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Research Papers

Optimization of Support Locations of Beam and Plate Structures Under Self-Weight by Using a Sprung Structure Model

[+] Author and Article Information
Gang-Won Jang

School of Mechanical Engineering, Kunsan National University, Kunsan, Jeonbuk 573-701, Korea

Ho Seong Shim

 Samsung Corning Precision Glass, Asan City, Chungnam 336-840, Korea

Yoon Young Kim1

School of Mechanical and Aerospace Engineering and National Creative Research Initiatives Center for Multiscale Design, Seoul National University, Shinlim-Dong, San 56-1, Kwanak-Gu, Seoul 151-742, Koreayykim@snu.ac.kr

1

Corresponding author.

J. Mech. Des 131(2), 021005 (Jan 06, 2009) (11 pages) doi:10.1115/1.3042154 History: Received March 19, 2008; Revised October 24, 2008; Published January 06, 2009

To find support locations minimizing uneven deformation is an important design issue in a large plate under self-weight. During the imprinting process of LCD panels, for instance, a large variation in the deflection of an LCD panel due to its self-weight deteriorates the quality of nanoscale imprinted lines. Motivated by this need, this research aims to develop an efficient gradient-based optimization method of finding optimal support locations of beam or plate structures under self-weight. To use a gradient-based algorithm, the support locating problem is formulated with continuous design variables. In this work, a beam or plate structure is assumed to be supported by a set of distributed springs, which are attached to all nodes of the discretized model of a given structure. The spring stiffness is made to vary continuously as a function of the design variable in which unsupported and supported states of a structure are represented with springs having limit stiffness values. Because elastically supported structures exhibit considerably different structural behaviors from structures without elastic supports, it is difficult to select an objective function fulfilling the design goal and ensuring convergence to distinct supported-unsupported states without ambiguous intermediate states. To address this issue, an extensive study is conducted and an appropriate objective function is then suggested. An optimization formulation using the objective function is presented and several numerical problems are considered to check the validity and usefulness of the developed formulation.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A beam-spring model for the support optimization problem

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Figure 2

Optimized supports by using the formulation in Eq. 3 for the case of (a) two, (b) four, (c) six, and (d) eight supports

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Figure 3

Comparison of (a) the L2 norm of deflection in Eq. 4 and (b) the mean compliance of supporting springs for a beam with four supports (the gray level indicates the magnitude of the L2 norm or the mean compliance)

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Figure 4

The deformed shapes of a uniformly loaded beam with (a) rigid supports (or supported with springs having infinite stiffness) and (b) elastic springs

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Figure 5

Comparison of the L2 norm of deflection and the mean compliance fcompt: (a) L2 norm for two supports, (b) fcompt for two supports, (c) L2 norm for four supports, and (d) fcompt for four supports (the gray level indicates the magnitude of the L2 norm or the mean compliance)

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Figure 6

Support locations obtained by the proposed formulation using Eq. 8 for (a) jiter(iteration number)=10, (b) jiter=20, (c) jiter=30, and (d) jiter=70 (converged with fL2=5.183×10−6 m2)

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Figure 7

Iteration history of the proposed approach for the four-support beam problem

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Figure 8

Optimized support locations by the proposed formulation in case of (a) two supports (fL2=3.582×10−5 m2), (b) six supports (fL2=4.753×10−8 m2), and (c) eight supports (fL2=1.773×10−9 m2)

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Figure 9

A subframe structure for which optimal support locations are to be determined

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Figure 10

Four-support locations for the subframe problem for (a) jiter(iteration number)=30(α=1), (b) jiter=50(α=10−1), (c) jiter=110α=10−2), and (d) for the converged state (α=10−4)

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Figure 11

Iteration history of the subframe problem with four supports (for plotting purposes, the initial value of the objective with a reset α value is scaled to be the same as the last value of the objective with the previous α value)

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Figure 12

Optimized support locations for the subframe in Fig. 9 in case of (a) six and (b) eight supports

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Figure 13

(a) A plate-spring model for the support location optimization problem of a plate (thickness: 0.01 m) and (b) its discretized model

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Figure 14

Comparison of (a) fL2 and (b) fcompt for various combinations of four-support locations in a rectangular plate of Fig. 1

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Figure 15

Optimized support locations for (a) 4, (b) 8, (c) 12, and (d) 14 supports

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Figure 16

A support optimization problem for a G-shaped plate (thickness: 0.01 m)

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Figure 17

Convergence history of five-support locations for the G-shaped plate problem after (a) 35 iterations (α=1), (b) 70 iterations (α=10−2), (c) 100 iterations (α=10−4), (d) 130 iterations (α=10−6), and (e) 150 iterations (α=10−7)

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Figure 18

Optimized support locations in the G-shaped plate for (a) three and (b) four supports

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