1. FEARLESS engineering Introduction of Thin Shell Simulation Ziying Tang

2. FEARLESS engineering Overview Introduction Previous works “Discrete Shells” “Real-time Simulation of Thin Shells”

3. 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.

4. 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

5. 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

6. 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

7. FEARLESS engineering “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

8. FEARLESS engineering 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

9. FEARLESS engineering Focus Focus on inextensible shells which are characterized by mostly isometric deformation: –Possibly significant deformation in bending but unnoticeable deformation in membrane modes.

10. 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)

11. FEARLESS engineering Shell modeling A simple, physically-motivated shell model can be expressed by the sum of membrane and flexural energies: W M is the membrane energy W B is the flexural energy k B is the bending stiffness

12. FEARLESS engineering Membrane Energy The membrane energy can be expressed as: W L is the stretching energy W A is the shearing energy k L is the stretching stiffness k A is the shearing stiffness || || is the deformed edge length || || is the deformed area || || is the undeformed edge length || || is the undeformed area

13. 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.

14. FEARLESS engineering 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

15. FEARLESS engineering Bending Energy Bending energy –The squared difference of mean curvature Mean curvature –The mean curvature calculated at point p is: Tr(S) denotes the sum of diagonal elements of the shape operator evaluated at p.

16. FEARLESS engineering Bending Energy Bending energy –The squared difference of mean curvature S and S bar are the shape operators evaluated over the deformed and undeformed surfaces, respectively H and H bar are the mean curvatures represents a diffeomorphism which is a map between topological space that is differentiable and has a differentiable inverse.

17. FEARLESS engineering Flexural Energy Continuous flexural energy –Integrate over reference domain Discrete flexural energy –Discrete the integral over the piecewise linear mesh: h e is a third of the average of the heights of the two triangles incident to the edge e

18. FEARLESS engineering 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

19. FEARLESS engineering Results Computation time –Few minutes to few hours on 2Ghz Pentium 4 Video – Beams Video –Hat

20. FEARLESS engineering Results

21. 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

22. UT DALLAS Erik Jonsson School of Engineering & Computer Science FEARLESS engineering Thank you~ Questions?

23. FEARLESS engineering Introduction of Thin Shell Simulation Ziying Tang

24. FEARLESS engineering Overview Introduction Previous works “Discrete Shells” “Real-time Simulation of Thin Shells”

25. 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

26. FEARLESS engineering Objectives Simulate thin shells undergoing large deformation Satisfactory physical model runs in real-time Simulate large rotational deformation

27. 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

28. FEARLESS engineering Elastic energy Elastic energy of a thin shell is sum of the membrane and flexural energies: where k A, k L and k B are material constants for stretch, shear, and flexural stiffness

29. FEARLESS engineering Governing equation Elastic force 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 represents the external forces acting on the n nodes.

30. 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.

31. FEARLESS engineering Modal rotations-1 ω A is the 3D rotation vector: the orientational change When the rotation is infinitesimal, the rotation matrix R A can be approximated by

32. FEARLESS engineering Modal rotations-2 Use notation Equating the derivative of the function with respect to ω A to zero,

33. FEARLESS engineering Modal rotations-3 q i and thus ω A are functions of the displacement u A. Differentiating both sides with respect to u A and evaluating the derivative for the undeformed state, we get:

34. FEARLESS engineering Modal rotations-4 Approximate ω A (u A ) 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.

35. FEARLESS engineering Integration of rotational parts A concept of local coordinate is employed The governing equation ：

36. FEARLESS engineering Experimental results-1 Real-time deformation of a large mesh

37. FEARLESS engineering Experimental results-2 Simulation of flat and V-beams deforming in the gravity field.

38. FEARLESS engineering Experimental results-3 Constraint-driven animation of a character consisting of four thin shells (the hat,body, and two legs).

39. FEARLESS engineering 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.

40. UT DALLAS Erik Jonsson School of Engineering & Computer Science FEARLESS engineering Thank you~~ Questions?