Emily Whiting John Ochsendorf Frédo Durand Massachusetts Institute Of Technology, USA

2. Procedural Modeling ofStructurally-Sound Masonry Buildings Emily Whiting John OchsendorfFrédo Durand Massachusetts Institute Of Technology, USA

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

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

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

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

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

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

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

10. related work design by optimization 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] [Smith et al. 2002] [Pottmannet al. 2008]

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

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

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

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

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

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

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

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

19. analysis overview • conditions for stability • static equilibrium • masonry compression-only for eachblock

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

21. f i+1 linear system of equations static equilibrium • f i each block • weight, wj Aeq· f + w = 0 geometrycoefficients forces weights,torques

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

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

24. Aeq· f + w = 0 static equilibriumfni ≥ 0 compressionAfr· f ≤ 0 friction summary model of feasibility unknownforces, f Stable solution existsUnstable no solution exists

25. Aeq· f + w = 0 static equilibriumfni ≥ 0 compressionAfr· f ≤ 0 friction summary model of feasibility Problembinary,solution f exists yes/no unknownforces, f Stable solution existsUnstable no solution exists

26. summary model of feasibility Problembinary,solution f exists yes/no Our Solution measure infeasibility • Aeq· f + w = 0 static equilibriumfni ≥ 0 compressionAfr· f ≤ 0 friction tension required to stand how much “glue”

27. measure of infeasibility Our Solution measure infeasibility minf • Aeq· f + w = 0 static equilibriumfni ≥ 0compressionAfr· f ≤ 0 friction tension required to stand how much “glue” relax constraint

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

29. measure of infeasibility Quadratic program minf s.t. Aeq· f +w = 0 static equilibrium fni+ ≥ 0, fni-≥ 0 allow tension Afr· f ≤ 0 friction

30. measure of infeasibility Quadratic program scalar outputy minf s.t. Aeq· f +w = 0 static equilibrium fni+ ≥ 0, fni-≥ 0 allow tension Afr· f ≤ 0 friction

31. measure of infeasibility Quadratic program scalar outputy minf s.t. Aeq· f +w = 0 static equilibrium fni+ ≥ 0, fni-≥ 0 allow tension Afr· f ≤ 0 friction y = 0feasible • y > 0measure of infeasibility

32. measure of infeasibility

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

34. optimization loop Update Parameters parameters Analysis ProceduralModel feasible? model fromoutputparameters

35. nested optimizations Update Parameters parameters pi+1 ProceduralModel • quadratic program feasible? minimum tension at parameterspi model fromoutputparameters

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

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

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

39. arch example p0 archthickness arch thickness column width columnwidth feasible region zero tension

40. Results

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

42. results saintechapelle unstable model frominput parameters tension forces

43. results saintechapelle 486 blocks, 17 sec/iter unstable model frominput parameters 4 parameter optimization

44. results saintechapelle 486 blocks 40 sec/iter unstable model frominput parameters 10 parameter optimization

45. results Bezier curves unstable model frominput parameters 6 parameter optimization

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

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

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

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

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