1. Elastic Deformation Using Boundary Element Methods
Maxim Garber

2. ArtDefo: Accurate Real Time Deformable Objects
Video
ArtDefo: Accurate Real Time Deformable Objects
Doug L. James and Dinesh K. Pai
University of British Columbia (in Canada )
presented at SIGGRAPH’99
Realtime
Haptic interaction
Implemented in Java3D
Linear Elastic Deformation
valid as long as deformations aren’t too big
Homogeneous Material

3. Outline This presentation goes backwards.
ArtDefo – a real time implementation of BEM
BEM – a method for solving linear elasticity
Linear Elasticity – the nasty math
[We start with the results, and then dig into the theory]

4. Outline ArtDefo – a real time implementation of BEM. BEM
Linear Elasticity

5. ArtDefo (0) Deformable object is represented as a polygonal mesh.
Rendered using a subdivision surface.
Object can be deformed by contact with one or more polygonal probes.

6. ArtDefo (1)
At each point xi of the boundary mesh we need to compute a displacement u(xi) for deformation and force p(xi), for haptics.
Let u and p be the concatenated force and displacement vectors for all n boundary nodes xi.
u and p have size 3*n

7. [We’ll see where this comes from later]
ArtDefo (2)
Assume we have a linear system:
Hu = Gp
that relates the displacements and forces.
[We’ll see where this comes from later]
H and G are dense 3n* 3n matrices, where n is the number of points on the boundary.
Think of radiosity form factors

8. ArtDefo (3) Boundary Conditions
Displacement conditions, set u(xi) at some boundary point xi, to represent collision.
Force conditions, not usually used, so force is free to change with deformation.
Once boundary conditions are specified we collect the unknowns in Hu = Gp to get a system:
Av = z
where A, v and z depend on the specific boundary value problem (BVP)….so they change as you change contact points or contact displacements 

9. ArtDefo (4) Interactive Simulation Changing Boundary Values Recall:
(Hu = Gp) => (Av = z)
for a particular BVP
ArtDefo (4)
Interactive Simulation
Changing Boundary Values
happens when contact point is the same, but displacement amount changes.
Take A0 for a particular BVP and precompute A0-1H and A0-1G.
Compute several such reference solutions.
Generate new solution, v, by:
uj and pj are changes in the specific boundary values for column j

10. ArtDefo (5) Interactive Simulation Recall: (Hu = Gp) => (Av = z)
for a particular BVP
ArtDefo (5)
Interactive Simulation
Changing Boundary Conditions
happens when probe makes contact (or breaks contact) with a boundary point.
Use coherence in contacts, i.e. running your finger along the surface, replaces one displacement boundary condition with another.
Use the Sherman-Morrison-Woodbury Formula to compute As-1 for a new BVP from precomputed matrix Ao-1.
[we’re not going into this any further, but its good to know that this trick exists]

11. Why Use This Instead of FEM?
Only solve for forces and displacements on the surface of the object
this is all you need for graphics and haptics
no need to generate mesh of entire volume
matrices as smaller
More accurate than FEM
forces are solved for just like displacements instead of using difference formulas

12. Why Not Use This For Everything?
FEM has sparser matrices which permit faster methods of solution in some cases.
FEM gives you internal information
engineering analysis
simulating fracture
FEM is more general
BEM is restricted to homogenous materials, although a small number of different regions could be modeled with internal boundaries.

13. Outline ArtDefo BEM – a method for solving linear elasticity

14. BEM (1) Let  = the object volume,  = the object boundary
Assume that we have an integral equation for linear elasticity:
[We’ll see where this comes from later]

15. BEM (2) Lets analyze this thing:
This equation must hold at every point x on 
u(y) and p(y) are the familiar displacements and forces at a point y
We have three 3x3 matrix functions :
c(x) is a smoothness function
u*(x,y) and p*(x,y) are tensor fields representing the known fundamental solutions, and can be precomputed
b(y) are the body forces, e.g. gravity, that act at every point of 
The integral over  can be entirely precomputed, although in this paper body forces are ignored, so this last term goes away.

16. BEM (3) To solve this equation:
Discretize  into a mesh, with nodal points at the centroids of the mesh elements.
Consider u and p as 3n-vectors of displacements and forces at all of the nodal points
Interpolate values at points x between the nodes:
u(x) = (ux, uy, uz)T = (x)u
p(x) = (px, py, pz)T = (x)p
where (x) is the 3 by n interpolation matrix
(linear interpolation)

17. BEM(4) Discritizing and solving for all N nodal points xi gives:
where ci = c(xi), ui = u(xi) and later pi = p(xi)
The integrals over  have been broken up into sums of integrals over each element j
The u(y) and p(y) terms have been replaced by interpolated values (y)u, and (y)p.

18. This is what we need for the ArtDefo formulation
BEM (5)
If we take the matrices and
and isolate the components that multiply against each uj and pj in u and p we get:
Finally, we roll in the ciui terms:
to get
This is what we need for the ArtDefo formulation

19. Lets go to the overheads!
Outline
ArtDefo
BEM
Linear Elasticity – the physics and math
Lets go to the overheads!

20. Conclusions The calculus behind BEM is: Not Trivial
Navier Equations
Theorems of Betti
Integral Equations
Loaded with Nasty Notation (some of which I simplified for this presentation)
Tensor notation
Einstein index notation

21. Conclusions The calculus behind BEM is: Well Established
there are books on the subject
Mostly Avoidable
in most cases you just need the final result to implement it

22. References
Papers
James, D. L., Pai, D. K. “ArtDefo: Accurate Real Time Deformable Objects”, Computer Graphics SIGGRAPH’99, July 1999
Hunter, P. and Pullin. A., “FEM/BEM Notes”, Department of Engineering Science, The University of Auckland, New Zealand

23. References Text Books (in order of usefulness)
Paris, F. and Cañas, J. “Boundary Element Method Fundamentals and Applications” Oxford University Press, 1997
main reference for the math overheads
Kythe, P. K., “An Introduction to Boundary Element Methods”, CRC Press, 1995
some minor help when the book above was not clear
Kreysig, E., “Advanced Engineering Mathematics” 7th Edition, John Wiley & Sons, 1993
very minor help with some math and notation

24. Nasty Notation
This is notation that I found in most text books on the subject of BEM. I got rid of most of this in the math I presented.
Einstein Subscript Notation
repeated subscript implies summation
comma in subscript implies partial derivative

25. Example: Who can figure this out?
Nasty Notation
This is notation that I found in most text books on the subject of BEM. I got rid of most of this in the math I presented.
Kronecker Delta Function
ij = 1, if i = j ij = 0, otherwise
Example: Who can figure this out?