FEARLESS engineering Introduction of Thin Shell Simulation Ziying Tang.

FEARLESS engineering

Introduction of Thin Shell Simulation

Ziying Tang


Overview

• Introduction• Previous works• “Discrete Shells”• “Real-time Simulation of Thin Shells”


Introduction

• What is thin-shell model?– Thin, flexible objects– High ratio of width to thickness (>100)– Curved undeformed configuration– Examples:

• Leaves, hats, papers, cans etc.


Introduction

• What’s the difference between thin shell and thin plate?– Shells are naturally curved (in the unstressed

state)– Plates are naturally flat– Cannot model thin shells using plate formulations


Introduction

• Thin shells – Remarkably difficult to simulate

• Degeneracy in one dimension (thinness)• Cannot straightforward tessellation• Cannot model as a 3D solid

• Thin plates– Cloth modeling– Mass-spring networks (diagonal springs)

• Calculate forces for shearing, stretching, bending• Unfortunately, insensitive to sign of dihedral angle


Previous works

• Continuum-based approaches– Kirchoff-Love constitutive equations– Cirak et al. 2000, Subdivision surfaces– Seth Green et al. 2002, Subdivision-based multilevel

methods for large scale engineering simulation of thin shells

– Grinspun et al. 2002, CHARMS

• Complex, challenging, costly to simulate


“Discrete Shells”

• Eitan Grinspun , Anil N. Hirani , Mathieu Desbrun , Peter Schröder, Discrete shells, Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation, July 26-27, 2003, San Diego, California


Contribution

• A small change to a cloth simulator yields thin shell simulation– A minor change to the bending energy

• Capture same characteristic behaviors as more complex models

• Very simple, easy to implement


Focus

• Focus on inextensible shells which are characterized by mostly isometric deformation:– Possibly significant deformation in bending but

unnoticeable deformation in membrane modes.


Discrete Shell Model

• 2-manifold triangle mesh• Governed by

– Membrane energies (intrinsic)• Stretching – length preserving• Shearing – area preserving

– Flexural energies (extrinsic)• Bending – angle preserving

• Deformation defined by piecewise-affine deformation map– Mapping of every face (resp. edge, vertex) of the

undeformed to the deformed surface (resp. edge, vertex)


Shell modeling

• A simple, physically-motivated shell model can be expressed by the sum of membrane and flexural energies:

M B BW W k W WM is the membrane energyWB is the flexural energykB is the bending stiffness


Membrane Energy

• The membrane energy can be expressed as:

M L L A AW k W k W

WL is the stretching energy WA is the shearing energykL is the stretching stiffness kA is the shearing stiffness

21L eW || e || / || e || || e ||

2A 1A

W || A|| / || A || || A || e|| || is the deformed edge length || || is the deformed area

|| || is the undeformed edge length || || is the undeformed areae A

A


Bending Energy

• When a surface bends, an extrinsic deformation, flexural energy comes.

• Invariant under rigid-body transformation• Bending energy intuition

– Measure of the difference in curvature

• Curvature– Differential of the Gauss map. Shape Operator

does this.


Gauss Map and Shape Operator

• Gauss Map– Maps from surface to the unit sphere, mapping

each surface point to its unit surface normal.

• Shape operator– Derivative of the Gauss map: measure the local

curvature at a point on a smooth surface


Bending Energy

• Bending energy – The squared difference of mean curvature

• Mean curvature – The mean curvature calculated at point p is:

1

2H p Tr S p

Tr(S) denotes the sum of diagonal elements of the shape operator evaluated at p.


Bending Energy

• Bending energy– The squared difference of mean curvature

2 24Tr * S Tr S H H

S and S bar are the shape operators evaluated over the deformed andundeformed surfaces, respectivelyH and H bar are the mean curvatures represents a diffeomorphism which is a map between topological spacethat is differentiable and has a differentiable inverse.


Flexural Energy

• Continuous flexural energy– Integrate over reference domain

• Discrete flexural energy– Discrete the integral over the piecewise linear

mesh:

24 H H dA

2B e e ee

W x || e || / h he is a third of the average of the heights of the two triangles incident to the edge e


Implementation

• Implementation– Take working code for a cloth simulator (eg.,Baraff)– Replace the bending energy

• Hurdles– Cloth simulators generally work with flat planes

• Doesn’t work for any surface which cannot be unfolded into a flat sheet

• Solution: Simply express the undeformed configuration in 3D coordinates

22

B e B e e ee e

W x W x || e || / h


Results

• Computation time– Few minutes to few hours on 2Ghz Pentium 4

• Video – Beams• Video –Hat


Results


Conclusion

• First work to geometrically derive a discrete model for thin shells aimed at computer animation

• Simple implementation• Separation of membrane and bending energies• Captures characteristic behaviors of shells

– Flexural rigidity– Crumpling

UT DALLASUT DALLAS Erik Jonsson School of Engineering & Computer Science


Thank you~

Questions?


Introduction of Thin Shell Simulation

Ziying Tang


Overview

• Introduction• Previous works• “Discrete Shells”• “Real-time Simulation of Thin Shells”


“Real-time Simulation of Thin Shells”

• Min Gyu Choi, Seung Yong Woo, and Hyeong-Seok Ko, “Real-Time Simulation of Thin Shells ”, Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation, pages 349-354, 2007


Objectives

• Simulate thin shells undergoing large deformation

• Satisfactory physical model runs in real-time• Simulate large rotational deformation


Review

• Thin shells– 2D flexible objects– Representing shells as 2D meshes

• Dynamics of thin shells– Discrete model by Grinspun et al.

• Model warping– Simulate large rotational deformation for 3D solid


Elastic energy

• Elastic energy of a thin shell is sum of the membrane and flexural energies:

A A L L B BE k E k E k E

where kA, kL and kB are material constants for stretch, shear, and flexural stiffness

21L eE || e || / || e || || e ||

2A 1A

E || A|| / || A || || A ||

2B e e ee

E || e || / h


Governing equation

• Elastic force E u

K u uu

u(t) is a 3n-dimensional vector representing the displacement of n node

• The governing equation that describes the dynamic movements of a thin shell can be written as

where M and C are the mass and damping matrices,and F is a 3n-dimensional vector that representsthe external forces acting on the n nodes.


Modal displacements

• When there is a small rotational deformation

• Solving a eigenvalue problem

• The columns of Φ form a basis of the 3n dimensional space, then:

Φ is the modal displacement matrix, the i-the column: the i-th mode shape. q(t) is a vector containing the corresponding modal amplitudes as its components.


Modal rotations-1

ωA is the 3D rotation vector: the orientational change

When the rotation is infinitesimal, the rotation matrix RA can be approximated by


Modal rotations-2

Use notation

Equating the derivative of the function with respect to ωA to zero,


Modal rotations-3

qi and thus ωA are functions of the displacement uA.Differentiating both sides with respect to uA and evaluating the derivative for the undeformed state, we get:


Modal rotations-4

• Approximate ωA(uA) with first-order Taylor expansion

0• The rotation vector of a mesh node by taking average of the rotation vectors of the triangles sharing the node

• Assemble the Jacobians of all the triangles to form the global matrix W such that Wu gives the 3n dimensional composite vector w.

Ψ is the modal rotation matrix.


Integration of rotational parts

• A concept of local coordinate is employed

• The governing equation：


Experimental results-1

Real-time deformation of a large mesh



Simulation of flat and V-beams deforming in the gravity field.



Constraint-driven animation of a character consisting of four thin shells (the hat,body, and two legs).


Conclusion

• proposed a real-time simulation technique for thin shells.

• Developed a novel procedure to find the rotational components of deformation in terms of the modal amplitudes.

• Stable even when the time step size was h = 1/30 second, and produced visually convincing results.

UT DALLASUT DALLAS Erik Jonsson School of Engineering & Computer Science


Thank you~~

Questions?

FEARLESS engineering Introduction of Thin Shell Simulation Ziying Tang.

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