Hadrien Courtecuisse

Raffaella Trivisonne

Laurent Goffin

Stéphane Bordas

Michel De Mathelin

Domain Decomposition for real time Simulation of needle insertion Yinoussa Adagolodjo

1. Planned trajectory 2. Deformation due to insertion 3. Camera capture the surface of the gel 4. FE Simulation to compute the deformed trajectory Interaction Model:  Constraints are imposed using Lagrangian Multipliers (LM) [2]  Re-meshing is avoided  Principle of virtual works: the displacement of a virtual point inside a tetrahedral element is given as a linear relation J of the degrees of freedom q:

= ∆ = ∆ where n denotes the needle and t the deformable object.

Time integration

The dynamic equation of simulated bodies is given by: ̈= -, ̇+ , where is the inertia matrix, the internal forces, the external forces and the constraint forces given by LM .

Implicit time integration is used with backward Euler scheme:  Stability, accuracy, interaction between models  First order Taylor expression



  = ∆ ̇, is the derivative of the constraints, A the implicit fem matrix, the constraint violation and W is known as the Delasus operator Contribution: Domain Decomposition to compute W in Real Time Simulation  is a bloc-tri-diagonal matrix that can be inverted in real time using Thomas algorithm.[2]  is a large matrix • The trajectory of the needle is known before the simulation • The LM are applied on few degrees of freedom in real-time using blas library for any mesh resolution  Few DOF are impacted by the LM along the planed trajectory: 87 DOF for 1512 nodes meshes 105 DOF for 3872 nodes meshes 130 DOF for 7800 nodes meshes Deformable Model Corotational Model [1]:  Linear relation between stress tensor and deformation tensor (Hooke's law)  Geometrical non linarites filtering: = Beam Model :  FEM corotational beam model  Elasticity and bending stiffness  Topology is a sequence of segments  Stiffness matrix is a block-tri-diagonal % of DOF impacted nodes by LM Number of DOF of the FEM mesh Size of -

Computation time (ms) to invert -