2. Emily Whiting John Ochsendorf Frédo Durand Massachusetts Institute Of Technology, USA Procedural Modeling of Structurally-Sound Masonry Buildings 2

3. virtual environments models require visual realism important to interact physically with surroundings state of the art simple models or react in scripted ways architectural models 3

4. structurally stable will look more realistic suitable for physical simulations –react to external forces architectural models our result 4

5. structurally stable will look more realistic suitable for physical simulations –react to external forces earthquake simulation architectural models our result 5

6. Generate models that are structurally sound Inverse Statics Procedural modeling quickly generates complex architectural models Masonry material goal 6 unstable input stable output

7. Focus is on visual realism, mainly for detail in façades our contribution: our contribution: introduce physical constraints related work procedural modeling Parish et al. [2001] Wonka et al. [2003] Müller et al. [2006] Müller et al. [2007] Lipp et al. [2008] [Muller et al. 2006] 7

8. [http://www.csiberkeley.com/] related work structural analysis Elastic Finite Element analysis wrong physical model for masonry not deformable 8 elastic material stress profile output is visualization solves forward problem not inverse

9. related work structural analysis geometric configuration rigid block assemblage [Heyman 1995] linear constraint formulation [Livesley 1978, 1992; RING software] elastic material masonry vs. 9 analyze material stress wrong physical model for masonry not deformable

10. Non-Structural Architectural free-form surfaces [Pottmann et al. 2008] Variational surface modeling [Welch and Witkin 1992] Layout design [Harada et al. 1995]Structural Structure optimization [Smith et al. 2002; Block et al. 2006] Tree modeling [Hart et al. 2003] Posing characters [Shi et al. 2007] related work design by optimization [Smith et al. 2002] [Pottmannet al. 2008] 10

11. procedural building generation analysis method for masonry inverse problem overview 11

12. procedural modeling [Muller et al. 2006] production rule input shape  production type (parameters) {output shapes} 12

13. procedural modeling input shape  production type (parameters) {output shapes} library of primitives production rule 13

14. procedural modeling input shape  production type (parameters) {output shapes} library of primitives production rule 14 production subdivision, scale, translation, …

15. procedural modeling input shape  production type (parameters) {output shapes} library of primitives production rule typical parameters height thickness of columns, walls, arches window size angle of flying buttresses 15 production subdivision, scale, translation, …

16. procedural modeling A  Repeat(“x”,0.2){B}B  Subdiv(“y”) {“wall”|C|”wall”} C  Subdiv(“y”){D|”arch”} A D  Subdiv(“x”){E} E  S(0.2,1,1){“wall”} 16

17. blocks: mass interfaces: contact surfaces between blocks Output procedural modeling 17

18. procedural building generation analysis method for masonry inverse problem overview 18

19. conditions for stability static equilibrium masonry compression-only analysis overview for each block

20. 20 conditions for stability static equilibrium masonry compression-only analysis overview requires tension feasible for each block

21. linear system of equations static equilibrium weights, torques geometry coefficients forces each block 21 A eq · f + w = 0 weight, w j f i f i+1

22. masonry compression-only positive normal forces no “glue” holding blocks together 22 normal force

23. linearized as pyramid friction cone 23 normal force friction force

24. summary model of feasibility Stablesolution exists Unstableno solution exists unknown forces, f A eq · f + w = 0 static equilibrium f n i ≥ 0 compression A fr · f ≤ 0 friction 24

25. summary model of feasibility Stablesolution exists Unstableno solution exists unknown forces, f A eq · f + w = 0 static equilibrium f n i ≥ 0 compression A fr · f ≤ 0 friction Problem binary,solution f exists yes/no 25

26. Problem binary,solution f exists yes/no tension required to stand how much “glue” Our Solution measure infeasibility summary model of feasibility A eq · f + w = 0 static equilibrium f n i ≥ 0 compression A fr · f ≤ 0 friction 26

27. tension required to stand how much “glue” Our Solution measure infeasibility measure of infeasibility A eq · f + w = 0 static equilibrium f n i ≥ 0 compression A fr · f ≤ 0 friction 27 relax constraint min f

28. f n i = f n i+ – f n i- where f n i+ ≥ 0 f n i- ≥ 0 tension split into positive, negative components normal force variable transformation compression 28 e.g. for compression forces f n i+ > 0 f n i- = 0

29. measure of infeasibility s.t. min f 29 A eq · f +w = 0 static equilibrium f n i+ ≥ 0, f n i- ≥ 0 allow tension A fr · f ≤ 0 friction Quadratic program

30. measure of infeasibility A eq · f +w = 0 static equilibrium f n i+ ≥ 0, f n i- ≥ 0 allow tension A fr · f ≤ 0 friction s.t. min f Quadratic program scalar output y 30

31. measure of infeasibility A eq · f +w = 0 static equilibrium f n i+ ≥ 0, f n i- ≥ 0 allow tension A fr · f ≤ 0 friction s.t. min f Quadratic program y = 0 feasible y > 0 measure of infeasibility scalar output y 31

32. measure of infeasibility 32

33. procedural building generation analysis method for masonry inverse problem overview 33

34. Procedural Model feasible? Analysis parameters optimization loop Update Parameters model from output parameters 34

35. Procedural Model feasible? parameters nested optimizations Update Parameters model from output parameters quadratic program minimum tension at parameters p i 35 p i+1

36. nested optimizations quadratic program minimum tension at parameters p i 36 p i+1 update parameters y(pi)y(pi)

37. nested optimizations quadratic program minimum tension at parameters p i 37 p i+1 y(pi)y(pi) find parameters for feasible structure, want y(p*) = 0

38. update parameters find parameters for feasible structure, want y(p*) = 0 nested optimizations quadratic program minimum tension at parameters p i 38 p i+1 y(pi)y(pi) nonlinear program arg min p y(p) MATLAB active-set algorithm, gradients with finite differencing

39. p0p0 arch example column width arch thickness column width arch thickness feasible region zero tension 39

40. Results 40

41. typical parameters building height thickness of columns, walls, arches window size angle of flying buttresses 41

42. results sainte chapelle tension forces unstable model from input parameters 42

43. results sainte chapelle 486 blocks, 17 sec/iter 4 parameter optimization unstable model from input parameters 43

44. results sainte chapelle 486 blocks 40 sec/iter 10 parameter optimization unstable model from input parameters 44

45. results Bezier curves 6 parameter optimizationunstable model from input parameters 45

46. results tower 32 parameter optimization 96 blocks,12 sec/iter unstable model from input parameters 46

47. results tower with safety factor unstable model from input parameters 32 parameter optimization 96 blocks,12 sec/iter 47

48. manually modify fixed parameters re-optimize free parameters to retain stability usage scenarios exploration Example user changes roof span automatically update angle of flying buttress 48

49. Load models into dynamic simulation Bullet Physics Engine [http://www.bulletphysics.com/] usage scenarios dynamics 49

50. ground shake Bullet Physics Engine [http://www.bulletphysics.com/] usage scenarios dynamics 50

51. Bullet Physics Engine [http://www.bulletphysics.com/] usage scenarios dynamics projectile 51

52. blocks removed Bullet Physics Engine [http://www.bulletphysics.com/] usage scenarios dynamics 52

53. Inverse analysis method Procedural modeling to specify design parameters Measure of infeasibility Optimization scheme to generate stable models summary stable buildings 53

54. Singapore-MIT Gambit Game Lab NSERC Canada Phillippe Siclait Sylvain Paris Yeuhi Abe Jovan Popovic Eugene Hsu 54 thanks...

55. Inverse analysis method Procedural modeling to specify design parameters Measure of infeasibility Optimization scheme to generate stable models summary stable buildings 55

57. extra slides 57

58. ground shake ∆ ground velocity= 4 m/s time step= 1/60 s model width~ 10 m Bullet settings: restitution (bounce) = 0.0 friction coefficient= 0.895 Bullet Physics Engine [http://www.bulletphysics.com/] usage scenarios dynamics 58

59. model#blocks#params#iterstime/iter Cluny 986 45794579 10 5 4 9 45.7 s 57.3 s 70.0 s 106.6 s arch 10260.1 s Sainte Chapelle 486 3 5 7 10 49684968 12.5 s 26.5 s 29.3 s 40.1 s tower 9632612.5 s barrel vault 140180.6 s performance 59