topology optimization ? ? state-of-the-art and future ... · ole sigmund, mechanical engineering,...
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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)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Topology Optimization 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 Denmark
Topology Optimization Applications
Acoustics
Nano-photonics
Small antennas
Energy harvesting
Structural coloursFluids
Extreme materials
Cloaking
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Applications in Architecture/Design
Mutsuro Sasaki : Qatar National Convention Centre
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
SIMP-approach Bendsøe (1989), Zhou and Rozvany (1991), Mlejnek (1992)
Stiffness interpolation:
E
1
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Sensitivity analysis – adjoint method
Augmentedobjective function:
Differentiate:
Collect U’ terms:
Objective function Equilibrium (FEM)
Adjoint problem:
Final sensitivity:
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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 Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
The ”TopOpt3d App”
(NB! Only, Appstore, iOS and PC – see www.topopt.dtu.dk)Stats: November 2015:iOS: 2600, web: 730
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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 Denmark
Length-scale control and robustness
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Compliant mechanism design
Sensitivity filtering Density filtering
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Projection method Guest et al (2004)
Filtering Projection
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Local geometry control
ErosionSigmund (2007)
”Volume preserving”Xu et al (2010)
DilationGuest et al (2004)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Robust formulation
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 Denmark
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)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Robust electrostatic actuator design
Qian and Sigmund, CMAME, (2012)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Ultra high resolution TopOpt(overcoming the Duplo problem)
?
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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.”
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
+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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
+100M design variables
10
101
0.02
0.05
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Results: 135 million elements
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Material design and non-linearities
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Manufacturing using Direct Ink Writing
Ahn et al., JoVE 58 (2011)Compton & Lewis, Adv. Mater. 26(34), 5930-5935 (2014)
Nozzle
InkSyringe
Air
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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 Denmark
Optimized 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 Denmark
All designs printed row- and columnwise
Scale bars: 5 mm
Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Complete set of realized designs
Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Deformation pattern for = -0.8
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Numerics vs experiments
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 Denmark
Parameterization for any [-0.8, 0.8]
Clausen, Wang, et al., Adv. Mater. 27(37), 5523-5527 (2015)
Poisson’s ratio ν
Aver
age
erro
r
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
3D Poisson’s ratio -0.8 Small deformation:
Finite deformation:
Wang et al., 2016, to be submitted
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Thermofluidics
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Cooling fins for LED lamps
HYPERCOOL – Cool Danish Design
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Intuitive designs by industrial designer
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Thermofluidic equations
Incompressible Navier-Stokes equation for porous flow
Convection-diffusion equation
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Optimization of fluid mixing
Andreasen; Gersborg & OS, IJNMF, 2009, 61, 498-513
Mixing Crossing The chess board
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Natural convection cooler problem
hot
cold
cold
cold ?
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Natural convection cooler problemG
r
Alexandersen et al., IJNMF, 2014, 76, 699-721
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
3d results – temperature distribution
Gr=103
Gr=105 Gr=106
Gr=104
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
3d results – velocity magnitude
Alexandersen, Sigmund and Aage, Submitted, 2015
Gr=103
Gr=105 Gr=106
Gr=104
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
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
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Probability-based topology optimization
Karhunen-Loeve expansion
Schevenels, Lazarov & Sigmund, CMAME, 2011, 200, 3613-3627
MCStochastic collocationStochastic perturbation
Random geometry errors
Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denmark
Probability-based topology optimization
Schevenels, Lazarov & Sigmund, CMAME, 2011, 200, 3613-3627
Deterministic design Probabilistic design