1. Adaptive resolution of 1D mechanical B-spline Julien Lenoir, Laurent Grisoni, Philippe Meseure, Christophe Chaillou

2. Problem Real-time physical simulation of a knot Fixed resolution simulation Goal: adaptive resolution simulation

3. Related work 1D model and knot tying  [Wang et al 05] Mass-spring model, not adaptive  [Brown et al 04] Non physics based model, ‘follow the leader’ rules, not adaptive Generality on multiresolution physical model  Discrete model: [Luciana et al 95, Hutchinson et al 96, Ganovelli et al 99]  Continuous model: [Wu et al 04, Debunne et al 01, Grinspun et al 02,Capell et al 02]

4. Outline Physical simulation of 1D B-spline Geometric subdivision of a B-spline Mechanical multiresolution Results Side effect Conclusion

5. Physical model: Lagrange formalism Variational formulation + Mechanical system defined via DOF = Energy minimization relatively to DOFs Physical simulation of 1D B-spline Geometric model: B-spline q k =(q k x,q k y, q k z ) position of the k th control points b k are the spline base functions t is the time, s the parametric abscissa

6. Physical simulation of 1D B-spline Physical Model:  Definition of the DOFs:

7. Physical simulation of 1D B-spline Physical Model:  Definition of the DOFs:  Lagrange equations applied to B-spline: Kinetic energy Energies not derived from a potential (Collisions, Frictions…) Potential energies (Deformations, Gravity…)

8. Physical simulation of 1D B-spline Physical Model:  Definition of the DOFs:  Lagrange equations applied to B-spline:  Generalized mass matrix: Gather the and terms

9. Physical simulation of 1D B-spline Continuous deformation energies  Stretching [Nocent01]: Green-Lagrange tensor allows large deformations Piola-Kirchhoff elasticity constitutive law  Bending: in progress…  Twisting: not treated (need to extend the model to a 4D model)

10. Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λ i ):  Direct integration into the mechanical system:

11. Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λ i ):  Direct integration into the mechanical system:  λ i links a scalar constraint g(s,t) to the DOFs

12. Physical simulation of 1D B-spline Constraints by Lagrange multipliers (λ i ):  Direct integration into the mechanical system:  λ i links a scalar constraint g(s,t) to the DOFs  L and E are determined via the Baumgarte scheme: Time step => Possible violation but no drift

13. Physical simulation of 1D B-spline Resulting physical simulation: - 6 constraint equations - 33 DOF Lack of DOF in some area Geometric problem

14. Geometric subdivision of a B-spline Exact insertion in NUB-spline (Oslo algorithm): NUBS of degree d Knot vectors: insertion The simplification of BSplines is often an approximation

15. Mechanical multiresolution Insertion and suppression:  Reallocate the data structure: pre-allocation  Shift the pre-computed data and re-compute the missing part (example:, ) Continuous stretching deformation energies:  Pre-computed terms:  4D array  Sparse  Symmetric Avoid multiple computation Storage in an 1D array

16. Mechanical multiresolution Criteria for insertion:  Geometric problem => geometric criteria  Problem appears in high curvature area  Fast curvature evaluation based on control points Criteria for suppression:  Segment rectilinear

17. Results Adaptive resolution Low resolution High resolution

18. Geometric property:  Multiple insertion at the same location decrease locally the continuity  => ‘Degree’ insertions = C -1 local continuity = Cutting Mechanical property:  Dynamic cutting without anything special to handle Side effect

19. Example of multiple cutting:

20. Conclusion Real-time adaptive 1D mechanical model Continuous model (in time and space) => Stable over time => Can handle sliding constraint [Lenoir04] Dynamic cutting appears as a side effect Future works:  Enhance the deformation energies: Better bending + Twisting (4D model, cosserat [Pai02])  Handle length constraint