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
Simulation Haptic interaction with physical model Echographic image generation Timestep: 0.01s Octane 175Mhz
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
Newton Iteration Full Newton-Rapson method: Reevaluation of Jacobian Faster convergence Modified Newton-Rapson method: Constant Jacobian Slower Convergence
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
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
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