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.