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Published byClemence Harrington Modified over 9 years ago
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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 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
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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: Kinetic energy Energies not derived from a potential (Collisions, Frictions…) Potential energies (Deformations, Gravity…)
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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 (λ i ): Direct integration into the mechanical system:
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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
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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
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Physical simulation of 1D B-spline Resulting physical simulation: - 6 constraint equations - 33 DOF Lack of DOF in some area Geometric problem
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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
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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 computation Storage in an 1D array
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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
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Results Adaptive resolution Low resolution High resolution
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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
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Example of multiple cutting:
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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
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