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FEARLESS engineering
Introduction of Thin Shell Simulation
Ziying Tang
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FEARLESS engineering
Overview
• Introduction• Previous works• “Discrete Shells”• “Real-time Simulation of Thin Shells”
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FEARLESS engineering
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.
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FEARLESS engineering
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
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FEARLESS engineering
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
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FEARLESS engineering
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
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“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
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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
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Focus
• Focus on inextensible shells which are characterized by mostly isometric deformation:– Possibly significant deformation in bending but
unnoticeable deformation in membrane modes.
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FEARLESS engineering
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)
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FEARLESS engineering
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
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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
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FEARLESS engineering
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.
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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
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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.
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FEARLESS engineering
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.
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FEARLESS engineering
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
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FEARLESS engineering
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
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FEARLESS engineering
Results
• Computation time– Few minutes to few hours on 2Ghz Pentium 4
• Video – Beams• Video –Hat
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FEARLESS engineering
Results
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FEARLESS engineering
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
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UT DALLASUT DALLAS Erik Jonsson School of Engineering & Computer Science
FEARLESS engineering
Thank you~
Questions?
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FEARLESS engineering
Introduction of Thin Shell Simulation
Ziying Tang
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FEARLESS engineering
Overview
• Introduction• Previous works• “Discrete Shells”• “Real-time Simulation of Thin Shells”
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FEARLESS engineering
“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
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Objectives
• Simulate thin shells undergoing large deformation
• Satisfactory physical model runs in real-time• Simulate large rotational deformation
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FEARLESS engineering
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
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FEARLESS engineering
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
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FEARLESS engineering
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.
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FEARLESS engineering
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.
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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
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FEARLESS engineering
Modal rotations-2
Use notation
Equating the derivative of the function with respect to ωA to zero,
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FEARLESS engineering
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:
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FEARLESS engineering
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.
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FEARLESS engineering
Integration of rotational parts
• A concept of local coordinate is employed
• The governing equation:
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Experimental results-1
Real-time deformation of a large mesh
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Experimental results-2
Simulation of flat and V-beams deforming in the gravity field.
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Experimental results-3
Constraint-driven animation of a character consisting of four thin shells (the hat,body, and two legs).
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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.
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UT DALLASUT DALLAS Erik Jonsson School of Engineering & Computer Science
FEARLESS engineering
Thank you~~
Questions?