1. i MAGIS is a joint project of CNRS - INPG - INRIA - UJF iMAGIS-GRAVIR / IMAG Controlling Anisotropy in Mass-Spring Systems David Bourguignon and Marie-Paule Cani iMAGIS-GRAVIR

2. iMAGIS-GRAVIR / IMAG Motivation Simulating biological materials –elastic –anisotropic –constant volume deformation Efficient model  mass-spring systems (widely used) A human liver with the main venous system superimposed

3. iMAGIS-GRAVIR / IMAG Mass-Spring Systems Mesh geometry influences material behavior –homogeneity –isotropy

4. iMAGIS-GRAVIR / IMAG Mass-Spring Systems Previous solutions –homogeneity  Voronoi regions [Deussen et al., 1995] –isotropy/anisotropy  parameter identification: simulated annealing, genetic algorithm [Deussen et al., 1995; Louchet et al., 1995]  hand-made mesh [Miller, 1988; Ng and Fiume, 1997] Voronoi regions v3v3 v2v2 v1v1

5. iMAGIS-GRAVIR / IMAG Mass-Spring Systems No volume preservation  correction methods [Lee et al., 1995; Promayon et al., 1996]

6. iMAGIS-GRAVIR / IMAG New Deformable Model Controlled isotropy/anisotropy  uncoupling springs and mesh geometry Volume preservation Easy to code, efficient  related to mass-spring systems

7. iMAGIS-GRAVIR / IMAG Elastic Volume Element Mechanical characteristics defined along axes of interest Forces resulting from local frame deformation Forces applied to masses (vertices) I1’I1’ I1I1 e1e1 e3e3 I3I3 I3’I3’ e2e2 I2I2 I2’I2’ I1’I1’ I1I1 e1e1 AB C    BarycenterIntersection points

8. iMAGIS-GRAVIR / IMAG Forces Calculations f1f1 I1’I1’ I1I1 e1e1 f1’f1’ f3f3 I1’I1’ I1I1 e1e1 e3e3 I3I3 I3’I3’ f1f1 f1’f1’ f3’f3’ Stretch: Axial damped spring forces (each axis) Shear: Angular spring forces (each pair of axes)

9. iMAGIS-GRAVIR / IMAG F’ 1 Animation Algorithm F C =  F 1 +  ’ F’ 1 +... FCFC Example taken for a tetrahedral mesh: 4 point masses 3 orthogonal axes of interest F1F1 I1’I1’ I1I1 e1e1 2. Determine local frame deformation 3. Evaluate resulting forces 4. Interpolate to get resulting forces on vertices x I =  x A +  x B +  x C AB C    I 1. Interpolate to get intersection points

10. iMAGIS-GRAVIR / IMAG Animation Algorithm x I =  x A + (1 –  )  x B + (1 –  )(1 –  ) x C +  (1 –  ) x D   A B C D I Interpolation scheme for an hexahedral mesh: 8 point masses 3 orthogonal axes of interest

11. iMAGIS-GRAVIR / IMAG Volume preservation Extra radial forces Tetra mesh: preserve sum of the barycenter-vertex distances Hexa mesh: preserve each barycenter-vertex distance With volume forces Mass-spring system Without volume forces Tetrahedral Mesh

12. iMAGIS-GRAVIR / IMAG Results Comparison with mass-spring systems: –no more undesired anisotropy –correct behavior in bending Orthotropic material, same parameters in the 3 directions

13. iMAGIS-GRAVIR / IMAG Results Control of anisotropy  same tetrahedral mesh  different anisotropic behaviors

14. iMAGIS-GRAVIR / IMAG Results HorizontalDiagonalHemicircular

15. iMAGIS-GRAVIR / IMAG Results Concentric Helicoidal (top view) RandomConcentric Helicoidal

16. iMAGIS-GRAVIR / IMAG Results Performance issues: benchmarks on an SGI O2 (MIPS R5000 CPU 300 MHz, 512 Mb main memory)

17. iMAGIS-GRAVIR / IMAG Conclusion and Future Work Same mesh, different behaviors  but different meshes, not the same behavior ! Soft constraint for volume preservation Combination of different volume element types with different orders of interpolation

18. iMAGIS-GRAVIR / IMAG Conclusion and Future Work Extension to active materials  human heart motion simulation  non-linear springs with time-varying properties Angular maps of the muscle fiber direction in a human heart

19. iMAGIS-GRAVIR / IMAG