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Research Papers

Are Circular Shaped Masks Adequate in Adaptive Mask Overlay Topology Synthesis Method?

[+] Author and Article Information
Anupam Saxena1

 Indian Institute of Technology, Kanpur 208016, Indiaanupams@iitk.ac.in

1

Present address: RWTH Aachen, Deutschland.

J. Mech. Des 133(1), 011001 (Dec 29, 2010) (11 pages) doi:10.1115/1.4002973 History: Received March 15, 2010; Revised November 01, 2010; Published December 29, 2010; Online December 29, 2010

Previous versions of the material mask overlay strategy (MMOS) for topology synthesis primarily employ circular masks to simulate voids within the design region. MMOS operates on the photolithographic principle by appropriately positioning and sizing a group of negative masks and thus iteratively improves the material layout to meet the desired objective. The fundamental notion is that a group of circular masks can represent a local void of any shape. The question whether masks of more general shapes (e.g., any two-dimensional closed, nonself intersecting polygon) would offer significant enhancements in efficiently attaining the appropriate topological features in a continuum remains. This paper investigates the performance of two other mask types; elliptical and rectangular masks are compared with that of the circular ones. These are the respective modest representatives of closed curves and their polygonal approximations. First, two mean compliance minimization examples under resource constraints are solved. Thereafter, compliant pliers are synthesized using the three mask types. It is observed that the use of elliptical or rectangular masks do not offer significant advantages over the use of circular ones. On the contrary, the examples suggest that less number of circular masks are adequate to model the topology design procedure more efficiently. Thus, it is postulated that using generic simple closed curves or polygonal masks will not introduce significant benefits over circular ones in the MMOS based topology design algorithms.

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Figures

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Figure 1

Working of the material mask overlay strategy. Circular masks (thick circles) are used for material removal. Hexagonal cells inside a circular mask are void (white), while those outside the masks are all filled (black) with the desired material. Circular masks of different sizes can be positioned appropriately to yield local voids of a variety of shapes. Key: F=fixed boundary, I=input port, and Δ=expected output deformation.

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Figure 2

Algorithm for dynamic altering of number, position, and size of the overlaying masks

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Figure 9

(a) Design domain of the second problem, (b) design of stiff beam 2 obtained using circular masks, (c) design obtained using elliptical masks, and (d) design obtained using rectangular masks

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Figure 10

Specifications for the design of compliant pliers

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Figure 11

Three solutions of compliant pliers are evolved independently and simultaneously using circular masks. (a) The solutions are shown in three columns. In each column, the top figure shows the binary 0-1 solution. Circular masks are shown using dashed green lines with the number of masks used mentioned. The second figure shows the full pliers topology with smoothened boundaries. The deformed profile is depicted, and the output displacement Δ is specified. The two figures below exhibit the histograms pertaining to distribution of stresses and relative rotations, respectively. (b) Variation of the number of circular masks with the iterations.

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Figure 12

Three solutions of compliant pliers are evolved independently and simultaneously using elliptical masks. (a) The solutions are shown in three columns. In each column, the top figure shows the binary 0-1 solution. Elliptical masks (dashed green lines) with the number of masks used are shown. The second figure shows the full pliers topology with smoothened boundaries. The third and fourth figures in each column exhibit the histograms pertaining to distribution of stresses and relative rotations, respectively. (b) Variation of the number of elliptical masks with the iterations.

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Figure 13

Three solutions of compliant pliers are evolved independently and simultaneously using rectangular masks. The layout is similar to those in Figs.  1213.

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Figure 14

Convergence histories for the stiff beam 2 example using different mask types, varying their number adaptively

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Figure 3

Symmetric mutations about the line of symmetry (vertical in this case). Mask mutations, addition, and deletion are all performed in pairs.

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Figure 4

Relative rotation between two contiguous hexagonal cells

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Figure 5

Design domain for stiff beam 1

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Figure 6

Designs of stiff beam 1 (Fig. 5) obtained using circular masks. (a) A constant number of 100 circular masks are used. (b) The continuum in (a) is shown after post-processing. (c) The number of masks is allowed to vary with the iterations. The initial number of masks used is 100. (d) The continuum in (c) is shown after boundary smoothing. (e) Variation of SE with the number of function evaluations. (f) Variation of the number of masks with the number of iterations. The abscissa is to be multiplied by 8.

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Figure 7

Designs of stiff beam 1 obtained using elliptical masks. (a) A constant number of 100 circular masks are used. (b) The continuum in (a) is shown after boundary smoothing. (c) Solution obtained by allowing the number of masks to vary with the iterations (initial number is 100). (d) The continuum in (c) shown after boundary smoothing. (e) Variation of SE with the number of function evaluations. Red line: for fixed number of masks as 100; blue line: the number of masks is adaptively varied. (f) Variation of the number of masks with the iterations.

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Figure 8

Stiff beams (Fig. 5) obtained using rectangular masks. (a) Solution with 100 masks. (b) Solution in (a) after boundary smoothing. (c) Solution obtained with varying masks (initial number is 100). (d) The continuum in (c) shown after boundary smoothing. (e) History of variation of SE. Red line: for fixed number of masks as 100; blue dotted line: the number of masks is adaptively varied. (f) History of variation of the number of masks with the iterations.

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