Thin shells and thin plates Thin, flexible objects Shells are naturally curved Thin Shells Plates are naturally flat y & Curvature Based & Curvature-Based Energy CS176 – Intro to Computer Graphics Research 2 Physics of membranes Physics of membranes S. Helfrich (FU Berlin), P. Canham (U.W. Ontario) S. Helfrich (FU Berlin), P. Canham (U.W. Ontario) CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 3 4 Engineering Mathematics Civil/mechanical/aeronautical design T. J. Willmore’s surfaces CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 5 6 1
Mathematics Mathematics T. J. Willmore’s surfaces T. J. Willmore’s surfaces CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 7 8 Related work Euler’s elastica Researchers in graphics: Early formulation of elastic curves Terzopoulos, Bridson, Breen, etc. ad-hoc models for cloth Bobenko & Suris, Pai B b k & S i P i discrete models of elastic curves Bernoulli began generalization to [Choi and Ko] surfaces CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 9 10 Chladni’s vibrating plates Chladni’s vibrating plates Plate vibrated by Plate vibrated by violin bow violin bow Prize for explanation: Sand settles on Sand settles on 1kg of gold, nodal curves nodal curves 1808,1811,1815 CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 11 12 2
Problem setup Problem setup What is the What is the deformation energy ? deformation energy ? undeformed deformed undeformed deformed body body body body deformation deformation x x x x CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 13 14 Problem setup Problem setup Energy is a non-negative scalar What is the function deformation energy ? undeformed deformed body body deformation [T. L. Brown. Making truth ] x x CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 15 16 Problem setup Problem setup Energy is a non-negative scalar Internal forces push “downhill” function CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 17 18 3
Plates Thin plate energy Germain’s argument: bending energy must be a symmetric even fct of principal curvatures Germain Poisson Navier CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 19 20 Thin plate energy Thin plate energy Poisson’s linearization Navier’s equation assuming small displacements, to find minimizer for linearized approx. curvature by 2 nd derivatives energy, solve a PDE CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 21 22 Thin plate energy Axiomatic approach Navier’s equation Energy should be: to find minimizer for linearized symm., even fct of princ. curvatures energy, solve a PDE extrinsic measure smooth w.r.t. change in shape h h i h invariant under rigid-body motion simple to compute easy to understand CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 23 24 4
Axiomatic Shells What about mass/spring? Diagonal springs don’t work “Simplest” answer to desiderata reference configuration is curved incorrect energy minima deformation Derivation: extrinsic change in shape operator CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 25 26 Discrete shells Discrete shells Elastic energy = Elastic energy = Gradient gives forces: CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 27 28 Pimp a cloth simulator Pimp a cloth simulator Have a cloth simulator handy? Have a cloth simulator handy? reuse all the existing code reuse all the existing code retrofit the bending term retrofit the bending term precompute ref. quantities offline l precompute ref. quantities offline l CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 29 30 5
Modeling Paper Are we done? Paper sheet Discrete shells are nice and simple. curled What else is out there? creased pinned d Kirchhoff Love Karman Koiter R h h CS176 – Intro to Computer Graphics Research CS176 – Intro to Computer Graphics Research 31 32 6
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