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.

References

References
1.
Bendsøe
,
M. P.
, and
Kikuchi
,
N.
,
1988
, “
Generating Optimal Topologies in Structural Design Using a Homogenization Method
,”
Comput. Methods Appl. Mech. Eng.
,
71
(
2
), pp.
197
224
.10.1016/0045-7825(88)90086-2
2.
Qian
,
X.
,
2013
, “
Topology Optimization in B-Spline Space
,”
Comput. Methods Appl. Mech. Eng.
,
265
, pp.
15
35
.10.1016/j.cma.2013.06.001
3.
Bendsøe
,
M. P.
,
1989
, “
Optimal Shape Design as a Material Distribution Problem
,”
Struct. Multidiscip. Optim.
,
1
(
4
), pp.
193
202
.10.1007/BF01650949
4.
Zhou
,
M.
, and
Rozvany
,
G.
,
1991
, “
The COC Algorithm. Part II: Topological, Geometrical and Generalized Shape Optimization
,”
Comput. Methods Appl. Mech. Eng.
,
89
(
1
), pp.
309
336
.10.1016/0045-7825(91)90046-9
5.
Bendsøe
,
M. P.
, and
Sigmund
,
O.
,
2003
,
Topology Optimization: Theory, Methods, and Applications
,
Springer
,
New York
.
6.
Bourdin
,
B.
,
2001
, “
Filters in Topology Optimization
,”
Int. J. Numer. Methods Eng.
,
50
(
9
), pp.
2143
2158
.10.1002/nme.116
7.
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.
8.
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
.10.1007/BF01279647
9.
Borrvall
,
T.
,
2001
, “
Topology Optimization of Elastic Continua Using Restriction
,”
Arch. Comput. Methods Eng.
,
8
(
4
), pp.
351
385
.10.1007/BF02743737
10.
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
.10.1007/s00158-010-0594-7
11.
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
.10.1002/nme.3072
12.
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
.10.1007/s00158-009-0443-8
13.
Kumar
,
A. V.
, and
Parthasarathy
,
A.
,
2011
, “
Topology Optimization Using B-Spline Finite Elements
,”
Struct. Multidiscip. Optim.
,
44
(4), pp.
1
11
.10.1007/s00158-011-0650-y
14.
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
.10.1007/978-3-540-68443-5_8
15.
Chen
,
J.
,
Shapiro
,
V.
,
Suresh
,
K.
, and
Tsukanov
,
I.
,
2007
, “
Shape Optimization With Topological Changes and Parametric Control
,”
Int. J. Numer. Methods Eng.
,
71
(
3
), pp.
313
346
.10.1002/nme.1943
16.
Piegl
,
L. A.
, and
Tiller
,
W.
,
1997
,
The NURBS Book
,
Springer
,
New York
.10.1007/978-3-642-59223-2
17.
Farin
,
G.
,
2001
,
Curves and Surfaces for CAGD: A Practical Guide
,
Morgan Kaufmann
,
Burlington, MA
.
18.
Pysparse, Accessed Sept. 24, 2014. Available at: http://pysparse.sourceforge.net
19.
Svanberg
,
K.
,
1987
, “
The Method of Moving Asymptotes: A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
359
373
.10.1002/nme.1620240207
20.
Hunter
,
W.
,
2009
, “
Predominantly Solid-Void Three-Dimensional Topology Optimisation Using Open Source Software
,” Ph.D. thesis, University of Stellenbosch, Stellenbosch, South Africa.
21.
CDM Lab, “
Matlab Source Code for Topology Optimization in B-Spline Space
,” Accessed Oct. 10, 2014. Available at: http://cdm.me.wisc.edu/code/btop85.htm
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