Topology Optim ization ? ? State-of-the-Art and Future - - PowerPoint PPT Presentation

Topology Optim ization

State-of-the-Art and Future Perspectives

Ole Sigmund TopOpt-Group (www.topopt.dtu.dk)

• Dept. of Mechanical Engineering

Technical University of Denmark (DTU)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

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Design domain Optimal material redistribution Interpretation FE-Discretization

Topology Optim ization in Aerospace

Bendsøe and Kikuchi (1988)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Topology Optim ization Applications

Wind turbines (SUZLON and FE-Design GmbH) Automotive industry (Fabian Duddeck ) Reconstructive surgery (Paulino/Sinn-Hanlon) Micromachines (DTU Nanotech) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Topology Optim ization Applications

Acoustics Nano-photonics Small antennas Energy harvesting Structural colours Fluids Extreme materials Cloaking

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Applications in Architecture/ Design

Mutsuro Sasaki : Qatar National Convention Centre Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Discrete topopt form ulation

• Combinations:

! ( )! ! N N M M  N=10, M=5 => 252 N=20, M=10 => 185.000 N=40, M=20 => 1.4·109 N=100, M=50 => 1029

0/1 Integer problem

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

SI MP-approach

Bendsøe (1989), Zhou and Rozvany (1991), Mlejnek (1992)

Stiffness interpolation: E 

1 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Sensitivity analysis – adjoint m ethod

Augmented

• bjective function:

Differentiate: Collect U’ terms:

Objective function Equilibrium (FEM)

Adjoint problem: Final sensitivity:

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

The Topology Optim ization Process

Finite Element Analysis (Elastic, Thermal, Electrical, etc.) Sensitivity Analysis Regularization (filtering) Optimization (material redistribution) Initialize FEM ρe converged? Plot results STOP yes no

Sensitivity analysis by adjoint method Mathematical Programming, Method of Moving Asymptotes (MMA) by Svanberg (1987)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Regularization by low -pass filtering

Neighborhood: R Sensitivity filtering (Sigmund 1997, Sigmund&Maute 2012) Mesh refinement Checkerboards Density filtering (Bruns&Tortorelli/Bourdin 2001) PDE-based filtering (Lazarov&Sigmund 2011)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

The ”TopOpt App”

The ”TopOpt App”: AppStore (iOS) Google Play (Android) Web-version: www.topopt.dtu.dk

See w w w .topopt.dtu.dk for more Stats: November 2015: Android: 4900, iOS: 9000, web: 9700 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

The ”TopOpt3 d App”

(NB! Only, Appstore, iOS and PC – see w w w .topopt.dtu.dk)

Stats: November 2015: iOS: 2600, web: 730

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

TopOpt Rhino plugin

Århus Architect School, Technion and DTU By Amir, Maier, Søndergaard, Aage, et al.

Download at www.grasshopper3d.com/group/topopt (2000 downloads by December 2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Public Codes

99 Line basic Matlab (Including FE, grad’s, OC)

OS, A 99 line topology optimization code written in MATLAB, SMO, 2 0 0 1 , 21, 120-127

88 line advanced Matlab (+advanced filters)

Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. & OS, Efficient topology optimization in MATLAB using 88 lines of code, SMO, 2 0 1 1 , 43, 1-16

On multigrid-CG for efficient topology optimization

Amir, O.; Aage, N. & Lazarov, B.S., SMO, 2 0 1 4 , 49, 815-829

Topology optimization using PETSc:

An easy-to-use, fully parallel, open-source topology optimization framework

Aage, N; Andreassen, E. & Lazarov, B.S., 2 0 1 5 , SMO, 51, 565-572

Freely downloadable from www.topopt.dtu.dk

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Challenges and goals

Methods

• Manufacturing limitations/uncertainties
• Feature control – advanced geometry control
• Adaption to Additive Manufacturing (AM)
• Super large scale

Applications

• Extremal material design
• Non-linearities
• Multiphysics
• Wave propagation
• Multiscale

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Length-scale control and robustness

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Com pliant m echanism design

Sensitivity filtering Density filtering

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Projection m ethod Guest et al (2004)

Filtering Projection

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Local geom etry control

Erosion Sigmund (2007) ”Volum e preserving” Xu et al (2010) Dilation Guest et al (2004)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Robust form ulation

Sigmund, Acta Mechanica Sinica, 25, 227-239 (2009) Wang, Lazarov and Sigmund, SMO, 43, 767-784 , (2011)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Robust topopt form ulation

Uniform over/under etching

Blue print Over etched Under etched Unique length scale control: c.f. Wang, Lazarov and Sigmund, SMO, (2011), Qian and Sigmund, CMAME, (2012) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Robust electrostatic actuator design

Qian and Sigmund, CMAME, (2012) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Robust electrostatic actuator design

Qian and Sigmund, CMAME, (2012) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Ultra high resolution TopOpt

( overcom ing the Duplo problem )

?

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Previous w ork in aircraft w ing design

Rao et al., JAST, 2 0 0 9 , 61, 402 Kenway et al., AIAA, 2 0 1 4 Dunning et.al., AIAA, 2 0 1 4 Stanford et al., JA, 2 0 1 5 , 52, 1298 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Previous w ork in aircraft w ing design

Stanford & Dunning, Journal of Aircraft, 2 0 1 5 , 52, 1298-1311: “… the resulting structure typically bears no resem blance to traditional rib/ spar netw orks, which may indicate one of two things. The first is that the appropriate physics, load cases, and/ or constraint boundaries w ere not included in the

• ptimization problem, and if they had been, the resulting topology would qualitatively

approach a lattice of ribs and spars. The second is that the design problem was properly defined, and that the non-traditional topology m ay present an interesting new direction for efficient w ing structures.”

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

+ 1 0 0 M design variables

The code:

• PETSc based – highly scalable
• Solver: F-GMRES with MG preconditioner.
• Open source (topopt.dtu.dk)
• Includes filters, MMA, IO.
• Comes with minimum compliance example
• Aage; Andreassen & Lazarov, SMO, 2 0 1 5 , 51, 565-572

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

+ 1 0 0 M design variables

10 10 1 0.02 0.05

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

NASA Com m on Research Model

Geometry and pressure load data from NASA: Meshing by structured slices: ~1 billion elements (1216 x 256 x 3456)… … largest element size ~ 8 m m

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Results: 1 3 5 m illion elem ents

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Material design and non-linearities

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Material w ith negative Poisson’s ratio ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Output displacements Input displacements

• FE on one cell with periodic B.C.
• Minimize Poisson’s ratio
• Constraint on bulk modulus and symmetry

?

Sigmund (1995)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

3 D Manufacturing and testing

Andreassen, Lazarov & OS, MoM, 2 0 1 4 , 69, 1-10 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Negative therm al expansion coefficient 02 . 4

*

  

∆T

?

1 1

, E 

2 2

, E 

Air

Sigmund&Torquato, APL, 1 9 9 6 , 69, 3203-3205 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

3 d negative therm al expansion

Produced by Erik Andreassen Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Finite deform ations

Wang et al., J. Mech. Phys. Solids, 2 0 1 4 , 69, 156-174 Clausen et al., Adv. Mater., 2 0 1 5 , 27(37), 5523-5527

a b a b

Wang et al., CMAME, 2 0 1 4 , 276, 453-472

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Manufacturing using Direct I nk W riting

Ahn et al., JoVE 58 (2011) Compton & Lewis, Adv. Mater. 26(34), 5930-5935 (2014)

Nozzle Ink Syringe Air

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Design adapted to Direct I nk W riting

Designs printed row - and colum n-w ise

Uniform features desired

⇒

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Uniform feature design using superellipses

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Optim ized designs for  -0 .8 :0 .2 :0 .8

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

All designs printed row - and colum nw ise

Scale bars: 5 mm

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Com plete set of realized designs

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Deform ation pattern for  = -0 .8

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Num erics vs experim ents

0% 5% 10% 15% 20%

0.8 0.8

0% 5% 10% 15% 20%

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Param eterization for any  [ -0 .8 , 0 .8 ]

Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)

Poisson’s ratio ν Average error

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

3 D Poisson’s ratio -0 .8

Small deformation: Finite deformation:

Wang et al., 2 0 1 6 , to be submitted Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Therm ofluidics

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Cooling fins for LED lam ps

HYPERCOOL – Cool Danish Design

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

I ntuitive designs by industrial designer

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Therm ofluidic equations

Incompressible Navier-Stokes equation for porous flow Convection-diffusion equation

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Optim ization of fluid m ixing

Andreasen; Gersborg & OS, IJNMF, 2 0 0 9 , 61, 498-513

Mixing Crossing The chess board

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Natural convection cooler problem

hot cold cold cold

?

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Natural convection cooler problem

Gr

Alexandersen et al., IJNMF, 2014, 76, 699-721 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

3 d results – tem perature distribution

Gr=103 Gr=105 Gr=106 Gr=104

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

3 d results – velocity m agnitude

Alexandersen, Sigmund and Aage, Submitted, 2 0 1 5

Gr=103 Gr=105 Gr=106 Gr=104

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Conclusions

• TO is efficient in solving wide classes of engineering design

problems

• Here mostly concentrated on solids – lots of application in

fluids, thermofluidics, electromagnetics, nano-optics, etc.

• We are at the verge of being able to skip the post-processing

step and send TO results directly to (additive) manufacturing

• Still several interesting challenges:
• Large scale
• Non-linearities
• Multiphysics
• Multiscale
• Taking advantage of new manufacturing possibilties

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Further reading

TopOpt background

• OS & Maute, K., Topology optimization approaches: A comparative review, 2 0 1 3 , 48, 1031-1055
• OS, On the usefulness of non-gradient approaches in topology optimization, SMO, 2 0 1 1 , 43, 589-596
• Schevenels, Lazarov & OS, Robust TopOpt account. f. spat. varying man. err., CMAME, 2 0 1 1 , 200, 3613-3627
• Wang, Lazarov & OS, On projection methods, convergence and robust formulations in TopOpt, SMO, 2 0 1 1 ,

43, 767-784 Codes

• Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. & OS, Efficient TopOpt in MATLAB using 88 lines of

code, SMO, 2 0 1 1 , 43, 1-16

• Aage; Andreassen & Lazarov, B.S., TopOpt using PETSc: An easy-to-use, fully parallel, open-source topopt

framework, SMO, 2 0 1 5 , SMO, 51, 565-572 Material Design

• Andreassen, E.; Lazarov, B. & OS, Design of manuf. 3D extremal elastic microstr., MoM, 2 0 1 4 , 69, 1-10
• Wang, F.; OS & Jensen, J., Design of materials with prescribed nonlinear properties, JMPS, 2 0 1 4 , 69, 156-174
• Clausen; Wang; Jensen; OS & Lewis, Topology Optimized Architectures with Programmable Poisson's Ratio
• ver Large Deformations, Advanced Materials, 2 0 1 5 , 27, 5523-5527

Fluid Applications

• Andreasen, C. S.; Gersborg, A. R. & OS, TO of laminar fluid mixers, IJNMF, 2 0 0 9 , 61, 498-513
• Alexandersen, J.; Aage, N.; Andreasen, C. & OS, TO for natural convection probls, IJNMF, 2 0 1 4 , 76, 699-721

See w w w .topopt.dtu.dk for more

”TopOpt” App ”TopOpt3D” App ”TopOpt Game” (iOS, Android,web)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Localized random variations

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Probability-based topology optim ization

Karhunen-Loeve expansion

Schevenels, Lazarov & Sigmund, CMAME, 2 0 1 1 , 200, 3613-3627

MC Stochastic collocation Stochastic perturbation Random geometry errors

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Probability-based topology optim ization

Schevenels, Lazarov & Sigmund, CMAME, 2 0 1 1 , 200, 3613-3627

Determ inistic design Probabilistic design

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark

Sm ooth boundaries