Raquel Ramos Pinho, Jo

1
TRANSITIONAL OBJECTS SHAPE SIMULATION BY
LAGRANGES EQUATION AND FINITE ELEMENT METHOD

• Raquel Ramos Pinho, João Manuel R. S. Tavares
• FEUP Faculty of Engineering, University of
  Porto, Portugal
• LOME Laboratory of Optics and Experimental
  Mechanics

ASM 2004 IASTED International Conference on
Applied Simulating and Modelling June 28-30,
2004 Rhodes, Greece
2
Introduction

• Modeling and Simulating Objects Deformation
• The transition between objects shape is
  simulated attendingto their physical attributes
  by solving Lagranges Equation and using the
  Finite Element Method.
• (The given objects are represented in images.)
• Applications
• Objects 3D reconstruction from 2D images
  (slices)
• Estimate the strain energy involved in the given
  deformation
• Compare/Identify objects
• ...

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
3
Introduction

• Main Objectives
• Simulate the displacement field between two given
  objects
• Compare/quantify deformations using the computed
  strain energy values.
• Approach
• Objects points are used as nodes
• Physical models are built using the Finite
  Element Method
• Nodes (some) are matched by Modal Analysis.

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
4
Foundations

• Finite Element Method
• - Objects expected behavior simulated by
  selection of an elastic virtual material
• - Sclaroffs Isoparametric Element
• Gaussian's interpolants
• Independent of the nodes order
• Modeled object behaves like an elastic membrane
  (2D) or a rubbery blob (3D)
• - Linear Axial Element
• Shallow models with 1D edges
• Nodes correct order must be predetermined
• Damping matrix linear combination of the Mass
  and Stiffness matrices (Rayleighs Damping)
• Nodes matched by Modal Analysis pairs of nodes
  with similar displacements in their modal spaces
  are considered matched.

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
5
Resolution of the Dynamic EquilibriumEquation
Input
Modal Matching
Physical modelsbuilt using FEM
Data (pixels) as nodes of a finite element model
Eigenmodes are computed
Displacements are analyzed in each modal space
Resolution of the generalized eigenvalue/vector
problem
Mass and stiffness matrices ( and )of each
model are assembled
Transitional Shape Deformation Simulation and
Strain Energy Evaluation
Estimates
Damping matrix, C
Dynamic Equilibrium Equation Resolution
Global and Local Strain Energy estimation
Applied charges on matched or unmatched nodes, R
Transitional deformation done according to
physical principals and obtained shapes can be
represented by
Displacement field is obtained
- applied charges intensities - global strain energy - local strain energy
Initial displacement , and velocity
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Resolution of the Dynamic Equilibrium Equation

• Mode Superposition Method used to solve
  Lagranges Equation
• Transforms the original system into a set of
  uncoupled equations
• Involved computational effort can be reducedwhen
  a modes subset is used
• Estimates
• Applied chargeson unmatched nodes(to apply on
  objects that dont have all nodes matched)
• Initial Displacement and Velocity
• Initial displacement considered proportional to
  the total displacement
• Initial velocity considered proportional to the
  initial displacement.

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
7
Implementation

• This approach was implemented on an previously
  existing platform that can be used to develop and
  test image and computer graphics algorithms.

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work

• Features
• Programming Language C
• Development tool Microsoft Visual C
• Operating systems Microsoft Windows
• Modular development
• Public libraries incorporated (e.g. Newmat, VTK).

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Experimental Results (2D)
Examples
Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
Target contour
Modal Matching between the given contours
Initial contour
Estimated transitional deformation represented by
applied charges intensities
... by local strain energy values
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Experimental Results (3D)
Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
Modal Matching done with Sclaroffs isoparametric
element 42 nodes successfully matched
with Linear Axial Elements 9 nodes matched
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Experimental Results (3D)
Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
Initial Surface
Transitional Objects Shape Simulation
Target Surface
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Experimental Results (Analysis)

• Physics based approach the obtained transitional
  objects shape simulation is coherent with the
  expected physical behavior.
• Sclaroffs Isoparametric Element vs. Axial Linear
  Elements
• It is easier to match objects using Sclaroffs
  isoparametric element
• Using Sclaroffs isoparametric element, less
  steps are needed to approach the target shape
  (convergence obtained quicker).

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
12
Conclusions

• We have presented a physical methodology that can
  be used to do transitional objects shape
  simulation and quantify the deformation involved.
• As we only used the objects nodes position in
  each image we had to estimate some parameters.
  Experimentations showed that we adopted adequate
  solutions.
• Analyzing the experimental results we also
  confirmed that the objects estimated behavior
  match our expectations.
• PHYSICAL TRANSITIONAL SIMULATION
• Sclaroffs Isoparametric Element is easier to use
  and generally obtains better results than Linear
  Axial Elements.

Contents
Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
13
Future Work
Contents

• In the future, new models can be developed to
  simulate the involved charges (always considering
  the possibility of unmatched nodes)
• and the validation of the proposed approach in
  real applications must be done.

Introduction Foundations Resolution of the
Dynamic Equilibrium Equation Implementation Experi
mental Results Conclusions Future Work
14
The End!
Thank You!