# Subdivision Surfaces A New Paradigm for Thin Shell Finite

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"Subdivision Surfaces A New Paradigm for Thin Shell Finite"

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

Fehmi Cirak
Michael Ortiz
Peter Schröder

Presented by: Scott Kircher
Context

Subdivision Surface FEM for Thin Shells         2
Overview

•   Thin Shell Theory
•   FEM basics
•   The Problem with Thin Shells
•   Subdivision Surfaces
•   Applying Subdivision Surfaces to FEM
•   Results
•   Conclusion
Subdivision Surface FEM for Thin Shells          3
Thin Shell Theory
Undeformed
Deformed

“Shell-director”

Parameter Domain
Subdivision Surface FEM for Thin Shells                4
Thin Shell Theory

• Restrict to Kirchoff-Love thin shells
– Shell-director is unit normal of middle surface
• Restrict to Linearized Kinematics
x( , ) x( , )  u( , )
1    2               1       2                 1   2

• u is a “displacement field”

Subdivision Surface FEM for Thin Shells           5
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)…

Subdivision Surface FEM for Thin Shells   6
Thin Shell Theory

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

 x              x x x 
 ij   i 1
 j i j
                  
x                                 2
 1

1  x              u u x 
x                       x
 1                     2                    2 i               j i j
                  
x
 2
Subdivision Surface FEM for Thin Shells      7
Thin Shell Theory

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

Exact form is complicated.
 ij       Think of it as something
related to mean curvature

Subdivision Surface FEM for Thin Shells           8
Thin Shell Theory

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

W  A  (α  α)  B  (β  β)
A and B are constant rank-4 contravarient tensors having to do with
the shape of the undeformed surface, and surface properties

Subdivision Surface FEM for Thin Shells    9
Thin Shell Theory

• Integrating W over surface gives internal
energy

      int
 W
M
• This means u must have square integrable
first and second derivatives
Subdivision Surface FEM for Thin Shells   10
Thin Shell Theory

• External energy is potential energy of

• Total energy is sum of internal and external
energies
• Minimizing this energy yields equilibrium
displacement field

Subdivision Surface FEM for Thin Shells   11
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

Subdivision Surface FEM for Thin Shells     12
Finite Element Method

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

Subdivision Surface FEM for Thin Shells   13
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

Subdivision Surface FEM for Thin Shells   14
The Problem with Thin Shells

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

Subdivision Surface FEM for Thin Shells       15
The Problem with Thin Shells

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

•Or at least… it wasn’t understood how
to do this properly at the time the paper
was written.

Subdivision Surface FEM for Thin Shells     16
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

Subdivision Surface FEM for Thin Shells   17
The Problem with Thin Shells

– 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

Subdivision Surface FEM for Thin Shells     18
Subdivision Surfaces

• Subdivision surfaces are generalizations of
splines
– Piecewise low-degree polynomial curves
p2

p1
S r (t )   p i N r (t  i)
i

p0                             [Loop97]
Subdivision Surface FEM for Thin Shells              19
Subdivision Surfaces

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

Mesh points converge
to actual smooth curve                                    [Loop97]
Subdivision Surface FEM for Thin Shells   20
Subdivision Surfaces

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

S r ,n (u, v)   p ij N r ,n (u  i, v  j )
ij

N r ,n ( x, y)  N r ( x) N n ( y)

[Loop97]

Subdivision Surface FEM for Thin Shells          21
Subdivision Surfaces

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

[Loop97]

Subdivision Surface FEM for Thin Shells   22
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

Subdivision Surface FEM for Thin Shells   23
Loop Subdivision

• Loop generalized quartic triangular spline
subdivision to arbitrary topology surfaces

What if not valence 6?
Subdivision Surface FEM for Thin Shells        24
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
 3
8N , N  3
w
3
 ,N 3
 16          [Warren95]

Subdivision Surface FEM for Thin Shells           25
Loop Subdivision

• Example

Extraordinary point (N=7)
Subdivision Surface FEM for Thin Shells   26
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

Subdivision Surface FEM for Thin Shells   27
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

Subdivision Surface FEM for Thin Shells   28
Subdivision Surfaces & FEM

• How to integrate?
– On regular patches, it’s a triangular spline
– Described by a known set of basis functions

Subdivision Surface FEM for Thin Shells   29
Subdivision Surfaces & FEM

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

Subdivision Surface FEM for Thin Shells   30
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

Subdivision Surface FEM for Thin Shells   31
Results

Subdivision Surface FEM for Thin Shells         32
Results

Subdivision Surface FEM for Thin Shells         33
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
Subdivision Surface FEM for Thin Shells    34

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