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Research Papers: Design Automation

Efficient Filtering in Topology Optimization via B-Splines1

[+] Author and Article Information
Mingming Wang

Mechanical, Materials and Aerospace
Engineering Department,
Illinois Institute of Technology,
Chicago, IL 60616
e-mail: mwang11@hawk.iit.edu

Xiaoping Qian

Department of Mechanical Engineering,
University of Wisconsin,
Madison, WI 53706
e-mail: qian@engr.wisc.edu

An earlier version of this paper appeared in 2014 ASME International Design Engineering Technical Conferences.

Matlab implementation of the 2D version is available at http://cdm.me.wisc.edu/code/btop85.htm.

2Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 9, 2014; final manuscript received December 3, 2014; published online January 9, 2015. Assoc. Editor: Shinji Nishiwaki.

J. Mech. Des 137(3), 031402 (Mar 01, 2015) (10 pages) Paper No: MD-14-1228; doi: 10.1115/1.4029373 History: Received April 09, 2014; Revised December 03, 2014; Online January 09, 2015

This paper presents a B-spline based approach for topology optimization of three-dimensional (3D) problems where the density representation is based on B-splines. Compared with the usual density filter in topology optimization, the new B-spline based density representation approach is advantageous in both memory usage and central processing unit (CPU) time. This is achieved through the use of tensor-product form of B-splines. As such, the storage of the filtered density variables is linear with respect to the effective filter size instead of the cubic order as in the usual density filter. Numerical examples of 3D topology optimization of minimal compliance and heat conduction problems are demonstrated. We further reveal that our B-spline based density representation resolves the bottleneck challenge in multiple density per element optimization scheme where the storage of filtering weights had been prohibitively expensive.

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References

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Bendsøe, M. P., and Sigmund, O., 2003, Topology Optimization: Theory, Methods, and Applications, Springer, New York.
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Sigmund, O., 1994, “Design of Material Structures Using Topology Optimization,” Danish Center for Applied Mathematics and Mechanics, Technical University of Denmark, Lyngby, Denmark, Report No. S69.
Haber, R. B., Jog, C. S., and Bendsøe, M. P., 1996, “A New Approach to Variable-Topology Shape Design Using a Constraint on Perimeter,” Struct. Multidiscip. Optim., 11(1), pp. 1–12. [CrossRef]
Borrvall, T., 2001, “Topology Optimization of Elastic Continua Using Restriction,” Arch. Comput. Methods Eng., 8(4), pp. 351–385. [CrossRef]
Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B. S., and Sigmund, O., 2011, “Efficient Topology Optimization in Matlab Using 88 Lines of Code,” Struct. Multidiscip. Optim., 43(1), pp. 1–16. [CrossRef]
Lazarov, B. S., and Sigmund, O., 2011, “Filters in Topology Optimization Based on Helmholtz-Type Differential Equations,” Int. J. Numer. Methods Eng., 86(6), pp. 765–781. [CrossRef]
Nguyen, T. H., Paulino, G. H., Song, J., and Le, C. H., 2010, “A Computational Paradigm for Multiresolution Topology Optimization (MTOP),” Struct. Multidiscip. Optim., 41(4), pp. 525–539. [CrossRef]
Kumar, A. V., and Parthasarathy, A., 2011, “Topology Optimization Using B-Spline Finite Elements,” Struct. Multidiscip. Optim., 44(4), pp. 1–11. [CrossRef]
Chen, J., and Shapiro, V., 2008, “Optimization of Continuous Heterogeneous Models,” Heterogeneous Objects Modelling and Applications (Lecture Notes in Computer Science), Vol. 4889, A.Pasko, V.Adzhiev, and P.Comninos, eds., Springer, Berlin, Germany, pp. 193–213. [CrossRef]
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Figures

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Fig. 1

Checkerboard appear in (a) can be suppressed either with fewer knot intervals (b) or high B-spline degrees (c). (a) p = 2, D28 × 56 × 112, (b) p = 2, D7 × 14 × 28, and (c) p = 8, D28 × 56 × 112. The cross sections of three respective designs are shown in the second row: (d) cross section of (a), (e) cross section of (b), and (f) cross section of (c).

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Fig. 2

Obtaining bivariate basis function N(x, y) through tensor product Bp(x)Bq(y). (a) Univariate basis functions B(x) and B(y) and (b) bivariate basis functions N(x, y).

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Fig. 3

Representing density distribution in 3D by B-splines. The intensity in B-spline coefficients corresponds to material density values.

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Fig. 4

Optimized density representation from degree p = 3, q = 2 B-splines of 30 × 10 knot intervals with analysis done by 30 × 10 quadrilateral linear elements [2]. (a) Distribution of 33 × 12 B-spline coefficients ρi (small circle) and (b) 3D view of density distribution ρ˜(x) and the B-spline coefficients ρi.

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Fig. 5

The CPU time (a) and memory usage (b) of density filter with different filter radius R for four tasks in each optimization iteration

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Fig. 6

Problem specification of (a) cantilever beam and (b) heat conduction

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Fig. 7

Topology optimization results using different degrees of B-splines at 14 × 18 × 56 knot intervals. (a) p2, I326, c6470, (b) p6, I583, c5884, and (c) p10, I942, c6325.

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Fig. 8

3D results optimized with degree 2 B-splines of different B-spline knot intervals. (a) Δ3, I460, c5230 (b) Δ6, I326, c6470, and (c) Δ12, I378, c7238.

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Fig. 9

3D results optimized with the density filter with analysis mesh 28 × 36 × 112. (a) R1.5, I452, c5324, (b) R3.5, I367, c6389, and (c) R5.0, I675, c7357.

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Fig. 10

The plot of CPU time (a) and memory usage (b) of B-spline representation for cantilever beam problem with different number of knot intervals

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Fig. 11

Topology optimization results of heat conduction problem with degree 2 B-splines of different knot interval numbers. The analysis mesh is 100 × 100 × 100. (a) p2, D100 × 100 × 100, memory: 84 MB, (b) p2, D80 × 80 × 80, memory: 84 MB, (c) p2, D60 × 60 × 60, memory: 84 MB, and (d) p2, D40 × 40 × 40, memory: 84 MB.

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Fig. 12

The plot of CPU time (a) and memory usage (b) of B-spline representation with different number of knot intervals for heat conduction problems

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Fig. 13

Illustration of multiple density Q4/M25 [12]. (a) MTOP scheme.

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Fig. 14

Density filter with analysis mesh: 14 × 18 × 36. R∕h = 2. (a) M = 1, c = 5656, (b) M = 23, c = 5777, (c) M = 33, c = 6479, and (d) M = 43, c = 7543.

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Fig. 15

Using quadratic B-splines with analysis mesh 42 × 54 × 108. (a) M = 1, c = 3697, (b) M = 23, c = 3637, (c) M = 33, c = 3884, and (d) M = 43, c = 4126.

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