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(1)

Topology Optimization for Computational Fabrication

Jun Wu, Niels Aage, Sylvain Lefebvre, Charlie Wang

(2)

Part 4: Topology Optimization for Appearance and Structure Synthesis

Sylvain Lefebvre

Inria

Topology Optimization for Computational Fabrication

(3)
(4)

Textures in Computer Graphics

(5)

Authoring textures

(6)

Authoring textures

Too much content to be done entirely manually

(7)

Texture Synthesis

Three main directions

By-example synthesis

Procedural synthesis

Simulation (e.g. erosion)

We will see both in the context of fabrication

(8)

Texture Synthesis

Three main directions

By-example synthesis

Procedural synthesis

Simulation (e.g. erosion)

We will see both in the context of fabrication

(9)

Texture synthesis: color formulation

Exemplar

Assumption (MRF):

(color field)

(10)

Volume Texture Synthesis

Solid Texture Synthesis [Kopf07]

(11)

On-surface texture synthesis

(12)

On-surface texture synthesis, the easier way

(13)

On-surface texture synthesis, the easier way

(14)

On-surface texture synthesis, the easier way

Distortion!

(15)

On-surface texture synthesis, the easier way

(16)

On-surface texture synthesis, the easier way

(17)

On-surface texture synthesis, the easier way

Blending

(18)

On-surface texture synthesis, the easier way

Selection

(19)

On-surface texture synthesis, the easier way

Transition error! Selection

(20)

On-surface texture synthesis, the easier way

Select Best

(21)

On-surface texture synthesis, the easier way

Random planes

Select Best

(22)

On-surface texture synthesis, the easier way

Random planes

Select Best

(23)

On-surface texture synthesis, the easier way

Plane choices

(24)

On-surface texture synthesis, the easier way

Shifts + Rotations

(25)

Labelling Problem

Surface neighborhood (2D)

Distortion error Transition error

(26)

Multiresolution Synthesis

• Upsample, jitter, correction [Lefebvre and Hoppe 2005]

(27)

Results

thing:168602 (Steelyd) thing:5506 (chylld)

Time 28.6s Time 14.7s Time 18.7s

(28)

Texture as structure?

Model + appearance + structure

Texture Synthesis? ???

(29)

Texture synthesis: structure formulation

Exemplar

(density field)

(30)

Printability

1. Connected components 2. Minimum thickness

3. No weak part (rigidity)

1. 2. 3.

(31)

Struts

Key ideas for structure synthesis

Pattern is stochastic

Exhibits degrees of freedom

Use pattern itself to locally reinforce structure

Synthesized

(32)

Key ideas for structure synthesis

Pattern is stochastic

Exhibits degrees of freedom

Use pattern itself to locally reinforce structure

Exemplar specifies local geometry

Large scale arrangement can be optimized ‘orthogonally’

Combination with topology optimization?

(33)

Key ideas for structure synthesis

Pattern is stochastic

Exhibits degrees of freedom

Use pattern itself to locally reinforce structure

Exemplar specifies local geometry

Large scale arrangement can be optimized ‘orthogonally’

Combination with topology optimization?

(34)

Pipeline

(35)

Pipeline

(36)

Pipeline

(37)

How to evaluate weak parts?

Similar to SIMP method, we consider ‘weak’ and ‘strong’ material

Issues:

Voxel grid is huge (~ 5M voxels)

Weak and strong hard to converge

We need 20-30 iterations synthesis/analysis

Too expensive

Approximate the pattern

(38)

Abstract Pattern Graph

(39)

Physical Simulation

Basic idea: replace graph by finite elements

In 2D: Quad & Triangle In 3D: Hex & Wedge

Local planarity assumption

Soft Rigid

(40)

Edge Selection Process

Solid

Empty

Selected

(41)

Simulation on the Final Mesh

153.9 KPa

30.5 KPa

Stress 99th%

(42)

Results – Structure + Color

ttotal: ttotal: ttotal:

(43)

From surface structure to final mesh

(44)

Results - Printouts

ttotal: ttotal: ttotal:

(45)
(46)

Other recent references

Designing Structurally-Sound Ornamental Curve Networks J. Zehnder, S. Coros, B. Thomaszewski, SIGGRAPH 2016

Stenciling: Designing Structurally-Sound Surfaces with Decorative Patterns C. Schumacher, B. Thomaszewski, M. Gross, SGP 2016

Synthesis of Filigrees for Digital Fabrication

W. Chen, X. Zhang, S. Xin, Y. Xia ,S. Lefebvre and W. Wang, SIGGRAPH 2016

All these works use a different point of view: discrete element distributions

(47)

Key ideas for structure synthesis

Pattern is stochastic

Exhibits degrees of freedom

Use pattern itself to locally reinforce structure

Exemplar specifies local geometry

Large scale arrangement can be optimized ‘orthogonally’

Combination with topology optimization?

(48)

Our Goal

Exemplar

Synthesize shapes under structural and appearance objectives

(49)

Local geometry

 ( )

E

Local geometry minimise

Example shape

Synthesized shape

) ( p N

p q

) (q N

)) (

), (

(

min q D N q N p

p

q

(50)

Structural properties

dx u

g. g ( )

rigidity

 ( )

E

minimise

g(

compliance

(51)

Gravity

Structural properties

dx u

g. g ( )

 ( )

E

minimise compliance

?

rigidity

(52)

Gravity

Structural properties

dx u

g. g ( )

 ( )

E

minimise

compliance rigidity

(53)

Challenge

)

0(

E

p

q D(N (p),N (q)) min

local geometry

minimise

dx u

g. g( )

rigidity

() E1

minimise

?

?

Gravity

(54)

Challenge

)

0(

E

p

q D(N (p),N (q)) min

local geometry

minimise

dx u

g. g( )

rigidity

() E1

minimise

?

?

Gravity

(55)

Minimize G(x) + λ C(x)

x15 x1.4 x1.3

Weighted sum

λ = 1 λ = 50 λ = 300

Ratio compliance

(56)

Appearance + rigidity

. max

) (

.u dx C

g g 



rigidity

() E1

such that

)

0(

E

p

q D(N (p),N (q)) min

appearance

minimise

Gravity

(57)

Solver

- Linear elasticity (FEM) - Derivatives C(x)

Appearance objective Compliance constraint Volume constraint

- Derivatives sum(x)

Gradient-based Optimization GCMMA [Svanberg95]

- Neighborhood matching [Barnes09, Busto10, Kaspar15]

- Derivatives A(x)

Not great due to combinatorial matching

(58)

Compliance Relaxation

α = 1.2, Vmax= 30%

α = 1.2, Vmax= 35% α = 1.2, Vmax= 40%

α = 1.4, Vmax= 30% α = 1.6, Vmax= 30%

(59)

Multiresolution

Compliance optimization

Appearance and compliance optimization

Level 0 Level 1 Level 2

Copt 0 Copt 1 Copt 2

(60)

Fabricated Objects

Contour extraction

(61)

Fabricated Objects: Shelves

(62)

Fabricated Objects: Tables

(63)

Fabricated Objects: Phone Stands

(64)

3D Structures

(65)

Fabricated Objects: Chairs

(66)

Texture Synthesis

Three main directions

By-example synthesis

Procedural synthesis

Simulation (e.g. erosion)

We will see both in the context of fabrication

(67)

Texture Synthesis

Three main directions

By-example synthesis

Procedural synthesis

Simulation (e.g. erosion)

We will see both in the context of fabrication

(68)

Foams in nature

Coral reef Metallic foam (chemical reaction)

(69)

Challenges: scale, fabricability, mechanical properties

Data size

4 GB (.ply)

Fabrication

Mechanical properties

?

(70)

Standard approach: periodic structures

(71)

Homogenisation

Representative Volume Element (RVE)

Homegenisation

Homogenized elasticity tensor [Andreassen and Andreasen 2014]

(72)

Drawbacks

[Pannetta et al. SIGGRAPH 2015]

Base Element

(73)

Periodic grid

Mapping?

Hard problem

Graded properties:

Possible, but transitions?

?

(74)

Procedural Voronoi Foams

Aperiodic, stochastic, stationary Mimics nature.

Trivially scales.

O(1) time + memory.

Fabricable.

Few pockets, connected, thickness ok.

Controllable elasticity

(75)

Procedural synthesis

Slice Fill with

structure F(x,y)

F(x,y) called in every slice ‘pixel’

(76)

Procedural synthesis

Target density Neighboring seeds Bisectors Voronoi edges

F(x,y): is q=(x,y) inside?

Local computations, O(1)

(77)

Gradation (stackless)

(78)

Gradation (stackless)

(79)

Elasticity control

(80)

Homogenisation

Young’s modulus

(81)

Crusty Knight

Results

(82)

Articulated Finger

Results

(83)

Cute Octopus

Results

(84)

Anisotropy

Results

(85)

Performances

(86)

Thank you for your attention!

Questions?

Sylvain Lefebvre

sylvain.Lefebvre@inria.fr

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