Subdivision Surfaces: A New Paradigm for Thin Shell Finite Element Analysis

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Subdivision Surfaces A New Paradigm for Thin
Shell Finite Element Analysis

• Fehmi Cirak
• Michael Ortiz
• Peter Schröder

Presented by Scott Kircher
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Context
3
Overview

• Thin Shell Theory
• FEM basics
• The Problem with Thin Shells
• Subdivision Surfaces
• Applying Subdivision Surfaces to FEM
• Results
• Conclusion

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Thin Shell Theory
Undeformed
Deformed
Shell-director
Parameter Domain
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Thin Shell Theory

• Restrict to Kirchoff-Love thin shells
• Shell-director is unit normal of middle surface
• Restrict to Linearized Kinematics
• u is a displacement field

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Thin Shell Theory

• Aside Why use Kirchoff-Love theory?
• Not assuming an infinitely thin sheet means we
  would need volume elements for FEM

They would be either very skinny (causing
stability issues)
Or they would be very small (causing efficiency
issues)
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Thin Shell Theory

• Membrane strain tensor formulated in terms of
  first partial derivatives of u

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Thin Shell Theory

• Bending strain tensor formulated in terms of
  first and second partial derivatives of u

Exact form is complicated. Think of it as
something related to mean curvature
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Thin Shell Theory

• Strain energy per unit area formulated in terms
  of membrane and bending strains

A and B are constant rank-4 contravarient tensors
having to do with the shape of the undeformed
surface, and surface properties
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Thin Shell Theory

• Integrating W over surface gives internal energy
• This means u must have square integrable first
  and second derivatives

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Thin Shell Theory

• External energy is potential energy of applied
  static loads
• Total energy is sum of internal and external
  energies
• Minimizing this energy yields equilibrium
  displacement field

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Finite Element Method

• FEM produces approximate solution to system of
  PDEs
• Domain is discretized into elements
• Approximate solution is a continuous function
  that satisfies the PDEs at each node

Node
Element
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Finite Element Method

• Approximate solution is continuous function
• Linear combination of some basis functions
• FEM characterized by basis functions with local
  support

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Finite Element Method

• In our case
• Nodal displacements will define deformed surface
• Deformed surface has some energy
• Minimize this energy by moving the nodes
• i.e. Solve some big sparse linear system

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The Problem with Thin Shells

• Whats wrong with previous methods?
• Conventional FEM on triangle meshes usually use
  purely local basis functions
• Surface within a triangle is based on the
  triangles nodes only

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The Problem with Thin Shells

• If only displacement is stored at each node,
  interpolant is trilinear
• Obviously only C0
• Whats the right Bending strain energy??

• Or at least it wasnt understood how to do this
  properly at the time the paper was written.

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The Problem with Thin Shells

• Could try higher-degree polynomials
• Store 1st and 2nd derivatives at nodes
• More complex and leads to well known problems
• Inability to handle discontinuous element
  properties
• Spurious oscillations of the solution

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The Problem with Thin Shells

• What can be done about this?
• Lots of approaches tried. None had great
  performance (according to the authors)
• New approach
• Let the surface be a subdivision surface!
• (Mostly) C2 smooth surface uniquely defined from
  arbitrary manifold-like triangle mesh, with nodal
  displacements only

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Subdivision Surfaces

• Subdivision surfaces are generalizations of
  splines
• Piecewise low-degree polynomial curves

p2
p1
p0
Loop97
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Subdivision Surfaces

• Splines can be evaluated by direct evaluation of
  their basis polynomials
• or through subdivision

Mesh points converge to actual smooth curve
Loop97
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Subdivision Surfaces

• Splines can represent surfaces as well
• Tensor product of univariate splines

Loop97
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Subdivision Surfaces

• Surfaces also can be evaluated by repeated
  subdivision of their control mesh

Loop97
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Subdivision Surfaces

• Spline subdivision works for regular control
  meshes (every vertex of proper valence)
• Impossible to have regular meshes of arbitrary
  topology
• Subdivision surfaces extend splines to
  2-manifolds of arbitrary topology

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Loop Subdivision

• Loop generalized quartic triangular spline
  subdivision to arbitrary topology surfaces

What if not valence 6?
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Loop Subdivision

• Quartic triangular splines are C2
• So Loop surfaces are too...
• Except at extraordinary points (valence ? 6)
• Make C1 at such points by modifying vertex weights

Warren95
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Loop Subdivision

• Example

Extraordinary point (N7)
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Subdivision Surfaces FEM

• Initial control mesh defines some smooth
  subdivision limit surface
• Deformed surface is defined by the displaced
  control mesh
• Add per-node displacement vector to each control
  mesh node

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Subdivision Surfaces FEM

• Every control mesh triangle defines a
  triangular element on the limit surface
• Can (numerically) integrate over these elements
  to get energy per triangle

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Subdivision Surfaces FEM

• How to integrate?
• On regular patches, its a triangular spline
• Described by a known set of basis functions

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Subdivision Surfaces FEM

• What about integrating on irregular patches?
• Subdivide until quadrature point is in a regular
  patch

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Subdivision Surfaces FEM

• Numerical integration yeilds total energy in
  terms of nodal displacement field
• They use one-point quadrature (just evaluate at
  barycenter of each triangle)
• Solve for minimum energy displacement field

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Results
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Results
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Conclusion

• Subdivision surfaces yield a simple and efficient
  FEM for thin shells
• Also would mesh well with the design workflow
• CAD app and FEM app can use exact same
  representation
• Should have demonstrated on more complex
  topologies