Modeling interaction with deformable objects in real-time

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Modeling interaction with deformable objects in
real-time

• Diego dAulignac
• GRAVIR/INRIA Rhone-Alpes
• France

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Keyhole Surgery
Surgery involves soft tissues
simulation
Need to model deformation
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Liver Model
Boux et al., ISER, 2000
Heterogenous
Non-linear
skin
Parenchyma
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Echography
In collaboration with TIMC laboratory in
Grenoble, France
Echographic images at sample points
Interpolation (translation, rotation, deformation)
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Thigh Model
Presented at IROS 1999
In collaboration with UC Berkeley
Identification (error minimization)
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Integration
2nd order non-linear differential equation
Convert to 1st order system
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Explicit Integration
Runge-Kutta method with s stages
Order of consistency vs. stages
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Linear Stability
Im

• At least 2 solutions
• Design better computer
• Design better algorithm

Re
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Simulation

• Achitecture
• SGI Onyx2
• Compexity
• 370 facets
• 1151 tetrahedrons
• 3399 springs
• Frequency
• 150Hz

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

• Haptic interaction with physical model
• Echographic image generation

Timestep 0.01s Octane 175Mhz
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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

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Newton Iteration

• Full Newton-Rapson method
• Reevaluation of Jacobian
• Faster convergence

• Modified Newton-Rapson method
• Constant Jacobian
• Slower Convergence

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Iterative Solution
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 lt
epsilon through Newton-Rapson iteration

• Divergence
• If objects are very soft
• Undercorrection

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

• Static
• Have clearly defined boundary conditions
• No liver throwing contest
• Dynamic
• Control of viscosity and inertia
• Transient response

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Future Directions

• Multi-grid methods
• More rapid propagation
• Parallelisation
• Divide into sub-regions
• e.g. Block Jacobi iteration

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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