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REVIEW ARTICLES

Topology optimization of continuum structures: A review*

[+] Author and Article Information
Hans A Eschenauer

Research Center for Multidisciplinary Analyses and Applied Structural Optimization, FOMAAS, University of Siegen, D-57068 Siegen, Germany; esch@fb5.uni-siegen.de

Niels Olhoff

Institute of Mechanical Engineering, Aalborg University, DK-9220 Aalborg East, Denmark; no@ime.auc.dk

Appl. Mech. Rev 54(4), 331-390 (Jul 01, 2001) (60 pages) doi:10.1115/1.1388075 History:
Copyright © 2001 by ASME
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Figures

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Topological mapping/transformation
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Topological properties of two-dimensional domains
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Conceptual processes of topology optimization
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Illustration of a mixed boundary value problem
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Elastic body subjected to external forces
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Microstructures for 2D continuum topology optimization problems: a) Perforated microstructure with rectangular holes in square unit cells, and b) Layered microstructure constructed from two different isotropic materials
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Construction of first-, second-, and third-rank microstructures by successive layering along different directions
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Model of a spatial rank three laminate with βi denoting the i-th length scale
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Relative stiffness vs volume density for the SIMP material model for different values of the penalization power p
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Microstructures of material and void realizing the material properties of the SIMP model with p=3 for a base material with Poisson’s ratio ν=1/3151
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Plane element with an underlying first-rank microstrucrure
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Second-rank microstructure with orthogonal layers
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The domain Ω with individual sub-domains Ω1
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Relative energy Erel=Ūiso*/Ūaniso*vs the volume density ρ of material for the entire range of principal stress ratios ω and η
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Example of finite element mesh refinement 330: a) Admissible design domain, loading and support conditions for example problem, and b,c,d) Topology results obtained for different mesh sizes
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Design domain, load, and support conditions for the 2D example problems
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Optimum solution to example problems 271; a) Solution by usage of second-rank 2D microstructures with orthogonal layers. The direction of the principal material stiffness is also shown; b) Solution based on layered microstructure of arbitrary rank governed by the moment formulation
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Cubic design domain subjected to four point loads P. (Material volume fraction=0.08): a) Design domain with loading and support conditions, b) Quadrupod solution to the problem with material densities less than 0.8 removed
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Oblong box-shaped design domain with clamped ends and a concentrated bending moment M acting in the center (Olhoff et al. 330): (Material volume fraction=0.3): a) Design domain with loading and support conditions; b) Topology solution represented with all material densities present; and c) Solution depicted with densities below 0.5 removed
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Solutions for a beam problem 327 with perimeter control (from 239): a)P̄=30 L, coarse mesh; b)P̄=30 L, fine mesh; c)P̄=24 L, coarse mesh; d)P̄=24 L, fine mesh; e)P̄=22 L, coarse mesh; f)P̄=22 L, fine mesh: and g)P̄=18 L, coarse mesh
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a) Shape and topology optimization by boundary variation and inserting holes; and b) Shape and topology optimization by material removal
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a) Sketch of roadway, loading and supports of a bridge; and b) Admissible design domain
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a) Optimum topology of bridge for maximum stress constraint; and b) Optimum topology of bridge for maximum deflection constraint
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a) Admissible design domain for a piston rod; and b) Optimal design for extreme piston position in upper dead center
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Examples of application of the SKO-method (rectangular design domain, different load cases, and boundary conditions): a) Cantilevered structure with tip load; b) Bridge structure with single load acting at the center; and c) Supporting structure for a prescribed (hatched) area subject to uniformly distributed load
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Bracket for a gearshift guide control
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Structural growth in growth cones by network topologies
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Flow chart of the MD method
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Variation of the topology and domain by inserting a circular hole (bubble): a) Global variation; b) Local variation; and c) Total (combined) variation
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Flowchart of the bubble method
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Hole positioning in plates with different supports
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Sphere in an infinite body
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Optimization of a cantilever disk within three topology classes
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Panel truss back-up structure of a 30m-Millimeter-Radio Telescope MRT (Version 1: h=150 mm; Version 2: h=250 mm)
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Topology classes with two different initial domains: a) Version 1; b) Version 2
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Deformation of the intermediate results of version 2 (magnifying factor 2000)
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Structural model of a handsaw grip
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Optimization history in single topology classes
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a) Principle sketch of a wing rib of an aeroplane; b) Load case 1: Pull-out after gliding flight; and c) Load case 2: Tank pressure (2 bar)
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Step-wise positioning and shape optimization of a wing rib in three topology classes; Load case 1: Pull-out after gliding flight
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Step-wise positioning and shape optimization of a wing rib in two topology classes; Load case 2: tank pressure
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a) Initial domain and domain after perforation; b) Truncated domain
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a) Design domain and boundary conditions for the cube; b) Optimal layouts obtained after 28, 37, and 50 iterations. Volumes represent 7, 3, and 1%, respectively, of the initial volume.
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Honeycomb microstructures with Poisson’s ratio equal to 0.75. Three different base cells. with: a) 21×12 elements, b) 63×36 elements, and c)15×15 elements, are used, Sigmund 378
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Negative Poisson’s ratio materials [a): −0.8; b): −0.6] obtained from base cells with 40×40 elements and enforcement of vertical symmetry 378
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Design of a material with negative thermal expansion coefficient. Left: base cell (design domain) with topology optimized bi-material microstructure that contracts when heated. The gray and the black material phases have a high and a low positive thermal coefficient, respectively, and the white sub-domains are void. Middle: thermal displacement of microstructure when heated. Right: Aggregate negative thermal expansion material 382
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Design domains for a micro gripper (left) and a micro displacement inverter and amplifier (right) 379
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Micro gripping mechanisms. Top left: optimum microgripper topology for one output port and its displacement pattern (bottom left). Top right: optimum microgripper topology for two output ports and its displacement pattern (bottom left). 379
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Micro displacement inverters 379
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Numerical simulation of the density distribution in a femoral head 87358: a) Finite element mesh, b) Calculated density distribution, and c) X-ray photograph
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Cantilever disk subjected to a) a vertical single load, and b) a single load acting under an angle of 45°
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a) Design domain of a cube and two intersecting loads (top view) and b) topology of the cube

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