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Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model

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This paper reports an efficient approach for uncertain topology optimization in which the uncertain optimization problem is equivalent to that of solving a deterministic topology optimization problem with multiple load cases. Probabilistic and fuzzy property of the directional uncertainty of the applied loads is considered in the topology optimization; the cloud model is employed to describe that property which can also take the correlations of the probability and fuzziness into account. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are utilized to obtain the final optimal solution. The proposed method is suitable for linear elastic problems with uncertain applied loads, subject to volume constraint. Several numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. In-depth discussions are also given on the effects of considering the probability and fuzziness of the directions of the applied loads on the final layout.
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RESEARCH PAPER
Layout optimization of continuum structures considering
the probabilistic and fuzzy directional uncertainty of applied
loads based on the cloud model
Jie Liu
1
&Guilin Wen
1
&Yi Min Xie
2
Received: 3 May 2015 /Revised: 9 August 2015 /Accepted: 13 August 2015 /Published online: 18 September 2015
#Springer-Verlag Berlin Heidelberg 2015
Abstract This paper reports an efficient approach for uncer-
tain topology optimization in which the uncertain optimiza-
tion problem is equivalent to that of solving a deterministic
topology optimization problem with multiple load cases.
Probabilistic and fuzzy property of the directional uncertainty
of the applied loads is considered in the topology optimiza-
tion; the cloud model is employed to describe that property
which can also take the correlations of the probability and
fuzziness into account. Convergent and mesh-independent
bi-directional evolutionary structural optimization (BESO) al-
gorithms are utilized to obtain the final optimal solution. The
proposed method is suitable for linear elastic problems with
uncertain applied loads, subject to volume constraint. Several
numerical examples are presented to demonstrate the capabil-
ity and effectiveness of the proposed approach. In-depth dis-
cussions are also given on the effects of considering the prob-
ability and fuzziness of the directions of the applied loads on
the final layout.
Keywords Topology optimization .Bi-directional
evolutionary structural optimization (BESO) .Probabilistic
and fuzzy uncertainty .Cloud model .Directional uncertainty
1 Introduction
Topology optimization (Bendsøe and Kikuchi 1988; Suzuki
and Kikuchi 1991; Sigmund and Petersson 1998; Xie and
Steven 1993; Duysinx and Bendsøe 1998;Wangetal.2003;
Kaya et al. 2010; Zuo et al. 2013; Leary et al. 2014; Vicente
et al. 2015;Guest2015) is a critical field in engineering.
Generally, it tends to search for the optimum distribution or
layout of material with a bounded domain, called the design
domain. This layout should optimize specified objectives or
target vectors with a set of constraints.Past three decades have
witnessed extensive studies of the topology optimization.
Among current topology, the evolutionary structural optimi-
zation (ESO) method plays a significant role (Xie and Steven
1993,1996,1997), which is based on the simple concept of
gradually removing inefficient material from a structure. As its
improved algorithm, bi-directional evolutionary structural op-
timization (BESO) method (Zuo et al. 2013; Querin et al.
1998;HuangandXie2009,2007)allowsmaterialtobere-
moved and added simultaneously according to sensitivity
numbers. Unfortunately, the sensitivity numbers of the void
elements are difficult to be estimated because there is hardly
any information available for void elements which are not
included in the finite element analysis. A soft-kill BESO
method (Huang and Xie 2007) utilizing the material interpo-
lation scheme with penalization is developed later to over-
come the problem.
Uncertainties are everywhere in the world, especially in the
engineering structures. Thus, although the field of topology
optimization is maturing, great efforts are still required when
accounting for uncertainties in the design process, such as
variations in the applied loads, geometrical dimensions, ma-
terial properties, among which uncertainty in applied loads is
often a governing issue. Formal incorporation of loading un-
certainty into the design optimization framework has recently
*Guilin Wen
glwen@hnu.edu.cn
Jie Liu
jliu@hnu.edu.cn
Yi Min Xie
mike.xie@rmit.edu.au
1
State Key Laboratory of Advanced Design and Manufacturing for
Vehicle Body, Hunan University, Changsha 410082, Peoples
Republic of China
2
Centre for Innovative Structures and Materials, School of Civil,
Environmental and Chemical Engineering, RMIT University, GPO
Box 2476, Melbourne 3001, Australia
Struct Multidisc Optim (2016) 53:81100
DOI 10.1007/s00158-015-1334-9
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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Book
Contents: Preface.- Introduction.- Basic Evolutionary Structural Optimization.- ESO for Multiple Load Cases and Multiple Support Environments.- Structures with Stiffness or Desplacement Contraints.- Frequency Optimization.- Optimization Against Buckling.- ESO for Pin- and Rigid-Jointed Frames.- ESO for Shape Optimization and the Reduction of Stress Concentrations.- ESO Computer Program Evolve97.- Author Index.- Subject Index.
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