a simple geometric model for elastic...

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


“As rigid as possible” work exceedingly well

Müller et al.. 2005

ContributionsThe geometry of elasticity

unifies previous approaches honest-to-goodness elasticity low cost/high performance solver


/ 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


Polar Decomp.

closest rotation

Many Different Roads…Similar core concept for all

how far is from (locally) rigid?

Initial (reference)configuration

deformedconfiguration


closest rotation

Many Different Roads…Similar core concept for all measure deviation from local rigidity measure deviation from local rigidity

seek minimizers



seek minimizers



seek minimizers


Non-linear since R depends on f


seek minimizersSorkine & Alexa, 2007

Liu et al., 2008


alternating solver

while( !converged){compute R from dfPoisson solve for f

}


seek minimizersSorkine & Alexa, 2007

Liu et al., 2008


alternating solverNon-linear since R depends on f

while( !converged){compute R from dfPoisson solve for f

}


seek minimizers w. Newton solver


Second variation Laplace Variation of R


seek minimizers w. Newton solver


Second variation Laplace Variation of R

Discrete SettingPiecewise linear mesh triangles

planar morphing, parameterization

triangles


mappededge

orig.edge

near.rot.

Discrete SettingPiecewise linear mesh triangles triangles

tetrahedraElasticity


mappededge

orig.edge

near.rot.

mappededge

orig.edge

near.rot.

Turning the CrankFor Newton solver gradient

fixed cotan weights(Laplace-Beltrami)

gradient: just differentiate

rotated area resp. volume gradients


Turning the CrankFor Newton solver gradient gradient: just differentiate

Hessian… 1st term easy (LB); 2nd more interesting


antisymmetric part of backward rotated variations

weight




multiply precomputed 3x3 matrices with current rotation and Y matrix




Wait! It gets BetterAnother orthogonal split







Shear modulus

Bulk modulus



non-linear Biot strain large displacement


small strain

1938




small strain




small strain




small strain

Dynamics:Time SteppingVariational integrator quadrature applied to HP principle

Kharevych et al. 2006

quadrature applied to HP principle time stepping == energy minimization

deformed conf.






potentialenergy

kineticenergy

affine termsdepending on k




small modification to existing solver


Dynamics SimulationConservation properties linear & angular momentum linear & angular momentum


Dynamics SimulationImportance of variable R


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

Dynamics SimulationVariational damping (stepsz. indep.)


factor 5 difference in step size; also note momentum conservation

2D ApplicationsParameterization (Liu et al. 2008)


As-Similar-As-Possible

2D ApplicationsParameterization (Liu et al. 2008)


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


pof integration changes

along path


Kilian et al. 2007


trajectory satisfies


point along optimal pathfor a given t


Kilian et al. 2007




replace with


Kilian et al. 2007





Kilian et al. 2007




non-linear springs pulling from ends

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!


conservation properties at little cost

lots of extension possibilities…

a simple geometric model for elastic...

Documents