1. Modeling interaction with deformable objects in real-time Diego dAulignac GRAVIR/INRIA Rhone-Alpes France

2. Keyhole Surgery Surgery involves soft tissues Need to model deformation simulation

3. Liver Model [Boux et al., ISER, 2000] HeterogenousNon-linear skinParenchyma

4. Echography In collaboration with TIMC laboratory in Grenoble, France Interpolation (translation, rotation, deformation) Echographic images at sample points

5. Thigh Model Identification (error minimization) In collaboration with UC Berkeley Presented at IROS 1999

6. Integration 2nd order non-linear differential equation Convert to 1st order system

7. Explicit Integration Runge-Kutta method with s stages Order of consistency vs. stages

8. Linear Stability Im Re At least 2 solutions: Design better computer Design better algorithm

9. Simulation Achitecture –SGI Onyx2 Compexity –370 facets –1151 tetrahedrons –3399 springs Frequency –150Hz

10. Implicit Integation linearisation Semi-implicit euler Implicit euler (non-linear system) A-stable … but not B-stable If you know your history, then you would know where you are coming from. Bob Marley Over-damped case

11. Simulation Haptic interaction with physical model Echographic image generation Timestep: 0.01s Octane 175Mhz

12. Static Resolution Principle of virtual work: internal and external forces are balanced Linear case: Pre-inversion (if enough space) No large strain No rotation No material non-linearity Non-linear case: Stiffness matrix changes with displacement

13. Newton Iteration Full Newton-Rapson method: Reevaluation of Jacobian Faster convergence Modified Newton-Rapson method: Constant Jacobian Slower Convergence

14. Calculate forces on nodes Evaluate stiffness matrix K? Iteratively solve linear system for displacements u Ku = f by successive over- relaxation (SOR) until residual forces < epsilon through Newton-Rapson iteration Iterative Solution Divergence If objects are very soft Undercorrection

15. Result 1157 tetraheadrons Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 1157 tetraheadrons Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 60 iterations/sec on SGI Octane 175Mhz Pseudo-dynamic

16. Static vs. Dynamic Static –Have clearly defined boundary conditions –No liver throwing contest Dynamic –Control of viscosity and inertia –Transient response

17. Future Directions Multi-grid methods –More rapid propagation Parallelisation –Divide into sub-regions –e.g. Block Jacobi iteration

18. Conclusions «Soft» soft-tissues may be simulated using explicit integration «Stiff» soft-tissues benefit from implicit methods Static analysis –well defined boundary conditions –transient response negligable