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External forces. Soft bodies are also exposed to external forces due to collisions.
It is not in the scope of this chapter to deal with all the theory of collision response.
We will briefly mention some of the approaches to computing forces due to col-
lision. The most common approach is to apply penalty forces . By definition, the
computed force depends on the penetration depth information between two col-
liding objects. The resulting force tries to minimize or reduce such penetration.
Usually, a skin depth is added to the objects so there is always a virtual penetra-
tion. Thus, the force f is usually computed using the penetration distance d and
the velocity of the object's penetration v in a mass-spring model: f = kd + b v ,
where k and b are respectively the stiffness and damping of the spring. This way
of computing forces is fast and straightforward, but as in all mass-spring systems,
it is difficult to choose its parameters and even more difficult to relate them to the
real elasticity of the object. Furthermore, the forces are locally applied, ignoring
the inertia of the object.
The other approach is based on a constraint scheme. The approach is com-
monly used in rigid-body dynamics, where no penetrations are formulated as a
linear complementary problem (LCP). The general idea is to identify the contact
points between two objects; for example, a box colliding with another box will
have four pairs of contact points, one for each corner. Each corner contact defines
a constraint where we want to keep a zero penetration distance. This is known
as a constrained optimization problem. Some commercial software use this ap-
proach combined with impulses to skip one level of integration, but its use with
soft bodies might become very lengthy. This is because the inherent nature of a
deformable body may lead to an excessive number of constraints.
10.4 Solving the Linear System
In the previous sections, we have seen how to build a linear system using the
finite element method (see Equation (10.17)). That system has to be solved in
every iteration. In this section, we will propose different ways to solve that linear
system. It is not our intention to describe all these methods deeply; instead, we
are going to propose some well-known techniques and give cues about when and
how to use them.
Before choosing a method to solve our linear system, we need to know its
properties. Our equation system has the following form:
S x = b,
(10.22)
where S is the coefficient matrix, x is the unknown vector and b is the constant
terms vector. Matrix S is symmetric, positive definite, and sparse. We recommend
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