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Research Papers

Lagrangian Description Based Topology Optimization—A Revival of Shape Optimization

[+] Author and Article Information
Weisheng Zhang, Jian Zhang

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China

Xu Guo

State Key Laboratory of Structural Analysis for
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China
e-mail: guoxu@dlut.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 19, 2015; final manuscript received December 30, 2015; published online January 28, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(4), 041010 (Jan 28, 2016) (16 pages) Paper No: JAM-15-1685; doi: 10.1115/1.4032432 History: Received December 19, 2015; Revised December 30, 2015

Unlike in the previous treatment where shape and topology optimization were carried out essentially in an Eulerian framework, the aim of the present work is to show how to perform topology optimization based on a Lagrangian framework, which is seamlessly consistent with classical shape optimization approaches, with use of a set of moving morphable components (MMCs). It is hoped that the present work may light up the revival of classical shape optimization in structural design and optimization and inspire some subsequent works along this direction. Some representative examples are also provided to illustrate the effectiveness of the proposed solution framework.

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Figures

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Fig. 1

Topology change cannot be achieved by homeomorphism (shape optimization)

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Fig. 2

The idea of MMCs-based topology optimization

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Fig. 3

Structural components of linearly varying thicknesses as basic building blocks of topology optimization

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Fig. 4

Geometry description of the kth structural component of linearly varying thicknesses. In this figure, ns and ts represent the outward normal vector and the tangent vector associate with the kth component.

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Fig. 5

The representation of geometry of structure (shape and topology) through the union of structural components

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Fig. 6

Structural components with different material properties

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Fig. 7

The calculation of ϕk(x,y) for the kth component

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Fig. 8

X-FEM analysis based on the fixed mesh

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Fig. 9

Calculation of boundary integral in Eq. (3.3) through Gauss quadrature

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Fig. 10

The short beam example

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Fig. 11

The initial design for the short beam example

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Fig. 12

Optimal topology of the short beam example (load imposed at point A). (a) Component plot and (b) contour plot.

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Fig. 13

Some intermediate iteration steps of the short beam example (load imposed at point A)

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Fig. 14

Optimal topology of the short beam example (load imposed at point B). (a)Componentplot and (b) contour plot.

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Fig. 15

Some intermediate iteration steps of the short beam example (load imposed at point B)

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Fig. 17

The initial designs for the MBB example

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Fig. 18

Some intermediate iteration steps of the MBB example in case A

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Fig. 19

Some intermediate iteration steps of the MBB example in case B

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Fig. 20

The compliant mechanism example

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Fig. 21

The initial designs for the compliant mechanism example

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Fig. 22

Some intermediate iteration steps of the compliant mechanism example in case A

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Fig. 23

Some intermediate iteration steps of the compliant mechanism example in case B

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Fig. 24

A schematic illustration of the concept of component derivative

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Fig. 25

The sketch of ∂Ωk−u, ∂Ωk−d,∂Ωk−l, ∂Ωk−r and their intersections with ∂Ω

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