1. TRANSITIONAL OBJECT’S SHAPE SIMULATION BY LAGRANGE’S 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
Good morning.
My name is João Tavares, I am a professor at Faculty of Engineering of University of Porto, from Portugal, and I am the second author of this presentation with title “Transitional Object’s Shape Simulation by Lagrange’s Equation and Finite Element Method”.
~ ASM 2004 ~
IASTED International Conference on Applied Simulating and Modelling
June 28-30, 2004
Rhodes, Greece
João Tavares

2. Introduction Applications:
Modeling and Simulating Objects Deformation:
The transition between object’s shape is simulated attending to their physical attributes by solving Lagrange’s 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
Experimental Results
Conclusions
Future Work
The work that I am here presenting is in the area of Modeling and Simulating Objects Deformation.
We simulate the transition between objects’ shape.
This is done in a physical manner by solving Lagrange’s Equation and using The Finite Element Method.
To do so we use, only the objects’ images.
This work may be applied to do objects 3D reconstruction when 2D images (also called slices) are given.
It also can be used to estimate the strain energy involved, and therefore it may be used in objects comparison and identification.
João Tavares

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
Experimental Results
Conclusions
Future Work
The main purposes of our work were the simulation of the displacement field between two given objects, and then the comparison and quantification of the represented transitional deformation using the estimated strain energy values.
In doing so we used some of the objects’ points as nodes, we applied the Finite Element Method to build the objects physical models, and then we used modal matching to match some of the models nodes.
João Tavares

4. Foundations Finite Element Method:
- Objects expected behavior simulated by selection of an elastic virtual material;
- Sclaroff’s 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 (Rayleigh’s 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
Experimental Results
Conclusions
Future Work
As we used the finite element method, the choice of a virtual material allows the simulation of the objects’ physical behavior.
We used and compared two different finite elements to build each objects’ model: the Sclaroff’s isoparametric element and the linear axial element.
Sclaroff’s element is defined by its Gaussian’s interpolants, and the obtained model does not depend on the nodes’ order. In this case the modeled objects behave like elastic membranes (if they are 2D) or like rubbery blobs (in the case they are 3D).
When Linear Axial Elements are used, shallow models are built (their edges are 1D), and the nodes correct order must be determined (which has been done with Delauney’s algorithm).
As I have mentioned before, the used objects are represented in images, and we have no other information about them. So to solve Lagrange’s Equation (also known as the Dynamic Equilibrium Equation), we had to estimate the damping matrix. To do so, we used Rayleigh’s damping that can be estimated as a linear combination of stiffness and mass matrices.
We also needed to determine some correspondences between the nodes of the given objects. By modal analysis, pairs of nodes that have similar modal displacements are considered matched.
João Tavares

5. Resolution of the Dynamic Equilibrium Equation
Input
Physical models built using FEM
Modal Matching
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
In this slide we have a diagram of the adopted methodology.
As input we have the data pixels (used as nodes of a finite elements model).
With the FEM (Sclaroff’s element or linear axial elements) we can determine the mass and stiffness matrices for each model.
Then each models’ eigenmodes are computed by solving a generalized eigenvalue/vector problem.
Pairs of nodes that have similar modal displacements are considered matched.
To proceed and solve the Lagrange’s Equation we estimate the damping matrix (with Rayleigh’s damping), and also have to estimate the applied charges on matched or unmatched nodes, as well as the initial displacement and velocity vectors.
When the Lagrange’s Equation is solved the displacement field is estimated, the global strain energy or local strain energy can be evaluated.
And the obtained 2D transitional shapes can be image represented according to the applied charges intensities, or global or local strain energy values.
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
João Tavares

6. Resolution of the Dynamic Equilibrium Equation
Mode Superposition Method used to solve Lagrange’s Equation:
Transforms the original system into a set of uncoupled equations;
Involved computational effort can be reduced when a modes subset is used;
Estimates:
Applied charges on unmatched nodes (to apply on objects that don’t 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
Experimental Results
Conclusions
Future Work
To solve the Lagrange’s Equation in this work we used the Mode Superposition Method that transforms the original system of equations into an uncoupled set
We chose this method because the involved computational effort can be reduced as we can use a subset of modes.
As I said before the applied charges had to be estimated. In our work we considered the applied charges on matched nodes to be proportional to the distance between them. For the unmatched nodes (B in this image) we used the neighborhood criterion: we apply the estimated global rigid transformation (obtaining B1) and attract this transformed point to the neighbor nodes that are within a predefined distance (for example A’ and C’) obtaining (B’).
The initial displacement we defined as proportional to the total displacement, and the initial velocity as proportional to the initial displacement.
João Tavares

7. Implementation Features:
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
Experimental 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).
Our work was implemented on an previously existing software platform that can be used to develop and test image and computer graphics algorithms.
Some features of this platform are here described: programming language c++; Development tool Microsoft Visual C++; Operating systems: Microsoft Windows; Modular development; Some Public libraries incorporated (e.g. Newmat, VTK).
João Tavares

8. Experimental Results (2D)
Examples:
Contents
Introduction
Foundations
Resolution of the Dynamic Equilibrium Equation
Implementation
Experimental Results
Conclusions
Future Work
Initial contour
Target contour
Modal Matching between the given contours
Some of the 2D results are hereby presented:
Consider these initial and target contours, matched by modal analysis, the estimated transitional deformation obtained with the approach we present can be represented by the global and local strain energy involved.
Please notice that the higher values of gray are due to higher intensities of charges and energy (left and right, respectively).
So, the applied charges decrease as the shape approaches its target (as I said before we estimated them like this – proportional to the total displacement).
On the other hand the strain energy values increase as the target shape is approached (in the next example we will see why this happens).
Estimated transitional deformation represented by applied charges intensities
... by local strain energy values
João Tavares

9. Experimental Results (3D)
Contents
Introduction
Foundations
Resolution of the Dynamic Equilibrium Equation
Implementation
Experimental Results
Conclusions
Future Work
Modal Matching done with Sclaroff’s isoparametric element
42% nodes successfully matched
… with Linear Axial Elements
9% nodes matched
Consider these surfaces given from two real heart images.
The initial surface (bottom) has hundred and twenty nodes and the target surface (upper) has hundred and seven nodes.
When we use Sclaroff’s isoperimetric element we obtained forty-five nodes successfully matched (figure of the left). But when we use Linear Axial Elements we only were able to match nine nodes (at the right).
From our experimentations we have noticed that Sclaroff’s element generally gets more nodes matched than models of grouped linear axial elements.
João Tavares

10. Experimental Results (3D)
Contents
Introduction
Foundations
Resolution of the Dynamic Equilibrium Equation
Implementation
Experimental Results
Conclusions
Future Work
Initial Surface
Transitional Object’s Shape Simulation
Here we find an example of 3D transitional objects simulation when the previous initial and target surfaces are given.
Analyzing the strain energy values along the simulated deformation, we notice that it increases as the system evolves, and its values are proportional to the global displacement (%). This is explained because in this examples we considered that the initial velocity and acceleration were null.
The results obtained in this example were obtained with Sclaroff’s isoparametric element; if linear axial elements were used, it would be necessary more steps to get the same approach of the target surface.
Target Surface
João Tavares

11. Experimental Results (Analysis)
Physics based approach: the obtained transitional object’s shape simulation is coherent with the expected physical behavior.
Sclaroff’s Isoparametric Element vs. Axial Linear Elements:
It is easier to match objects using Sclaroff’s isoparametric element;
Using Sclaroff’s isoparametric element, less steps are needed to approach the target shape (convergence obtained quicker).
Contents
Introduction
Foundations
Resolution of the Dynamic Equilibrium Equation
Implementation
Experimental Results
Conclusions
Future Work
We noticed that the obtained results were coherent with the results we physically expected (for example if the applied charges intensities get higher, then the displacement between steps is bigger).
When comparing the results obtained by each of the finite elements used, we noticed that:
- It is easier to match objects using Sclaroff’s isoparametric element;
- Using Sclaroff’s isoparametric element, less steps are needed to approach the target shape; that is the target shape is obtained quicker.
João Tavares

12. PHYSICAL TRANSITIONAL SIMULATION
Conclusions
We have presented a physical methodology that can be used to do transitional object’s 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
Sclaroff’s 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
Experimental Results
Conclusions
Future Work
So, concluding, in our work we have used a physical approach to do transitional object’s shape simulation and for quantify the involved deformation.
To do so we only considered the objects nodes position in each image, and so we had to do some estimates (such as the applied charges intensities). With our experimentations we noticed that the obtained results are coherent with the physical expected behavior, which shows that we have adopted adequate solutions in our estimates.
When comparing the two type of finite elements used, we noticed that generally Sclaroff’s element is easier to use (because it does not require the nodes ordering) and obtains better results than Linear Axial Elements (more nodes are matched and the convergence to target surface is quicker).
João Tavares

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
Experimental Results
Conclusions
Future Work
To apply the methodology proposed, it was necessary to estimate the implicit applied charges. The adopted solution can be improved through the search of alternative models to represent these charges (always attending to the possibility of unmatched nodes).
Another future task is the necessary validation of this methodology with real application. For example, in the domain of computer graphics this methodology may be used to simulate virtual reality, namely in the case of collision between deformable objects or to interpolate objects’ data.
João Tavares

14. The End! Thank You! And with this, I have finished.
Thank you for your attention.
João Tavares