Image Processing Reference

In-Depth Information

Computer Graphics or Computer Vision applications and introduce some of the simplified versions

that have proved effective in these fields.

2.1.1 THE FINITE ELEMENTMETHOD

FEM
Bathe
[
1982
],
Zienkiewicz
[
1989
] is the method of choice to accurately simulate the deforma-

tions of structures such as beams, plates, shells, and 3D volumes under various loads. The structure

of interest is represented by a discrete set of elements, such as segments, triangles, or tetrahedra, that

are linked by their nodes. Following the laws of mechanics, mass, damping, and stiffness matrices

are built for each element separately. These matrices typically depend on physical parameters, such

as Young's modulus, Poisson's ratio, shear modulus, and thickness of the structure. They are then

assembled to write the equations of motion that govern the deformations of the whole structure as

M

u

¨

+

D

u

˙

+

Ku

=

f
,

(2.1)

where
u
is the unknown vertex displacement,
M
,
D
, and
K
are the mass, damping, and stiffness

matrices respectively, and
f
represents the external forces. This models the full dynamical behav-

ior, which can be simplified by ignoring the terms depending on temporal derivatives when only

attempting to compute static deformations.

When considering only small deformations of a materially linear object, that is deformations

that are only barely visible, the matrices of Eq.
2.1
can be assumed to be independent of the de-

formation, and the system can be solved directly. However, almost by definition, both Computer

Vision and Computer Graphics are concerned by much larger deformations that are clearly visible.

This introduces geometrical nonlinearities that can be compounded with the fact that the material

may be subject to either hyper-elasticity or plasticity phenomena. Consequently, the stiffness ma-

trices become functions of the displacements, and the whole problem becomes much more complex

because they have to be recomputed very often. This results not only in an additional computational

burden but often also in instabilities due to buckling or the appearance of critical points that yield

different solutions.

Many resolution methods have been proposed over the years to overcome these difficulties.

The best known ones are the following
Zienkiewicz
[
1989
]:

The Total Lagrangian approach.
The solution is computed from a reference configuration

that remains unchanged throughout the computation.

The Updated Lagrangian approach.
It operates along the same lines as the Total Lagrangian

approach, except for the fact that the reference configuration is replaced by the current solution

every so often.

The Corotational Approach.
It involves decomposing large deformations into rigid trans-

formations of the elements and small deformations, which allows for stable resolution.

The last two are the most commonly used today. They are successful in Mechanical Engineering but

require both tremendous computational power, which makes them ill-suited for real-time Computer

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