a simple geometric model for elastic...
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![Page 1: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/1.jpg)
A Simple Geometric Model for Elastic
DeformationsIsaac Chao (Caltech)
Multi-Res Modeling Group
Ulrich Pinkall (TUB)Patrick Sanan (Caltech)
Peter Schröder (IAS TU München, Caltech)
Geometry ProcessingElastic models common editing geometric shapes
Sorkine & Alexa. 2007
editing geometric shapes morphing shape space interpolation many others…
Kilian et al. 2007
Liu et al. 2008
Alexa et al.. 2000
Multi-Res Modeling Group
“As rigid as possible” work exceedingly well
Müller et al.. 2005
![Page 2: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/2.jpg)
ContributionsThe geometry of elasticity
unifies previous approaches honest-to-goodness elasticity low cost/high performance solver
Multi-Res Modeling Group
/ g p improves all applications focus today on elasticity proper
Many Different Roads…Similar core concept for all
how far is from (locally) rigid?
Initial (reference)configuration
deformedconfiguration
Multi-Res Modeling Group
Polar Decomp.
closest rotation
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Many Different Roads…Similar core concept for all
how far is from (locally) rigid?
Initial (reference)configuration
deformedconfiguration
Multi-Res Modeling Group
closest rotation
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizers
Multi-Res Modeling Group
![Page 4: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/4.jpg)
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizers
Multi-Res Modeling Group
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizers
Multi-Res Modeling Group
Non-linear since R depends on f
![Page 5: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/5.jpg)
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizersSorkine & Alexa, 2007
Liu et al., 2008
Multi-Res Modeling Group
alternating solver
while( !converged){compute R from dfPoisson solve for f
}
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizersSorkine & Alexa, 2007
Liu et al., 2008
Multi-Res Modeling Group
alternating solverNon-linear since R depends on f
while( !converged){compute R from dfPoisson solve for f
}
![Page 6: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/6.jpg)
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizers w. Newton solver
Multi-Res Modeling Group
Second variation Laplace Variation of R
Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity
seek minimizers w. Newton solver
Multi-Res Modeling Group
Second variation Laplace Variation of R
![Page 7: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/7.jpg)
Discrete SettingPiecewise linear mesh triangles
planar morphing, parameterization
triangles
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mappededge
orig.edge
near.rot.
Discrete SettingPiecewise linear mesh triangles triangles
tetrahedraElasticity
Multi-Res Modeling Group
mappededge
orig.edge
near.rot.
mappededge
orig.edge
near.rot.
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Turning the CrankFor Newton solver gradient
fixed cotan weights(Laplace-Beltrami)
gradient: just differentiate
rotated area resp. volume gradients
Multi-Res Modeling Group
Turning the CrankFor Newton solver gradient gradient: just differentiate
Hessian… 1st term easy (LB); 2nd more interesting
Multi-Res Modeling Group
antisymmetric part of backward rotated variations
weight
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Turning the CrankFor Newton solver gradient gradient: just differentiate
Hessian… 1st term easy (LB); 2nd more interesting
Multi-Res Modeling Group
multiply precomputed 3x3 matrices with current rotation and Y matrix
Turning the CrankFor Newton solver gradient gradient: just differentiate
Hessian… 1st term easy (LB); 2nd more interesting
Multi-Res Modeling Group
![Page 10: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/10.jpg)
Wait! It gets BetterAnother orthogonal split
Multi-Res Modeling Group
Wait! It gets BetterAnother orthogonal split
Multi-Res Modeling Group
![Page 11: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/11.jpg)
Wait! It gets BetterAnother orthogonal split
Multi-Res Modeling Group
Wait! It gets BetterAnother orthogonal split
Shear modulus
Bulk modulus
Multi-Res Modeling Group
![Page 12: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/12.jpg)
Wait! It gets BetterAnother orthogonal split
non-linear Biot strain large displacement
Multi-Res Modeling Group
small strain
1938
Wait! It gets BetterAnother orthogonal split
non-linear Biot strain large displacement
Multi-Res Modeling Group
small strain
![Page 13: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/13.jpg)
Wait! It gets BetterAnother orthogonal split
non-linear Biot strain large displacement
Multi-Res Modeling Group
small strain
Wait! It gets BetterAnother orthogonal split
non-linear Biot strain large displacement
Multi-Res Modeling Group
small strain
![Page 14: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/14.jpg)
Dynamics:Time SteppingVariational integrator quadrature applied to HP principle
Kharevych et al. 2006
quadrature applied to HP principle time stepping == energy minimization
deformed conf.
Multi-Res Modeling Group
Dynamics:Time SteppingVariational integrator quadrature applied to HP principle
Kharevych et al. 2006
quadrature applied to HP principle time stepping == energy minimization
Multi-Res Modeling Group
potentialenergy
kineticenergy
affine termsdepending on k
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Dynamics:Time SteppingVariational integrator quadrature applied to HP principle
Kharevych et al. 2006
quadrature applied to HP principle time stepping == energy minimization
small modification to existing solver
Multi-Res Modeling Group
Dynamics SimulationConservation properties linear & angular momentum linear & angular momentum
Multi-Res Modeling Group
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Dynamics SimulationImportance of variable R
Multi-Res Modeling Group
R variable during time step R recomputed at beginning of each step
Dynamics SimulationNumerical damping (stepsize dep.)
Multi-Res Modeling Groupsmall time step smaller time step
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Dynamics SimulationVariational damping (stepsz. indep.)
Multi-Res Modeling Group
factor 5 difference in step size; also note momentum conservation
2D ApplicationsParameterization (Liu et al. 2008)
Multi-Res Modeling Group
As-Similar-As-Possible
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2D ApplicationsParameterization (Liu et al. 2008)
Multi-Res Modeling Group
discrete conformal
Shape SpaceGeodesic interpolation least distortion path
Kilian et al. 2007
least distortion path
distance squaredfrom M0 to M1
all possiblepaths nasty since domain
f i t ti h
Multi-Res Modeling Group
pof integration changes
along path
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Shape SpaceGeodesic interpolation least distortion path
Kilian et al. 2007
least distortion path
trajectory satisfies
Multi-Res Modeling Group
point along optimal pathfor a given t
Shape SpaceGeodesic interpolation least distortion path
Kilian et al. 2007
least distortion path
trajectory satisfies
Multi-Res Modeling Group
replace with
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Shape SpaceGeodesic interpolation least distortion path
Kilian et al. 2007
least distortion path
trajectory satisfies
Multi-Res Modeling Group
Shape SpaceGeodesic interpolation least distortion path
Kilian et al. 2007
least distortion path
trajectory satisfies
Multi-Res Modeling Group
non-linear springs pulling from ends
![Page 21: A Simple Geometric Model for Elastic Deformationscourses.cms.caltech.edu/cs177/notes_fa11/GeomElastic.pdf · Multi-Res Modeling Group Ulrich Pinkall (TUB) Patrick Sanan (Caltech)](https://reader034.vdocuments.net/reader034/viewer/2022050119/5f4f917f19504476da3a77f8/html5/thumbnails/21.jpg)
Conclusion“As Rigid as Possible” honest to goodness elasticity honest-to-goodness elasticity smooth differential geometry pict. Newton solver pays off handsomely second term in Hessian essential!
Multi-Res Modeling Group
conservation properties at little cost
lots of extension possibilities…