube.fe, qmound.fe, and
ringblob.fe
contains some examples of named quantities and instances. The first two
are quantity versions of cube.fe and mound.fe. These
illustrate the most general and useful methods, namely
facet_vector_integral, facet_scalar_integral, and edge_vector_integral,
rather than the faster but more specialized methods such as facet_area.
My advice is that the user stick to the old implicit methods for
area, volume, and gravitational energy, and use named quantities only
for specialized circumstances.
Chapter contents:
A "method" is a way of calculating a scalar
value from some particular type of element (vertex, edge, facet, body).
Each method is implemented internally as a set of functions for
calculating the value and its gradient as a function of vertex
positions. The most common methods also have Hessian functions.
Methods are referred to by their names.
See
Implemented methods for a list of available methods.
Adding a new method involves writing C routines to calculate the value
and the gradient (and maybe the Hessian)
as functions of vertex coordinates, adding the function declarations
to quantity.h, and adding a
structure to the method declaration array in quantity.c. All the
other syntax for invoking it from the datafile is already in place.
A "method instance" is the sum of a particular method applied
to a particular set of geometric elements.
Some methods (like facet_area)
are completely self-contained. Others (like facet_vector_integral)
require the user to specify some further information. For these,
each instance has a specification of this further information.
Method instances are defined in the datafile, and may either be
unnamed parts of named quantity
definitions or separate
named method instances for inclusion in named quantities.
The separate named version is useful if you want to inspect
instance values for the whole surface or individual elements.
An instance total value can be printed with the A
commands, or may be
referred to as "instancename.value" in commands. The instance name
itself may be used as an element attribute. For example, supposing there
is an instance named moment, which applies to facets.
Then typical commands would be
print moment.value
print facet[3].moment
list facet where moment > 0.1
Every method
instance has a "modulus", which is multiplied times the basic
method value to give the instance value. A modulus of 0
causes the entire instance calculation to be omitted whenever quantities
are calculated. The modulus may be set in the
datafile or with the
A command or by assignment. Example commands:
print moment.modulus
moment.modulus := 1.3
A method instance may be declared to use
a different modulus for each element by specifying an element extra
attribute to use for that purpose. The extra attribute has to have already
been declared. Example:
define facet attribute mymod real
quantity myquant energy method facet_area global element_modulus mymod
Of course, it is up to the user to properly initialize the values of the
extra attribute.
A "named quantity" is the sum total of various
method instances,
although usually just one instance is involved.
The instances need not apply to the same type of element;
for example, both facet and edge integrals may be needed to
define a volume quantity.
Each named quantity is one of four types:
- "energy" quantities which are added to the total energy of the surface;
- "fixed" quantities that are constrained to a
fixed target value (by Newton steps at each iteration); and
- "conserved" quantities are like fixed, but the value is irrelevant. The
quantity gradient is used to eliminate a degree of freedom in motion.
Rarely used, but useful to eliminate rotational degree of freedom, for example.
Will not work with optimizing parameters, since
they do gradients by differences.
- "info_only" quantities whose values are merely reported to the user.
This type is initially set in a quantity's
datafile declaration. A quantity can be toggled between fixed and
info_only with the "fix quantityname" and
"unfix quantityname" commands.
The value of a quantity may be displayed with the
A or v commands,
or as an expression "quantityname.value". Furthermore, using the
quantity name as an element attribute evaluates to the sum of all
the applicable component instance values on that element. For example,
supposing there is a quantity named vol, one could do
print vol.value
print facet[2].vol
histogram(facet,vol)
Each quantity has a "modulus", which is just a scalar multiplier for
the sum of all instance values. A modulus of 0 will turn off calculation
of all the instances. The modulus can be set in the
datafile declaration, with the
A command, or by assignment:
quantityname.modulus := 1.2
Each fixed quantity has a target value, to which the Evolver attempts to
constraint the quantity value. Each time an iteration is done
( g command or the various
Hessian commands), Newton's Method
is used to project the surface to the constrained values. The target
value can be displayed with the A or
v commands, or as "quantityname.target".
It can be changed with the A command or
by assignment. Example:
print qname.target
qname.target := 3.12
A quantity can have a constant value added to it, similar to the
body attribute volconst.
This quantity attribute is also called volconst. It is
useful for adding in known values of say integrals that are omitted
from the actual calculation. It can be set in the quantity's
datafile definition, or
by an assignment command.
Each fixed quantity has a Lagrange multiplier associated to it.
The Lagrange multiplier of a constraint is the rate of energy change
with respect to the constraint target value. For a volume constraint,
the Lagrange multiplier is just the pressure. Lagrange multipliers are
calculated whenever an iteration step is done. They may be displayed
with the v command in the "pressure"
column, or as an expression "quantityname.pressure".
A fixed quantity can have a tolerance attribute, which
is used to judge convergence.
A surface is deemed converged when the sum of all ratios of quantity
discrepancies to tolerances is less than 1. This sum also includes
bodies of fixed volume. If the tolerance is not set or is negative,
the value of
the variable target_tolerance is used, which has a default value of 0.0001.
The sample datafile column.fe
contains some examples of named quantities and instances.
It is planned that eventually all
energies and global constraints will be converted to named quantity system.
However, existing syntax will remain valid wherever possible.
Starting Evolver with the -q option will do this conversion now.
Some methods, those that logically depend on the orientation
of the element, can be applied with a relative orientation.
When applied to individual elements in the datafile, a negative
orientation is indicated by a '-' after the instance name.
When applied at runtime with the set command,
the orientation will be negative if the element is generated
with negative orientation,
i.e. set body[1].facet method_instance qqq.
The methods currently implementing this feature are:
edge_vector_integral, string_gravity, facet_vector_integral,
facet_2form_integral, facet_volume, facet_torus_volume,
simplex_vector_integral, simplex_k_vector_integral,
edge_k_vector_integral, gravity_method, and full_gravity_method.
The currently implemented methods are listed here, grouped
somewhat by nature. Within each group, they are more or less
in order of importance.
0-dimensional
1-dimensional
2-dimensional
2-D Curvatures
General dimensions
Knot energies
Weird and miscellaneous
The descriptions below of the individual methods give a
mathematical definition of the method, what type of element
it applies to, definition parameters,
which types of models it applies to, any
restrictions on the dimension of ambient space, and whether
the method has a Hessian implemented. Unless specifically noted,
a method has the gradient implemented, and hence may be used for
an energy or a constraint. The definition parameters are usually
scalar or vector integrands (see the
datafile declaration
for full syntax). Some methods also depend on global
variables as noted. The sample datafile declarations given
are for simple cases;
full syntax is given elsewhere.
Remember in the samples that for quantities not declared global,
the quantity has to be individually applied to the desired elements.
0-dimensional
Named method.
Description: Function value at a vertex. This actually produces a sum
over vertices, but as a mathematician, I think of a sum over vertices
as a point-weighted integral.
Element: vertex.
Parameters: scalar_integrand.
Models: linear, quadratic, Lagrange, simplex.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity point_value energy method vertex_scalar_integral
scalar_integrand: x^2 + y^2 - 2x + 3
1-dimensional
Named method.
Description: Length of edge.
Quadratic model uses Gaussian quadrature of order integral_order_1D.
Element: edge.
Parameters: none.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity len energy method edge_length global
Named method.
Description: Length of edge, multiplied by the edge density.
Quadratic model uses Gaussian quadrature of order integral_order_1D.
Element: edge.
Parameters: none.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity len energy method density_edge_length global
Named method.
Description: Integral of a scalar function over arclength.
Uses Gaussian quadrature of order integral_order_1D.
Element:
Parameters:
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Type: edge. Parameters: scalar_integrand.
Example datafile declaration:
quantity edge_sint energy method edge_scalar_integral
scalar_integrand: x^2 - 3*y + 4
Named method.
Description: Integral of a vectorfield over an oriented edge.
Uses Gaussian quadrature of order integral_order_1D.
Element: edge.
Parameters: vector_integrand.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Orientable: yes.
Example datafile declaration:
quantity edge_vint energy method edge_vector_integral
vector_integrand:
q1: 0
q2: 0
q3: z^2/2
Named method.
Description: Integral of a scalar function of
position and tangent over an edge. The components of the tangent vector
are represented by continuing the coordinate indices. That is, in 3D the
position coordinates are x1,x2,x3 and the tangent components are x4,x5,x6.
For proper behavior, the integrand should be homogeneous of degree 1
in the tangent components.
Uses Gaussian quadrature of order integral_order_1D.
Element: edge.
Parameters: scalar_integrand.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
the edge length in 3D could be calculated with this quantity:
quantity arclength energy method edge_general_integral
scalar_integrand: sqrt(x4^2 + x5^2 + x6^2)
Named method.
Description: For calculating the area of a body in the string model.
Implemented as the exact integral of -y dx over the
edge. Valid for torus model, but not general symmetry groups.
You may have to set the quantity volconst attribute in the
torus model, since the area calculation is ambiguous up
to one torus area.
Element: edge.
Parameters: none.
Models: linear, quadratic, Lagrange.
Ambient dimension: 2.
Hessian: yes.
Example datafile declaration:
quantity cell1_area fixed = 1.3 method edge_area
Named method.
Description: For 2D torus string model body area calculations.
Contains adjustments for torus wraps.
You may have to set the quantity volconst attribute in the
torus model, since the area calculation is ambiguous up
to one torus area.
Element: edge.
Parameters: none.
Models: torus; string; linear,quadratic,Lagrange.
Ambient dimension: 2.
Hessian: no.
Example datafile declaration:
quantity cell_area fixed = 1.3 method edge_torus_area
Named method.
Description: To calculate the gravitational potential energy
of a body in the string model. Uses differences in body
densities. Does not use gravitational constant G as modulus
(unless invoked as internal quantity by convert_to_quantities).
Element: edge.
Parameters: none.
Models: string linear, quadratic, lagrange.
Ambient dimension: 2.
Hessian: yes.
Orientable: yes.
Example datafile declaration:
quantity cell_grav energy modulus 980*8.5 method string_gravity
Named method.
Description:
One would often like to require edges to have fixed length.
The total length of some set of edges may be constrained by
defining a fixed quantity. This is used to fix the total
length of an evolving knot, for example. But to have one
constraint for each edge would be impractical, since projecting
to n constraints requires inverting an n x n matrix.
Instead there is a Hooke's Law energy available to encourage
edges to have equal length. Its form per edge is
E = | L - L_0| ^p
where L is the edge length,
L_0 is the equilibrium length, embodied as an adjustable
parameter `hooke_length', and the power p is an adjustable
parameter `hooke_power'. The default power is p = 2, and
the default equilibrium length is the average edge length
in the initial datafile. You will want to adjust this,
especially if you have a total length constaint.
A high modulus will decrease the hooke component of the total
energy, since the restoring force is linear in displacement and the
energy is quadratic (when p=2). As an extra added bonus,
a `hooke_power' of 0 will give
E = -\log|L-L_0|.
See hooke2_energy for individual edge
equilibrium lengths.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
parameter hooke_length 0.3 // will apply to all edges
parameter hooke_power 2 // the default
quantity slinky energy method hooke_energy global
Named method.
Description: Same as hooke_energy,
but each edge has an equilibrium length
extra attribute `hooke_size'
(which the user need not declare).
If the user does not set hooke_size by the time the method is
first called, the value will default to the current length.
Hooke_size is not automatically adjusted by refining.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
parameter hooke_power 2 // the default
define edge attribute hooke_size real
quantity slinky energy method hooke2_energy global
...
read
r;r;set edge hooke_size length
Named method.
Description: Same as hooke2_energy,
but uses an elastic model instead of a spring. The energy is
energy = 0.5*(length-hooke_size)^2/hooke_size.
The exponent can be altered from 2 by setting the parameter hooke3_power.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
parameter hooke3_power 2 // the default
quantity slinky energy method hooke3_energy global
...
read
r;r;set edge hooke_size length
Named method.
Description:
Energy of edges as springs with
equilibrium length being average of lengths of neighbor edges.
Actually, the energy is calculated per vertex,
E = ({L_1 - L_2 \over L_1 + L_2})^2
where L_1 and L_2 are the lengths of the edges adjacent
to the vertex. Meant for loops of string. (by John Sullivan)
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity slinky energy method local_hooke_energy global
Named method.
Description:
Energy of an edge is edge length times square of angle between
normals of adjacent facets. Actually, e = (1 - cos(angle))*length.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity bender energy method dihedral_hooke global
Named method.
Description: Integral of squared curvature in
string model. Assumes two edges per vertex, so don't use with triple
points. Value zero at endpoint of curve. Calue is calculated as if
the exterior angle at the vertex is evenly spread over the adjacent
half-edges. More precisely, if s1 and s2 are the adjacent edge lengths
and t is the exterior angle, value = 4*(1 - cos(t))/(s1+s2).
Other powers of the curvature can be specified by using the
parameter parameter_1 in the instance definition.
Element: vertex.
Parameters: parameter_1.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity sq energy method sqcurve_string global
parameter_1 3
Named method.
Description: In the string model with a
Riemannian metric, this is the
length of an edge.
Element: edge.
Parameters: none.
Models: linear,quadratic,simplex.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
string
space_dimension 2
metric
1+x^2 y
y 1+y^2
quantity mel energy method metric_edge_length global
Named method.
Description: Edge length in Klein hyperbolic plane model. Does not
depend on klein_metric
being declared. Vertices should be inside unit sphere.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 2.
Hessian: no.
Example datafile declaration:
quantity kleinlen energy method klein_length global
Named method.
Description: Edge length, modelling the edge as a circular
arc through three points, hence useful only in the
quadratic model.
If not in the quadratic model, it evaluates as the
edge_length method.
The presence of this quantity has the side effect of automatically
toggling circular_arc_draw,
causing edges to display as circular arcs in the quadratic model.
Element: edge.
Parameters: none.
Models: quadratic.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity arclen energy method circular_arc_length global
Named method.
Description: Area between an edge and the y axis, with the edge modelled
as a circular arc through three points. Useful in the
quadratic model;
in other models it is the same as
edge_area.
Element: edge.
Parameters: none.
Models: quadratic.
Ambient dimension: 2.
Orientable: yes.
Hessian: yes.
Example datafile declaration:
quantity arcarea energy method circular_arc_area global
Named method.
Description: Edge length, modelling the edge as a spherical
great circle arc between its two endpoints, which are assumed to
lie on an arbitrary radius sphere centered at the origin.
This method is meant for modelling string networks on spheres, and
is suitable for use with the
length_method_name feature for substituting the default edge length
calculation method. Note that this method is an exact spherical calculation
in the linear model,
so there is no need to refine edges or use higher order models for accuracy.
Edges are graphed as spherical arcs (actually, lots of segments).
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: yes.
Example datafile declaration:
parameter rad = 2
constraint 1
formula: x^2 + y^2 + z^2 = rad^2
length_method_name "spherical_arc_length"
spherical_arc_area_n, spherical_arc_area_s
Named method.
Description: Area on a sphere between an edge (considered as a great circle
arc) and the north (or south) pole. This is an exact calculation in
the linear model. Meant for calculating the areas of facets in the
string model with the string network confined to a sphere of arbitrary
radius centered at the origin. There are two versions of this method,
since calculation of facet areas by means of edges necessarily has
a singularity somewhere on the sphere. Spherical_arc_area_n has its
singularity at the south pole, and spherical_arc_area_s has its
singularity at the north pole. Thus spherical_arc_area_s will work
accurately for facets not including the north pole in there interiors;
a facet including the north pole will have its area calculated as the
negative complement of its true area, so a body defined using it could
get the correct area by using a volconst of a whole sphere area. If
the singular pole falls on an edge or vertex, then results are
unpredictable. With these caveats, these methods are suitable for
use with the
area_method_name feature for substituting the default edge area
method. If you do a facet as an explicit quantity, you are responsible
for applying or unapplying the quantity after topology changes!!
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Orientable: yes.
Hessian: yes.
Example datafile declaration:
parameter rad = 2
constraint 1
formula: x^2 + y^2 + z^2 = rad^2
area_method_name "spherical_arc_area_s"
2-dimensional
Named method.
Description: Area of facet. Does not multiply by facet density;
density_facet_area does that.
Quadratic model uses Gaussian cubature of order integral_order_2D.
Beware that this is an approximation to the area, and if the facets
in the quadratic or Lagrange model
get too distorted, it can be a bad approximation. Furthermore,
facets can distort themselves in seeking the lowest numerical area.
By default, changing the model to
quadratic or Lagrange will set an appropriate integral_order_2D.
Element: facet.
Parameters: none.
Models: linear, quadratic, Lagrange, simplex.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity farea energy method facet_area global
Named method.
Description: Area of facet, multiplied by its density.
Otherwise same as facet_area.
Element:
Parameters:
Models: linear, quadratic, Lagrange, simplex.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity farea energy method density_facet_area global
Named method.
Description: Integral of z dx dy over an oriented facet. Valid
in the torus domain. Not valid for other symmetry groups.
You may have to set the quantity volconst attribute in the
torus model, since the volume calculation is ambiguous up
to one torus volume.
Element: facet.
Parameters: none.
Models: linear, quadratic, Lagrange.
Ambient dimension: 3.
Hessian: yes.
Orientable: yes.
Example datafile declaration:
quantity vol fixed = 1.3 method facet_volume
Named method.
Description: Integral of a scalar function over facet area.
Uses Gaussian cubature of order integral_order_2D.
Element: facet.
Parameters: scalar_integrand.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity fint energy method facet_scalar_integral global
scalar_integrand: x^2+y^2
Named method.
Description:
Integral of a vectorfield inner product with the surface normal over a facet.
The normal is the right-hand rule normal of the facet as defined in the
datafile.
Uses Gaussian cubature of order integral_order_2D.
Element: facet.
Parameters: vector_integrand.
Models: linear, quadratic, Lagrange, simplex.
Ambient dimension: any.
Hessian: yes.
Orientable: yes.
Example datafile declaration, for volume equivalent:
quantity fvint energy method facet_vector_integrand
vector_integrand:
q1: 0
q2: 0
q3: z
Named method.
Description:
Integral of a 2-form over
a facet. Meant for ambient dimensions higher than 3.
Uses Gaussian cubature of order integral_order_2D.
Element: facet. Parameters: form_integrand (components in lexicographic order).
Models: linear, Lagrange, simplex.
Ambient dimension: any.
Hessian: yes.
Orientable: yes.
Example datafile declaration in 4D:
quantity formex energy method facet_2form_integral
form_integrand:
q1: x2 // 12 component
q2: 0 // 13 component
q3: x4 // 14 component
q4: 0 // 23 component
q5: 0 // 24 component
q6: x3*x2 // 34 component
Named method.
Description: Integral of a scalar function
of position and normal vector over a facet.
Uses Gaussian cubature of order integral_order_2D. The components of the normal vector
are represented by continuing the coordinate indices. That is, in 3D the
position coordinates are x1,x2,x3 and the normal components are x4,x5,x6.
For proper behavior, the integrand should be homogeneous of degree 1
in the normal components.
Element: facet.
Parameters: scalar_integrand.
Models: linear, quadratic, Lagrange.
Ambient dimension: any.
Hessian: yes.
Example: The facet area could be calculated with this quantity:
quantity surfacearea energy method facet_general_integral
scalar_integrand: sqrt(x4^2 + x5^2 + x6^2)
Named method.
Description: For 3D soapfilm model, calculates body volume integral
for a facet, with corrections for edge wraps.
You may have to set the quantity volconst attribute in the
torus model, since the volume calculation is ambiguous up
to one torus volume.
Element: facet.
Parameters: none.
Models: linear,quadratic,lagrange.
Ambient dimension: 3.
Hessian: yes.
Orientable: yes.
Example datafile declaration:
quantity body_vol energy method facet_torus_volume
Named method.
Description: Gravitational energy, integral of
p z^2/2 dxdy over a facet, where p is difference in adjacent body densities.
Note: this method uses the gravitational constant as the modulus
if invoked as full_gravity_method.
Just gravity_method does not automatically use the
gravitational constant.
Element: facet.
Parameters: none.
Models: linear, quadratic, Lagrange.
Ambient dimension: 3.
Hessian: yes.
Orientable: yes.
Example datafile declaration:
quantity grav energy modulus 980*8.5 method gravity_method
Named method.
Description: Area of facet. In quadratic model, it is
an upper bound of area, by the Schwarz Inequality. For the paranoid.
Same as facet_area in linear model.
Sets integral_order_2D to 6, since it doesn't work well with less.
Using the density_facet_area_u name automatically
incorporates the facet tension, but facet_area_u doesn't.
Element: facet.
Parameters: none.
Models: linear, quadratic.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity area_u energy method facet_area_u global
Named method.
Description: Implementation of gap energy, which is designed to keep edges from short-cutting curved
constraint surfaces. This method serves the same purpose as
declaring a constraint convex.
Automatically incorporates the
gap_constant set in the datafile
or by the k command.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity gappy energy method gap_energy global
Named method.
Description: For a
Riemannian metric, this is the
area of a facet.
Element: edge.
Parameters: none.
Models: linear,quadratic,simplex.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
metric
1+x^2 0 z
0 1+y^2 0
z 0 1+z^2
quantity mfa energy method metric_facet_area global
Named method.
Description: Facet area in Klein hyperbolic 3D space model.
Does not
depend on klein_metric
being declared in the datafile.
Vertices should be inside the unit sphere.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity kleinarea energy method klein_area global
Named method.
Description: Same as the facet_tension
method, but the Hessian is modified to be guaranteed positive definite,
after the scheme of Polthier and Pinkall
[PP]. The energy is taken to be the
Dirichlet integral of the perturbation from the current surface, which
is exactly quadratic and positive definite. Hence the hessian command
always works, but final convergence may be slow (no faster than regular
iteration) since it is only an approximate Hessian. Also see
the dirichlet command.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity dirarea energy method dirichlet_area global
Named method.
Description: Same as the facet_tension
method, but the Hessian is modified to be guaranteed positive definite,
after the scheme of Renka and Neuberger.
[RN]. Hence the hessian command
always works, but final convergence may be slow (no faster than regular
iteration) since it is only an approximate Hessian. Also see
the sobolev command.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity sobarea energy method sobolev_area global
Named method.
Description: Same as the facet_area
method, but the Hessian can be adjusted various ways by setting
the variables fgagfa_coeff, gfa_2_coeff, gfagfa_coeff, and gfaafg_coeff.
This will make sense if you look at the Dirichlet section of the
Technical Reference chapter of the printed manual. The default
values of the coefficients are -1, 1, -1, and 0 respectively.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity parea energy method pos_area_hess global
Named method.
Description: Area of the facet projected to unit sphere. The vertices of
the facet are assumed to be on the unit sphere.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
constraint 1 formula: x^2 + y^2 + z^2 = 1
quantity spharea energy method spherical_area global
Named method.
Description: Square of the Laplacian of z viewed as a function of (x,y).
Meant for the calculation of two-dimensional Stokes flow of a fluid (i.e.
slow steady-state flow where inertia is not significant) by having the
Evolver surface be the graph of the velocity potential and minimizing
the viscous dissipation, which is the square of the Laplacian of z.
Boundary conditions are handled by declaring a vertex attribute "stokes_type"
of type integer, and assigning each boundary vertex one of these values:
- 0 - vertex is not on a wall; treat as if on a mirror symmetry plane.
- 1 - vertex is on a slip wall.
- 2 = vertex is on a nonslip wall; normal derivative of potential is zero.
Boundary values of z should be set to constants between 0 and 1 on
various sections of boundary that represent walls.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: yes.
Example datafile declaration:
quantity dissip energy method stokes2d global
Named method.
Description: The Laplacian of z viewed as a function of (x,y). This is
auxiliary to the stokes2d method. It is the
same Laplacian, unsquared, with the same boundary rules. Meant for
calculating pressures and such after stokes2d energy has been minimized.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: yes.
Example datafile declaration:
quantity laplac info_only method stokes2d_laplacian global
Surface curvature functions
Named method.
Description: Integral of signed scalar mean curvature of a 2D surface.
The computation is exact, in the sense that
for a polyhedral surface the mean curvature is concentrated
on edges and singular there, but the total mean curvature for
an edge is the edge length times its dihedral angle.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity mci energy method mean_curvature_integral
Named method.
Description: Integral of squared mean curvature of a surface.
There are several methods implemented for calculating the integral
of the squared mean curvature of a surface. The older methods
sq_mean_curvature,
eff_area_sq_mean_curvature, and
normal_sq_mean_curvature, are now
deprecated, since they don't have Hessians and the newer methods
star_sq_mean_curvature,
star_eff_area_sq_mean_curvature,
star_normal_sq_mean_curvature, and
my current favorite
star_perp_sq_mean_curvature, do
have Hessians and can now handle incomplete facet stars around vertices.
But read the following for general remarks on squared curvature also.
The integral of squared mean curvature in the soapfilm model
is calculated for this method as follows:
Each vertex v has a star of facets around it of area A.
The force F due to surface tension on the vertex is the gradient of area,
Since each facet has 3 vertices, the area associated with v is
A/3. Hence the average mean curvature at v is
h = (1/2)(F/(A/3)),
where the 1/2 factor comes from the "mean" part of "mean curvature".
This vertex's contribution to the total integral is then
E = h2A/3 = (3/4)F2/A.
Philosophical note: The squared mean curvature on a triangulated
surface is technically infinite, so some kind of approximation scheme
is needed. The alternative to locating curvature at vertices is to
locate it on the edges, where it really is, and average it over the
neighboring facets. But this has the problem that a least area
triangulated surface would have nonzero squared curvature, whereas
in the vertex formulation it would have zero squared curvature.
Practical note: The above definition of squared mean curvature
seems in practice to be subject to instablities. One is that
sharp corners grow sharper rather than smoothing out. Another is
that some facets want to get very large at the expense of their
neighbors. Hence a couple of alternate definitions have been added.
Curvature at boundary: If the edge of the surface is a free boundary
on a constraint, then the above calculation gives the proper curvature
under the assumption the surface is continued by reflection across
the constraint. This permits symmetric surfaces to be represented
by one fundamental region. If the edge of the surface is a fixed edge
or on a 1-dimensional boundary, then there is no way to calculate the
curvature on a boundary vertex from knowledge of neighboring facets.
For example, the rings of facets around the bases of a catenoid and
a spherical cap may be identical. Therefore curvature is calculated
only at interior vertices, and when the surface integral is done,
area along the boundary is assigned to the nearest interior vertex.
However, including IGNORE_FIXED or IGNORE_CONSTRAINTS in the method declaration
will force the calculation of energy even at fixed points or
ignoring constraints respectively.
If the parameter or vertex attribute h_zero is defined,
then the value per vertex
is the same as for the following method,
eff_area_sq_mean_curvature.
Element: vertex.
Parameters: IGNORE_CONSTRAINTS, IGNORE_FIXED.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity sqc energy method sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature of a surface, with
a slightly different definition from
sq_mean_curvature or
normal_sq_mean_curvature.
The area around a vertex is taken to
be the magnitude of the gradient of the volume. This is less than the
true area, so makes a larger curvature. This also eliminates the
spike instability, since a spike has more area gradient but the
same volume gradient. Letting N be the volume gradient at vertex v,
h = (1/2)(F/N)),
and
E = h2 A/3 = (3/4)(F·F/N·N)A.
The facets of the surface must be consistently oriented for this
to work, since the evolver needs an `inside' and `outside' of the
surface to calculate the volume gradient. There are still possible
instabilities where some facets grow at the expense of others.
If the parameter or vertex attribute
h_zero is defined, then the value per vertex
is
E = (h-h0)2 A/3 = (3/4)(F·N/N·N-2h0)2A.
This does not reduce to the non-h_zero formula when h_zero
has the value zero, but is actually a pretty good formula in its own
right (see star_perp_sq_mean_curvature
.
If the vertex is on one or several constraints, the F and N are
projected to the constraints, essentially making the constraints act as
mirror symmetry planes.
WARNING: For some extreme shapes,
Evolver may have problems detecting consistent local surface orientation.
The assume_oriented toggle lets Evolver assume that the facets
have been defined with consistent local orientation.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity effsq energy method eff_area_sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature of a surface, with
a slightly different definition from
sq_mean_curvature or
eff_area_sq_mean_curvature.
To alleviate the instability of eff_area_sq_mean_curvature,
normal_sq_mean_curvature considers the area around the
vertex to be the component of the volume gradient parallel to the
mean curvature vector, rather than the magnitude of the volume gradient.
Thus
h = (1/2)(F·F/N·F)
E = h2A/3 = (3/4)(F·F/N·F)2 A.
This is still not perfect, but is a lot better.
WARNING: For some extreme shapes,
Evolver may have problems detecting consistent local surface orientation.
The assume_oriented toggle lets Evolver assume that the facets
have been defined with consistent local orientation.
If the parameter or vertex attribute h_zero is defined,
then the value per vertex is
E = (h-h0)2 A/3 = (3/4)(F·F/N·F - 2h0)2A
If the vertex is on one or several constraints, the F and N are
projected to the constraints, essentially making the constraints act as
mirror symmetry planes.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity nsq energy method normal_sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature over a surface.
This is a different implementation of
sq_mean_curvature, and it has a Hessian.
This method no longer requires a complete circle of vertices around a vertex;
boundary edges are treated as if they are on mirror symmetry planes, which
is usually true.
The positive orientation of the surface is
determined by the positive orientation of the first facet of the vertex's
internal facet list.
This method does not do prescribed mean curvature with
the h_zero parameter.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity starsq energy method star_sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature over a surface.
This is a different implementation of
eff_area_sq_mean_curvature, and it has a Hessian.
This method no longer requires a complete circle of vertices around a vertex;
boundary edges are treated as if they are on mirror symmetry planes, which
is usually true.
The positive orientation of the surface is
determined by the positive orientation of the first facet of the vertex's
internal facet list.
This method does not use the h_zero parameter.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity seffsq energy method star_eff_area_sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature over a surface.
This is a different implementation of
normal_sq_mean_curvature which is more suitable for parallel
calculation and has a Hessian.
This method no longer requires a complete circle of vertices around a vertex;
boundary edges are treated as if they are on mirror symmetry planes, which
is usually true.
The positive orientation of the surface is
determined by the positive orientation of the first facet of the vertex's
internal facet list.
This method can use the h_zero parameter or vertex attribute
for prescribed mean curvature.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity stnsq energy method star_normal_sq_mean_curvature global
Named method.
Description: Integral of squared mean curvature over a surface.
This is my current favorite implementation of squared mean curvature.
It is an implementation specifically
designed to agree with the mean curvature computed as the gradient of
area when normal motion is on (either the
normal_motion toggle for 'g' iteration, or Hessian with
hessian_normal). Thus if you
get zero squared mean curvature with this method, then switch to area energy,
the hessian will report exact convergence. Likewise if you do prescribed
curvature and then convert to area minimization with a volume constraint.
This method has a Hessian.
This method does not require a complete circle of vertices around a vertex;
boundary edges are treated as if they are on mirror symmetry planes, which
is usually true.
This method can use the h_zero parameter or vertex attribute
for prescribed mean
curvature. The actual formula for the energy at a vertex is
h = (1/2)(F·N/N·N)
E = (h-h0)2 A/3 = (3/4)(F·N/N·N - 2h0)2A
where F is the area gradient at the vertex, N is the volume gradient,
and A is the area of the adjacent facets.
If the vertex is on one or several constraints, the F and N are
projected to the constraints, essentially making the constraints act as
mirror symmetry planes.
The positive orientation of the surface is
determined by the positive orientation of the first facet of the vertex's
internal facet list.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
quantity stnsq energy method star_perp_sq_mean_curvature global
Named method.
Description:
This computes the total Gaussian curvature of a surface with
boundary. The Gaussian curvature of a polyhedral surface may
be defined at an interior vertex as the angle deficit of the
adjacent angles. But as is well-known, total Gaussian curvature
can be computed simply in terms of the boundary vertices, which
is what is done here.
The total Gaussian curvature is implemented as the total geodesic
curvature around the boundary of the surface.
The contribution of a boundary vertex is
E = (\sum_i \theta_i) - pi.
For reasons due to the Evolver's internal architecture, the sum is actually
broken up as a sum over facets, adding the vertex angle for each facet
vertex on the boundary and subtracting pi for each boundary edge.
The total over all boundary vertices is
exactly equal to the total angle deficit
of all interior vertices plus 2*pi*chi, where chi is the
Euler characteristic of the surface.
Boundary vertices are deemed to be those that are fixed or on a
parametric boundary. Alternately, one may define a vertex extra attribute
gauss_bdry_v and an edge extra attribute gauss_bdry_e
and set them nonzero on the relevant vertices and edges; this overrides
the fixed/boundary criterion.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity gint energy method gauss_curvature_integral global
Named method.
Computes the angle deficit around vertices to which this method is applied.
The angle deficit is 2*pi minus the sum of all the adjacent angles of facets.
No compensation is made for vertices on the boundary of a surface; you just
get big deficits there. Deficits are counted as positive, following the
convention for gaussian curvature.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity total_deficit energy method star_gauss_curvature global
Named method.
Description:
Computes the integral of the squared Gaussian curvature. At each
vertex, the Gaussian curvature is calculated as the angle defect
divided by one third of the total area of the adjacent facets. This is
then squared and weighted with one third of the area of the adjacent
facets. This method works only on closed surfaces with no singularities
due to the way it calculates the angle defect.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity sqg energy method sq_gauss_curvature global
Simplex model methods
Named method.
Description: Integral of a vectorfield over a (n-1)-dimensional
simplicial facet in n-space. Vectorfield is dotted with normal of
facet; actually the side vectors of the simplex and the integrand
vector are formed into a determinant.
Element: facet.
Parameters: vector_integrand.
Models: simplex.
Ambient dimension: any.
Hessian: no.
Orientable: yes.
Example datafile declaration, for 4-volume under a 3D surface in 4D:
quantity xvint energy method simplex_vector_integral
vector_integrand:
q1: 0
q2: 0
q3: 0
q4: x4
Named method.
Description: Integral of a simple (n-k)-vector over an oriented k-dimensional
simplicial facet in n-space.
The vector integrand lists the components of each of the k vectors
sequentially. Evaluation is done by forming a determinant whose
first k rows are k vectors spanning the facet, and last (n-k) rows are vectors
of the integrand.
Element: facet.
Parameters: k_vector_order, vector_integrand.
Models: simplex.
Ambient dimension: any.
Hessian: yes.
Orientable: yes.
Example datafile declaration, for 3D surface in 5D:
quantity kvec energy method simplex_k_vector_integral
k_vector_order 3
vector_integrand:
q1: 0 // first vector
q2: 0
q3: 0
q4: 0
q5: x4
q6: 0 // second vector
q7: 0
q8: 0
q9: x3
q10: 0
Named method.
Description: Integral of a simple (n-k)-vector over an oriented
k-dimensional simplicial edge in n-space.
The vector integrand lists the components of each of the k vectors
sequentially. Evaluation is done by forming a determinant whose
first k rows are k vectors spanning the edge, and last (n-k) rows are vectors
of the integrand.
Element: edge.
Parameters: k_vector_order, vector_integrand.
Models: linear, quadratic, simplex.
Ambient dimension: any.
Hessian: yes.
Orientable: yes.
Example datafile declaration, for 3D edges of a 4D surface in 5D:
quantity kvec energy method edge_k_vector_integral
k_vector_order 3
vector_integrand:
q1: 0 // first vector
q2: 0
q3: 0
q4: 0
q5: x4
q6: 0 // second vector
q7: 0
q8: 0
q9: x3
q10: 0
Named method.
Description:
An ``electrostatic'' energy in which vertices
are endowed with equal charges. Inverse power law of potential is adjustable
via the global parameter `knot_power', default value 2 (which is
not electrostatic, but the knot theorists like it). If the
extra attribute
`node_charge' is defined for vertices, then that value is used for
the vertex charge. Use of this energy is not restricted to knots;
it has been used to embed complicated network graphs in space.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: yes.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method knot_energy global
Named method.
Description: A knot energy where vertex charge
is proportional to neighboring edge length. This simulates an
electrostatic charge uniformly distributed along a wire.
Inverse power law of potential is adjustable via the
global parameter `knot_power' (default 2).
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method uniform_knot_energy global
Named method.
Description: Supposed to approximate
the part of uniform_knot_energy
that is singular in the continuous limit.
Element: vertex.
Parameters:
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knottenorm energy method uniform_knot_energy global
method uniform_knot_energy_normalizer global
Named method.
Description: Calculates internal knot energy to normalize
singular divergence of integral of
uniform_knot_energy.
Actually a synonym for
uniform_knot_energy_normalizer.
No gradient.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knottenorm energy method uniform_knot_energy global
method uniform_knot_energy_normalizer1 global
Named method.
Description: Calculates internal knot energy to normalize
singular divergence of integral of
uniform_knot_energy
a different way from uniform_knot_energy_normalizer.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knottenorm energy method uniform_knot_energy global
method uniform_knot_energy_normalizer2 global
Named method.
Description:
Between pairs of edges, energy is inverse square power of distance
between midpoints of edges. Can also be called just
edge_knot_energy. See also
edge_knot_energy_normalizer.
(by John Sullivan)
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity knotten energy method edge_edge_knot_energy global
Named method.
Description: Calculates internal knot energy to normalize
singular divergence of integral of
edge_edge_knot_energy.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity knotten energy method edge_edge_knot_energy global
method edge_knot_energy_normalizer global
Named method.
Description: Another normalization of edge_knot_energy, which I
don't feel like deciphering right now.
Element: edge.
Parameters: none.
Models: string linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity kenergy energy method edge_knotenergy global
method simon_knot_energy_normalizer global
Named method.
Description: Charge on vertex is proportional to area of neighboring facets.
Meant for knotted surfaces in 4D. Power law of potential is adjustable
via the global parameter `knot_power'. See also
facet_knot_energy_fix.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method facet_knot_energy global
Named method.
Description: Provides adjacent vertex correction to
facet_knot_energy.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method facet_knot_energy global
method facet_knot_energy_fix global
Named method.
Description: Energy between pair of edges
given by formula suggested by Greg Buck.
Power law of potential is adjustable
via the global parameter `knot_power'.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method buck_knot_energy global
Named method.
Description:
This energy is due to Gregory Buck. It tries to
eliminate the need for a normalization term by projecting
the energy to the normal to the curve. Its form is
E_{e_1e_2} = {L_1L_2 \cos^p\theta\over |x_1 - x_2|^p}
where x_1,x_2 are the midpoints of the edges and \theta
is the angle between the normal plane of edge e_1 and
the vector x_1 - x_2. The default power is 2.
Power law of potential is adjustable
via the global parameter `knot_power'.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method proj_knot_energy global
Named method.
Description: This energy is due to Peter Doyle, who says it is equivalent
in the continuous case to the insulating wire with power 2.
Its form is
E_{e_1e_2} = {L_1L_2 (1 - \cos\alpha)^2 \over |x_1 - x_2|^2},
where x_1,x_2 are the midpoints of the edges and \alpha
is the angle between edge 1 and the circle through x_1
tangent to edge 2 at x_2. Only power 2 is implemented.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity knotten energy method circle_knot_energy global
Named method.
Description:
This is the 2D surface version of the circle energy.
Its most general form is
E_{f_1f_2} = { A_1A_2(1 - \cos\alpha)^p \over |x_1 - x_2|^q},
where A_1,A_2 are the facet areas, x_1,x_2 are the barycenters
of the facets, and \alpha is the angle between f_1 and the sphere
through x_1 tangent to f2 at x_2. The energy is conformally
invariant for p = 1 and q = 4. For p=0 and q=1, one gets
electrostatic energy for a uniform charge density. Note that
facet self-energies are not included. For electrostatic energy,
this is approximately 2.8A^{3/2} per facet.
The powers p and q are Evolver variables surface_knot_power and
surface_cos_power respectively.
The defaults are p=1 and q=4.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter surface_knot_power 1 // the default
parameter surface_cos_power 4 // the default
quantity knotten energy method sphere_knot_energy global
Named method.
Description: Another weird way to
calculate a nonsingular energy between midpoints of pairs of edges.
(by John Sullivan)
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity knotten energy method sin_knot_energy global
Named method.
Description: For string model. The energy evaluates to zero, but the force
calculated is the mean curvature vector rotated to the binormal direction.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity curbi energy method curvature_binormal global
Named method.
Description: Third derivative of curve position as
function of arclength, squared.
Element: vertex.
Parameters: none.
Models: string, linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity ddd energy method ddd_gamma_sq global
Named method.
Description:
Between pairs of edges, energy is inverse square power of distance
between closest points of edges.
Energy = 1/d^2 * |e1||e2|
This should be roughly the same as edge_edge_knot_energy, but distances
are calculated from edge midpoints there.
This is not a smooth function,
so we don't try to compute a gradient. DO NOT use as an energy;
use just for info_only quantities.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity eminknot info_only method edge_min_knot_energy global
Named method.
Description: Calculates the average crossing number of an edge with
respect to all other edges, averaged over all projections. Knot stuff.
No gradient, so use just in info_only quantities.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity true_cross info_only method true_average_crossings global
Named method.
Description: For calculating the writhe of a link or knot.
No gradient, so use just in info_only quantities.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity twrithe info_only method true_average_crossings global
Named method.
Description: Another average crossing number calculation.
No gradient, so use just in info_only quantities.
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity twister info_only method twist global
Named method.
Description: An average crossing number calculation. This one does
have a gradient. Suggested by Hermann Gluck.
Programmed by John Sullivan.
Between pairs of edges, energy is inverse cube power of distance
between midpoints of edges, times triple product of edge vectors
and distance vector.
E = 1/d^3 * (e1,e2,d)
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity writhy energy method writhe global
Named method.
Description: Calculates forces as function of
mean and Gaussian curvatures at vertices. Function may be
changed by user by altering teix.c. No energy, just forces.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity curfun energy method curvature_function global
Named method.
Description: To calculate the average
crossing number in all projections of a knot. (by John Sullivan)
Element: edge.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity across energy method average_crossings global
Weird and miscellaneous
Named method.
Description: Method version of wulff
energy. If Wulff filename is not given in top section of datafile,
then the user will be prompted for it.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
wulff "crystal.wlf"
quantity wolf energy method wulff_energy global
Named method.
Description: To calculate the linear elastic strain energy
energy for facets based on the Cauchy-Green strain matrix.
Let S be Gram matrix of unstrained facet (dots of sides).
Let Q be the inverse of S.
Let F be Gram matrix of strained facet.
Let C = (FQ-I)/2, the Cauchy-Green strain tensor.
Let v be Poisson ratio.
Then energy density is
(1/2/(1+v))(Tr(C^2) + v*(Tr C)^2/(1-(dim-1)*v))
Each facet has extra attribute poisson_ratio and
extra attribute array form_factors[3] = {s11,s12,s22}, which
are the entries in S. That is, s11 = dot(v2-v1,v2-v1),
s12 = dot(v2-v1,v3-v1), and s22 = dot(v3-v1,v3-v1).
If form_factor is not defined by the user, it will be created by Evolver,
and the initial facet shape will be assumed to be unstrained.
For a version of this method that gives compression zero energy, see
relaxed_elastic_A.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: yes.
Example datafile declaration:
quantity lastic energy method linear_elastic global
Named method.
Description: A variation of the
linear_elastic method.
To calculate the linear elastic strain energy
energy for facets based on the Cauchy-Green strain matrix.
Let S be Gram matrix of unstrained facet (dots of sides).
Let Q be the inverse of S.
Let F be Gram matrix of strained facet.
Let C = (FQ-I)/2, the Cauchy-Green strain tensor.
Let v be Poisson ratio.
Then energy density is
(1/2/(1+v))(Tr(C^2) + v*(Tr C)^2/(1-(dim-1)*v))
Each facet has extra attribute poisson_ratio and
each vertex has two extra coordinates, the coordinates of
the unstrained surface in a plane. Hence the surface must
be set up as five dimensional.
For a version of this method that gives compression zero energy, see
relaxed_elastic.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 5.
Hessian: yes.
Example datafile declaration:
space_dimension 5
quantity lastic energy method linear_elastic_B global
Named method.
Description: Energy of a facet is the square of the facet area.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity asquare energy method area_square global
Named method.
Description: Calculates the linear elastic strain energy
energy for facets based on the Cauchy-Green strain matrix,
with compression counting for zero energy, simulating, say, plastic film.
The effect is to permit wrinkling.
Let S be the Gram matrix of unstrained facet (dots of sides).
Let Q be the inverse of S.
Let F be Gram matrix of strained facet.
Let C = (FQ-I)/2, the Cauchy-Green strain tensor.
Let v be Poisson ratio.
Then the energy is
(1/2/(1+v))(Tr(C^2) + v*(Tr C)^2/(1-(dim-1)*v))
Each facet has extra attribute poisson_ratio and
extra attribute array form_factors[3] = {s11,s12,s22}, which
are the entries in S. That is, s11 = dot(v2-v1,v2-v1),
s12 = dot(v2-v1,v3-v1), and s22 = dot(v3-v1,v3-v1).
If form_factor is not defined by the user, it will be created by Evolver,
and the initial facet shape will be assumed to be unstrained.
The compression is detected by doing an eigenvalue analysis of the strain
tensor, and discarding any negative eigenvalues.
Facets which are stressed in one or two dimensions can
be separately counted by the relaxed_elastic1_A (one stress direction, and one
wrinkle direction) and
relaxed_elastic2_A (two stressed directions) methods, which are meant to be
used in info_only mode.
For a sample datafile, see mylarcube.fe.
For a version of this method that gives compression positive energy, see
linear_elastic.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: yes.
Example datafile declaration:
quantity lastic energy method relaxed_elastic_A global
Named method.
Description: A variation of the
linear_elastic method.
Calculates the linear elastic strain energy
energy for facets based on the Cauchy-Green strain matrix,
with compression counting for zero energy, simulating, say, plastic film.
The effect is to permit wrinkling.
Let S be Gram matrix of unstrained facet (dots of sides).
Let Q be the inverse of S.
Let F be Gram matrix of strained facet.
Let C = (FQ-I)/2, the Cauchy-Green strain tensor.
Let v be Poisson ratio.
Then energy density is
(1/2/(1+v))(Tr(C^2) + v*(Tr C)^2/(1-(dim-1)*v))
Each facet has extra attribute poisson_ratio and
each vertex has two extra coordinates, the coordinates of
the unstrained surface in a plane. Hence the surface must
be set up as five dimensional.
The compression is detected by doing an eigenvalue analysis of the strain
tensor, and discarding any negative eigenvalues. The eigenvalues may
be separately accessed by the relaxed_elastic1_A (lower eigenvalue) and
relaxed_elastic2_A (higher eigenvalue) methods, which are meant to be
used in info_only mode.
For a sample datafile, see mylarcube.fe.
For a version of this method that gives compression zero energy, see
linear_elastic_B.
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 5.
Hessian: yes.
Example datafile declaration:
space_dimension 5
quantity lastic energy method relaxed_elastic global
Named method.
Description:
Craig Carter's energy.
pre>
Given bodies $B_1$ and $B_2$ in $R^3$, define the energy
E = \int_{B_1}\int_{B_2} {1 \over |z_1 - z_2|^{p} } d^3 z_2 d^3 z_1
This reduces to
E = {1\over (3-p)(2-p)}\sum_{F_2\in\partial B_2}\sum_{F_1\in\partial B_1}
N_1 \cdot N_2 \int_{F_2}\int_{F_1}{1\over |z_1 - z_2|^{p-2}}
d^2 z_1 d^2 z_2.
And if we crudely approximate with centroids $\bar z_1$ and $\bar z_2$,
E = {1\over (3-p)(2-p)}\sum_{F_2\in\partial B_2}\sum_{F_1\in\partial B_1}
{A_1 \cdot A_2 \over |\bar z_1 - \bar z_2|^{p-2}},
where $A_1$ and $A_2$ are unnormalized area vectors for the facets.
The power p is set by the variable carter_power (default 6).
Element: facet.
Parameters: none.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
parameter carter_power 6 // the default
quantity craig energy method carter_energy global
Named method.
Description: This energy is the gradient^2 of the
knot_energy method,
assuming the points are constrained to the unit sphere.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
parameter knot_power 2 // the default
quantity knotten energy method knot_energy global
Named method.
Description: For all point pairs (meant to be on a sphere),
E = (pi - asin(d/2))/d,
where d is chord distance. For point
packing problems on the sphere.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
constraint 1 formula: x^2+y^2+z^2 = 1
quantity jms energy method johndust global
Named method.
Description: Hmm. Looks like this one calculates integrals of
components of a stress tensor. The scalar_integrand value
is set as an integer standing for which component to do (a kludge).
See the function stress_integral in method3.c for details.
Does not have a gradient, so should be used for just info_only quantities.
Element: facet.
Parameters: scalar_integrand.
Models: linear.
Ambient dimension: 3.
Hessian: no.
Example datafile declaration:
quantity stressy info_only method stress_integral global
scalar_integrand: 3
Named method.
Description: Not actually an energy, but a
kludge to put inertia on vertices.
Uses extra velocity coordinates to represent vertex in phase space.
Invocation actually transfers computed forces from space coordinates
to velocity coordinates, so forces become acceleration instead of
velocity.
Element: vertex.
Parameters: none.
Models: linear.
Ambient dimension: any.
Hessian: no.
Example datafile declaration:
quantity jeremy energy method ackerman global
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