A physical simulation of objects’ behaviour by finite element method.

A Physical Simulation of Objects’ Behaviour by Finite Element Method, Modal Matching and Dynamic Equilibrium Equation LOME - Laboratório de Óptica e Mecânica Experimental DEMEGI - Departamento de Engenharia Mecânica e Gestão Industrial FEUP - Faculdade de Engenharia da Universidade do Porto, R. Dr. Roberto Frias, 4200-465, Porto, Portugal E-mail: {rpinho, tavares}@fe.up.pt Keywords: Deformable Objects, Movement/Defor- mation Simulation, Dynamic Equilibrium Equation, Finite Element Method, Modal Analysis. Abstract
This paper presents a physical approach to simulate
image represented objects’ behaviour. The Finite Element Method (FEM) is employed to physically model the given objects, then modal analysis is used to match some objects’ nodes (by solving the related eigenvalue/vector problem and analysing each node
displacement in the respective modal space (Sclaroff,
Tavares)), and finally the dynamic equilibrium equation is solved to estimate the object’s displacement field. To solve the Dynamic Equilibrium Equation different integration methods can be used, therefore the obtained
results may differ. In this paper we briefly present the
used approach and focus on the results obtained by
three numerical integration methods: Central
Difference, Newmark’s and Mode Superposition
Although their might be some exceptions, from
(Cook). The foremost method has first order precision,
several experimental examples considered we have
as the mass and stiffness matrixes are not diagonal and
noticed that the closest approaches to the target shape
the damping effect is non-negligible, and we used an
are obtained with the Central Difference method.
algorithm where the velocity is delayed in half time
This might be explained by the existence of
step. On the other hand, with Newmark’s method the
numerical damping in the Central Difference Method,
equation resolution can be unconditionally stable, with
no numerical damping but with second order precision.
unconditionally stable method. We have also verified
The latter method was solved either with the Central
that the results obtained by Newmark’s method and
Difference Method (usual algorithm used because the
by Mode Superposition method (combined either
Mode Superposition transformed mass and stiffness
with Central Difference or with Newmark’s method)
matrixes are diagonal) or with Newmark’s Method.
do not differ significantly; but with the latter, the
For an experimental result, we can consider the initial
computational cost is lower because the number of
surface α represented in figure 1, obtained from a real
used modes can be reduced without a considerable
pedobarography image (Tavares), and the target
surface β in figure 2, obtained from α by applying a
Some References
rigid transformation, with all nodes (124) successfully
R. Cook, D. Malkus, M. Plesha, “Concepts and
matched. With all mentioned integration methods, four
Applications of Finite Element Analysis”, Wiley,
intermediate shapes can be obtained: The Central
Difference method’s last shape approaches the target
S. Sclaroff, A. Pentland, “Modal Matching for
surface in less than 700 pixels (which means than in
average each node is less than 6 pixels away from its
Transactions on Pattern Analysis and Machine
final position), figure 3. The closest approach of the
target surface obtained by Newmark’s method is at
J. Tavares, J. Barbosa, A. J. Padilha, “Matching
1600 pixels from β , figure 4. When the Mode
Image Objects in Dynamic Pedobarography”,
Superposition method is used with 75% of the model’s
RecPad'2000 - 11th Portuguese Conference on
modes, the Central Difference’s last shape is 1800
Pattern Recognition, Porto, Portugal, 2000.
pixels from β , figure 5, while with the Newmark’s

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