Topology Optimization for Computational Fabrication Jun Wu Depart. - PowerPoint PPT Presentation

Topology Optimization for Computational Fabrication Jun Wu Depart. of Design Engineering, TU Delft www.jun-wu.net

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3 http://www.wikiwand.com/en/Delftware

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Computational Design and Fabrication Group • Charlie Wang (CUHK->TU Delft), Jun Wu (TU Munich->Denmark->Delft) • Generative design | Soft robots | 3D printing and robot manufacturing Rob Scharff RoboFDM, Wu et al. 2017 5

Outline • Basics of Topology Optimization • Topology Optimization for Additive Manufacturing 6

Bone Chair by Joris Laarman 7

Back Seat 8

9 www.jorislaarman.com

Topology Optimization Examples Frustum Inc. Airbus APWorks, 2016 Qatar national convention 10

Full-scale aircraft wing design Aage et al., Nature 2017 11

Classes of Structural optimization: Sizing, Shape, Topology Initial Optimized Sizing Shape Topology 12

A Toy Problem Design the stiffest shape, by placing 𝟕𝟏 Lego blocks into a grid of 𝟑𝟏 × 𝟐𝟏 • 10 20 13

A Toy Problem: Possible Solutions • Number of possible designs 200! 60! 200−60 ! = 7.04 × 10 51 𝐷 200,60 = – • Which one is the stiffest? Design C Design A Design B 14

A Toy Problem: Possible Solutions • Which one is the stiffest? Design C Design A Design B

A Toy Problem: Possible Solutions • Which one is the stiffest? Design C Design A Design B 16

Topology Optimization Animation 𝑔 17

Topology Optimization 1 1 1 2 𝑙𝑣 2 𝑑 = 2 𝑔𝑣 = 2 𝑉 𝑈 𝐿𝑉 Minimize: 𝑑 = Elastic energy 𝑙𝑣 = 𝑔 Subject to: 𝐿𝑉 = 𝐺 Static equation 𝑙 𝑣 𝑔 18

Topology Optimization 1 2 𝑉 𝑈 𝐿𝑉 Minimize: 𝑑 = Elastic energy Subject to: 𝐿𝑉 = 𝐺 Static equation 𝜍 𝑗 = 1 (solid) 0 (void) , ∀𝑗 𝜍 𝑗 ∈ [0 , 1] Design variables g = 𝜍 𝑗 − 𝑊 0 ≤ 0 Volume constraint 𝑗 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 19

Topology Optimization Compute 1 2 𝑉 𝑈 𝐿𝑉 Minimize: 𝑑 = displacement (KU=F) Subject to: 𝐿𝑉 = 𝐺 𝜍 𝑗 ∈ [0,1], ∀𝑗 Sensitivity g = 𝜍 𝑗 − 𝑊 0 ≤ 0 𝑗 analysis Update 𝜍 𝑗 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Converged? 1 1 1 1 No 1 1 1 1 1 1 1 1 1 1 Yes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20

Topology Optimization Animation 𝑔 21

Demo • www.topopt.dtu.dk 22

Topology Optimization 90% Compute 1 2 𝑉 𝑈 𝐿𝑉 Minimize: 𝑑 = displacement (KU=F) Subject to: 𝐿𝑉 = 𝐺 𝜍 𝑗 ∈ [0,1], ∀𝑗 Sensitivity g = 𝜍 𝑗 − 𝑊 0 ≤ 0 𝑗 analysis Update 𝜍 𝑗 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Converged? 1 1 1 1 No 1 1 1 1 1 1 1 1 1 1 Yes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 23

Geometric Multigrid: Solving 𝐿𝑣 = 𝑔 Successively compute approximations 𝑣 𝑛 to the solution u = lim 𝑛→∞ 𝑣 𝑛 • • Consider the problem on a hierarchy of successively coarser grids to accelerate convergence ℎ ≈ 𝑔 ℎ ℎ ≈ 𝑔 ℎ Ω ℎ Relax 𝐿 ℎ 𝑣 Relax 𝐿 ℎ 𝑣 Residual 𝑠 ℎ = 𝑔 ℎ − 𝐿 ℎ 𝑣 ℎ ← 𝑣 ℎ + 𝑓 ℎ ℎ Correct 𝑣 Restrict Interpolate 𝑠 2ℎ = 𝑆 ℎ 𝑓 ℎ = 𝐽 2ℎ ℎ 𝑓 2ℎ 2ℎ 𝑠 ℎ Ω 2ℎ Ω 4ℎ W. Briggs, A multigrid tutorial, 2000 Solve 𝐿 4ℎ 𝑓 4ℎ = 𝑠 4ℎ 24 ⋮

Memory-Efficient Implementation on GPU • On-the-fly assembly – Avoid storing matrices on the finest level • Non-dyadic coarsening (i.e., 4:1 as opposed to 2:1) – Avoid storing matrices on the second finest level Ω ℎ Ω 2ℎ Ω 4ℎ Wu et al., TVCG’2016 Dick et al., SMPT’2011 25 ⋮

High-Resolution Design Resolution: 621 × 400 × 1000 #Element 14.2m Time: 12 minutes 26

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Negative Poisson's ratio Negative thermal expansion Electric actuator Larsen et al. 1997 Sigmund &Torquato 1996 Sigmund 2000 Natural convection Fluid flow 29 Alexandersen et al. 2016 Maute & Pingen

A General Formulation Minimize: 𝑑(𝜍) Solve state Subject to: 𝜍 𝑗 ∈ [0,1], ∀𝑗 equation 𝑕 𝑗 (𝜍) ≤ 0 Sensitivity analysis Update 𝜍 𝑗 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Converged? 1 1 1 1 No 1 1 1 1 1 1 1 1 1 1 Yes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 30

Outline • Basics of Topology Optimization • Topology Optimization for Additive Manufacturing 31

Additive Manufacturing: Complexity is free Joshua Harker Scott Summit TU Delft & MX3D, 2015 32

Complexity is free? … Not really! • Printer resolution: Minimum geometric feature size • Layer-upon-layer: Supports for overhang region • Shell-infill composite Tiny details Supports Infill Ralph Müller Concept Laser GmhH mpi.fs.tum.de Paul Crompton 33

Outline • Basics of Topology Optimization • Topology Optimization for Additive Manufacturing – Geometric feature control by density filters – Geometric feature control by alternative parameterizations 34

Test case • Messerschmidt-Bölkow-Blohm (MBB) beam 35

Test case 36

Geometric feature control by density filters (An incomplete list) Reference Minimum feature size, Guest’04 Coating structure, Clausen’15 Self- supporting design, Langelaar’16 Porous infill, Wu’16 37

Infill in 3D Printing: Regular Structures 3dplatform.com www.makerbot.com 38

Infill in Bone: Porous Structures 39

Can we apply the principle of bone to 3D printing? 40

Topology Optimization Applied to Design Infill No similarity in structure Infill in the bone Topology optimization 41

Topology Optimization Applied to Design Infill • Materials accumulate to “important” regions The total volume 𝜍 𝑗 𝑤 𝑗 ≤ 𝑊 • 0 does not restrict local material 𝑗 distribution Infill by standard Infill in the bone topology optimization 42

Bone-like Infill in 2D Cross-section of a human femur 43

Approaching Bone-like Structures: The Idea • Impose local constraints to avoid fully solid regions 1 2 𝑉 𝑈 𝐿𝑉 c = Min: 𝜍 𝑗 = 0.0 𝐿𝑉 = 𝐺 s.t. : 𝜍 𝑗 ∈ [0,1], ∀𝑗 𝜍 𝑗 ≤ 𝑊 0 𝑗 𝜍 𝑗 = 0.6 𝜍 𝑗 ≤ 𝛽, ∀𝑗 𝜍 𝑗 = 1.0 𝑘∈𝛻 𝑗 𝜍 𝑘 𝜍 𝑗 = 𝑘∈𝛻 𝑗 1 𝛻 𝑗 Local-volume measure 44

Constraints Aggregation (Reduce the Number of Constraints) 1 𝑗=1,…,𝑜 𝜍 𝑗 max ≤ 𝛽 𝑞 ≤ 𝛽 𝜍 𝑗 ≤ 𝛽, ∀𝑗 𝑞 𝑞→∞ 𝜍 𝑞 = 𝜍 𝑗 lim 𝑗 Too many constraints! A single constraint A single constraint But non-differentiable and differentiable Approximated with 𝑞 = 16 45

Optimization Process: The same as in the standard topopt • Impose local constraints to avoid fully solid regions Compute 1 2 𝑉 𝑈 𝐿𝑉 c = Min: displacement (KU=F) 𝐿𝑉 = 𝐺 s.t. : 𝜍 𝑗 ∈ [0,1], ∀𝑗 Sensitivity 𝜍 𝑗 ≤ 𝑊 0 𝑗 analysis 𝜍 𝑗 ≤ 𝛽, ∀𝑗 Update 𝜍 𝑗 𝑘∈𝛻 𝑗 𝜍 𝑘 𝜍 𝑗 = 𝑘∈𝛻 𝑗 1 Converged? 𝛻 𝑗 No Local-volume measure Yes 46

A Test Example 47

Effects of Filter Radius and Local Volume Upper Bound R=6 𝛽, 𝑑 = (0.6, 76.9) (0.5, 96.0) 0.4, 130.0 R=12 (0.6, 73.9) (0.5, 91.2) 0.4, 119.8 48

2D Animations

Robustness wrt. Force Variations • Porous structures are significantly stiffer (126%) in case of force variations Local volume constraints Total volume constraint c = 30.54 c = 36.72 c’= 45.83 c’ =36.23 50

Robustness wrt. Material Deficiency • Porous structures are significantly stiffer (180%) in case of material deficiency Local volume constraints Total volume constraint c = 93.48 c = 76.83 c’= 134.84 c’ =242.77 51

Bone-like Infill in 3D Infill in the bone Optimized bone-like infill Wu et al., TVCG’2017 52

FDM Prints 53

Chair 54

Video 55

Geometric feature control by density filters (An incomplete list) Reference Minimum feature size, Guest’04 Coating structure, Clausen’15 Self- supporting design, Langelaar’16 Porous infill, Wu’16 56

Concurrent Shell-Infill Optimization Wu et al., CMAME 2017 57

Outline • Basics of Topology Optimization • Topology Optimization for Additive Manufacturing – Geometric feature control by density filters – Geometric feature control by alternative parameterizations 58

Geometric feature control by alternative parameterizations (An incomplete list) 1 1 1 1 1 1 1 1 1 1 1 1 1 Voxel, Prévost’13 1 1 1 1 1 1 1 1 1 1 Voronoi cells, Lu’14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Skin- frame, Wang’13 Reference: Voxel discretization 59 Offset surfaces, Musialski’15 Ray representation, Wu’16 Adaptive rhombic, Wu’16