Adaptive resolution of 1D mechanical Bspline

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Adaptive resolution of 1D mechanical B-spline

• Julien Lenoir, Laurent Grisoni, Philippe Meseure,
  Christophe Chaillou

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Problem

• Real-time physical simulation of a knot

Fixed resolution simulation
Goal adaptive resolution simulation
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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

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Outline

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

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Physical simulation of 1D B-spline

• Geometric model B-spline

qk(qkx,qky,qkz) position of the kth control
points bk are the spline base functions t is the
time, s the parametric abscissa

• Physical model Lagrange formalism
• Variational formulation
• Mechanical system defined via DOF
• Energy minimization relatively to DOFs

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Physical simulation of 1D B-spline

• Physical Model
• Definition of the DOFs

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Physical simulation of 1D B-spline

• Physical Model
• Definition of the DOFs
• Lagrange equations applied to B-spline

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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
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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)

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Physical simulation of 1D B-spline

• Constraints by Lagrange multipliers (?i)
• Direct integration into the mechanical system

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Physical simulation of 1D B-spline

• Constraints by Lagrange multipliers (?i)
• Direct integration into the mechanical system
• ?i links a scalar constraint g(s,t) to the DOFs

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Physical simulation of 1D B-spline

• Constraints by Lagrange multipliers (?i)
• Direct integration into the mechanical system
• ?i links a scalar constraint g(s,t) to the DOFs
• L and E are determined via the Baumgarte
  scheme

gt Possible violation but no drift
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Physical simulation of 1D B-spline
Resulting physical simulation - 6 constraint
equations - 33 DOF
Lack of DOF in some area
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Geometric subdivision of a B-spline

• Exact insertion in NUB-spline (Oslo algorithm)

NUBS of degree d
Knot vectors

• The simplification of BSplines is often an
  approximation

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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 computationStorage in an 1D array
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Mechanical multiresolution

• Criteria for insertion
• Geometric problem gt geometric criteria
• Problem appears in high curvature area
• Fast curvature evaluation based on control points
• Criteria for suppression
• Segment rectilinear

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Results
Low resolution
Adaptive resolution
High resolution
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Side effect

• Geometric property
• Multiple insertion at the same location decrease
  locally the continuity
• gt Degree insertions C-1 local continuity
  Cutting
• Mechanical property
• Dynamic cuttingwithout anythingspecial to
  handle ?

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Side effect

• Example of multiple cutting

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Conclusion

• Real-time adaptive 1D mechanical model
• Continuous model (in time and space)gt Stable
  over timegt 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