Game Development Reference
In-Depth Information
Figure 10.1. On the left, the object at its rest position. In the middle, the arrows show a
constant displacement field. On the right, the fields are more complex due to deformations
of the object.
The mathematical modeling of a continuum object consists of analyzing the
behavior of a set of infinitesimal volumetric elements, known as material points.
The positions of these material points at time t defines the configuration ,orgeo-
metrical state, of the body at that time t . Hence, the behavior of any object can be
described by analyzing the evolution of its configuration throughout time. Often,
the undeformed body is considered to be the configuration at t =0(also known
as the equilibrium or reference configuration, or initial or rest shape). This initial
configuration is used as a reference for all the subsequent configurations that rep-
resent the deformations of the body. The coordinates, or position x 0 R
3 ,ofa
point in the undeformed shape of the object are called material coordinates .
When forces are applied, the body deforms, changing its configuration; in
other words, any given point x 0 moves to a new location x
3 . The new loca-
tion in the deformed configuration is known as the spatial or world coordinates
of the object, and the displacement of the point is given by the vector
R
u = x
x 0 .
The displacements of all the points of the continuum body represent the dis-
placement field of the body. A pure translation of the body, also known as a
rigid body translation, creates a constant displacement field, while a complex de-
formation creates an arbitrary field (see Figure 10.1 ) . It would be impossible to
simulate the displacements of all the points of the continuum body. Instead, as
will be shown later in Section 10.3.1, we discretize the object into a set of finite
adjacent elements, normally tetrahedrons or hexahedrons, and map the displace-
ment field to their corners. Thus, we don't simulate the displacements of all the
points of the continuum body, we only compute the vertices' displacements of a
finite element set, as we will see in Section 10.3.