emily whiting john ochsendorf frédo durand massachusetts institute of technology, usa

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Emily Whiting John Ochsendorf Frédo DurandMassachusetts Institute Of Technology, USA

Procedural Modeling ofStructurally-Sound Masonry Buildings

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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

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structurally stable• will look more realistic• suitable for physical simulations

– react to external forces

architectural models

our result

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structurally stable• will look more realistic• suitable for physical simulations

– react to external forces

earthquake simulation

architectural models

our result

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Generate models that are structurally sound

• Inverse Statics

• Procedural modelingquickly generates complex architectural models

• Masonry material

goal

unstable input stable output

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Focus is on visual realism, mainly for detail in façades

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]

8 [http://www.csiberkeley.com/]

related work structural analysis

Elastic Finite Element analysis

wrong physical model for masonrynot deformable

elastic material

stress profileoutput is visualizationsolves forward problem not inverse

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related work structural analysis

geometric configuration

rigid block assemblage [Heyman 1995]

linear constraint formulation[Livesley 1978, 1992; RING software]

elastic material

masonry

vs.

analyze material stress

wrong physical model for masonrynot deformable

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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]

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procedural building generation

analysis method for masonry

inverse problem

overview

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procedural modeling

[Muller et al. 2006]

production ruleinput shape production type (parameters) {output shapes}

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procedural modeling

input shape production type (parameters) {output shapes}

library of primitives

production rule

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procedural modeling

input shape production type (parameters) {output shapes}

library of primitives

production rule

production• subdivision, scale, translation, …

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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

production• subdivision, scale, translation, …

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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”}

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• blocks: mass• interfaces: contact

surfaces between blocks

Output

procedural modeling

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procedural building generation

analysis method for masonry

inverse problem

overview

conditions for stability

• static equilibrium

• masonry compression-only

analysis overview

0

0

torques

forcesfor eachblock

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conditions for stability

• static equilibrium

• masonry compression-only

analysis overview

requires tension

feasible

0

0

torques

forcesfor eachblock

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linear system of equations

static equilibrium

weights,torques

geometrycoefficients

forces

each block0

0

torques

forces

Aeq· f + w = 0

weight, wj

f if i+1

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masonry

0inf

compression-onlypositive normal forces

inf

no “glue” holding blocks together

normal force

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linearized as pyramid

friction cone

in

it

it fff 21 ,

inf i

tf 1

itf 2

normal forcefriction force

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summary model of feasibility

Stable solution existsUnstable no solution exists

unknownforces, f

Aeq· f + w = 0 static equilibrium

fni ≥ 0 compression

Afr· f ≤ 0 friction

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summary model of feasibility

Stable solution existsUnstable no solution exists

unknownforces, f

Aeq· f + w = 0 static equilibrium

fni ≥ 0 compression

Afr· f ≤ 0 friction

Problembinary,solution f exists yes/no

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Problembinary,solution f exists yes/no

tension required to stand

how much “glue”

Our Solutionmeasure infeasibility

summary model of feasibility

Aeq· f + w = 0 static equilibrium

fni ≥ 0 compression

Afr· f ≤ 0 friction

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tension required to stand

how much “glue”

Our Solutionmeasure infeasibility

measure of infeasibility

Aeq· f + w = 0 static equilibrium

fni ≥ 0 compression

Afr· f ≤ 0 friction

tension

relax constraint

minf

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fni = fn

i+ – fni- where fn

i+ ≥ 0 fni- ≥ 0

tension

split into positive, negative components

normal force variable transformation

compression

inf

e.g. for compression forces fni+ > 0

fni- = 0

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measure of infeasibility

2)( inf

s.t.

minf

Aeq· f +w = 0 static equilibrium

fni+ ≥ 0, fn

i-≥ 0 allow tension

Afr· f ≤ 0 friction

Quadratic program

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measure of infeasibility

2)( inf

Aeq· f +w = 0 static equilibrium

fni+ ≥ 0, fn

i-≥ 0 allow tension

Afr· f ≤ 0 friction

s.t.

minf

Quadratic program

scalar output y

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measure of infeasibility

2)( inf

Aeq· f +w = 0 static equilibrium

fni+ ≥ 0, fn

i-≥ 0 allow tension

Afr· f ≤ 0 friction

s.t.

minf

Quadratic program

y = 0 feasibley > 0 measure of infeasibility

scalar output y

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measure of infeasibility

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procedural building generation

analysis method for masonry

inverse problem

overview

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ProceduralModel

feasible?Analysis

parameters

optimization loop

Update Parameters

model fromoutput

parameters

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ProceduralModel

feasible?

parameters

nested optimizations

Update Parameters

model fromoutput

parameters

quadratic program

minimum tension at parameters

pi

pi+1

nested optimizations

quadratic program

minimum tension at parameters

pi

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pi+1 update parameters

y(pi)

update parameters

nested optimizations

quadratic program

minimum tension at parameters

pi

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pi+1

y(pi)

find parameters for feasible structure, want y(p*) = 0

update parameters

find parameters for feasible structure, want y(p*) = 0

nested optimizations

quadratic program

minimum tension at parameters

pi

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pi+1

y(pi)

nonlinear programarg minp y(p)

MATLAB active-set algorithm, gradients with finite differencing

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p0

arch example

column widtharch thickness

columnwidth

archthickness

feasible regionzero tension

2)( tension

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Results

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typical parameters

• building height• thickness of columns,

walls, arches• window size• angle of flying buttresses

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results sainte chapelle

tension forcesunstable model frominput parameters

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results sainte chapelle 486 blocks, 17 sec/iter

4 parameter optimizationunstable model frominput parameters

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results sainte chapelle 486 blocks40 sec/iter

10 parameter optimizationunstable model frominput parameters

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results Bezier curves

6 parameter optimizationunstable model frominput parameters

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results tower

32 parameter optimization

96 blocks,12 sec/iter

unstable model frominput parameters

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results tower

with safety factorunstable model from

input parameters 32 parameter optimization

96 blocks,12 sec/iter

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• manually modify fixed parameters• re-optimize free parameters to retain stability

usage scenarios exploration

Exampleuser changes roof span

automatically update angle of flying buttress

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Load models into dynamic simulation

Bullet Physics Engine[http://www.bulletphysics.com/]

usage scenarios dynamics

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ground shake

Bullet Physics Engine [http://www.bulletphysics.com/]

usage scenarios dynamics

51Bullet Physics Engine [http://www.bulletphysics.com/]

usage scenarios dynamicsprojectile

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blocksremoved

Bullet Physics Engine [http://www.bulletphysics.com/]

usage scenarios dynamics

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• Inverse analysis method• Procedural modeling to specify design parameters• Measure of infeasibility• Optimization scheme to generate stable models

summary stable buildings

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Singapore-MIT Gambit Game LabNSERC Canada

Phillippe SiclaitSylvain Paris

Yeuhi AbeJovan PopovicEugene Hsu

thanks...

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• Inverse analysis method• Procedural modeling to specify design parameters• Measure of infeasibility• Optimization scheme to generate stable models

summary stable buildings

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extra slides

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ground shake

∆ ground velocity = 4 m/stime step = 1/60 smodel width ~ 10 m

Bullet settings:restitution (bounce) = 0.0friction coefficient =

0.895

Bullet Physics Engine [http://www.bulletphysics.com/]

usage scenarios dynamics

model #blocks #params #iters time/iter

Cluny 986

4579

10549

45.7 s57.3 s70.0 s106.6 s

arch 10 2 6 0.1 s

SainteChapelle 486

35710

4968

12.5 s26.5 s29.3 s40.1 s

tower 96 32 6 12.5 s

barrel vault 140 1 8 0.6 s

performance

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