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Figure 10.4. Internal force computation. (a) Rest position. (b) Deformed configuration.
(c) Unrotated configuration. (d) Final internal forces. This illustration can be extended to
tetrahedral elements.
of the tetrahedron by splitting the deformation into translations and rotations. We
sketch the method as in [Muller and Gross 04] (see Figure 10.4 ):
1. The coordinates x of the object at the deformed state are rotated back to
an unrotated state using R 1
e
x ,where R e is the rotation of the tetrahedron
given by
R e ( t )
0 3 × 3
0 3 × 3
0 3 × 3
0 3 × 3 R e ( t )
0 3 × 3
0 3 × 3
R e, 12 × 12 ( t )=
.
0 3 × 3
0 3 × 3 R e ( t )
0 3 × 3
0 3 × 3
0 3 × 3
0 3 × 3 R e ( t )
2. In the unrotated state the displacements are given by u e = R 1
x e
x e ( t 0 ).
e
Hence, the forces in the unrotated state are given by
f e unrotated
= K e R e x
x 0 .
3. Finally, the forces in the unrotated state are rotated back to the deformed
state, i.e.,
f e = R e K e R e x
x 0
= R e K e R e x
R e K e x 0 .
Note that K e x 0 can be precomputed since K e is constant during the sim-
ulation, allowing us to accelerate the simulation. For the entire object, we
get
f = K g x + f g 0 ,