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<p>
So far we have discussed breps with intrinsic and embedded dimension
equal to 3.  The <em>embedded dimension</em> of a brep is the
dimension in which it is embedded and is equal to the number of
coordinates in each of its control points.  QMG 2.0 supports either
2 or 3 for this value.  A brep's intrinsic dimension is the highest-dimensional
face that it owns.  For instance, a brep with topological vertices and
edges but no surfaces or chambers would have intrinsic dimension equal
to 1.  The intrinsic dimension can never exceed the embedded dimension.
The mesh generator requires that the intrinsic and embedded dimensions
be equal, although other routines in QMG 2.0 allow the intrinsic
dimension to be lower.

<p>
If the intrinsic and
embedded dimension of the brep are both <em>d</em> 
(where either <em>d</em>=2 or <em>d</em>=3)
then the term <em>region</em> is used to denote the topological 
entities
of dimension <em>d</em>.  
Thus, for a three-dimensional brep, its chambers
are its regions.  For a two-dimensional brep, its surfaces (subsets
of the plane) are its regions.  Regions are completely specified
by their boundaries, so they do not have any
geometric
entities 
associated with them in a brep definition.

<p>
Thus, the data items that make up a brep are as follows.
<ul>
<li>
Two integers: its embedded and intrinsic dimension.
<li>
Its global property-value pairs.
<li>
A table of control points.
<li>
Its faces (topological entities).  Faces are listed in increasing
order of dimension (i.e., vertices first, etc).
Each topological entity has five data items associated with
it:
<ol>
<li>
The face name, which is a string.
<li>
Property-value pairs of the face (a list of ordered pairs of strings).
<li>
The boundary of the face, which is a list of faces of one lower
dimension.  This list may be empty if the face is closed
(such as
a sphere).   Each bounding face in this list is
accompanied by the specification of its orientation with respect
to the parent face: inward, outward, or no orientation specified.
This orientation information is currently not used by QMG.
<li>
The list of faces of two or more dimensions lower that are
internal boundaries.  See below.
<li>
The geometric entities making up the face. 
Regions cannot have geometric entities; on the other hand,
every other face must have at least one geometric entity.
A geometric entity
is specified by 
<ul>
<li>
its type (vertex, curve, triangle, or quadrilateral),
<li>
its degree (for a curve or triangle) or degree-pair (for 
a quad), and
<li>
its list of control points (integer indices into the control point
table) in the correct order described above.
</ul>
</ol>
</ul>

<h2><a name = "geotopdiff">On the difference between geometric and topological entities</a></h2>

The preceding section detailed geometric and topological entities, but the
reader may wonder why we bothered with this distinction.  Why not have
geometric entities only?  There are two advantages to this system:
<ul>
<li>
Property-value pairs are associated with topological entities.  This
means that many geometric entities can be grouped together with a single
finite element boundary condition (for example).
<li>
The mesh generator respects boundaries between topological entities
but not between geometric entities.  This has important consequences
in practice, and you should design your domains with this
fact in mind.
</ul>
For example, consider the following regular 30-gon.

<p>
<center>
<img src=thirtygon.jpg alt="picture of 30-gon">
</center>

Two possible ways to represent this in QMG would be:
<ul>
<li>
as a domain with a 
single topological edge as its boundary.  That edge
is made up of
30 linear Bezier curves, or
<li>
as a domain with 30 topological edges on the boundary.  Each topological
edge made up of one linear Bezier curve.
</ul>
In the first case, the topological edge is not 
<em>G</em><sup>1</sup>, but nonetheless QMG permits small
divergences from <em>G</em><sup>1</sup> as mentioned earlier.
There are also many other options for representing
the 30-gon; for example, there could be 
five edges each made of six linear curves, etc.

<p>
Here is the mesh generated by QMG (coarse mesh requested) for the first
option:

<p>
<center>
<img src=thirtygon-mesh1.jpg alt="picture of first mesh of 30-gon">
</center>

<p>
Here is the mesh generated by QMG (coarse mesh requested) for the
second option:

<p>
<center>
<img src=thirtygon-mesh2.jpg alt = "picture of second mesh of 30-gon">
</center>

Notice the two meshes are quite different.  Depending on your application, 
one or the other might be more appropriate.



<h2><a name = "internal-boundaries">Internal boundaries in breps</a></h2>
As mentioned early, a brep face can have bounding faces of one lower
dimension.
A brep face can also have <em>internal boundaries</em>,
which are subfaces of lower dimension that are inside the face
itself, i.e., they have the interior of the face on both sides
of them.

<p>
Internal boundaries usually serve one of two purposes in a scientific
computation: they act as demarcations between separate regions
of the domain (for instance, the domain may represent an object
that is a composite of several materials) or they act as
singular boundaries, for example, in modeling a region
with a slit or crack.
Holes in a domain are not internal boundaries&#8212;they
are classified as ordinary boundaries.

<p>
Internal boundaries are useful because the mesh generator will
respect them in its mesh (i.e. mesh elements will not cross through
them).  They can also be the site of boundary conditions for the
finite element package.

<p>
There are three kinds of internal boundaries supported by the QMG
package, multi-region, repeated boundary, and low-dimensional
boundary.

<p>
A <em>multi-region</em> brep means that there are multiple regions
in the brep.  (Recall that <em>region</em> means a topological entity
of dimension equal to the embedded dimension.)
The boundary between two regions is a type of internal boundary.  The
region on either side sees it as an ordinary boundary.

<p>
A <em>repeated boundary</em> means that a region has a boundary
face listed twice in its list of boundaries.  For example,
suppose a 3D chamber lists a topological surface twice in
its list of bounding faces.  This makes the surface act like
a crack or fissure inside the chamber.  Lower-dimensional
faces can also have repeated boundaries.  This is sometimes
necessary to attach a region's repeated boundary to an exterior boundary.  
For instance,
to represent a cube with a crack, such that the crack meets 
an exterior boundary, the crack itself would be a repeated
boundary of the region, and then its edge where it meets the
exterior boundary would be a repeated boundary of the exterior
boundary.

<p>
Finally, a face can have one or more <em>low-dimensional boundaries</em>,
which act like low dimensional cracks.  In particular, a chamber
can have edges or vertices as low-dimensional boundaries, and
a surface can have vertices as a low-dimensional boundaries.

<p>
Multi-region breps are generally used for the case of internal
boundaries that demarcate the brep into several zones.
The mesh generator lists the elements that it generates
according to the region in which they lie.
See the description of meshes below.
The finite element program can take advantage of these labels during
matrix assembly.  For example, it can use different functions to
compute conductivity and source terms for the two sides 
of the internal boundary corresponding to different materials.

<p>
In the case of a single-region brep with repeated boundaries, 
there is no concept of one side of
the internal boundary versus the other since the fissure may cut
only partway through the domain.  Thus, all elements have the
same label.

<p>
The various types of internal boundaries can be freely
mixed in the same brep. 

<p>
The mesh generator will produce a single layer of nodes
along either an repeated boundary or an
internal boundary in a multi-region domain.  In some applications
a double layer of nodes is desirable.  It is possible
to produce a double layer of nodes along a repeated boundary
with the
<a href="ref.html#gmdouble"><code>gmdouble</code></a>
function.

<h2><a name="meshes">Overview of simplicial complexes</a></h2>

A simplicial complex is a collection of simplices of one specific
dimension embedded in a space of the same or higher dimension.  (Thus,
simplicial complexes in QMG cannot contain
simplices of several different dimensions in the same complex).
Simplicial complexes are the output of the mesh generator and are an
input to the finite-element program and to the graphics programs.

<p> 
As with a brep, a simplicial complex has two integers associated
with it, the embedded dimension and the intrinsic dimension.  
Also like a brep, a simplicial complex can have property-value pairs.
For example, one possible property name is <strong>geo_global_id</strong>.
The mesh generator stores the brep's ID in this field during
mesh generation.  The field is omitted in
a mesh generated for a brep without a global ID.
The rest of the data is lists of
vertices and elements. 

<p>
The first list is the vertex real-space coordinates.  Each vertex has
a global ID number, which must be a nonnegative integer.  Global ID
numbers must be unique within the mesh but
do not have consecutive or in order.  With each global ID is
a tuple of two or three real-space coordinates of the node.  (The number
of real-space coordinates per node equals the embedded dimension.)


<p>
Next are lists of vertices and simplicial faces associated with brep
topological 
entities.  Let <em>d</em> be the embedded dimension of the mesh. 
Let <em>B</em> be the brep that gave rise to the mesh.
Then each brep topological entity <em>F</em> of dimension 0 to 
<em>d</em>&#8722;1 has a list of vertices lying on it.
For each vertex on <em>F</em>, the following information is
stored: its global ID number, the curve or patch index of
the geometric entity containing the vertex (omitted when <em>d</em>=0)
and the parametric coordinates on that entity (omitted when <em>d</em>=0).
Mesh vertices lying on lower-dimensional subfaces of 
<em>F</em> are also listed as lying on <em>F</em> since they
will have a different patch-index and parametric coordinates
with respect to <em>F</em>.  Thus, the same mesh vertex may appear in
many different lists associated with various brep faces.

<p>
Each brep topological entity of dimension 1 to <em>d</em> has
a list of simplex faces lying on it.  This list is made of tuples.
A brep entity of dimension <em>k</em> contains
tuples made up of <em>k</em>+1 vertices.  These vertices are listed
via their global ID numbers.  For faces of dimension 1 to
<em>d</em>&#8722;1, all vertices listed in any of these
tuples must also be listed as a vertex of the face as in
the previous paragraph.
The full-dimensional simplex faces (i.e.,
the case <em>k</em>=<em>d</em>) are the actual simplices of the
mesh.

<h3> Two geometric properties of meshes</h3>
<p>
<a name="aspect_ratio">An important property of a simplex 
is its aspect ratio</a>.  The aspect
ratio of a simplex can be defined in several ways, which are
all equivalent up to constant multiples.  In the QMG package,
aspect ratio of a simplex is defined as the ratio of the length of its
longest edge divided by its shortest altitude.

<p>
Small aspect ratios are desirable because large aspect ratios
cause a loss of accuracy in the finite element approximation, 
and can also lead to condition number problems for the
assembled stiffness matrix.

<p>
<a name="orientation">
All full-dimensional simplices should to have the correct
orientation.</a> Let <em>v</em><sub>0</sub>, ..., 
<em>v<sub>d</sub></em> 
be the vertices of a simplex. 
The <em>correct orientation</em> is defined as follows.  Form
the <em>d</em>&#215;<em>d</em> 
matrix whose <em>i</em>th row is <em>v<sub>i</sub></em>&#8722;<em>v</em><sub>0</sub> 
(a <em>d</em>-vector).  The determinant
of this matrix must be positive.  In two dimensions, this means
that the vertices are listed in counterclockwise order.  In three
dimensions this means that if you curl your right-hand around 
<em>v</em><sub>0</sub>, <em>v</em><sub>1</sub>,<em>v</em><sub>2</sub>, 
then your thumb is pointing toward <em>v</em><sub>3</sub>.  

<p>


<h2><a name="asciirep">Ascii format for objects</a></h2>

Both Matlab and Tcl/Tk versions of QMG support Ascii format.  In
particular, the <a href="ref.html#gm_read"><code>gm_read</code></a>
and <a href="ref.html#gm_write"><code>gm_write</code></a>
functions use this format.  Furthermore, in Tcl/Tk, conversion
to Ascii format is automatic for any operation that causes
string format to be released.  (See <a href="#tclrep">below</a>
for more information.)

<p>
Ascii format consists of free-form plain text with parentheses
as delimiters.  Free-form means that linebreaks are usually not significant,
except that they act like spaces.  Comments are marked with a '#'
sign.  A comment may start at any position in the line, and is
in effect until the end of the current line.

<p>
The Ascii format for a brep follows the following grammar.  In this grammar,
normal-weight text denotes field names and boldface denotes literals.  
The symbol
:= indicates substitution for a field name.  
The vertical bar symbol | means
either/or, and curly braces {} indicate optional data.  Parentheses when
indicated are literals (i.e., parentheses occur in the corresponding place
in the actual brep).
<ul>
<li>
asciiBrep := code intrinsicDim embeddedDim globalPropVal controlPoints
vertexList {edgeList {surfaceList {chamberList}}}
<li>
code := <strong> brep_v2.0 </strong>
<li>
intrinsicDim := <strong>0</strong> | <strong> 1 </strong> |
<strong> 2 </strong> | <strong> 3 </strong>
[Intrinsic dim must be less than or equal to embedded dim.]
<li>
embeddedDim := <strong>2</strong> | <strong> 3 </strong>
<li>
globalPropVal := ( prop<sub>1</sub> val<sub>1</sub> ... 
prop<sub><em>m</em></sub> val<sub><em>m</em></sub> )
[The property/value pairs are arbitrary strings.  If
property or value strings contain
space, then there must
be an outer pair of parentheses to delimit the string.
Currently, the only global property value with significance to
QMG 2.0 is <strong>geo_global_id</strong>.]
<li>
controlPoints := ( entry<sub>1</sub> ... entry<sub><em>m</em></sub> )
[The number of entries <em>m</em> must be a multiple of the
embedded dimension <em>e</em>.
Each entry in this list is a floating-point number
possibly in scientific notation.  There are a total of
<em>m</em>/<em>e</em> control points, with each control point's coordinates being denoted
by <em>e</em> consecutive entries in this list.]
<li>
vertexList := faceList; edgeList := faceList; surfaceList := faceList;
chamberList := faceList. [The number of such lists must equal
the intrinsic dimension plus 1.  For example, if the intrinsic dimension
is 1, then the brep must have a vertexList and an edgeList but
cannot have a surfaceList nor a chamberList.]

<li>
faceList := (name<sub>1</sub> propVal<sub>1</sub> 
boundary<sub>1</sub> lowdim<sub>1</sub> geom<sub>1</sub> ...
name<sub><em>n</em></sub> 
propVal<sub><em>n</em></sub> 
boundary<sub><em>n</em></sub> lowdim<sub><em>n</em></sub> 
geom<sub><em>n</em></sub>)
[In other words, each brep face has five entries in its
face list: name, propVal, child, lowdim, geom.
The name field is an arbitary string.  A name field with spaces
should be enclosed in parentheses.  The remaining
fields are as follows.]

<li> propVal := (prop<sub>1</sub> name<sub>1</sub> ... 
prop<sub><em>m</em></sub> name<sub><em>m</em></sub>)
[As above, these can be arbitrary string, although
parentheses should be used for strings with spaces.]

<li> boundary := (childname<sub>1</sub> ... 
childname<sub><em>m</em></sub>)
[These are names of bounding faces.  That is, these names refer
to the names appear in the 'name' field of the lower-dimensional
faces.  Names appearing in this list must be faces of exactly one
lower dimension.  Each childname can optionally be preceded
by the symbol + or - to indicate orientation.  Orientations
are not used by QMG 2.0.]

<li> lowdim := (childname<sub>1</sub> ... 
childname<sub><em>m</em></sub>)
[These are the names of the low-dimensional bounding faces.
Names appearing in this list must be faces of dimension lower by