ArticlePublisher preview available

Topology optimization of dynamic stress response reliability of continuum structures involving multi-phase materials

DOI:10.1007/s00158-018-2105-1
Authors:
Lei Zhao
Lei Zhao
Bin Xu at Northwestern Polytechnical University
Bin Xu
Yongsheng Han at Northwestern Polytechnical University
Yongsheng Han
Yi Min Xie at RMIT University
Yi Min Xie
Read publisher preview
Download citation
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

This paper proposes a methodology for maximizing dynamic stress response reliability of continuum structures involving multi-phase materials by using a bi-directional evolutionary structural optimization (BESO) method. The topology optimization model is built based on a material interpolation scheme with multiple materials. The objective function is to maximize the dynamic stress response reliability index subject to volume constraints on multi-phase materials. To solve the defined topology optimization problems, the sensitivity of the dynamic stress response reliability index with respect to the design variables is derived for iteratively updating the structural topology. Subsequently, an optimization procedure based on the BESO method is developed. Finally, a series of numerical examples of both 2D and 3D structures are presented to demonstrate the effectiveness of the proposed approach.
Figures - available from: Structural and Multidisciplinary Optimization
This content is subject to copyright. Terms and conditions apply.
  • A preview of this full-text is provided by Springer Nature.
  • Learn more
Preview content only
Content available from Structural and Multidisciplinary Optimization
This content is subject to copyright. Terms and conditions apply.
RESEARCH PAPER
Topology optimization of dynamic stress response reliability
of continuum structures involving multi-phase materials
Lei Zhao
1
&Bin Xu
1
&Yongsheng Han
1
&Yi Min Xie
2,3
Received: 11 May 2018 /Revised: 18 September 2018 /Accepted: 19 September 2018 /Published online: 26 October 2018
#Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
This paper proposes a methodology for maximizing dynamic stress response reliability of continuum structures involving multi-
phase materials by using a bi-directional evolutionary structural optimization (BESO) method. The topology optimization model
is built based on a material interpolation scheme with multiple materials. The objective function is to maximize the dynamic
stress response reliability index subject to volume constraints on multi-phase materials. To solve the defined topology optimi-
zation problems, the sensitivity of the dynamic stress response reliability index with respect to the design variables is derived for
iteratively updating the structural topology. Subsequently, an optimization procedure based on the BESO method is developed.
Finally, a series of numerical examples of both 2D and 3D structures are presented to demonstrate the effectiveness of the
proposed approach.
Keywords Dynamic stress response reliability .BESO method .Multi-phasematerials .Material interpolationscheme .Topol ogy
optimization
1 Introduction
There are several ways to reduce structural vibration, includ-
ing vibration control and structural dynamic optimization.
Structural topology optimization has been an important design
tool for obtaining lighter and more efficient structures. Since
the first widespread use of numerical topology optimization
proposed by Bendsøe and Kikuchi (1988), topology optimi-
zation has attracted many researchers from different fields.
Several topology optimization methods were developed, in-
cluding the solid isotropic material with penalization (SIMP)
(Bendsøe 1989;BendsøeandSigmund2003), level set meth-
od (Sethian and Wiegmann 2000; Mei and Wang 2004), the
evolutionary structural optimization (ESO) (Xie and Steven
1993,1996a), bi-directional evolutionary structural optimiza-
tion (BESO) (Huang and Xie 2007,2010), and moving
morphable components (MMC) methods (Zhang et al. 2016;
Guo et al. 2016) and several others.
Much of the previous research on structural dynamic opti-
mization has mainly dealt with dynamic behaviors. Xie and
Steven proposed an ESO method to solve a wide range of
frequency optimization problems, which include maximizing
or minimizing a chosen frequency of a structure, and maxi-
mizing the gap of arbitrarily given two frequencies and so on
(Xie and Steven 1996b). Pedersen proposed a method for
maximizing the first eigenfrequency based on SIMP method
(Pedersen 2000). Sigmund and Jensen optimized periodic ma-
terials and structures to either minimize the structural response
along boundaries (wave damping) or maximize the response
at certain boundary locations (waveguiding) (Sigmund and
Jensen 2003). Du and Olhoff dealt with topology optimization
problems formulated directly with the design objective of
minimizing the sound power radiated from the structural sur-
face(s) into a surrounding acoustic medium (Du and Olhoff
Responsible Editor: Emilio Carlos Nelli Silva
*Bin Xu
xubin@nwpu.edu.cn
1
School of Mechanics, Civil Engineering and Architecture,
Northwestern Polytechnical University, Xi’an 710072, People’s
Republic of China
2
Centre for Innovative Structures and Materials, School of
Engineering, RMIT University, GPO Box 2476, Melbourne 3001,
Australia
3
XIE Archi-Structure Design (Shanghai) Co., Ltd., Shanghai 200433,
China
Structural and Multidisciplinary Optimization (2019) 59:851–876
https://doi.org/10.1007/s00158-018-2105-1
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Up to now, most dynamic topology optimizations dealt with the dynamic compliance minimization or dynamic displacement response minimization of continuum structure [13][14][15][16]. Recently, Zhao et al. proposed a methodology for maximizing dynamic stress response reliability of continuum structures involving multi-phase materials on the macro scale [17]. Long et al. investigated a method for stress-constrained topology optimization of continuum structure under harmonic load excitations using sequential quadratic programming [18]. ...
... Moreover, the number of the cellular materials used in topology optimization is limited. For these reasons, the material interpolation scheme based on SIMP method is adopted in this paper [17]. Therefore, the i-th elemental elastic matrix D s i and mass density ρ s i on the macro scale can be expressed as ...
Concurrent design of composite macrostructure and cellular microstructure with respect to dynamic stress response under random excitations
Article
This paper proposes a concurrent topology optimization method of macrostructural material distribution and periodic microstructure considering dynamic stress response under random excitations. The optimization problem is the minimization of the dynamic stress response of the macrostructure subject to volume constraints in both macrostructure and microstructure. To ensure the safety of the macrostructure, a new relaxation method is put forward to establish a relationship between the dynamic stress limit and the mechanical properties of microstructure. The sensitivities of the dynamic stress response with respect to the design variables in two scales, i.e., macro and micro scales, are derived. Then, the aforementioned optimization problem is solved by the bi-directional evolutionary structural optimization (BESO) method. Finally, several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.
... Besides the SIMP method and the level set method, multi-material topology optimization using the phase field method (Zhou and Wang 2006;Tavakoli 2014) and the ESO method (Radman et al. 2014;Zhao et al. 2019) have also been presented. There are also some special issues in the multi-material structural design that have been widely studied, such as the integrated design of the structural topology and the embedded components Kang et al. 2016), topology optimization of the coated structures (Clausen et al. 2015;Wang and Kang 2018b), concurrent design of multiple microstructural topologies and their macroscale distributions (Xu et al. 2015;Wang and Kang 2019), etc. ...
A multi-material topology optimization method based on the material-field series-expansion model
Article
Full-text available
Multi-material topology optimization is an important issue in the structural and multidisciplinary design. Compared with single-material topology optimization, the multi-material design usually involves more design variables and poses higher requirement for the convergence and efficiency of the topology optimization method. This paper proposes a new multi-material topology optimization strategy based on the material-field series-expansion (MFSE) model. For a structure composed of m different phases of solid materials, m individual material fields are introduced to describe the topology distribution in the multi-material representation model. Herein, each material field is expressed as a linear combination of the eigenvectors and corresponding expansion coefficients based on a reduced series expansion. Thus, the number of design variables can be significantly reduced. Moreover, a new type of smooth Heaviside projection on the material-field function is introduced in the MFSE model, which releases the bound constraints of the material field from the optimization formulation. In this way, the efficiency of the MFSE method is further improved when solving multi-material design problems. Several 2D and 3D numerical examples are presented to show the validity and efficiency of the proposed multi-material method.
A novel dynamic reliability-based topology optimization (DRBTO) framework for continuum structures via interval-process collocation and the first-passage theories
Article
In view of the general inertia and damping features as well as the inevitable uncertainty factors in engineering structures, a novel dynamic reliability-based topology optimization (DRBTO) strategy is investigated for time-variant mechanical systems with overall consideration of material dispersion and loading deviation effects. The static interval-set model is first utilized to quantify multi-source uncertainty inputs and the transient interval-process model is then established to characterize unknown-but-bounded response results, which can be readily solved through the proposed interval-process collocation approach combined with a classical Newmark difference scheme. Different from the traditional deterministic design framework, the present DRBTO scheme will directly consider new reliability constraints, for which the non-probabilistic time-variant reliability (NTR) index is mathematically deduced using the first-passage principle. In addition, the issues related to uncertainty-oriented design sensitivity and filtering method are discussed. The usage and effectiveness of DRBTO are demonstrated with three numerical examples.
An incremental form interpolation model together with the Smolyak method for multi-material topology optimization
Article
An incremental form interpolation model integrated with the anisotropic Smolyak method and the Non-Uniform Rational B-Spline (NURBS) method is proposed for multiple material topology optimization. Only a single category of design variables is exploited to characterize the material properties for any number of material types and the Smolyak method embedded within the NURBS method is devised to represent the topology density field while keeping the number of polynomials to a minimum, leading to a significant reduction in the number of design variables. An arbitrary number of volume and/or mass constraints can be implemented with the introduction of the characterization function to indicate the existence information for each material. Finally, the effectiveness of the proposed methodology is illustrated by three numerical examples.
A new three-dimensional topology optimization method based on Moving Morphable Components (MMCs)
Article
Full-text available
In the present paper, a new method for solving three-dimensional topology optimization problem is proposed. This method is constructed under the so-called Moving Morphable Components (MMC) based solution framework. The novel aspect of the proposed method is that a set of structural components is introduced to describe the topology of a three-dimensional structure and the optimal structural topology is found by optimizing the layout of the components explicitly. The standard finite element method with ersatz material is adopted for structural response analysis and the shape sensitivity analysis only need to be carried out along the structural boundary. Compared to the existing methods, the description of structural topology is totally independent of the finite element/finite difference resolution in the proposed solution framework and therefore the number of design variables can be reduced substantially. Some widely investigated benchmark examples, in the three-dimensional topology optimization designs, are presented to demonstrate the effectiveness of the proposed approach.
A unified aggregation and relaxation approach for stress-constrained topology optimization
Article
Full-text available
In this paper, we propose a unified aggregation and relaxation approach for topology optimization with stress constraints. Following this approach, we first reformulate the original optimization problem with a design-dependent set of constraints into an equivalent optimization problem with a fixed design-independent set of constraints. The next step is to perform constraint aggregation over the reformulated local constraints using a lower bound aggregation function. We demonstrate that this approach concurrently aggregates the constraints and relaxes the feasible domain, thereby making singular optima accessible. The main advantage is that no separate constraint relaxation techniques are necessary, which reduces the parameter dependence of the problem. Furthermore, there is a clear relationship between the original feasible domain and the perturbed feasible domain via this aggregation parameter.
A Moving Morphable Void (MMV)-based explicit approach for topology optimization considering stress constraints
Article
Topology optimization considering stress constraints has received ever-increasing attention in recent years for both of its academic challenges and great potential in real-world engineering applications. Traditionally, stress-constrained topology optimization problems are solved with approaches where structural geometry/topology is represented in an implicit way. This treatment, however, would lead to problems such as the existence of singular optima, the risk of low accuracy of stress computation, and the lack of direct link between optimized results and computer-aided design/engineering (CAD/CAE) systems. With the aim of resolving the aforementioned issues straightforwardly, a Moving Morphable Void (MMV)-based approach is proposed in the present study. Compared with existing approaches, the distinctive advantage of the proposed approach is that the structural geometry/topology is described in a completely explicit way. This feature provides the possibility of obtaining optimized designs with crisp and explicitly parameterized boundaries using much fewer numbers of degrees of freedom for finite element analysis and design variables for optimization, respectively. Several numerical examples provided demonstrate the effectiveness and advantages of the proposed approach.
Topological design of structures under dynamic periodic loads
Article
Most of forces acted on real structures are dynamic and topology optimization of dynamic structures has aroused wide attention over the past years. Due to the complexity of dynamic behavior, achieving clear 0/1 optimal topology of dynamic structures is still challenging. This paper aims to develop a topology optimization algorithm of dynamic structures under periodic loads based on the bi-directional evolutionary structural optimization (BESO) method. To minimize the dynamic compliance under the single or multiple excitation frequencies, four typical topology optimization problems are proposed for different scenarios. To solve the defined topology optimization problems, sensitivity analysis with regard to the variation of design variables is conducted for iteratively updating the structural topology. Since BESO uses discrete design variables, the resulting solid-void solutions show unambiguous topologies of dynamic structures. Various 2D and 3D numerical examples are given to demonstrate the capability of the proposed method for obtaining optimal designs of dynamic structures under periodic loads.
Topology optimization of thin plate structures with bending stress constraints
Article
This paper presents the topology optimization of thin plate structures with bending stress constraints. To avoid the stress singularity phenomena, the qp-relaxation is used for local stress interpolation. The local stress constraints are aggregated into a single global constraint based on the p-norm stress measure. The framework of the topology optimization is constructed using the commercial finite element software ANSYS. In the presented work, the volume of the structure is minimized with the global stress constraint. Numerical examples are demonstrated to validate the proposed topology optimization method.
Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons
Article
  • Jul 2016
In the present paper, an explicit topology optimization approach based on moving morphable components (MMC) with curved skeletons (central lines) is proposed. This is achieved by constructing the topology description function (TDF) which describes the geometry of a structural component with curved skeleton explicitly in an elegant way. The proposed method has very flexible geometry modeling capability and represents a substantial improvement of the geometry modeling capability of existing MMC-based approaches. Numerical examples demonstrate the effectiveness of the proposed approach.
Topology optimization of continuum structures with uncertain-but-bounded parameters for maximum non-probabilistic reliability of frequency requirement
Article
A method for the non-probabilistic reliability optimization on frequency of continuum structures with uncertain-but-bounded parameters is proposed. The objective function is to maximize the non-probabilistic reliability index of frequency requirement.The corresponding bi-level optimization model is built, where the constraints are applied on the material volume in the outer loop and the limit state equation in the inner loop. The non-probabilistic reliability index of frequency requirement is derived by the analytical method for the continuum structure with the uncertain elastic module and mass density. Further, the sensitivity of the non-probabilistic reliability index with respect to the design variables is analyzed. The topology optimization in the outer loop is performed by a bi-directional evolutionary structural optimization (BESO) method, where the numerical techniques and the optimization procedure of BESO method are presented. Numerical results show that the proposed BESO method is efficient, and convergent optimal solutions can be achieved for a variety of optimization problems on frequency non-probabilistic reliability of continuum structures.
An efficient approach for reliability-based topology optimization
Article
  • May 2015
This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15-30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.