1. Overview Class #7 (Thurs, Feb 6) Black box approach to linear elastostatics Discrete Green’s function methods –Three parts: What are Green’s 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. Green’s Functions for Interactive Elliptic PDEs A RT D EFO : 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 Green’s 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. We’ll 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) Green’s function matrix General solution to RBVP (bar  specified BV)

7. Example: Displacement Constrained Model (white dots indicate “fixed” vertices)

8. Corresponding Green’s 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

9. Anatomy of a Green’s Function GF column corresponding to j th node,  j Block influence coefficient describes effect of j th SBV on i th UBV.

10. Anatomy of a Green’s Function GF corresponding to a single vertex…

11. Boundary Value Notation Various model descriptions/spaces possible Variables defined at n nodes/vertices: x=(x 1,x 2,…,x n ) T Continuous displacement u(x) and traction p(x) fields, e.g., Discrete displacement u and traction p fields, e.g., u=(u 1,u 2,…,u n ) T, u k =u(x k ) p=(p 1,p 2,…,p n ) T, p k =p(x k ) Force relationship: f k =a k p k, a k =  k d  Sign convention: (u k,p k )  0

12. 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)

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

14. Example: BEM (from last class) Identification with BEM equations Hu=Gp (A RT D EFO paper)

15. Recap: Solving the BVP  A v = z, A large, dense Red: BV specified Yellow: BV unknown H u = G p H,G large & dense Specify boundary conditions

16. Green's Functions (GFs) Reference BVP (RBVP) Green’s function matrix Solutions to RBVP are

17. 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

18. Force-feedback Rendering

19. More generally... GFs: fundamental response of a linear system See whiteboard: If Lu=f + BVP then GF, G, satisfies LG=delta + 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 Green’s functions for a particular deformable object (& constraint configuration), hence –Numerical approx  “discrete Green’s functions”

20. Fast Capacitance Matrix Algorithms A RT D EFO : 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)

21. Exploiting BVP Equation Structure A (0) v (0) =z (0) A (1) v (1) =z (1) A (2) v (2) =z (2) A (3) v (3) =z (3)

22. Boundary Value Changes Only the value of the constraint changes Constraint type (position  force) doesn’t change

23. Boundary Value Changes BV changes only affect z in Av=z A v = z H u = G p = Traction-free BC are trivial: 0+0+0+...

24. Boundary Condition Type Changes Position  Force constraint type switching Intermediate BV changes

25. Boundary Condition Type Changes BC change swaps a block column of A = H u = G p A v = z = +

26. Sherman-Morrison-Woodbury Idea: Exploit coherence between BVPs If s-by-s capacitance matrix  = = Smaller matrix to invert and store!

27. Motivation: Changing BVP Type Traction  displacement constraint switching Example: single nonzero constraint:  Self-effect relationship:  Equivalent traction constraint:  Equivalent Green’s function (displ. constraint): Systematic formulation is CMA

28. Capacitance Matrix Algorithms Solving general BVP using RBVP’s 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]

29. CMA: Notation Updated capacitance node list, S S=(S 1,S 2,…,S s ) for s updates. Contact compliance matrix, C C = -E T  E Capacitance matrix E: dense  sparse row expansion e.g., S={k}, E=I :k  3n  3 E T : sparse  dense row extraction

30. CMA: Formulae Solution to any BVP in terms of  Direct solver with input/output sensitivity O( s 3 ) C -1 construction for s switched contacts O(s 2 + sn ) solve for s nonzero BC and n outputs Using Sherman-Morrison-Woodbury... v = v (0) + (E+(  E)) C -1 E T v (0) v (0) = [  (I-EE T ) - EE T ] v + B C = -E T  E = s -by- s capacitance matrix _

31. CMA: Formulae (cont’d)

32. Capacitance Matrix Algorithm (CMA) 1.Compute C -1 2.Compute v (0) 3.Compute s updated BVs: E T v = C -1 E T v (0)  3s 4.Add correction to v (0) to obtain v: v (0) += (E+(  E)) (C -1 E T v (0) ) (Simpler when v (0) = -v ) v = v (0) + (E+(  E)) C -1 E T v (0) v (0) = [  (I-EE T ) - EE T ] v + B C = -E T  E = s -by- s capacitance matrix _ _

33. Demo!

34. Early A RT D EFO Examples A RT D EFO : 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(s 2 s  ) cost for s  BC changes Effective updating of explicit matrix inverse

36. Capacitance Inverse Updating

37. Haptic Interaction