Overview Class

1
OverviewClass 7 (Thurs, Feb 6)

• Black box approach to linear elastostatics
• Discrete Greens function methods
• Three parts
• What are Greens functions?
• Precomputation
• Fast contact handling via low-rank updates
• Capacitance matrix algorithm
• Multiresolution extensions (later)

2
Linear Elastostatic Models (recap from last
class)

• Small-strain time-independent (static/equilibrium)
  deformation response
• Various origins, e.g., solid bodies, thin shells,
  abstract linear systems,
• Various surface representations and
  discretization possible, e.g., FEM, BEM, FVM,
  FDM, spectral,

3
Greens Functions for Interactive Elliptic PDEs
ARTDEFO Accurate Real Time Deformable Objects

In SIGGRAPH 99 Conference
Proceedings, ACM SIGGRAPH, 1999. (with Dinesh K.
Pai) A Unified Treatment of Elastostatic and
Rigid Contact for Real Time Haptics,
Haptics-e, The Electronic Journal of
Haptics Research (www.haptics-e.org), 2(1), 2001.
(w/ DKP)
4
GF Deformation Basis

• Greens functions are physically based basis
  functions adapted to
• particular geometry
• particular constraints
• GF matrix is an input-output model of the linear
  deformable system (for a particular BVP-type)
• Relates displacements to tractions, etc.
• Well focus on surface constraints surface GFs
• Also works for volumetric quantities
• displacement, stress, strain, strain-rate, etc.

5
Some Graphics References

• See webpage
• Cotin et al., 96/99.
• James Pai
• ARTDEFO Accurate Real Time Deformable Objects,
  In SIGGRAPH 99 Conference Proceedings, ACM
  SIGGRAPH, 1999.
• A Unified Treatment of Elastostatic and Rigid
  Contact for Real Time Haptics, Haptics-e, The
  Electronic Journal of Haptics Research
  (www.haptics-e.org), 2(1), 2001.
• Doug L. James and Dinesh K. Pai, Multiresolution
  Green's Function Methods for Interactive
  Simulation of Large-scale Elastostatic Objects,
  ACM Transactions on Graphics, Volume 22, No. 1,
  Jan. 2003.

6
Discrete Green's Functions (GFs)(in a
nutshell...)

• Reference BVP (RBVP)
• Greens function matrix
• General solution to RBVP (bar?specified BV)

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Example Displacement Constrained Model (white
dots indicate fixed vertices)
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Corresponding Greens Functions

• GF for this vertex is the response due to a
  vertex force in the x, y and z directions
• Use linear superposition to combine responses

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Anatomy of a Greens Function

• GF column corresponding to jth node, ?j

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Anatomy of a Greens Function

• GF corresponding to a single vertex

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Boundary Value Notation

• Various model descriptions/spaces possible
• Variables defined at n nodes/vertices
• x(x1,x2,,xn)T
• Continuous displacement u(x) and traction p(x)
  fields,
• e.g.,
• Discrete displacement u and traction p fields,
  e.g.,
• u(u1,u2,,un)T, uku(xk)
• p(p1,p2,,pn)T, pkp(xk)
• Force relationship fkakpk, ak??kd?
• Sign convention (uk,pk)?0

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Boundary Value Problem (BVP)

• Specified and unspecified nodal variables
• (?u, ?p) are complementary node sets specifying
  nodes with u or p constraints
• BVP Given and (?u, ?p) ? Compute v
• (Mixed nodal boundary conditions possible)

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Matrix BVP

• Linear models formally satisfy
• Boundary Value Eqn
• Matrix BVP
• b represents body force effects.

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Example BEM (from last class)

• Identification with BEM equations
• HuGp
• (ARTDEFO paper)

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Recap Solving the BVP
H u G p H,G large dense

• ? A v z, A large, dense

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Green's Functions (GFs)

• Reference BVP (RBVP)
• Greens function matrix
• Solutions to RBVP are

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Data-driven GF Formulation

• Excellent for interactive applications!
• Precompute GFs for speed
• Exploits linearity
• Avoids redundant work
• Optional boundary-only description for speed
• Black-box model definition

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Force-feedback Rendering
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More generally...

• GFs fundamental response of a linear system
• See whiteboard
• If Luf BVP then GF, G, satisfies LGdelta
  homog BC.
• In linear elasticity, there are formulae for
  free space solutions, and a few others.
• Survey of GFs for other physical phenomena
• We want Greens functions for a particular
  deformable object ( constraint configuration),
  hence
• Numerical approx ? discrete Greens functions

20
Fast Capacitance Matrix Algorithms
ARTDEFO Accurate Real Time Deformable Objects

In SIGGRAPH 99 Conference Proceedings, ACM
SIGGRAPH, 1999. (with Dinesh K. Pai) A Unified
Treatment of Elastostatic and Rigid Contact for
Real Time Haptics, Haptics-e, The Electronic
Journal of Haptics Research (www.haptics-e.org),
2(1), 2001. (w/ DKP)
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Exploiting BVP Equation Structure
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Boundary Value Changes
Constraint type (position?force) doesnt change
Only the value of the constraint changes
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Boundary Value Changes

• BV changes only affect z in Avz

• Traction-free BC are trivial

• 000...

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Boundary Condition Type Changes
Position ? Force constraint type switching
Intermediate BV changes
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Boundary Condition Type Changes

• BC change swaps a block column of A

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Sherman-Morrison-Woodbury

• Idea Exploit coherence between BVPs

• s-by-s capacitance matrix ????????

• Smaller matrix to invert and store!

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Motivation Changing BVP Type

• Traction?displacement constraint switching
• Example single nonzero constraint
• ? Self-effect relationship
• ? Equivalent traction constraint
• ? Equivalent Greens function (displ. constraint)

• Systematic formulation is CMA

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Capacitance Matrix Algorithms

• Solving general BVP using RBVPs GFs
• Low-rank updating techniques
• Long history in computing
• Sherman-Morrison-Woodbury et al. (50)
• Static reanalysis
• Contact mechanics Ezawa Okamoto 89
• Domain decomposition
• Real time simulation with precomputed GF Cotin
  et al. 96, JamesPai99

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CMA Notation

• Updated capacitance node list, S
• S(S1,S2,,Ss) for s updates.
• Contact compliance matrix, C
• C -ET?E
• Capacitance matrix
• E dense?sparse row expansion
• e.g., Sk, EIk??3n?3
• ET sparse?dense row extraction

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CMA Formulae

• Solution to any BVP in terms of ?
• Direct solver with input/output sensitivity
• O(s3) C-1 construction for s switched contacts
• O(s2sn) solve for s nonzero BC and n outputs

Using Sherman-Morrison-Woodbury... v v(0)
(E(?E)) C-1ET v(0) v(0) ?(I-EET) - EET v
B C -ET?E s-by-s capacitance matrix
_
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CMA Formulae (contd)
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Capacitance Matrix Algorithm (CMA)
_

• Compute C-1
• Compute v(0)
• Compute s updated BVs
• ET v C-1ET v(0) ??3s
• Add correction to v(0) to obtain v
• v(0) (E(?E)) (C-1ET v(0))
• (Simpler when v(0) -v )

_
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Demo!
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Early ARTDEFO Examples
ARTDEFO Accurate Real Time Deformable Objects

In SIGGRAPH 99 Conference
Proceedings, ACM SIGGRAPH, 1999. (with Dinesh K.
Pai)
35
Capacitance Inverse Updating

• Sequential inversion
• Use one C-1 to construct another
• Exploits temporal coherence between matrix BVP
• O(s2s?) cost for s? BC changes
• Effective updating of explicit matrix inverse

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Capacitance Inverse Updating
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Haptic Interaction