Bendsøe and Topology Optim ization in Aerospace Kikuchi (1988) Topology Optim ization ? ? State-of-the-Art and Future Perspectives Design domain Ole Sigmund TopOpt-Group (www.topopt.dtu.dk) FE-Discretization Dept. of Mechanical Engineering Technical University of Denmark (DTU) Interpretation Optimal material redistribution Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Topology Optim ization Applications Topology Optim ization Applications Small antennas Acoustics Automotive industry (Fabian Duddeck ) Wind turbines (SUZLON and FE-Design GmbH) Cloaking Energy harvesting Extreme materials Reconstructive surgery (Paulino/Sinn-Hanlon) Fluids Micromachines (DTU Nanotech) Nano-photonics Structural colours Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
Applications in Architecture/ Design Discrete topopt form ulation Mutsuro Sasaki : Qatar National Convention Centre 0/1 Integer problem N=10, M=5 => 252 • Combinations: N=20, M=10 => 185.000 N ! N=40, M=20 => 1.4·10 9 ( N M )! M ! N=100, M=50 => 10 29 Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark SI MP-approach Sensitivity analysis – adjoint m ethod Bendsøe (1989), Zhou and Rozvany (1991), Mlejnek (1992) Objective function Equilibrium (FEM) Stiffness interpolation: Augmented E objective function: Differentiate: Collect U’ terms: 1 Adjoint problem: Final sensitivity: Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
The Topology Optim ization Process Regularization by low -pass filtering Neighborhood: Initialize FEM R Finite Element Analysis Checkerboards (Elastic, Thermal, Electrical, etc.) Sensitivity filtering (Sigmund 1997, Sigmund&Maute 2012) Sensitivity Analysis Regularization (filtering) Sensitivity analysis by adjoint method Optimization (material redistribution) Density filtering (Bruns&Tortorelli/Bourdin 2001) no Mesh refinement ρ e converged? yes Plot results Mathematical Programming, STOP Method of Moving Asymptotes (MMA) PDE-based filtering (Lazarov&Sigmund 2011) by Svanberg (1987) Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark The ”TopOpt App” The ”TopOpt3 d App” The ”TopOpt App”: AppStore (iOS) Google Play (Android) Web-version: www.topopt.dtu.dk Stats: November 2015: Stats: November 2015: See w w w .topopt.dtu.dk for more (NB! Only, Appstore, iOS and PC – see w w w .topopt.dtu.dk ) Android: 4900, iOS: 9000, web: 9700 iOS: 2600, web: 730 Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
Århus Architect School, TopOpt Rhino plugin Public Codes Technion and DTU 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 By Amir, Maier, Søndergaard, Aage, et al. Freely downloadable from www.topopt.dtu.dk Download at www.grasshopper3d.com/group/topopt (2000 downloads by December 2015) Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark Challenges and goals Methods • Manufacturing limitations/uncertainties Length-scale control and robustness • 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 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
Com pliant m echanism design Projection m ethod Guest et al (2004) Filtering Projection Sensitivity filtering Density filtering Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark Local geom etry control Robust form ulation Erosion Sigmund (2007) ”Volum e preserving” Xu et al (2010) Dilation Guest et al (2004) Sigmund , Acta Mechanica Sinica, 25 , 227-239 (2009) Wang, Lazarov and Sigmund, S MO, 43 , 767-784 , (2011) Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
Robust topopt form ulation Robust electrostatic actuator design Uniform over/under etching Over etched Blue print Under etched Qian and Sigmund, CMAME , (2012) Unique length scale control: c.f. Wang, Lazarov and Sigmund, S MO , (2011), Qian and Sigmund, CMAME , (2012) Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark Robust electrostatic actuator design Ultra high resolution TopOpt ( overcom ing the Duplo problem ) ? Qian and Sigmund, CMAME , (2012) Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
Previous w ork in aircraft w ing design 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 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 m ay present an interesting new direction for efficient w ing structures .” Rao et al., JAST, 2 0 0 9 , 61 , 402 Dunning et.al., AIAA, 2 0 1 4 Kenway et al., AIAA, 2 0 1 4 Stanford et al. , JA, 2 0 1 5 , 52 , 1298 Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark + 1 0 0 M design variables + 1 0 0 M design variables 10 0.05 10 1 0.02 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 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
NASA Com m on Research Model Results: 1 3 5 m illion elem ents 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 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark Material w ith negative Poisson’s ratio Input displacements ? ? ? ? ? ? ? ? ? ? ? ? Material design and non-linearities ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Output 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 Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
3 D Manufacturing and testing Negative therm al expansion coefficient ∆ T E 1 1 , ? E 2 2 , Air * 4 . 02 Andreassen, Lazarov & OS, MoM, 2 0 1 4 , 69 , 1-10 Sigmund&Torquato, APL , 1 9 9 6 , 69 , 3203-3205 Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark 3 d negative therm al expansion Finite deform ations a a b b Wang et al., J. Mech. Phys. Solids, 2 0 1 4 , 69 , 156-174 Wang et al., CMAME, 2 0 1 4 , 276 , 453-472 Produced by Erik Andreassen Clausen et al., Adv. Mater., 2 0 1 5 , 27 (37), 5523-5527 Ole Sigmund, Mechanical Engineering, Solid Mechanics Ole Sigmund, Mechanical Engineering, Solid Mechanics Technical University of Denm ark Technical University of Denm ark
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