topology optimization ? ? state-of-the-art and future ... · stanford & dunning, journal of...

Topology OptimizationState-of-the-Art and Future Perspectives

Ole Sigmund

TopOpt-Group (www.topopt.dtu.dk)Dept. of Mechanical EngineeringTechnical University of Denmark (DTU)

Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark

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Design domain

Optimal material redistributionInterpretation

FE-Discretization

Topology Optimization in Aerospace Bendsøe and Kikuchi (1988)


Topology Optimization Applications

Wind turbines (SUZLON and FE-Design GmbH)

Automotive industry (Fabian Duddeck )

Reconstructive surgery (Paulino/Sinn-Hanlon)Micromachines (DTU Nanotech)


Topology Optimization Applications

Acoustics

Nano-photonics

Small antennas

Energy harvesting

Structural coloursFluids

Extreme materials

Cloaking


Applications in Architecture/Design

Mutsuro Sasaki : Qatar National Convention Centre


Discrete topopt formulation

• Combinations:!

( )! !N

N M M

N=10, M=5 => 252N=20, M=10 => 185.000N=40, M=20 => 1.4·109

N=100, M=50 => 1029

0/1 Integer problem


SIMP-approach Bendsøe (1989), Zhou and Rozvany (1991), Mlejnek (1992)

Stiffness interpolation:

E

1


Sensitivity analysis – adjoint method

Augmentedobjective function:

Differentiate:

Collect U’ terms:

Objective function Equilibrium (FEM)

Adjoint problem:

Final sensitivity:


The Topology Optimization 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)


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)


The ”TopOpt App”

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

See www.topopt.dtu.dk for moreStats: November 2015:Android: 4900, iOS: 9000, web: 9700


The ”TopOpt3d App”

(NB! Only, Appstore, iOS and PC – see www.topopt.dtu.dk)Stats: November 2015:iOS: 2600, web: 730


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 Denmark

Public Codes

99 Line basic Matlab (Including FE, grad’s, OC)OS, A 99 line topology optimization code written in MATLAB, SMO, 2001, 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, 2011, 43, 1-16

On multigrid-CG for efficient topology optimizationAmir, O.; Aage, N. & Lazarov, B.S., SMO, 2014, 49, 815-829

Topology optimization using PETSc:An easy-to-use, fully parallel, open-source topology optimization frameworkAage, N; Andreassen, E. & Lazarov, B.S., 2015, SMO, 51, 565-572

Freely downloadable from www.topopt.dtu.dk


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


Length-scale control and robustness


Compliant mechanism design

Sensitivity filtering Density filtering


Projection method Guest et al (2004)

Filtering Projection


Local geometry control

ErosionSigmund (2007)

”Volume preserving”Xu et al (2010)

DilationGuest et al (2004)


Robust formulation

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


Robust topopt formulation

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 Denmark

Robust electrostatic actuator design

Qian and Sigmund, CMAME, (2012)


Robust electrostatic actuator design

Qian and Sigmund, CMAME, (2012)


Ultra high resolution TopOpt(overcoming the Duplo problem)

?


Previous work in aircraft wing design

Rao et al., JAST, 2009, 61, 402

Kenway et al., AIAA, 2014

Dunning et.al., AIAA, 2014

Stanford et al., JA, 2015, 52, 1298


Previous work in aircraft wing designStanford & Dunning, Journal of Aircraft, 2015, 52, 1298-1311:

“… the resulting structure typically bears no resemblance to traditional rib/spar networks, which may indicate one of two things. The first is that the appropriate physics, load cases, and/or constraint boundaries were not included in the optimization 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 may present an interesting new direction for efficient wing structures.”


+100M 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, 2015, 51, 565-572


+100M design variables

10

101

0.02

0.05


NASA Common 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 mm


Results: 135 million elements


Material design and non-linearities


Material with negative Poisson’s ratio

? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?

Output displacements

Inpu

t di

spla

cem

ents

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

?

Sigmund (1995)


3D Manufacturing and testing

Andreassen, Lazarov & OS, MoM, 2014, 69, 1-10 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark

Negative thermal expansion coefficient

02.4*

∆T?

1 1,E 2 2,E

Air

Sigmund&Torquato, APL, 1996, 69, 3203-3205


3d negative thermal expansion

Produced by Erik AndreassenOle Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark

Finite deformations

Wang et al., J. Mech. Phys. Solids, 2014, 69, 156-174

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

a

b

a

b

Wang et al., CMAME, 2014, 276, 453-472


Manufacturing using Direct Ink Writing

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

Nozzle

InkSyringe

Air


Design adapted to Direct Ink Writing

Designs printed row- and column-wise Uniform features desired⇒

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


Uniform feature design using superellipses



Optimized designs for -0.8:0.2:0.8



All designs printed row- and columnwise

Scale bars: 5 mm



Complete set of realized designs



Deformation pattern for = -0.8


Numerics vs experiments

0% 5% 10% 15% 20%

0.8

0.8

0% 5% 10% 15% 20%



Parameterization for any [-0.8, 0.8]


Poisson’s ratio ν

Aver

age

erro

r


3D Poisson’s ratio -0.8 Small deformation:

Finite deformation:

Wang et al., 2016, to be submitted


Thermofluidics


Cooling fins for LED lamps

HYPERCOOL – Cool Danish Design


Intuitive designs by industrial designer


Thermofluidic equations

Incompressible Navier-Stokes equation for porous flow

Convection-diffusion equation


Optimization of fluid mixing

Andreasen; Gersborg & OS, IJNMF, 2009, 61, 498-513

Mixing Crossing The chess board


Natural convection cooler problem

hot

cold

cold

cold ?


Natural convection cooler problemG

r

Alexandersen et al., IJNMF, 2014, 76, 699-721


3d results – temperature distribution

Gr=103

Gr=105 Gr=106

Gr=104


3d results – velocity magnitude

Alexandersen, Sigmund and Aage, Submitted, 2015

Gr=103

Gr=105 Gr=106

Gr=104


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-processingstep and send TO results directly to (additive) manufacturing

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


Further readingTopOpt background• OS & Maute, K., Topology optimization approaches: A comparative review, 2013, 48, 1031-1055 • OS, On the usefulness of non-gradient approaches in topology optimization, SMO, 2011, 43, 589-596• Schevenels, Lazarov & OS, Robust TopOpt account. f. spat. varying man. err., CMAME, 2011, 200, 3613-3627 • Wang, Lazarov & OS, On projection methods, convergence and robust formulations in TopOpt, SMO, 2011,

43, 767-784Codes• Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. & OS, Efficient TopOpt in MATLAB using 88 lines of

code, SMO, 2011, 43, 1-16• Aage; Andreassen & Lazarov, B.S., TopOpt using PETSc: An easy-to-use, fully parallel, open-source topoptframework, SMO, 2015, SMO, 51, 565-572Material Design• Andreassen, E.; Lazarov, B. & OS, Design of manuf. 3D extremal elastic microstr., MoM, 2014, 69, 1-10 • Wang, F.; OS & Jensen, J., Design of materials with prescribed nonlinear properties, JMPS, 2014, 69, 156-174 • Clausen; Wang; Jensen; OS & Lewis, Topology Optimized Architectures with Programmable Poisson's Ratio

over Large Deformations, Advanced Materials, 2015, 27, 5523-5527 Fluid Applications• Andreasen, C. S.; Gersborg, A. R. & OS, TO of laminar fluid mixers, IJNMF, 2009, 61, 498-513 • Alexandersen, J.; Aage, N.; Andreasen, C. & OS, TO for natural convection probls, IJNMF, 2014, 76, 699-721

See www.topopt.dtu.dk for more

”TopOpt” App”TopOpt3D” App”TopOpt Game”

(iOS, Android,web)Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark

Localized random variations


Probability-based topology optimization

Karhunen-Loeve expansion

Schevenels, Lazarov & Sigmund, CMAME, 2011, 200, 3613-3627

MCStochastic collocationStochastic perturbation

Random geometry errors


Probability-based topology optimization

Schevenels, Lazarov & Sigmund, CMAME, 2011, 200, 3613-3627

Deterministic design Probabilistic design


Smooth boundaries

topology optimization ? ? state-of-the-art and future ... · stanford & dunning, journal of...

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