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

Source: http://www.cs.kuleuven.ac.be/conference/iccam2004/abstracts/pinho.pdf