OOF2: The Manual
In OOF2, before creating a finite element mesh, you must first
Skeleton defines only the
geometry of the mesh. It does not include
any information about
Fields or finite element
shape functions. All of that information is in the
class, which will be discussed later.
Skeleton is an intermediate step between the pixelized
and the finite element solution. It represents the finite
element discretization of the
Microstructure may contain
Skeletons, representing different discretizations. One
Skeleton, in turn, may generate many
Meshes, allowing different
physics or different solution methods to be tried in a single
Skeleton is constructed, it can be declared to be
periodic in the x or y directions, or neither, or both. If it
is periodic, then any modifications performed on one edge will
also apply to the opposite edge, Every
Segment on a
periodic edge will have a matching partner on the opposite
edge. All skeleton modifications will maintain the
periodicity of the skeleton.
Skeletons are composed of triangular and quadrilateral
elements, as shown in Figure 2.3. These are non-overlapping
polygons that completely cover the
will be converted directly into
Elements when a
Mesh is created.
Skeleton elements inherit their
the pixels beneath them in the
Skeleton geometry is to be a good approximation of the
Microstructure geometry, then all of the pixels lying beneath an
element should have the same assigned
homogeneity of a
Skeleton element is a
measure of how well the element achieves this goal. (See
The homogeneity is computed by finding the area of the
element that overlies each category of
that have different assigned
Materials or belong to
PixelGroups are in different categories. The
category claiming the largest area of the element is the
dominant category. The homogeneity is
defined as the ratio of the area of the dominant category to
the area of the element as a whole. A completely
homogeneous element has a homogeneity of 1.0. An element
made up of N equal components has a homogeneity of 1/N. The
Material assigned to an element is the
Material of its
dominant pixel category.
The color of the pixels in an
Many of the tools for modifying
Skeletons, such as
work by reducing an effective energy
of the mesh. This functional assigns
a number between 0 and 1 to each element. It is called an
energy because of its role in the
operation, where it plays the role of the energy in a
statistical mechanical simulated annealing process.
Skeleton modifications that use
will not consider the shape of
elements at all, and will result in homogeneous but badly
shaped elements. When , modifications will not
consider homogeneity, and will result in well shaped but
possibly inhomogeneous elements. When ,
there will be a trade-off between shape and homogeneity.
The homogeneity energy is simply one minus the homogeneity, so that it is minimized when an element is completely homogeneous.
Finite elements are usually better behaved (the resulting matrix equations are easier to solve) if the elements do not have sharp angles or high aspect ratios. The shape energy function returns 0 for equilateral triangular or square quadrilateral elements, and 1 for elements that are degenerate (ie, have an aspect ratio of 0 or three collinear vertices).
The explicit expression for triangular elements is
where is the area of the element and is the sum of the squares of the lengths of its sides.
For quadrilateral elements the shape energy is found by first computing a “quality factor”, , for each corner . is the area of the parallelogram defined by the two sides of the element that converge at node , divided by the sum of the squares of the sides, and normalized so that its value is 1 for a square. It's value is always less than 1 at a corner where the two converging edges have different lengths or meet at an acute or obtuse angle, and is zero in the degenerate cases when the edges are colinear or when the length of one edge is 0. The shape energy is defined to be
where is the minimum (worst) in the element, is the at the opposite corner, and is a small number. (The term is required to prevent pathologies that occur when the shape energy has no dependence on the position of one of the nodes. is set to 1.e-5 in the program, but its exact value is inconsequential.)
The nodes at the corners of an element are ordered. The perimeter of the element is traversed counterclockwise when moving from one node to the next. Any operation that breaks this ordering makes the element illegal. Elements with three collinear nodes are also illegal, as are non-convex quadrilaterals. (Such elements introduce singularities and instabilities in the finite element stiffness matrix.) Figure 2.5 illustrates how node motion may create illegal elements.
Skeleton tools will refuse to create illegal
elements. The one exception is the Move Node
toolbox, which allows the user to move nodes by hand.
Sometimes it may be necessary to temporarily make an illegal
element while moving a bunch of nodes.
Nodes may be moved when a
Skeleton is modified.
Different nodes have different degrees of mobility. The
Nodes at the four corners of a
Microstructure can never move. The
Nodes along the edges of a
Microstructure can move along the edge,
but cannot move into the interior. All the interior Nodes
can move freely (see Figure 2.6).
In addition, any Node may be explicitly pinned
to prevent it from moving at all.
Homogeneity can be computed on
Segments just as it can on
Elements. Analogous to the definition of Element homogeneity,
the homogeneity of a Segment is defined as the fraction of
the length of the segment that lies above that Segment's
dominant pixel type. See Figure 2.7 for a graphical
The components of a
may be placed into named groups. These groups form a
convenient way to save and recover sets of selected objects.
Groups are created and manipulated by the Skeleton Selection
Skeleton boundaries define the places where
conditions will be applied when solving equations on a
Mesh inherits its boundaries from its
There is no way to create boundaries in a
Boundaries may coincide with the perimeter of the
there is no requirement that they do so.
Boundaries are created and manipulated by the Skeleton Boundaries task page.
Edge boundaries are composed of directed sets of conjoined
Skeleton automatically contains edge boundaries named
boundary conditions may be applied at edge boundaries.