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... Deng and Chen [13] presented a robust concurrent topology optimization for two-and three-dimensional structure in order to reduce the compliance. Xu and Xie [14] proposed a concurrent multiscale topology optimization method for two-dimensional macro-and microstructures to minimize the displacement response mean square. Xu et al. [15] presented a concurrent multiscale topology optimization method aiming to maximize the macrostructural stiffness with multiphase material. ...

... Note that _ L is calculated under the conditions E 0 ijmn ¼ 0 and I kl 0 m;n ¼ 0. Here, we assume that the state equation (Eq. (14)) and the adjoint equation (Eq. (15)) in the macrostructure, the state equation (Eq. ...

... When the state Eqs. (14) and (16), the adjoint Eqs. (15) and (17), and the constraint Eq. (18) are all satisfied, the perturbation expansion of the Lagrange function is expressed as Eq. ...

In this study, we present a shape optimization approach for designing the shapes of periodic microstructures using the homogenization method and the H¹ gradient method. The compliance of a macrostructure is minimized under the constraint conditions of the total area of the microstructures distributed in the macrostructure, the elastic equation of the macrostructure and the homogenization equation of the unit cells. The shape optimization problem is formulated as a distributed-parameter optimization problem, and the shape gradient function involving the state and adjoint variables for both the macro- and micro-structures is theoretically derived. Clear and smooth boundary shapes of the unit cells can be determined with the H¹ gradient method. The proposed method is applied to multiscale structures, in which the numbers of domains with the microstructures are varied and the optimized shapes of the unit cells and the compliances obtained are compared. The numerical results confirm the effectiveness of the proposed method for creating the optimal shapes of microstructures distributed in macrostructures.

... As a result, it is significant to execute a concurrent design of macrostructure and microstructure. In order to further improve structural characteristics, Xu and Xie investigated the concurrent topology optimization of macrostructural material distribution and periodic microstructure under random excitations [22]. Wang et al. proposed a new multi-material level set topology description model for topology and shape optimization of structures involving multiple materials [23]. ...

... where ρ 22 and ρ 32 are the density of the second phase material in the base cell of cellular material 2 and 3, respectively. Without loss of generality, the second phase material in the base cell of cellular material 2 and 3 are different materials herein, that is, D 22 -D 32 and ρ 32 -ρ 22 . ...

... After substituting Eqs. (22) into (21), we have ...

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.

... Li et al. (2019) presented a concurrent multiscale optimization method for a two-dimensional elastic structure by varying the length-width ratio of the hole in the periodic cell. Although not concurrent multiscale optimization, the homogenization method has been applied for attractive metamaterial designs such as negative Poisson's ratio properties (Xu and Xie 2015;Jha and Dayyani 2021;Zhang and Khandelwal 2019;Ai and Gao 2017) and zero Poisson's ratio (Jha and Dayyani 2021). ...

... Many of the proposed methods are based on a gradient approach, but evolutionary methods are also used. For example, Xu and Xie (2015) presented a method using the bi-directional evolutionary structural optimization (BESO) method, in which two-dimensional structures were optimized with two types of distributed microstructures within the macrostructure. studied compliance minimization of a macrostructure by optimizing the microstructures using the BESO method. ...

We propose a novel shape optimization method for designing a multiscale structure with the desired stiffness. The shapes of the macro-and microstructures are concurrently optimized. The squared error norm between actual and target displacements of the macrostructure is minimized as an objective function. The design variables are the shape variation fields of the outer and interface shapes of the macrostructure and the shapes of holes in the microstructures. Subdomains with independent periodic microstructures are arbitrarily defined in the macrostructure in advance. Homogenized elastic tensors are calculated and applied to the correspondent subdomains. The shape gradient functions are theoretically derived with respect to each shape variation of the macro-and microstructures, and applied to the H 1 gradient method to determine the optimum shapes. The proposed method is applied to several numerical examples, including Poisson's ratio design and deformation control designs of an L-shaped bracket and a both ends fixed beam with holes. The results of the design examples confirm that the desired stiff or compliant deformation can be achieved while obtaining clear and smooth boundaries. The influence on the final results of the initial shape of the unit cell, the connectivity of adjacent microstructures, and interface optimization is also discussed.

... Concurrent robust design and optimization considering load uncertainties was investigated in [44]. Concurrent design of composite macrostructure and cellular microstructure under random excitations was studied in [45]. But there is litter work on the concurrent topological design of composite structures and their multi-phase materials. ...

... For three or more phases material design problem in the micro models, it is straightforward to interpolate the material properties between two neighboring phases (i.e. E j and E jþ1 ) [45,54], as ...

This paper presents a concurrent topology optimization approach for simultaneous design of composite structures and their periodic material microstructures with three or more phases. The effective properties of multi-phase materials are obtained via homogenization technique which serves as a bridge of the finite element models of the macrostructure and the material microstructure. The base materials of periodic microstructures used in each phase of the macrostructure are divided into several groups and sensitivity analysis are carried out on them one by one. Meanwhile, the sensitivity number at the macrostructure is derived which is coupled with the designed material properties. Then, the composite configurations of material microstructures and macrostructures are inversely optimized concurrently based on the bi-directional evolutionary structural optimization (BESO) algorithm. Several 2D and 3D numerical examples are presented to demonstrate the effectiveness of proposed design approach.

... With the established models for material microstructural design, one comes up naturally with the idea of concurrent or integrate designs of both material ans structure. In other words, by topology optimization one determines not only the optimal spatial material layout distribution at the macroscopic structural scale, but also the optimal local use of the cellular material at the microscopic scale, as schematically shown in Fig. 4. The most commonly applied strategy is designing a universal material microstructure at the microscopic scale either for a fixed (e.g., [64,106]) or concurrently changed (e.g., [26,52,124,128,144]) structure at the macroscopic scale. Obviously, such designs have not yet released the full potentiality of concurrent two-scale designs. ...

... Direct implementation of Algorithm 1 with discrete variables would result in the divergence of the design process [114]. In practical implementations (e.g., [64,124,125,128]), a simplified version of Algorithm 1, as summarized in Algorithm 2, is adopted. Algorithm 2 in fact avoids solving the ''slave'' material stiffness maximization problems (56) while treating both scale variables q and g x in an integral manner. ...

Research on topology optimization mainly deals with the design of monoscale structures, which are usually made of homogeneous materials. Recent advances of multiscale structural modeling enables the consideration of microscale material heterogeneities and constituent nonlinearities when assessing the macroscale structural performance. However, due to the modeling complexity and the expensive computing requirement of multiscale modeling, there has been very limited research on topology optimization of multiscale nonlinear structures. This paper reviews firstly recent advances made by the authors on topology optimization of multiscale nonlinear structures, in particular techniques regarding to nonlinear topology optimization and computational homogenization (also known as FE2) are summarized. Then the conventional concurrent material and structure topology optimization design approaches are reviewed and compared with a recently proposed FE2-based design approach, which treats the microscale topology optimization process integrally as a generalized nonlinear constitutive behavior. In addition, discussions on the use of model reduction techniques is provided in regard to the prohibitive computational cost.

... To directly optimize the material attribute (such as elastic tensor) of each element, free material optimization was proposed (Ringertz 1993;Bendsoe et al. 1994). Rodrigues et al. (2002) introduced the SIMP technique for hierarchical macrostructure and microstructure optimization, and many researchers have since done research of concurrent multiscale optimization (Li et al. 2019;Liu et al. 2020Liu et al. , 2008Xu and Xie 2015;Wang et al. 2018;Yan and Cheng 2020). Some work was performed to extend multiscale mechanical stiffness maximization to including thermal bulk conductivity extremum of microscale only (Yan et al. 2015). ...

Structural light weighting is vital for increasing energy efficiency and reducing CO2 emissions. Furthermore, for many applications, high heat conductivity is necessary to attain efficient energy transfer while increasing the product stiffness and reducing the weight. In recent years, with the development of 3D printing technology, attention has been directed toward porous materials that greatly contribute to weight reduction. As such, this educational research is aiming toward introducing the methodology of concurrent multiscale topology optimization attaining designs of lightweight, high heat conductive, and stiff porous structures utilizing multi-objective optimization method. The normalized multi-objective function is used in this research to maximize heat conductivity and stiffness. Therefore, the objective criteria are consisting of heat and mechanical compliance minimization. Utilizing the SIMP method, the multiscale sensitivity analysis, and optimization formulation were driven theoretically using adjoint method to reduce the computational cost and presented in a MATLAB code. 2D cases were studied, and a proper Pareto front was attained. The results showed good coupling of the macro and microscale design. The MATLAB code is explained and included in the appendix and it is intended for educational purposes.

... introduced a porous anisotropic material with a penalty (PAMP) model to optimize both the macro-structure and micromaterial simultaneously for maximum fundamental frequency, thermo-elastic, mechanical, thermal load coupling, and multi-objective problems (Niu et al. 2009;Yan et al. (2008); ; Yan et al. 2016;Deng et al. 2013). Based on the bi-directional evolutionary structural optimization (BESO) method, Huang et al. (2013) proposed a mathematical model for simultaneous two-scale topology optimization with different objectives, including minimizing static, dynamic compliance, and maximizing the fundamental frequency (Xu and Xie 2015;Xu et al. 2018). Xia and Breitkopf (2014) presented a model for concurrent topology optimization of materials and structures using nonlinear multi-scale analysis. ...

This paper proposes an efficient methodology for concurrent multi-scale design optimization of composite frames considering specific design constraints to obtain the minimum structure cost when the fundamental frequency is considered as a constraint. To overcome the challenge posed by the strongly singular optimum and the weakness of the conventional polynomial material interpolation (PLMP) scheme, a new area/moment of inertia–density interpolation scheme, which is labeled as adapted PLMP (APLMP) is proposed. The APLMP scheme and discrete material optimization approach are employed to optimize the macroscopic topology of a frame structure and microscopic composite material selection concurrently. The corresponding optimization formulation and solution procedures are also developed and validated through numerical examples. Numerical examples show that the proposed APLMP scheme can effectively solve the singular optimum problem in the multi-scale design optimization of composite frames with fundamental frequency constraints. The proposed multi-scale optimization model for obtaining the minimum cost of structures with a fundamental frequency constraint is expected to provide a new choice for the design of composite frames in engineering applications.

... The algorithm has been further improved in [48] with a new filter scheme and a gradual procedure. Xu and Xie [49] proposed a parallel framework for the distribution of macrostructure materials and periodic microstructure in composites. Gan and Wang [50] adopted the BESO method to optimize the dynamic and static characteristics of multi-material structures. ...

The synergy between different constituent materials can drastically improve the performance of composite structures. The optimal design of such structures for practical applications is complicated by the often-encountered non-deterministic loading conditions. This paper proposes an efficient method for robust multi-material topology optimization problems of continuum structures under load uncertainty. Specifically, the weighted sum of the mean and standard deviation of structural compliance is minimized under volume constraints for each material phase. Based on the theory of linear elasticity and using the orthogonal diagonalization of real symmetric matrices, the Monte Carlo sampling is separated from the topology optimization procedure and an efficient procedure for sensitivity analysis is established. By employing an alternating active-phase algorithm of the Gauss-Seidel version, the multi-material topology optimization problem is split into a series of binary topology optimization sub-problems, which can be easily solved using the modified SIMP model. Several 2D examples are presented to demonstrate the effectiveness of the proposed method.

... Moreover, by incorporating the macro-scale design concurrently in the two-scale topology optimization, influence of the shape and finite dimensional size of the macroscopic design domain as well as the practical boundary conditions can be considered directly [22]. Due to these excellent features, the concurrent two-scale topology optimization has seen many applications such as static compliance-and stress-constrained mechanical designs [31][32][33][34][35], stiffness-constrained thermoelastic structure designs [36,37] and also many multi-physics scenarios including acoustic-structural problems [21,22,[38][39][40][41]. Especially, the concurrent two-scale topology optimization method was developed to create band gap property successfully given a finite macroscopic design domain with detailed boundary conditions for the first time in our previous work [22]. ...

Phononic crystals (PnCs) have seen increasing popularity due to band gap property for sound wave propagation. As a natural bridge, topology optimization has been applied to the design of PnCs. However, thus far most of the existent works on topological design of PnCs have been focused on single micro-scale topology optimization of a periodical unit cell. Moreover, practical manufacturing of those designed structures has been rarely involved. This paper presents a quasi two-scale topology optimization framework suitable for additive manufacturing (AM) implementation to design 2D phononic-like structures with respect to sound transmission coefficient (STC). A designate topology is employed and subjected to sizing optimization in the micro-scale design. The thin-walled square lattice structures made of single metal material are selected as the infills for the design domain to guarantee material connectivity in the optimized design in order to facilitate fabrication by AM. The practical effective mechanical property of the lattice structures with different volume densities obtained by experimental measurement is employed in the topology optimization. The proposed framework is applied to the design of 2D phononic-like structures with different macroscopic shapes for the desired band gap feature. Numerical examples show the desired band gap containing a prescribed excitation frequency can be realized through the proposed quasi two-scale topology optimization method. Moreover, the optimized designs are reconstructed into CAD files with the thin-walled lattice infills. The reconstruction makes fabrication of the optimized designs feasible by practical AM process.

... Lining with this idea, concurrent optimization with diverse physical problems is widely discussed, e.g., dynamic problem (Niu et al. 2009), thermoelastic problem (Deng et al. 2013), and acoustic problem (Liang and Du 2019). Besides the SIMP method, bidirectional evolutionary structural optimization method with black-and-white material distributions also contributed intensively in two-scale optimization (Da 2019;Xu et al. 2016;Xu and Xie 2015;Yan et al. 2014;Yan et al. 2015). It is obvious that consideration of point-topoint material distribution in microstructure level is usually redundant and low-efficiency, homogeneous material design is always far from optimal performance. ...

Based on hybrid cellular automata (HCA), we present a two-scale optimization model for heterogeneous structures with non-uniform porous cells at the microscopic scale. The method uses the K-means clustering algorithm to achieve locally nonperiodicity through easily obtained elemental strain energy. This energy is used again for a two-scale topological optimization procedure without sensitivity analysis, avoiding drastically the computational complexity. Both the experimental tests and numerical results illustrate a significant increase in the resulting structural stiffness with locally nonperiodicity, as compared to using uniform periodic cells. The effects of parameters such as clustering number and adopted method versus classical Optimality Criteria (OC) are discussed. Finally, the proposed methodology is extended to 3D two-scale heterogeneous structure design.

... Yan and Huang et al. [8][9] introduced a two-scale topology optimization algorithm based on the bi-directional evolutionary structural optimization (BESO) method to concurrently design materials and structures for maximizing the structural stiffness and minimizing the material thermal conductivity. Xu et al. [10][11] extended the concurrent design method to the optimization problems under dynamic loadings. Chen et al. [12] presented a new MIST (moving isosurface threshold) formulation and algorithm for the concurrent design of structures and cellular materials in order to maximize the structural stiffness. ...

An Integrated structural and material topology optimization method considering optimal material orientation is presented based the on bi-direction evolutionary structural optimization (BESO) method. It is assumed that the macrostructure is composed of uniform cellular material but with different orientation. The homogenization method is used to calculate the effective material properties which builds a connection between material and structure. The continuous material orientation design variables and the discrete topology design variables are treated hierarchically in an iteration. The principal stress method is adopted and embedded to determine the optimal material orientation, meanwhile the topologies of the macrostructure and its material microstructure are concurrently optimized by using the BESO method. Numerical examples are conducted to demonstrate the effectiveness of the proposed optimization algorithm.

... The two-scale concurrent topology optimization framework [6], where the material microstructure is assumed to be the same throughout the macroscopic structure, is the focus of this work. This framework has been applied to many structural design problems, such as static compliance [6,32,33], eigenfrequency [34][35][36], frequency response [37][38][39], transient response [40], random vibration [41], thermoelastic performance [42,43], buckling load [44], as well as robust compliance [45,46]. However, it was found that the conventional coupled sensitivity analysis method employed in the existing concurrent topology optimization works is inefficient even for statics problems [47]. ...

This paper aims to develop an efficient concurrent topology optimization approach for minimizing the maximum dynamic response of two-scale hierarchical structures in the time domain. Compared with statics problems, the dynamic response problems usually involve many time steps, which may lead to intensive computational burdens in both dynamic response and sensitivity analyses. This study thus proposes an enhanced decoupled sensitivity analysis method for concurrent topology optimization of the time-domain dynamic response problems. The mode acceleration method is incorporated for efficient dynamic response analysis. The three-field density-based approach is employed for topology optimization of macrostructure and microstructure. A previously proposed aggregation functional is employed to approximate the maximum dynamic response of the structure. The method of moving asymptotes (MMA) is employed to update the design variables. Three numerical examples are presented to demonstrate the effectiveness of the proposed approach. Some discoveries regarding the concurrent topology optimization for dynamics problems are presented and discussed. Furthermore, the potential of the concurrent topology optimization formulation for designing lightweight structures under dynamic loads is also demonstrated.

... (33) is referred to as coupled method because the terms ∂D H /∂η j and ∂ρ H /∂η j , which are involved in computing ∂m 0 i /∂η j and ∂k 0 i /∂η j , are tightly coupled with λ i,n and u i,n when computing d J d /dη j . This coupled method has been applied to sensitivity analysis with respect to microscale design variables for frequency response and random vibration problems [50,52]. However, our recent study showed that even for static problems, the coupled method would be computationally inefficient [58]. ...

The purpose of this work is to develop an efficient concurrent topology optimization approach for minimizing frequency response of two-scale hierarchical structures over a given frequency interval. Compared with static problems, frequency response problems usually involve many load steps, which may lead to intensive computational burdens in both frequency response analysis and sensitivity analysis. This study thus proposes an enhanced decoupled sensitivity analysis method for frequency response problems, which is efficient even when plenty of frequency steps are involved and/or damping is considered. Furthermore, a combined method of modal superposition and model order reduction is incorporated for efficient frequency response analysis of two-scale hierarchical structures. A modified threshold Heaviside projection method is used to obtain black-and-white designs and the method of moving asymptotes (MMA) is employed to update the design variables. Several numerical examples are presented to demonstrate the effectiveness of the proposed approach.

... On one side, AM has already been used to produce ultralight and ultra-stiff structures [28], auxetic structures [29], functionally graded materials [30] and acoustic materials with a negative refraction index [31]. On the other side, topology optimization have been developed to design cellular structures [32] with a maximized energy absorption [33], prescribed properties [34] and multiscale structure [35]. ...

... The simultaneous design with multiple phases of material and composite structures is still limited. For instance, on the design of composite structures, Huang and Xie [1] use the BESO method to solve the problem of minimum compliance of structures with multiple materials, while Huang and Xie [2] study the problem of maximizing the frequency of multimaterial structures, this problem is in force in works by Xu and Xie [3], Liu et al. [4] and Long et al. [5], among others. ...

A hierarchical structure is a structure that can be described by different characteristic lengths, and is such that its layout in the smaller scale (microscale) affects its behavior in the bigger scale (macroscale). Each hierarchical level is treated as a continuous medium composed of one or more materials. The simultaneous design of multiphase composite structures aims at finding the optimal distribution of materials such that one or more structural parameters are maximized (or minimized). In this work, the Bi-directional Evolutionary Structural Optimization method, BESO hereinafter, is applied to the maximization of the fundamental frequency of a structure subjected to a constraint on the total volume of materials used. Numerical experiments are made in order to validate the implementation and confirm the efficacy of the method in optimizing the topology of the structure.

... Although this assumption reduces the design space, it simplifies the design and manufacturing process. The two-scale concurrent topology optimization framework has been used to improve the static compliance (Liu et al. 2008;Yan et al. 2014;Chen et al. 2017), structural frequency (Niu et al. 2009;Zuo et al. 2013;Liu et al. 2016), frequency response (Xu et al. 2015b;Vicente et al. 2016), transient response (Xu et al. 2016a), random vibration (Xu and Xie 2015a), buckling load (Cheng and Xu 2016), thermoelastic performance (Deng et al. 2013;Yan et al. 2016), as well as robust compliance (Guo et al. 2015;Deng and Chen 2017). ...

The conventional coupled sensitivity analysis method for concurrent topology optimization problems is computationally expensive for microscale design variables. This study thus proposes an efficient decoupled sensitivity analysis method for concurrent topology optimization based on the chain differentiation rule. Two numerical studies are performed to demonstrate the effectiveness of the decoupled sensitivity analysis method for concurrent topology optimization problems with single or multiple porous materials. It can be concluded from the results that the decoupled method is computationally much more efficient than the coupled method, while they are mathematically equivalent. The outstanding merits of the decoupled method are two-fold: (1) computational efficiency of sensitivity analysis with respect to the microscale design variables; and (2) applicability to concurrent topology optimization problems with single or multiple porous materials as well as with composite microstructure and multi-phase materials.

... Nowadays, the BESO algorithm has been developed to achieve multifunctional designs [31] and to maximize natural frequency with a given mass [32]. Also, Xu et al. [33][34][35] furthered the BESO method to concurrent topology optimization in regard to material distribution in macrostructure and periodic microstructure under harmonic, transient, and random excitations. Zhang and Sun [36] revealed the size effect of materials and structures in the integrated two-scale optimization approach. ...

The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously. Nodal displacement of macrostructure and effective thermal conductivity of microstructure are referred as the constraint functions, which means taking into account both the load-carrying capabilities and the thermal insulation properties. The effective properties of porous material derived from numerical homogenization are used for macro-structural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are utilized for the sensitivity analysis of the microstructure. Design variables in form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.

... Liu et al. [25] also proposed a concurrent topology optimization model to maximize natural frequency with a given mass. Xu et al. [26][27][28] extended the BESO method to concurrent topology optimization in regard to material distribution in macrostructure and periodic microstructure under harmonic, transient and random excitations. They also discussed the concurrent design of thermo-elastic structures composed of periodic multiphase materials [29]. ...

This paper introduces a two-scale concurrent topology optimization method for maximizing the frequency of composite macrostructure that are composed of periodic composite units (PCUs) consisting of two isotropic materials with distinct Poisson’s ratios. Interpolation of Poisson’s ratios of different constituent phases is used in PCU to exploit the Poisson effect. The effective properties of the composite are computed by numerical homogenization and integrated into the frequency analysis. The sensitivities of the eigenvalue of macro- and micro-scale density are derived. The design variables on both the macro- and micro-scales are efficiently updated by the well-established optimality criteria methods. Several 2D and 3D illustrative examples are presented to demonstrate the capability and effectiveness of the proposed approach. The effect of the micro-scale volume fraction and Poisson’s ratio of the constituent phases on the optimal topology are investigated. It is observed that higher frequency can be achieved at specific range of micro-scale level volume fraction for optimal composites than that obtained from structures made of individual base materials.

... Instead of maximizing the natural frequency, Vicente et al. [139] extended the BESO method for minimizing the frequency responses of two-scale systems subject to harmonic loads. Another study on frequency response designs using the BESO method was conducted by Xu and Xie [167], by which the displacement response mean square is minimized subject to random excitations. Xu et al. [169] further carried out simultaneous design for the maximization of dynamic strain energy of a two-scale structure under dynamic loading based on the equivalent static loads. ...

The evolutionary structural optimization (ESO) method developed by Xie and Steven (1993, [162]), an important branch of topology optimization, has undergone tremendous development over the past decades. Among all its variants , the convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) method developed by Huang and Xie (2007, [48]) allowing both material removal and addition, has become a widely adopted design methodology for both academic research and engineering applications because of its efficiency and robustness. This paper intends to present a comprehensive review on the development of ESO-type methods, in particular the latest con-vergent and mesh-independent BESO method is highlighted. Recent applications of the BESO method to the design of advanced structures and materials are summarized. Compact Malab codes using the BESO method for benchmark structural and material microstructural designs are also provided.

... In order to further improve structural characteristics or functions, researchers have increasingly paid more attention to the concurrent design of macrostructure and material microstructure. Various objective functions and constraints have been extensively investigated: the displacement response mean square [32], the dynamic compliance [33], the first natural frequency [34], the static mean compliance [35][36][37][38], the weight average of the static mean compliance and the overall conduction capability index [39]. Unfortunately, little attention has been devoted to the concurrent design of macrostructure and periodic microstructure considering transient analysis problem for some range of time. ...

... commonly applied strategy is designing a universal material microstructure at the microscopic scale either for a fixed [138,85] or simultaneously changed [39,183,165,162,70] structure at the macroscopic scale. Zhang et al. [176] made a step further by designing several different cellular materials for different layers of a layered structure. ...

... However, in the afore-mention research only single scale is considered in the topology optimization. Several techniques have been developed for the concurrent topology optimization problem dealing with both macro-structure and micro-structure [25][26][27][28][29][30][31]. However, most of these studies are confined to the optimization on static characteristics or that with one material. ...

This paper develops a concurrent topology optimization approach for designing two‐scale hierarchical structures under stress constraints without specifying the topology of the unit cell as a priori. Compared with traditional stress‐constrained topology optimization, the number of stress constraints involved in concurrent topology optimization is much larger and the sensitivity analysis for stress constraints is computationally more expensive. To address these challenges, a novel hierarchical aggregation strategy is proposed to handle the large number of stress constraints and an adjoint method is developed for efficient sensitivity analysis. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. It is also found that the structures obtained by traditional topology optimization usually perform better than the two‐scale structures obtained by concurrent topology optimization when only concentrated loads are considered, while the latter may exhibit better performance in the case that distributed loads are involved.

The demand for advanced manufacturing is gradually turning to the multi-functional integrated structure, so the effective and innovative functional structure topology optimization design method can maximize the performance of the structure. This paper proposes a parallel topology optimization design method for dynamic and static characteristics of multiphase materials. The inherent discrete design variables of bidirectional evolutionary structure optimization (BESO) can clearly distinguish the topological boundaries of multiphase materials without the intermediate density materials. The pre-processing density filtering scheme is introduced into the BESO method to eliminate the numerical calculation instability problem. Then, the improved multiphase materials interpolation scheme can overcome the local modal phenomenon of vibration problems. Finally, a series of typical numerical examples show that the method is effective, efficient and easy to implement in multiphase materials topology optimization with dynamic and static characteristics.

A two-scale concurrent topology optimization method based on the couple stress theory is proposed for maximizing structural fundamental eigenfrequency. Because of the fact that the classical mechanics theory cannot reveal the size effect because of neglecting the influence of microstructure, the theory of couple stress including the microscopic properties of materials can be used to describe the size effect in deformations. On the foundation of the couple stress theory, the two-scale optimization model for finding optimal configurations of macrostructures and their periodic composite material microstructures is built. And the fundamental eigenfrequency of the macrostructure is maximized. The effective macroscopic couple stress constitutive constants of macrostructures are calculated by the representative volume element method. And a modified solid isotropic material with a penalization model is used to effectively avoid the localized mode. The optimization algorithm based on the bidirectional evolutionary structural optimization method is proposed. The optimal results of numerical examples show that the optimal topologies and natural frequencies obtained by the couple stress theory may differ significantly from those obtained by the typical Cauchy theory. It is obvious that couple stress theory can effectively describe the size effect in topology optimization.

In areas that require high performance components, such as the automotive, aeronautics and aerospace industries, optimization of the dynamic behavior of structures is sought through different approaches, such as the design of materials specific to the application, for instance through structural topology optimization. The bi-directional evolutionary structural optimization (BESO) method, in particular, has been used for the simultaneous design of hierarchical structures, which means that the structural domain consists not only of the macrostructure but also of the microstructural topology of the materials employed. The purpose of this work is to apply the BESO method to solve two-dimensional multiscale problems in order to minimize the response of structures subjected to forced vibrations in a given frequency range. The homogenization method is applied to integrate the different scales of the problem. In particular, the material interpolation model for two materials is used. The BESO method is applied to different cases of optimization, in macroscale, microscale, and multiscale structural domains. Numerical examples are presented to validate the optimization and demonstrate the potential of this approach. The numerical examples show that the multiscale bi-material topology optimization method implemented here is able to produce structures and microstructures for optimization of the frequency domain response, satisfying prescribed volume constraints.

Phononic crystals have been paid plenty of attention due to the particular characteristics of band gap for elastic wave propagation. Many works have been focused on the design of the phononic crystals materials/structures through different methods including experimental and numerical approaches such as topology optimization. However, most of the works on topological design of the phononic materials/structures are on micro-scale topology optimization of the crystal unit cell based on the assumption of infinite periodicity. Finite design domain and corresponding boundary condition are seldom considered directly in the single micro-scale topology optimization of the crystal unit cell. This paper presents a concurrent two-scale topology optimization framework to design phononic-like structures with respect to the vibro-acoustic criterion, and the finite dimension and the boundary condition of the macro-scale design domain can be fully taken into consideration simultaneously. Accuracy of the proposed model and method to compute the wave band gap property of two-dimensional phononic structures is validated. Then the concurrent two-scale topology optimization approach is employed to design the phononic-like structures and tune the wave band gap property. Numerical examples show the advantage of the concurrent two-scale topology optimization over the single micro-scale design of the crystal unit cell. Many interesting features of the proposed approach are also revealed and discussed. The presented work shows that the concurrent two-scale topology optimization approach is promising to be a powerful tool in the design of vibro-acoustic phononic-like structures for achieving the desired band gap property.

Topology optimization has been regarded as a scientific and efficient tool to search the optimal material distribution with the best structural performance, subject to the prescribed constraints. It has been accepted a wide array of applications in many fields, like the biology, the mechanical, the medical and etc. However, the conventional works, where the topology optimization is performed on the basis of the homogenized materials, cannot maintain the high requirements of the ultra-lightweight, the specific properties and the integration of the functionals in the modern industrial products. How to explore the performance of material microstructures in improving the functional becomes more and more popular in the research field of the topology optimization, where material layouts and material properties are both considered in the multiscale design of structure-material. In the current work, the parametric level set method (PLSM) combined with the homogenization theory is firstly applied to study the design of mechanical metamaterials and optimize material microstructures. The topology optimization formulation for the multiscale design of structure-material is studied, which is later applied to discuss the single material microstructure, multiple microstructures and the dynamic, respectively.
Firstly, the topology optimization formulation for the rational design of mechanical metamterials is proposed based on the parmetric level set. An energy-based homogenization method (EBHM) is developed to evaluate the macroscopic effective properties of material microstructures, which can effectively remove several numerical difficulties of numerical homogenization method, such as the complexity of the theoretical derivations. We adopt the PLSM and the EBHM to develop the topology optimization formulation for the systematic design of mechanical metamaterials. Several numerical examples in 2D and 3D for the maximal bulk modulus, the maximal shear modulus and the negative Poisson’s ratio are studied to demonstrate the effectiveness.
Secondly, the topology optimization formulation for the multiscale design of structure-material with a kind of microstructures to maximize the stiffness performance is proposed. In the formulation for mechanical metamaterials, we introduce the conventional topology optimization considering the homogenized materials. In terms of the single kind of material microstructures, we employ the PLSM and the EBHM to develop the multiscale topology optimization formulation. The PLSM can ensure the smooth structural boundary and distinct material interface to improve the manufacturability, and the EBHM is beneficial to reduce the computational cost of the finite element analysis. The topologies at the macro and micro are concurrently optimized to improve the stiffness performance.
Then, the topology optimization formulation for the multiscale design of structure-material with multiple kinds of microstructures to maximize the stiffness performance is proposed. The macrostructural topology, the topologies of multiple kinds of microstructures and their overall distribution in the macrostructure should be simultaneously considered. A multiscale topology optimization formulation with two stages are proposed, where the first stage employes the variable thickness sheet method to construct the material distribution optimization model for seeking the optimal layout of material microstructures. In the second stage, the topologies of the macrostructure and multiple kinds of material microstructures are concurrently optimized based on the PLSM and the EBHM.
Later, the topology optimization formulation for the multiscale design of structure-material with multiple kinds of microstructures for the minimization of frequency responses is proposed, which should consider the macrostructural topology, the topologies of multiple kinds of microstructures and their overall distribution in the macrostructure. Based on the proposed multiscale topology optimization formulation for the stiffness, we propose the multiscale topology optimization formulation with two stages for the frequency responses. The quasi-static Ritz vector is applied to approximate the displacement responses to reduce the computational cost, and the kinematical connectors are pre-defined in microstructures to ensure the connectivity between adjacent microstructures, so that the macrostructure can have a reasonable loading transmission path.
Subsequently, we employ the ANSYS engineering software to simulate the mechanical metamaterials and present the auxetic behavior. The proposed materials design formulation and the multiscale topology optimization formulation are applied to the discussions of lattice materials in the aerospace and the main-bearing structures in the satellite, respectively. The effectiveness and the engineering practicability can be presented in the final designs.
Finally, the concluded remarks of the current work and the key contributions are both outlined in the final section, and we also provide some prospects for the future works.

In this paper, an efficient concurrent optimization method of macrostructures, and material microstructures and orientations is proposed for maximizing natural frequency. It is assumed that the macrostructure is composed of uniform material with the same microstructure but with various orientation. The bi-directional evolutionary structural optimization (BESO) method is applied to optimize the macrostructure and its material microstructure under a given weight constraint. Meanwhile, the optimality condition with respect to local material orientation is derived and embedded in the two-scale design of macrostructures and material microstructures. Numerical examples are presented to demonstrate the capability and effectiveness of the proposed optimization algorithm. The results show that the current design of macrostructures, material microstructures, and local material orientation greatly improves structural dynamic performance.

Concurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This paper proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of microstructure are calculated via the homogenization method. Efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bi‐directional evolutionary structural optimization (BESO) method is employed to obtain black‐and‐white designs for both macrostructure and microstructure. Several 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure. This article is protected by copyright. All rights reserved.

A concurrent optimization design method for the topologies of structures and materials and the material orientation is presented based on bi-direction evolutionary structural optimization (BESO) method. The macrostructure is assumed to be composed of a uniform cellular material but with different orientation. The homogenization technique is used to calculate the effective properties of the cellular material which builds a connection between material and structure. An analytical method, which is flexible to deal with the shear “weak” and “strong” materials, is proposed to solve the material orientation optimization problem. The optimization algorithm considering the simultaneous optimization of topologies of macrostructures and material microstructures, and material orientations is developed. Numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm and show that concurrent topology design of structures and materials with material orientation optimization can greatly improve the structural performance.

Most of the presented works in the field of vibro-acoustic topology optimization are focused on single-scale design of the structure or material so far, which cannot exert the potential of the material to the largest extent. Even though multi-scale topology optimization has been investigated increasingly in recent years, few works concern the topological design with respect to the vibro-acoustic criteria. In this paper, a concurrent multi-scale multi-material topology optimization method is presented for minimizing sound radiation power of the vibrating structure subjected to harmonic loading. The metamaterial consisting of different periodic microstructures and its distribution over the macrostructural domain are designable to reduce the sound radiation power. A general multi-scale multi-material interpolation model based on SIMP and PAMP is developed and applied to the concurrent topological design. The optimum distribution of the base materials at micro-scale and metamaterial associated with the optimized microstructures at macro-scale will be obtained concurrently. The homogenization method is employed to calculate the equivalent macro-scale material properties of the periodic microstructures. A high-frequency approximation formulation is introduced to simplify calculation of the sound power from the vibrating structure to its surrounding acoustic medium. The sensitivities of the sound power with respect to macro-scale and micro-scale topological densities are calculated by the adjoint method. The MMA method is employed to find the solution of the concurrent multi-scale vibro-acoustic topology optimization problem. Numerical examples are given to validate the accuracy of the established model and show the advantages of the multi-scale topology optimization in specific cases of vibro-acoustic design. Many interesting features of the concurrent vibro-acoustic multi-scale topological design have been revealed and discussed. In comparison with the single-scale microstructural design, the importance of simultaneous macro-structural level design to improve overall vibro-acoustic characteristics of the structure is proved by the examples.

Purpose
The optimal material microstructures in pure material design are no longer efficient or optimal when accounting macroscopic structure performance with specific boundary conditions. Therefore, it is important to provide a novel multiscale topology optimization framework to tailor the topology of structure and the material to achieve specific applications. In comparison with porous materials, composites consisting of two or more phase materials are more attractive and advantageous from the perspective of engineering application. This paper aims to provide a novel concurrent topological design of structures and microscopic materials for thermal conductivity involving multi-material topology optimization (material distribution) at the lower scale.
Design/methodology/approach
In this work, the effective thermal conductivity properties of microscopic three or more phase materials are obtained via homogenization theory, which serves as a bridge of the macrostructure and the periodic material microstructures. The optimization problem, including the topological design of macrostructures and inverse homogenization of microscopic materials, are solved by bi-directional evolutionary structure optimization method.
Findings
As a result, the presented framework shows high stability during the optimization process and requires little iterations for convergence. A number of interesting and valid macrostructures and material microstructures are obtained in terms of optimal thermal conductive path, which verify the effectiveness of the proposed mutliscale topology optimization method. Numerical examples adequately consider effects of initial guesses of the representative unit cell and of the volume constraints of adopted base materials at the microscopic scale on the final design. The resultant structures at both the scales with clear and distinctive boundary between different phases, making the manufacturing straightforward.
Originality/value
This paper presents a novel multiscale concurrent topology optimization method for structures and the underlying multi-phase materials for thermal conductivity. The authors have carried out the concurrent multi-phase topology optimization for both 2D and 3D cases, which makes this work distinguished from existing references. In addition, some interesting and efficient multi-phase material microstructures and macrostructures have been obtained in terms of optimal thermal conductive path.

Negative Poisson’s ratio (NPR) material attracts a lot of attentions for its unique mechanical properties. However, achieving NPR is at the expense of reducing Young’s modulus. It has been observed that the composite stiffness can be enhanced when blending positive Poisson’s ratio (PPR) material into NPR material. Based on the respective interpolation of Young’s modulus and Poisson’s ratio, two concurrent topology optimization problems with different types of constraints, called Problem A and B, are respectively discussed to explore the Poisson’s ratio effect in porous microstructure. In Problem A, the volume constraints are respectively imposed on macro and micro structures; in Problem B, besides setting an upper bound on the total available base materials, the micro thermal insulation capability is considered as well. Besides considering the influence of micro thermal insulation capability on the optimized results in Problem B, the similar and dissimilar influences of Poisson’s ratios, volume fractions in Problem A and B are also investigated through several 2D and 3D numerical examples. It is observed that the concurrent structural stiffness resulting from the mixture of PPR and NPR base materials can exceed the concurrent structural stiffness composed of any individual base material.

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.

Most studies on composites assume that the constituent phases have different values of stiffness. Little attention has been paid to the effect of constituent phases having distinct Poisson’s ratios. This research focuses on a concurrent optimization method for simultaneously designing composite structures and materials with distinct Poisson’s ratios. The proposed method aims to minimize the mean compliance of the macrostructure with a given mass of base materials. In contrast to the traditional interpolation of the stiffness matrix through numerical results, an interpolation scheme of the Young’s modulus and Poisson’s ratio using different parameters is adopted. The numerical results demonstrate that the Poisson effect plays a key role in reducing the mean compliance of the final design. An important contribution of the present study is that the proposed concurrent optimization method can automatically distribute base materials with distinct Poisson’s ratios between the macrostructural and microstructural levels under a single constraint of the total mass.

This article introduces thermal conductivity constraints into concurrent design. The influence of thermal conductivity on macrostructure and orthotropic composite material is extensively investigated using the minimum mean compliance as the objective function. To simultaneously control the amounts of different phase materials, a given mass fraction is applied in the optimization algorithm. Two phase materials are assumed to compete with each other to be distributed during the process of maximizing stiffness and thermal conductivity when the mass fraction constraint is small, where phase 1 has superior stiffness and thermal conductivity whereas phase 2 has a superior ratio of stiffness to density. The effective properties of the material microstructure are computed by a numerical homogenization technique, in which the effective elasticity matrix is applied to macrostructural analyses and the effective thermal conductivity matrix is applied to the thermal conductivity constraint. To validate the effectiveness of the proposed optimization algorithm, several three-dimensional illustrative examples are provided and the features under different boundary conditions are analysed.

A method for the multi-scale design of composite thermoelastic macrostructure and periodic microstructure with multi-phase materials is proposed. A concurrent topology optimization model of macrostructure and microstructure is established, where the objective is to maximize the macrostructural stiffness subject to volume constraints on the macro-material distribution and phase materials. Based on the material interpolation scheme of the solid isotropic material with penalization (SIMP), the sensitivity of the mean compliance of the composite macrostructure with respect to design variables on two scales, i.e., macro and micro scales, is derived. The optimization problem is solved using a bi-directional evolutionary structural optimization (BESO) method and the corresponding optimization procedure for the concurrent topology optimization is proposed. Several examples are presented to demonstrate the effectiveness of the proposed method.

This paper presents a hierarchical topology optimization method to simultaneously achieve the optimum structures and multiphase material cells for minimum system thermal compliance. Macro design variables and micro phase design variables are introduced independently, and coupled through elemental phase relative density. Based on uniform interpolation scheme with multiple materials, the sensitivities of thermal compliance with respect to the design variables on the two scales are derived. Correspondingly, the hierarchical optimization model of structures and multiphase material cells is built under prescribed volume fraction and mass constraints. The proposed method and computational model are validated by several 2D numerical examples. The superiority of multiphase materials in hierarchical optimization is presented through the comparison of single phase materials. The optimized results of periodic structure, hierarchical structure and traditional continuous structure are compared and analyzed. At last, the effects of volume fraction and mass constraints are discussed.

The present paper studies multi-objective design of lightweight thermoelastic structure composed of homogeneous porous material. The concurrent optimization model is applied to design the topologies of light weight structures and of the material microstructure. The multi-objective optimization formulation attempts to find minimum structural compliance under only mechanical loads and minimum thermal expansion of the surfaces we are interested in under only thermo loads. The proposed optimization model is applied to a sandwich elliptically curved shell structure, an axisymmetric structure and a 3D structure. The advantage of the concurrent optimization model to single scale topology optimization model in improving the multi-objective performances of the thermoelastic structures is investigated. The influences of available material volume fraction and weighting coefficients are also discussed. Numerical examples demonstrate that the porous material is conducive to enhance the multi-objective performance of the thermoelastic structures in some cases, especially when lightweight structure is emphasized. An “optimal” material volume fraction is observed in some numerical examples.

This investigation presents an optimization of laminated cylindrical panels based on fundamental natural frequency. Also, trends of change in optimum stacking sequence while the proportions of structures vary, are studied which can be insightful for design purposes. A displacement based finite element model is used, in order to extract fundamental natural frequencies of T300/5208 Carbon/Epoxy cylindrical panels. To obtain optimum designs, the Globalized Bounded Nelder–Mead (GBNM) algorithm is employed. Predictions are compared with the results of Genetic Algorithm (GA) method and show faster and more accurate convergence to the global optimum, while variables are continuous in GBNM and discrete in GA. Moreover, verification of novel convergence criteria to ameliorate local searcher in GBNM is examined.

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials
in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study
is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods
agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions
are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs
by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained
for a variety of design cases.

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.

this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors. The method is extended for the case of repeated eigenvalues, which leads to restrictions on the parameterization. For repeated eigenvalues of multiplicity two, these restrictions are formulated expicitly. Applications and limitations of the method are demonstrated by examples. Nomenclature

The aim of this study was to design isotropic periodic microstructures of cellular materials using the bidirectional evolutionary structural optimization (BESO) technique. The goal was to determine the optimal distribution of material phase within the periodic base cell. Maximizing bulk modulus or shear modulus was selected as the objective of the material design subject to an isotropy constraint and a volume constraint. The effective properties of the material were found using the homogenization method based on finite element analyses of the base cell. The proposed BESO procedure utilizes the gradient-based sensitivity method to impose the isotropy constraint and gradually evolve the microstructures of cellular materials to an optimum. Numerical examples show the computational efficiency of the approach. A series of new and interesting microstructures of isotropic cellular materials that maximize the bulk or shear modulus have been found and presented. The methodology can be extended to incorporate other material properties of interest such as designing isotropic cellular materials with negative Poisson's ratio.

More and more stringent structural performance requirements are imposed in advanced engineering application, only a limited number of works have been devoted to the topology optimization of the structures with random vibration response requirements. In this study, the topology optimization problem with the objective function being the structural weight and the constraint functions being structural random vibration responses is investigated. An approximate topological optimization model for suppressing ‘localized modes’ of vibrating Cauchy solids is established in this paper. Based on moving asymptotes approximate functions, approximated-approximations expressions of the dynamic responses are constructed. In order to control the change quantity of topologic design variables, new dynamic response constraint limits are formed and introduced into the optimization model at the beginning of each sub-loop iteration. Then, an optimization sequential quadratic programming is introduced, and a set of iteration formulas for Lagrange multipliers is developed. Two examples are provided to demonstrate that the proposed method is feasible and effective for obtaining optimal topology.

This paper describes a methodology for simultaneous topology and material optimization in optimal design of laminated composite beams with eigenfrequency constraints. The structural response is analyzed using beam finite elements. The beam sectional properties are evaluated using a finite element based cross section analysis tool which is able to account for effects stemming from material anisotropy and inhomogeneity in sections of arbitrary geometry. The optimization is performed within a multi-material topology optimization framework where the continuous design variables represent the volume fractions of different candidate materials at each point in the cross section. An approach based on the Kreisselmeier–Steinhauser function is proposed to deal with the non-differentiability issues typically encountered when dealing with eigenfrequency constraints. The framework is applied to the optimal design of a laminated composite cantilever beam with constant cross section. Solutions are presented for problems dealing with the maximization of the minimum eigenfrequency and maximization of the gap between consecutive eigenfrequencies with constraints on the weight and shear center position. The results suggest that the devised methodology is suitable for simultaneous optimization of the cross section topology and material properties in design of beams with eigenfrequency constraints.

This paper introduces a hierarchical concurrent design approach to maximizing the natural frequency of a structure. Multiple material phases are considered in the topology optimization performed on both the macro and micro scales. A general problem for composite structure and material design is formulated that contains the cellular design problem as a special case. The design of the macro structure and material micro structure is coupled. The designed material properties are applied to the analysis of the macro structure, while the macro structure displacement field is considered in the sensitivity analysis on the micro scale. The material edistribution is controlled by an optimality criterion for frequency maximization. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are employed to obtain the final optimal solution. Several numerical examples of composite structures and materials are presented to demonstrate the capability and effectiveness of the proposed approach. Results include various orthotropic or anisotropic composite materials, as well as vibration-resisting layouts of the macro structure. In-depth discussions are also given on the effects of the base material phases and the assignment of the volume fractions on each scale. (c) 2013 Elsevier Ltd. All rights reserved.

This paper introduces a topology optimization algorithm for the optimal design of cellular materials and composites with periodic microstructures so that the resulting macrostructure has the maximum stiffness (or minimum mean compliance). The effective properties of the heterogeneous material are obtained through the homogenization theory, and these properties are integrated into the analysis of the macrostructure. The sensitivity analysis for the material unit cell is established for such a two-scale optimization problem. Then, a bi-directional evolutionary structural optimization (BESO) approach is developed to achieve a clear and optimized topology for the material microstructure. Several numerical examples are presented to validate the proposed optimization algorithm and a variety of anisotropic microstructures of cellular materials and composites are obtained. The various effects on the topological design of the material microstructure are discussed.

The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.

This paper presents a novel framework for simultaneous optimization of topology and laminate properties in structural design of laminated composite beam cross sections. The structural response of the beam is evaluated using a beam finite element model comprising a cross section analysis tool which is suitable for the analysis of anisotropic and inhomogeneous sections of arbitrary geometry. The optimization framework is based on a multi-material topology optimization model in which the design variables represent the amount of the given materials in the cross section. Existing material interpolation, penalization, and filtering schemes have been extended to accommodate any number of anisotropic materials. The methodology is applied to the optimal design of several laminated composite beams with different cross sections. Solutions are presented for a minimum compliance (maximum stiffness) problem with constraints on the weight, and the shear and mass center positions. The practical applicability of the method is illustrated by performing optimal design of an idealized wind turbine blade subjected to static loading of aerodynamic nature. The numerical results suggest that the proposed framework is suitable for simultaneous optimization of cross section topology and identification of optimal laminate properties in structural design of laminated composite beams.

In recent years, the Evolutionary Structural Optimization (ESO) method has been developed into an effective tool for engineering design. However, no attempts have been made to incorporate random dynamic response constraints. The optimum design of structures with dynamic response constraints is of great importance, particularly in the aeronautical and automotive industries. This paper considers the extension and modification of the ESO method to control the structural random dynamic responses. The random dynamic theory is applied to build an expression of random dynamic response constraints considering engineering requirements. Based on the modal truncation method of eigenderivatives and some approximate process, a set of formulations for sensitivity numbers of mean square random dynamic responses is derived. The algorithm is implemented in optimization software. Several examples are provided to demonstrate the validity and effectiveness of the proposed method.

In this paper, a new algorithm for bi-directional evolutionary structural optimization (BESO) is proposed. In the new BESO method, the adding and removing of material is controlled by a single parameter, i.e. the removal ratio of volume (or weight). The convergence of the iteration is determined by a performance index of the structure. It is found that the new BESO algorithm has many advantages over existing ESO and BESO methods in terms of efficiency and robustness. Several 2D and 3D examples of stiffness optimization problems are presented and discussed.

Considering stress-related objective or constraint functions in structural topology optimization problems is very important from both theoretical and application perspectives. It has been known, however, that stress-related topology optimization problem is challenging since several difficulties must be overcome in order to solve it effectively. Traditionally, SIMP (Solid Isotropic Material with Penalization) method was often employed to tackle it. Although some remarkable achievements have been made with this computational framework, there are still some issues requiring further explorations. In the present work, stress-related topology optimization problems are investigated via a level set-based approach, which is a different topology optimization framework from SIMP. Numerical examples show that under appropriate problem formulations, level set approach is a promising tool for stress-related topology optimization problems.

Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combining with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient, and convergent solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.

This paper presents two computational models to design the periodic microstructure of cellular materials for optimal elastic properties. The material equivalent mechanical properties are obtained through a homogenization model. The two formulations address the problem of finding the optimal representative microstructural element for periodic media that maximizes either the weighted sum of the equivalent strain energy density for specified multiple macroscopic strain fields, or a linear combination of the equivalent mechanical properties. Constraints on material volume fraction and material symmetries are considered. The computational models are established using finite elements and mathematical programming techniques and tested in several numerical examples.

Two methods for calculating the derivatives of a repeated eigenvalue of viscously damped vibrating systems with respect to a parameter are given. The first method implements the subspace spanned by the eigenvectors corresponding to the repeated eigenvalue. The second method is based on an explicit formula that uses the characteristic equation directly, without explicitly employing the eigenvector data. Examples demonstrate the various results. Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

The design sensitivity analysis of a natural frequency of a vibration system for various gradient-based optimization algorithms was discussed. It was observed that the sensitivities of a repeated frequency were the extrema of the directional derivative in the related mode space. It was also observed that the primary modes corresponds to each design variation were not coincident with each other. The results show that the sufficient condition was presented to identify the maximum value of the lowest frequency when it was increased.

A practical methodology based on a topology group concept is presented for finding optimal topologies of trusses. The trusses are subjected to natural frequency, stress, displacement and Euler buckling constraints. Multiple loading conditions are considered, and a constant nodal mass is assumed for each existing node. The nodal cost as well as the member cost is incorporated in the cost function. Starting with a ground structure, a sequence of substructures with different node distribution, called topology group, is generated by using the binary number combinatorial algorithm. Before optimizing a certain topology, its meaningfulness should be examined. If a topology is meaningless, it is then excluded; otherwise, it is optimized as a sectional area optimization problem. In order to avoid a singular solution, the dimension of the structure for a given topology is kept unchanged in the optimization process by giving the member to be removed a tiny sectional area. A parabolic interpolation method is used to solve a non-linear constrained problem, which forms the part of the algorithm. The efficiency of the proposed method is demonstrated by two typical examples of truss.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

The variational asymptotic method for unit cell homogenization is used to find the sensitivity of the effective properties of periodically heterogeneous materials, within a periodic base-cell. The sensitivities are found by the direct differentiation of the variational asymptotic method for unit cell homogenization (VAMUCH) and by the method of adjoint variables. This sensitivity theory is implemented using the finite element method and the engineering program VAMUCH. The methodology is used to design the periodic microstructure of a material that allows obtaining prescribed constitutive properties. The microstructure is modeled as a 2D periodic structure, but a complete set of 3D material properties are obtained. Furthermore, the present methodology can be used to perform the micromechanical analysis and related sensitivity analysis of heterogeneous materials that have 3D periodic structures. The effective material properties of the artificially mixed materials of the microstructure are obtained by the density approach, in which the solid material and void are mixed artificially.

Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homog-enization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing tech-niques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the micro-structure according to the materials' future application, a feature clearly demonstrated by the presented results. The meth-odology can be extended to incorporate other material properties of interest as well as the design of composite materials.

The problem of determining highly localized buckling modes in perfectly periodic cellular microstructures of infinite extent is addressed. A double scale asymptotic technique is applied to the linearized stability problem for a periodic structure built from linearly elastic microstructures. The obtained stability condition for the microscale level is then used to establish a comparative analysis between different material distributions in the base cell subjected to the same strain field at the macroscale level. The idea is illustrated by some two-dimensional finite element examples and used to design materials with optimal elastic properties that are less prone to localized instability in the form of local buckling modes at the scale of the microstructure. Copyright © 2002 John Wiley & Sons, Ltd.

This paper presents an extension of the hierarchical model for topology optimisation to three-dimensional structures. The
problem addressed covers the simultaneous characterisation of the optimal topology of the structure and the optimal design
of the cellular material used in its construction. In this study, hierarchical suggests that the optimisation model works
at two interconnected levels, the global and local levels identified, respectively, with the structure and its material. The
class of cellular materials, defining the material microstructure, is restricted to single scale cellular materials, with
the cell geometry locally optimised for the given objective function and constraints. The model uses the asymptotic homogenisation
model to obtain the equivalent material properties for the specific local microstructures designed using a SIMP based approach.
The necessary optimality conditions for the hierarchical optimal design problem are discussed and approximated numerically
by a proper finite element discretisation of the global and local analysis and design problems. Examples to explore and demonstrate
the model developed are presented.

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures,
based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material
that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction.
We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary
conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model
is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid
isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization
algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during
the optimization procedure and provide precise topologies without increase in computational cost.
KeywordsCouple-stress-Size effect-Topology optimization-RVE-Cellular materials

An optimum design method to minimize the weight of a linear elastic structural system subjected to random excitations is
presented. It is focused on the constraints of the first passage failure and displacement response mean square (RMS) at certain
degree of freedoms. Constraints on natural frequencies and bounds of design variables are also considered in the optimization.
Both correlated and un-correlated generalized random excitations are considered in the present formulation. The sensitivities
of the expected number of crossings as well as the displacement RMS with respect to the design variables are also derived.
The present method is applicable to stationary Guassian random excitations. Computational examples show the feasibility and
efficiency of the proposed method.

An inverse homogenization problem for two-phase viscoelastic composites is formulated as a topology optimization problem. The effective complex moduli are estimated by the numerical homogenization using the finite element method. Sensitivity analysis shows that the sensitivity calculations do not require the solution of any adjoint problem. The objective function is defined so that the topology optimization problem finds microstructures of viscoelastic composites which exhibit improved stiffness/damping characteristics within the specified operating frequency range. Design constraints include volume fraction, effective complex moduli, geometric symmetry and material symmetry. Several numerical design examples are presented with discussions on the nature of the designed microstructures. From the designed microstructures, it is found that mechanism-like structures and wavy structures are formed to maximize damping while retaining stiffness at the desired level.

The layerwise optimisation (LO) approach is extended in this work to point-supported, symmetrically laminated rectangular plates. The plates considered rest on some elastic or rigid point supports distributed in different arrangements. The LO approach provides a simple design procedure for composite structural optimisation which may be used to maximise the fundamental frequencies of the plates. The design variables are taken for a set of fibre orientation angles in the symmetric layers. The vibration problem is solved by the Ritz method with consideration of strain energy stored in the elastic supports or of the Lagrange multiplier for rigid supports. In numerical results, the symmetric 8-layer plates with eight different sets of support positions are considered and the applicability of the LO approach to the problem is demonstrated by comparing the present optimum solutions with reference frequencies of laminated plates with typical lay-ups. The relation of the support location and the optimum fibre orientation angles is discussed.

In this work a computational procedure for two-scale topology optimization problem using parallel computing techniques is developed. The goal is to obtain simultaneously the best structure and material, minimizing structural compliance. An algorithmic strategy is presented in a suitable way for parallelization. In terms of parallel computing facilities, an IBM Cluster 1350 is used comprising 70 computing nodes each with two dual core processors, for a total of 280 cores. Scalability studies are performed with mechanical structures of low/moderate dimensions. Finally the applicability of the proposed methodology is demonstrated solving a grand challenge problem that is the simulation of trabecular bone adaptation.

After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.

This paper presents a new approach to structural topology optimization. We represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension. Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, a well-founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints. The result is a 3D topology optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation. We have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, isotropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

A design control optimization approach is used to determine optimal levels of ply thickness, fiber orientation angle and closed-loop control force for composite laminated doubly curved shells. The optimization objective is the minimization of the dynamic response of a shell subject to constraints on the thickness and control energy. A higher-order shell theory is used to formulate the control objective for various cases of boundary conditions. The dynamic response is expressed as the sum of the total elastic energy of the shell and a penalty functional of a closed-loop control force. Comparative examples are presented for symmetric (or antisymmetric) spherical and cylindrical shells with various cases of boundary conditions. The advantages of the present control optimization over some design and control approaches are examined. The effect of number of layers, aspect ratio and orthotropy ratio on the control process is demonstrated. The discrepancy between optimal results obtained using the classical, first-order and higher-order shell theories is studied.

This paper deals with the construction of materials with arbitrary prescribed positive semi-definite constitutive tensors. The construction problem can be called an inverse problem of finding a material with given homogenized coefficients. The inverse problem is formulated as a topology optimization problem i.e. finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters. Numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies. Numerical results show that arbitrary materials, including materials with Poisson's ratio −1.0 and other extreme materials, can be obtained by modelling the base cell as a truss structure. Furthermore, a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness. Only the two-dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three-dimensional case.