## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

To read the full-text of this research,

you can request a copy directly from the authors.

... To the authors' knowledge, so far, limited researches have focused on the multi-scale topology optimization of composite plate structures in a frequency range. In this case, dynamic compliance is often used as the design objective for vibration response design [27][28][29][30][31][32][33]. In these studies, dynamic compliance is calculated using proportional damping, which cannot accurately consider the variation of damping due to the change of damping material configurations. ...

... D MA is the elastic matrix that can be computed with Equation (28). ρ MA is the density that can be computed with Equation (29). Ω is the design domain of the macrostructure. ...

... The thin-walled metal panel is the non-design domain; thus K p is a constant. Using the material interpolation scheme proposed in Equations (28) and (29), and based on the stiffness K v in Equation (30) ...

This paper proposes a novel density-based concurrent topology optimization method to support the two-scale design of composite plates for vibration mitigation. To have exceptional damping performance, dynamic compliance of the composite plate is taken as the objective function. The complex stiffness model is used to describe the material damping and accurately consider the variation of structural response due to the change of damping composite material configurations. The mode superposition method is used to calculate the complex frequency response of the composite plates to reduce the heavy computational burden caused by a large number of sample points in the frequency range during each iteration. Both microstructural configurations and macroscopic distribution are optimized in an integrated manner. At the microscale, the damping layer consists of periodic composites with distinct damping and stiffness. The effective properties of the periodic composites are homogenized and then are fed into the complex frequency response analysis at the macroscale. To implement the concurrent topology optimization at two different scales, the design variables are assigned for both macro- and micro-scales. The adjoint sensitivity analysis is presented to compute the derivatives of dynamic compliance of composite plates with respect to the micro and macro design variables. Several numerical examples with different excitation inputs and boundary conditions are presented to confirm the validity of the proposed methodologies. This paper represents a first step towards designing two-scale composite plates with optional dynamic performance under harmonic loading using an inverse design method.

... 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. ...

... N e is the number of macrostructure elements. a s jk and a m;q pl are respectively the design variables in macro scale and micro scale, which are constructed based on the material interpolation scheme by the solid isotropic material with penalization (SIMP) technique [33]. V p is the volumes of element p in the micro base cell model. ...

... (1) Step 4: Update the design variables in macro and micro scales based on sensitivity analysis by BESO method [33]. ...

... To obtain the optimal structures with good physical characteristics, it is essential to optimize the material layout within the prescribed design domain. The performance measurements cover a wide range of physical properties, including mechanics [30][31][32], heat transfer [33], fluidthermal coupling [34,35], and damping [36,37]. Wang et al [38,39]，Nguyen and Choi [40] studied the multiscale design of functionally graded microstructures, and then fabricated the optimized structures by additive manufacturing. ...

... The optimized results are shown in Fig. 13, in which Fig. 13 (a) is the result with the objective of minimizing the sum of compliance at the first and fiftieth time steps, indicated as -Time step [1,50].‖ Fig. 13 (b) minimizes the sum of compliance at the first and sixtieth time steps, indicated as -Time step [1,60],‖ Fig. 13 (c) first and seventieth time steps, 31 indicated as -Time step [1,70],‖ Fig. 13 (d) first and eightieth time steps, named -Time step [1,80],‖ and Fig. 13 (e) first and ninetieth time steps, named -Time step [1,90].‖ Different material layouts were obtained with different objectives. ...

Recent advances in biomedical engineering have promoted the development of innovative metal implants that have integrated mechanical and biodegradable properties. Most of the existing implant designs were created by using a trial-and-error approach, which depends on the designer's experience. Alternatively, inverse design approaches, such as topology optimization, have evolved into an efficient design solution to optimize the structural and material layout within a given design domain. Here we introduce a novel topology optimization scheme to support the microstructural design of biodegradable metal matrix composite structures (BM−MCS). The effect of material degradation on the mechanical performance of the structure is considered by integrating a degradation simulation algorithm into the structural finite element (FE) analysis. The objective function is to minimize the structural compliance at macroscale in a certain number of time steps to realize sufficient structural integrity in the initial bone healing stage, and the different stiffness reduction properties can be adjusted by changing the volume ratio of the two base constitutive biodegradable materials. The sensitivity of the above objective function concerning design variables was derived with considering the time-dependent degradation of the biodegradable material. Several numerical design examples were presented and benchmarked with classical designs. Finally, several prototypes were fabricated by integrating additive manufacturing with casting technology. Collectively, this demonstrates the feasibility and effectiveness of the proposed inverse design method for additive manufacturing.

... Many methods have been developed in this field to achieve this purpose like the homogenization method (Ma et al. 1993), density-based approach (Olhoff and Du 2016), bidirectional evolutionary optimization method (Liu et al. 2017), and level set approach (Shu et al. 2011). And many studies adopt the eigenvalue (Olhoff 1977;Du and Olhoff 2007), vibration magnitude (Kang et al. 2012), dynamic compliance in single external frequency (Liu et al. 2015;Ma et al. 1993;Niu et al. 2018;Olhoff 2014;Shu et al. 2011;Silva et al. 2019, Xu et al. 2015Yoon 2010;Zhao et al. 2019;Zhu et al. 2018), and active input power (Silva et al. 2020) as the optimal objective functions aiming to broaden structural operating frequency range or suppress structural vibration. Also, the vibration control methodologies are combined with the topology optimization technique to achieve the best control performance. ...

... Besides reducing structural vibration, the topology optimization technique can also be used to pursue lighter structure under limited consumption of material. For the latter purpose, material microstructure optimization methods (Yi et al. 2000;Andreasen et al. 2014;Andreassen and Jensen 2014;Huang et al. 2015) and multi-scale topology optimization techniques in the frequency domain (Xu et al. 2015;Andreassen and Jensen 2016;Zhang et al. 2019;Zhao et al. 2019) and in the time domain (Zhao and Wang 2016) have been developed. ...

For fiber-reinforced composites, the combination of structural topology optimization and fiber angle design is an excellent way to suppress structural vibration more efficiently and acquire a lighter structure. This paper proposes a concurrent optimization model for fiber-reinforced composite structures, where the mathematical model is built by combining the polynomial interpolation scheme (PIS) and the Heaviside penalization of discrete material optimization (HPDMO). Therein, the former is adopted to avoid the localized mode phenomena due to the mismatch between element stiffness and mass. And the latter is used for multi-candidate fiber orientation design. The objective function is to minimize the integration of the dynamic compliance in a given frequency band subject to macro-volume constraint. In order to ease the heavy computational burden caused by hundreds of frequency steps in each iteration, too many freedom degrees, and discrete design variables in HPDMO, the modal acceleration method is used to obtain a high-precision estimation of displacement response, and a decoupled method suitable for multiphase fiber material sensitivity analysis is also developed. Finally, a modified threshold Heaviside projection is used to obtain a black-and-white design. Several numerical examples are presented to verify the validity and robustness of the proposed model.

... 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. ...

... Step 4: According to the stress state in the macrostructure, determine the optimal material orientation by using equation (10). ...

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 [12], where the macroscopic structure is assumed to be composed of a uniform material microstructure, is the focus of this work. This concurrent topology optimization framework has been applied to improve the static compliance [12,15,45], structural frequency [46- 48], frequency response [49,50], transient response [51], random vibration [52], buckling load [53], thermoelastic performance [54,55], as well as robust compliance [56,57]. When performing concurrent topology optimization for minimizing frequency response of damped structures, sensitivity analysis as well as frequency response analysis would become computationally expensive when many excitation frequencies are involved. ...

... When performing concurrent topology optimization for minimizing frequency response of damped structures, sensitivity analysis as well as frequency response analysis would become computationally expensive when many excitation frequencies are involved. However, the conventional coupled sensitivity analysis method presented in the existing works [49,50] is inefficient even for static problems [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.

... Based on the concept of topology optimization, Bendsøe and Sigmund [47] proposed a multi-phase material mixture model in SIMP, and such a model has been extended to various topology optimization problems [48][49][50][51]. Recently, different topology optimization methods were employed to design multi-phase material structures like level set-based methods [52][53][54][55] and the BESO method [56][57][58]. ...

... The key contribution of this work is to integrate the BESO method, homogenization and multi-phase material interpolation scheme to carry out multiscale topology optimization with the consideration of multi-phase material microstructures at the lower scale, which has been rarely examined in the literature yet according to our best knowledge. Comparing with only two phase materials employed at microscopic scale in [57], we designed the underlying multi-phase material microstructures for both the solid material phase and the compliant material phase of the macroscale structure. The effective constitutive parameters of three or more materials are evaluated by the numerical homogenization analysis [59,60]. ...

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.

... Hvejsel and Lund [49] proposed a generalized multi-material SIMP model and employed a large number of sparse linear constraints to ensure a clear topology. Multimaterial bi-directional evolutionary structural optimization (BESO) method [50][51][52][53][54][55] were introduced for ensuring that each element is exactly and fully covered by one phase. Alternatively, several boundary-based topology optimization algorithms have been developed for the clear and smooth description of boundary of multi-material designs, such as the "color" level set approach [56][57], the new Multi-Material Level Set (MM-LS) method [58], the piecewise constant level set model [59][60], and the MMC method [61]. ...

The rapid development of additive manufacturing (AM) offers new opportunities to fabricate multi-material structures, whose performance can be optimized by the integrated design of multiple materials distribution and their interface behaviors. However, the graded-interface assumption between different materials often caused some numerical difficulties during topology optimization, e.g., poor applicability in weak interface and difficulty in accurately controlling interface width. This work develops a new element-based topology optimization algorithm by explicitly considering strong, weak or intermediate interfaces, and the interfacial width can also be controlled precisely. Under the explicit expression of a graded interface, a linear multi-material interpolation scheme is proposed to gradually achieve realistic graded physical field within the interfacial zone, where the interdiffusion or reaction inevitably happens and leads to the gradual transition of the interface property. The compliance minimization of multi-material structures with different types of graded interfaces is formulated under multiple volume fraction constraints. The sensitivity of the objective function and constraints with respect to design variables are derived. Numerical examples demonstrate that the optimized designs resulting from the proposed method always achieve a lower compliance, compared with those of the traditional multi-material designs. A phase diagram is presented to describe the sensitivity of the topological design on the interface behavior.

... In the microstructural topology optimization design, the asymptotic homogenization [13,24], which actually plays a role of connection between the macro and micro quantiles, is utilized to predict the effective properties of the periodic microstructures. Later on, various novel materials with either extreme proprieties or prescribed properties [25][26][27][28][29][30][31] have been developed by incorporating the topology optimization methods with the numerical homogenization theory. The inverse homogenization approach explores novel material microstructures without any preliminary knowledge on their optimal topologies. ...

This paper presents a systematic optimization design method for the multiphase auxetic metamaterials with different deformation mechanisms in both 2D and 3D scenarios. In this method, the parametric color level set (PCLS) is developed to describe different material phases in the microstructures, in which at most 2L materials phases can be precisely represented by only L description functions without any overlaps. Furthermore, clear and smooth material interfaces can be guaranteed in the design, and multiple material usage constraints can be efficiently handled by the well-established gradient-based algorithm. The shape derivative theory is introduced to analyze the design sensitivities for the optimization problem. The effective elasticity properties of the multiphase composites are evaluated by the numerical homogenization method under periodic boundary conditions. Various symmetric conditions are defined and enforced to induce the re-entrant and chiral patterns in both 2D and 3D metamaterials. Several numerical examples are provided to demonstrate the features of the proposed method in tailoring different types of multiphase auxetics. The multiphase metamaterials with two and more material phases are discussed. It is shown that the presented design method can be used to devise both the re-entrant and chiral auxetic metamaterials with excellent auxetic and stiffness properties.

... Besides the SIMP method and the level set method, multi-material topology optimization using the phase field method (Zhou and Wang 2006;Tavakoli 2014) and the ESO method (Radman et al. 2014;Zhao et al. 2019) have also been presented. There are also some special issues in the multi-material structural design that have been widely studied, such as the integrated design of the structural topology and the embedded components Kang et al. 2016), topology optimization of the coated structures (Clausen et al. 2015;Wang and Kang 2018b), concurrent design of multiple microstructural topologies and their macroscale distributions (Xu et al. 2015;Wang and Kang 2019), etc. ...

Multi-material topology optimization is an important issue in the structural and multidisciplinary design. Compared with single-material topology optimization, the multi-material design usually involves more design variables and poses higher requirement for the convergence and efficiency of the topology optimization method. This paper proposes a new multi-material topology optimization strategy based on the material-field series-expansion (MFSE) model. For a structure composed of m different phases of solid materials, m individual material fields are introduced to describe the topology distribution in the multi-material representation model. Herein, each material field is expressed as a linear combination of the eigenvectors and corresponding expansion coefficients based on a reduced series expansion. Thus, the number of design variables can be significantly reduced. Moreover, a new type of smooth Heaviside projection on the material-field function is introduced in the MFSE model, which releases the bound constraints of the material field from the optimization formulation. In this way, the efficiency of the MFSE method is further improved when solving multi-material design problems. Several 2D and 3D numerical examples are presented to show the validity and efficiency of the proposed multi-material method.

... . It indicates that only m level set functions are used to represent total of 2 m -phases by the "color" level set method [18,19]. Besides, Huang et al. [20] extended the BESO method for achieving an optimal solution of multi-material structures, and the proposed method is further applied to the topological design of multi-material microstructures under multiple volume constraints [21][22][23][24][25]. Zhang et al. [26] proposed the moving morphable component (MMC) method to solve the multi-material topology optimization problem under multiple volume constraints using much fewer design variables and degrees of freedom. ...

... 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.

... The BESO studies for this problem were mainly focused on optimizing the structural dynamic characteristics, [14,[41][42][43], which have been proven to be effective. However, the more direct method for this problem is seldom studied, and only a few works focus on the optimization of structural global [44][45][46][47] and local [48] responses were reported. It should be noted that the approaches in the second category are much similar in solving the structural dynamic response, in which there might be a strong nonlinearity when the loading frequency gets close to the structural eigenfrequency. ...

The bi-directional evolutionary structural optimization (BESO) method has been widely studied and applied due to its efficient iteration and clear boundaries. However, due to the use of the discrete design variable, numerical difficulties are more likely to occur with this method, especially in cases with strong nonlinearity. This limits the application of the BESO method in certain cases, such as the suppression of structural dynamic frequency response under high-frequency excitation. In this work, a normalization strategy is proposed for the BESO-based topology optimization, by which the magnitude of the sensitivities can be efficiently unified to the same order to avoid the possible numerical instabilities caused by the nonlinearity. To validate its merit in applications, the normalization-based BESO (NBESO) method is proposed for minimizing the structural frequency response. By means of the weighted sum method, a normalized weighted sum method is also proposed for multi-frequency involved problems. A series of 2D and 3D numerical examples is presented to illustrate the advantages of the NBESO. The effectiveness of the NBESO for multi-frequency response suppression is also demonstrated, in which the frequency ranges below and above the eigenfrequency are involved, respectively.

... [20][21][22][23][24]. These studies show that CTO can evidently extend the design space than those using the single-scale based approaches, and further improve the structural performances, such as compliance [12], dynamic compliance [25], natural frequency [8,26] and frequency responses [27]. ...

This paper studied a robust concurrent topology optimization (RCTO) approach to design the structure and its composite materials simultaneously. For the first time, the material uncertainty with imprecise probability is integrated into the multi-scale concurrent topology optimization (CTO) framework. To describe the imprecise probabilistic uncertainty efficiently, the type I hybrid interval random model is adopted. An improved hybrid perturbation analysis (IHPA) method is formulated to estimate the expectation and stand variance of the objective function in the worst case. Combined with the bi-directional evolutionary structural optimization (BESO) framework, the robust designs of the structure and its composite material are carried out. Several 2D and 3D numerical examples are presented to illustrate the effectiveness of the proposed method. The results show that the proposed method has high efficiency and low precision loss. In addition, the proposed RCTO approach remains efficient in both of linear static and dynamic structures, which shows its extensive adaptability.

... [20][21][22][23][24]. These studies show that CTO can evidently extend the design space than those using the single-scale based approaches, and further improve the structural performances, such as compliance [12], dynamic compliance [25], natural frequency [8,26] and frequency responses [27]. ...

This paper studied a robust concurrent topology optimization (RCTO) approach to design the structure and its composite materials simultaneously. For the first time, the material uncertainty with imprecise probability is integrated into the multi-scale concurrent topology optimization (CTO) framework. To describe the imprecise probabilistic uncertainty efficiently, the type I hybrid interval random model is adopted. An improved hybrid perturbation analysis (IHPA) method is formulated to estimate the expectation and stand variance of the objective function in the worst case. Combined with the bi-directional evolutionary structural optimization (BESO) framework, the robust designs of the structure and its composite material are carried out. Several 2D and 3D numerical examples are presented to illustrate the effectiveness of the proposed method. The results show that the proposed method has high efficiency and low precision loss. In addition, the proposed RCTO approach remains efficient in both of linear static and dynamic structures, which shows its extensive adaptability.

... 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.

... Recently, based on discrete material and thickness optimization (DMTO, Sørensen and Lund 2013;Sørensen et al. 2014), Wu et al. (2019) realized the simultaneous design of ply orientation and thickness of laminated structures with adopting casting-based explicit parameterization to suppress the intermediate void across the thickness of the laminate. Based on the BESO algorithm, Xu et al. (2015) investigated concurrent topology optimization of composite macrostructure and multi-phase periodic microstructure for the minimum dynamic compliance. Liu et al. (2008) proposed the porous anisotropic material with penalization (PAMP) model to investigate the concurrent topology optimization of porous materials. ...

This paper proposes a methodology for simultaneous optimization of composite frame topology and its material design considering specific manufacturing constraints for the maximum fundamental frequency with a bound formulation. The discrete material optimization (DMO) approach is employed to couple two geometrical scales: frame structural topology scale and microscopic composite material parameter scale. The simultaneous optimization of macroscopic size or topology of the frame and microscopic composite material design can be implemented within the DMO framework. Six types of manufacturing constraints are explicitly included in the optimization model as a series of linear inequality or equality constraints. Sensitivity analysis with respect to variables of the two geometrical scales is performed using the semi-analytical sensitivity analysis method. Corresponding optimization formulation and solution procedures are also developed and validated through numerical examples. Numerical study shows that the proposed simultaneous optimization model can effectively enhance the frame fundamental frequency while including specific manufacturing constraints that reduce the risk of local failure of the laminated composite. The proposed multi-scale optimization model for the maximum fundamental frequency is expected to provide a new choice for the design of composite frames in engineering applications.

... Zhao et al. [64] proposed an enhanced decouple sensitivity analysis method for the concurrent dynamic optimization, where the macro and micro structures were both composed of a single material. Xu et al. [65,66] proposed a multiscale optimization formulation for two-phase composites under different dynamic loads. In the case, only the single material description model (replace the "void" phase with one solid phase) is used to describe two phases. ...

This paper proposes a new multiscale topology optimization method for the concurrent design of multiphase composite structures under a certain range of excitation frequencies. Distinguished from the existed studies, a general concurrent design formulation for the dynamic composite structures with more than two material phases is developed. The macro structure and its microstructures with multiple material phases are optimized simultaneously. The integral of the dynamic compliances over an interval of frequencies is formulated as the optimization objective, so as to minimize the frequency response within the concerned excitation range. The effective properties of the multiphase microstructures are evaluated by using the numerical homogenization method, which actually serves as a link to bridge the macro and micro finite element analyses. Furthermore, to describe the boundaries of multiple material phases for the microstructure, a parametric color level set method (PCLSM) is developed by using an efficient interpolation scheme. In this way, L level set functions can represent at most 2L material phases without any overlaps. Moreover, these “color” level sets are updated by directly using the well-established gradient-based algorithm, which can greatly facilitate the proposed method to solve the multi-material optimizations with multiple design constraints. Several 2D and 3D numerical examples are used to demonstrate the effectiveness of the proposed method in the concurrent design of the dynamic composite structures under the excitation frequency ranges.

... Then, the BESO is used to develop a concurrent topology optimization model to maximize the fundamental frequency [58]. Xu et al [59] performed the concurrent design of the composite macrostructures and multi-phase microstructure for minimizing the structural dynamic compliance. However, it should be noted that the previous works for the dynamic multiscale topology optimization only considered a unique microstructure configured in the macro domain. ...

In this paper, a new dynamic multiscale topology optimization method for cellular composites with multi-regional material microstructures is proposed to improve the structural performance. Firstly, a free-material distribution optimization method (FMDO) is developed to generate the overall configuration for the discrete element densities distributed within a multi-regional pattern. The macrostructure is divided into several sub regions, and each of them consists of a number of elements but with the same densities. Secondly, a dynamic topology optimization formulation is developed to perform the concurrent design of the macrostructure and material microstructures, subject to the multi-regional distributed element densities. A parametric level set method is employed to optimize the topologies of the macrostructure and material microstructures, with the effective macroscopic properties evaluated by the homogenization. In the numerical implementation, the quasi-static Ritz vector (QSRV) method is incorporated into the finite element analysis so as to reduce the computational cost in numerical analysis, and some kinematical connectors are introduced to make sure the connectivity between adjacent material microstructures. Finally, 2D and 3D numerical examples are tested to demonstrate the effectiveness of the proposed dynamic multiscale topology optimization method for the material-structural composites.

... g on this idea, Yan, Cheng, and Liu 2008, Deng, Yan, and Cheng 2013, Yan et al. 2014 Gu 2018 discussed the concurrent optimization of thermoelasticity and thermal conduction. Besides the discussion about the multifunctional applications, the concurrent approaches in considering multiphase materials are concerned as well in recent years. Especially Xu et al. Xu, Jiang, and Xie 2015 built the concurrent optimization models in regard to multiphase materials respectively under harmonic, random and mechanical-thermal coupled loads. Da et al. 2017 focused on the super multiphase materials problem in concurrent optimization. Besides the homogenization theory, other approaches are also employed in concurrent optimization ...

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.

... For example, for maximizing the fundamental frequency of a structure, Niu et al. [33], Zuo et al. [34] and Liu et al. [35] studied the topology optimization for concurrent design of composite structures and material designs. In order to min- imize the dynamic compliance, Xu et al. [36] built a concurrent topology optimization model of the macrostructure and multi-phase periodic microstructure by bidirectional evolutionary structural optimization (BESO) method. Vicente et al. [37] developed a concurrent topology optimization method to minimize the frequency response based on BESO method. ...

A new robust topology optimization method based on level sets is developed for the concurrent design of dynamic structures composed of uniform periodic microstructures subject to random and interval hybrid uncertainties. A Hybrid Dimensional Reduction (HDR) method is proposed to estimate the interval mean and the interval variance of the uncertain objective function based on a bivariate dimension reduction scheme. The robust objective function is defined as a weighted sum of the mean and standard variance of the dynamic compliance under the worst case. The sensitivity information of the robust objective function with respect to the macro and micro design variables can then be obtained after the uncertainty analysis. Several examples are used to validate the effectiveness of the proposed robust topology optimization method.

... 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.

... Sivapuram et al. (2016) studied using different heterogeneous materials in prescribed domains in macrostructure and formulating a new way to decompose macro and micro design problems in a linear way. Xu et al. (2015) studied concurrent design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance using BESO method. Gao and Ma (2015) studied a modified model for concurrent topology optimization, where the microstructure orientation is introduced as a new type of design variable. ...

This paper studies two-scale concurrent topology optimization with multiple micro heterogeneous materials subjected to volume constraints. In previous work on concurrent two-scale optimization, either only one material with optimal microstructure is assumed or multiple micro materials are included but are distributed in prescribed geometrical domains. Here the selection of micro heterogeneous materials is based on the criterion for principal stress orientation in the macro structure. To meet this requirement, an additional constraint, called misplaced material volume constraint, is introduced to constrain the volume fraction of material that is misplaced in macro structure to be less than a small parameter ε. This constraint comprises several piecewise smooth penalty functions, each of which is a proper modification of Heaviside function. One advantage of the misplaced material volume constraint is that, without much modification to the original formulation, the optimized macro material is distributed in line with the use criterion and the material microstructures automatically converge to different optimized topologies. Three numerical examples are presented to show the effectiveness of the proposed method.

... Sivapuram et al. (2016) studied using different heterogeneous materials in prescribed domains in macrostructure and formulating a new way to decompose macro and micro design problems in a linear way. Xu et al. (2015) studied concurrent design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance using BESO method. Gao and Ma (2015) studied a modified model for concurrent topology optimization, where the microstructure orientation is introduced as a new type of design variable. ...

... 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.

... Guo et al. (2015) proposed a robust concurrent optimization formula when the macrostructure was subject to uncertain loads. In other research with binary design variables, unambiguous topologies of macrostructure and microstructure were obtained using the BESO method in dealing with static and dynamic problems (Huang et al. 2013;Xu, Jiang, and Xie 2015;Vicente et al. 2016). To obtain the optimized design of porous materials, most of the aforementioned works focused on the enforcement of macrovolume and microvolume fraction constraints in the optimization procedure. ...

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.

... The design variables on the micro scale based on the SIMP method are defined in [35]. Then the equivalent elastic matrix and the equivalent mass density on the micro scale can be described as follows where I is an identity and udenotes the displacement fields of the unit cell caused by these uniform strain fields. ...

The topology optimization method of material microstructure of piezoelectric composite structures is proposed so as to improve the dynamic characteristics of the closed-loop system. The topology optimization model of material microstructure is built, where the design variable is constructed based on the SIMP interpolation scheme and the objective function is to maximize the damping dissipation velocity of the piezoelectric composite structure based on the independent modal control strategy, while the constraints are applied on the volume fractions of the phase materials. The sensitivity formulations of the damping dissipation velocity in modal space with respect to the design variables are derived. A bi-direction evolutionary structural optimization (BESO) method is developed to obtain a clear and optimal topology for material microstructure. The results of several numerical examples show that different initial designs of the base cell, different boundary conditions and the property of the disturbance have effect on the optimal solution. Also, the proposed method can effectively improve the active control performance and reduce structural weight.

... 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.

... Da et al. [22] assumed the macroscale structure is made of two different composite materials and each of them consists of three to four material phases at the microscopic scale. Xu et al. [168] have also extended the dynamic compliance design of two-scale structures to the multi-phase case considering macroscale composite structures and microscale multi-phase microstructures. Fig. 58(a) gives one representative stiffness maximization design of a thermoelastic composite structure. ...

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.

... Along a similar path optimal topologies that minimize the dynamic compliance due to a thermal action are found in [6] whereas Liu et al. [7] faces the subtle issue of minimizing the dynamic compliance in the presence of rotating harmonic loads characterized by a single frequency. Under the same hypothesis on the acting loads, Xu et al. [8] introduces a concurrent design strategy for the optimal design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance. A strategy for topology optimization of magnetorheological fluid layers in sandwich plates for semi-active vibration control is proposed in [9], where either a single-frequency harmonic load is considered or, to account for the possible variability of the frequency content of the excitation, an objective function is introduced that is the maximum of all possible dynamic compliances computed for each single harmonic load. ...

Dynamic compliance (structural and topology) optimization is a topic of active and fertile research carried on by several research groups aiming to extend to the dynamic regime by now well-consolidated approaches for static compliance optimization. Available approaches on this purpose are first divided into time-domain and frequency-domain strategies and then a comparison between the two is performed with respect to actual significance and CPU time of relevant optimal designs. For this paper sake reference is made to the optimal design of viscoelastic thin beams but the approach may be shown to apply with no modifications to other and possibly more complex systems such as 2D and 3D dynamic-compliance topology optimization. By extensive numerical investigations, it is shown that the frequency-domain approach to be preferred over time-domain schemes even though relevant computations happen to be heavier, mainly as far as the computation of the -norm of the system transfer function is concerned.

... Thus, among several other sub-areas, during the latest decades, topology optimization of continuum structures has been developed and applied in the following sub-areas of dynamic and vibro-acoustic topological design, & dynamic compliance and response (Ma et al. 1995;Min et al. 1999;Shu et al. 2011;Kang et al. 2012;Yang and Li 2013;Zhang and Kang 2014;Zhao and Wang 2015;Xu et al. 2015;Jung et al. 2015;Liu et al. 2015;Olhoff and Niu 2015), & vibro-acoustic response (Jog 2002a, b, Olhoff andDu 2006;Jensen 2007;Calvel and Mongeau 2007;Du and Olhoff 2007a;Niu et al. 2010;Yoon 2010;Nandy and Jog 2011;Du et al. 2011;Yang and Du 2012;Du and Yang 2015), and & eigenfrequencies, eigenfrequency gaps, and band-gaps (Sigmund 2001;Sigmund and Jensen 2003;Jensen 2003;Halkjaer et al. 2006;Olhoff 2007a, b, c, Olhoff et al. 2012). ...

This paper deals with topological design optimization of elastic, continuum structures without damping that are subjected to time-harmonic, design-independent external dynamic loading with prescribed excitation frequency, amplitude and spatial distribution. The admissible design domain, the boundary conditions, and the available amount(s) of material(s) for single- or bi-material structures are given. An important objective of such a design problem is often to drive the resonance frequencies of the structure as far away as possible from the given excitation frequency in order to avoid resonance and to reduce the vibration level of the structure. In the present paper, the excitation frequency is defined to be ‘low’ for positive values up to and including the fundamental resonance frequency of the structure, and to be ‘high’ for values beyond that. Our paper shows that it may be very important to consider different design paths in problems of minimum dynamic compliance in order to obtain desirable solutions for prescribed excitation frequencies, and a so-called ‘incremental frequency technique’ (IF technique) is applied for this. Subsequently, the IF technique is integrated into an extended, systematic method named as the ‘generalized incremental frequency method’ (GIF method) that is developed for gradient based dynamic compliance minimization for not only ‘low’, but also ‘high’ excitation frequencies of the structure. The GIF method performs search for and determination of the solution to the minimum dynamic compliance design problem, but this is subject to the complexity that problems with prescribed high excitation frequencies exhibit disjointed design sub-spaces. Each of these sub-spaces is associated with a local minimum value of the dynamic compliance, so in general the ‘global optimum solution’ will have to be selected as the ‘best’ solution from among a number of local candidate solutions. Illustrative examples of application of the IF technique and the GIF method are presented in the paper.

... V i is the volume of element i, V Ã q is the prescribed volume for phase material q and n is the total number of material phases. a ij is the design variable as defined in [26]. X, _ X and € X are the system displacement, velocity and acceleration vectors. ...

... 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.

Recent advances in biomedical engineering have promoted the development of innovative metal implants that have integrated mechanical and biodegradable properties. Most of the existing implant designs were created by using a trial-and-error approach, which depends on the designer’s experience. Alternatively, inverse design approaches, such as topology optimization, have evolved into an efficient design solution to optimize the structural and material layout within a given design domain. Here we introduce a novel topology optimization scheme to support the microstructural design of biodegradable metal matrix composite structures (BMMCS). The effect of material degradation on the mechanical performance of the structure is considered by integrating a degradation simulation algorithm into the structural finite element (FE) analysis. The objective function is to minimize the structural compliance at macroscale in a certain number of time steps to realize sufficient structural integrity in the initial bone healing stage, and the different stiffness reduction properties can be adjusted by changing the volume ratio of the two base constitutive biodegradable materials. The sensitivity of the above objective function concerning design variables was derived with considering the time-dependent degradation of the biodegradable material. Several numerical design examples were presented and benchmarked with classical designs. Finally, several prototypes were fabricated by integrating additive manufacturing with casting technology. Collectively, this demonstrates the feasibility and effectiveness of the proposed inverse design method for additive manufacturing.

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.

The bi-directional evolutionary structural optimization (BESO) method has been widely studied and applied for its efficient iteration and clear boundaries. However, due to the use of the discrete design variable, numerical difficulties are more likely to occur to this method, especially in the cases with strong nonlinearity. This limits the application of the BESO method in certain topics, such as the suppression of structural dynamic frequency response under high-frequency excitation. In this work, a normalization strategy is proposed for the BESO based topology optimization, by which the magnitude of the sensitivities can be efficiently unified to the same order to avoid the possible numerical instabilities caused by the nonlinearity. To validate its merit in applications, the normalization based BESO (NBESO) method is proposed for minimizing the structural frequency response. By means of the weight sum method, a normalized weight sum (NWS) method is also proposed for the multi-frequency involved problems. A series of 2D and 3D numerical examples are presented to illustrate the advantages of the NBESO. The effectiveness of the NBESO for the multi-frequency response suppression is also demonstrated, in which the frequency ranges below and above the eigenfrequency are involved, respectively.

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.

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.

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.

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 capabilities and operation of electromagnetic devices can be dramatically enhanced if artificial materials that provide certain prescribed properties can be designed and fabricated. This paper presents a systematic methodology for the design of dielectric materials with prescribed electric permittivity. A gradient-based topology optimization method is used to find the distribution of dielectric material for the unit cell of a periodic microstructure composed of one or two dielectric materials. The optimization problem is formulated as a problem to minimize the square of the difference between the effective permittivity and a prescribed value. The optimization algorithm uses the adjoint variable method (AVM) for the sensitivity analysis and the finite element method (FEM) for solving the equilibrium and adjoint equations, respectively. A Heaviside projection filter is used to obtain clear optimized configurations. Several design problems show that clear optimized unit cell configurations that provide the prescribed electric permittivity can be obtained for all the presented cases. These include the design of isotropic material, anisotropic material, anisotropic material with a non-zero off-diagonal terms, and anisotropic material with loss. The results show that the optimized values are in agreement with theoretical bounds, confirming that our method yields appropriate and useful solutions.

This paper introduces a two-scale topology optimization approach by integrating optimized structures
with the design of their materials. The optimization aims to find a multifunctional structure composed
of homogeneous porous material. Driven by the multi-objective functions, macrostructural stiffness
and material thermal conductivity, stiff but lightweight structures composed of thermal insulation materials
can be achieved through optimizing the topologies of the macrostructures and their material microstructure
simultaneously. For such a two-scale optimization problem, the effective properties of
materials derived from the homogenization method are applied to the analysis of macrostructure. Meanwhile,
the displacement field of the macrostructure under given boundary conditions is used for the sensitivity
analysis of the material microstructure. Then, the bi-directional evolutionary structural
optimization (BESO) method is employed to iteratively update the macrostructures and material microstructures
by ranking elemental sensitivity numbers at the both scales. Finally, some 2D and 3D numerical
examples are presented to demonstrate the effectiveness of the proposed optimization algorithm. A
variety of optimal macrostructures and their optimal material microstructures are obtained.

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 paper presents a bidirectional evolutionary structural optimization (BESO) method for designing periodic microstructures of two-phase composites with extremal electromagnetic permeability and permittivity. The effective permeability and effective permittivity of the composite are obtained by applying the homogenization technique to the representative periodic base cell (PBC). Single or multiple objectives are defined to maximize or minimize the electromagnetic properties separately or simultaneously. The sensitivity analysis of the objective function is conducted using the adjoint method. Based on the established sensitivity number, BESO gradually evolves the topology of the PBC to an optimum. Numerical examples demonstrate that the electromagnetic properties of the resulting 2D and 3D microstructures are very close to the theoretical Hashin-Shtrikman (HS) bounds. The proposed BESO algorithm is computationally efficient as the solution usually converges in less than 50 iterations. The proposed BESO method can be implemented easily as a post-processor to standard commercial finite element analysis software packages, e.g. ANSYS which has been used in this study. The resulting topologies are clear black-and-white solutions (with no grey areas). Some interesting topological patterns such as Vigdergauz-type structure and Schwarz primitive structure have been found which will be useful for the design of electromagnetic materials.

This paper aims to develop a level-set-based topology optimization approach for the design of negative permeability electromagnetic metamaterials, where the topological configuration of the base cell is represented by the zero-level contour of a higher-dimensional level-set function. Such an implicit expression enables us to create a distinct interface between the free space and conducting phase (metal). By seeking for an optimality of a Lagrangian functional in terms of the objective function and the governing wave equation, we derived an adjoint system. The normal velocity (sensitivity) of the level-set model is determined by making the Eulerian derivative of the Lagrangian functional non-positive. Both the governing and adjoint systems are solved by a powerful finite-difference time-domain algorithm. The solution to the adjoint system is separated into two parts, namely the self-adjoint part, which is linearly proportional to the solution of the governing equation; and the non-self-adjoint part, which is obtained by swapping the locations of the incident wave and the receiving planes in the simulation model. From the demonstrative examples, we found that the well-known U-shaped metamaterials might not be the best in terms of the minimal value of the imaginary part of the effective permeability. Following the present topology optimization procedure, some novel structures with desired negative permeability at the specified frequency are obtained.

In the topology optimization of structures, compliant mechanisms or materials, a density-like function is often used for material
interpolation to overcome the computational difficulties encountered in the large “0-1” type integer programming problem.
In this paper, we illustrate that a gradually formed continuous peak function can be used for material interpolation. One
of the advantages of introducing the peak function is that multiple materials can easily be incorporated into the topology
optimization without increasing the number of design variables. By using the peak function and the optimality criteria method,
we synthesize compliant mechanisms with multiple materials with and without the material resource constraint. The numerical
examples include the two-phase, three-phase, and four-phase materials where void is treated as one material. This new design method enables us to optimally juxtapose stiff and flexible materials in compliant
mechanisms, which can be built using modern manufacturing methods.

We present a topology optimization method for the design of periodic composites with dissipative materials for maximizing the loss/attenuation of propagating waves. The computational model is based on a finite element discretization of the periodic unit cell and a complex eigenvalue problem with a prescribed wave frequency. The attenuation in the material is described by its complex wavenumber, and we demonstrate in several examples optimized distributions of a stiff low loss and a soft lossy material in order to maximize the attenuation. In the examples we cover different frequency ranges and relate the results to previous studies on composites with high damping and stiffness based on quasi-static conditions for low frequencies and the bandgap phenomenon for high frequencies. Additionally, we consider the issues of stiffness and connectivity constraints and finally present optimized composites with direction dependent loss properties.

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.

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.

This research develops a new interpolation scheme, a separable stress interpolation (SSI) which allows stress-based topology optimization with multiple materials (STOMM). In common material interpolation function such as extended solid isotropic material with penalization (SIMP) for multiple materials, Young's modulus is interpolated from those of several solids to a smaller value with respect to design variables whose number is same as the number of considered materials. When the same interpolated Young's modulus is used for stress evaluation, it is found that the calculated stress norm becomes a small value when ones are assigned to the design variables of each element causing physically unacceptable layouts. In order to resolve this ill-posed issue for STOMM, we present the SSI scheme which computes the stress constraints of stacked elements separately. For a stable topology optimization process, the computational issues of the p-norm stress measure, the number of stress evaluation points inside an element, and the correction parameter for the approximated stress measure are addressed for STOMM. Furthermore, we present a new regional constraint method based on the sorting algorithm. The applicability and limitations of the newly developed framework are discussed in the context of its application to several stress-based topology optimizations with multiple materials.

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.

Different from the independent design of macrostructures or material microstructures, a two-scale topology optimization algorithm is proposed by using the bi-directional evolutionary structural optimization (BESO) method for the concurrent design of the macrostructure and its composite microstructure. It is assumed that the macrostructure is made of composite materials whose effective properties are calculated through the homogenization method. By conducting finite element analysis of both structures and materials, sensitivity numbers at the macro- and micro-scale levels are derived. Then, the BESO method is used to iteratively update the macrostructures and the composite microstructures according to the elemental sensitivity numbers at both scales. Some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm. A variety of optimal macrostructures and optimal material microstructures have been obtained.

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.

We present a method to design manufacturable extremal elastic materials. Extremal materials can possess interesting properties such as a negative Poisson’s ratio. The effective properties of the obtained microstructures are shown to be close to the theoretical limit given by mathematical bounds, and the deviations are due to the imposed manufacturing constraints. The designs are generated using topology optimization. Due to high resolution and the imposed robustness requirement they are manufacturable without any need for post-processing. This has been validated by the manufacturing of an isotropic material with a Poisson’s ratio of ν=-0.5ν=-0.5 and a bulk modulus of 0.2% times the solid base material’s bulk modulus.

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.

There are several well-established techniques for the generation of solid-void optimal topologies such as solid isotropic
material with penalization (SIMP) method and evolutionary structural optimization (ESO) and its later version bi-directional
ESO (BESO) methods. Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed
in this paper. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent
optimal solutions for structures with one or multiple materials. The results show that the optimal designs from the present
BESO method are independent on the degree of penalization. The resulted optimal topologies and values of the objective function
compare well with those of SIMP method.

Presents a simple heuristic optimization algorithm to determine weight-minimal laminate structures. The algorithm is based on an inverted form of growth strategy, where material is removed in areas with low stresses. In the presented algorithm the removal of material is performed in a layerwise manner in areas with low stress or in areas where the layer orientation angle differs significantly from the principal stress direction. In order to get a production-adapted structure, the layer orientation angles of the available individual plies are not modified and the material is removed only locally in the respective plies. Contrary to a more formal mathematical optimization no sensitivity analyses are needed by the procedure outlined so far and this keeps the corresponding numerical effort reasonably low. The structural analyses have been performed by the Finite Element Program ANSYS and the outlined heuristic algorithm has been implemented by ANSYS-macros. Several examples of rectangular laminate plates show the effectiveness of the present algorithm.

This paper studies topology optimization of a coupled opto-mechanical problem with the goal of finding the material layout which maximizes the optical modulation, i.e. the difference between the optical response for the mechanically deformed and undeformed configuration. The optimization is performed on a periodic cell and the periodic modeling of the optical and mechanical fields have been carried out using transverse electric Bloch waves and homogenization theory in a plane stress setting, respectively. Two coupling effects are included being the photoelastic effect and the geometric effect caused by the mechanical deformation.
For the studied objective and material choice it is concluded that the photoelastic effect and the geometric effect counteract each other, which yields designs which are fundamentally different if the optimization takes only one effect into account. When both effects are active a compromise is found; however, a strong regularization is needed in order to achieve reasonable 0–1 designs with a clear physical interpretation. Copyright

This paper presents two alternative approaches for realizing weight-minimal laminate structures by topology optimization. In both cases, topology optimization is performed in a layerwise manner such that the individual laminate plies are allowed to have their individual topologies. In the first approach, a heuristic optimization algorithm is applied: starting from an oversized initial design, this algorithm iteratively removes single layer material at locations where it is not seriously needed. In the second approach, a genetic algorithm has been developed that adapts the material distribution and the local reinforcement directions to the given structural needs. Two examples of laminate structures show the effectiveness of the proposed algorithms.

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two-scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so-called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale-related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 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.

A methodology based on topology optimization for the design of metamaterials with negative permeability is presented. The
formulation is based on the design of a thin layer of copper printed on a dielectric, rectangular plate of fixed dimensions.
An effective media theory is used to estimate the effective permeability, obtained after solving Maxwell’s equations on a
representative cell of a periodic arrangement using a full 3D finite element model. The effective permeability depends on
the layout of copper, and the subject of the topology optimization problem is to find layouts that result in negative (real)
permeability at a prescribed frequency. A SIMP-like model is invoked to represent the conductivity of regions of intermediate
density. A number of different filtering strategies are invoked to facilitate convergence to binary solutions. Examples of
designs for S-band applications are presented for illustration. New metamaterial concepts are uncovered, beyond the classical
split-ring inspired layouts.

Topology optimization is a promising method for systematic design of optical devices. As an example, we demonstrate how the method can be used to design a 90° bend in a two-dimensional photonic crystal waveguide with a transmission loss of less than 0.3% in almost the entire frequency range of the guided mode. The method can directly be applied to the design of other optical devices, e.g., multiplexers and wave splitters, with optimized performance. © 2004 American Institute of Physics.

The present paper deals with optimization of hybrid fiber reinforced plastic laminated plates subjected to impact loading. Finite element method (FEM) and genetic algorithm (GA) have been used to obtain optimum laminate in terms of minimizing the cost, weight or both cost and weight of graphite/epoxy (T300/5208)–aramid/epoxy (Kevlar 49) hybrid laminates while maximizing the strength. Impact induced delamination and matrix cracking have been used as failure criteria for the optimization of laminate. Fiber orientation, material and thickness in each lamina as well as number of lamina in the laminate have been used as design variables. Multi-objective approach has been used to achieve the optimum design of a laminate for combined normalized weighted cost and weight minimization. The results obtained from the integrated module show that GA with FEM can lead to a near optimal solution for both single as well multiple objective functions.

This paper describes a method to design the periodic microstructure of a material to obtain prescribed constitutive properties. The microstructure is modelled as a truss or thin frame structure in 2 and 3 dimensions. The problem of finding the simplest possible microstructure with the prescribed elastic properties can be called an inverse homogenization problem, and is formulated as an optimization problem of finding a microstructure with the lowest possible weight which fulfils the specified behavioral requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm. This implies that the optimal microstructure of a base cell is found from a truss or frame structure with 120 possible members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method, using Finite-Elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method.Numerical examples in two and three dimensions show that it is possible to design materials with many different properties using base cells modelled as truss or frame works. Hereunder is shown that it is possible to tailor extreme materials, such as isotropic materials with Poisson's ratio close to − 1, 0 and 0.5, by the proposed method. Some of the proposed materials have been tested as macro models which demonstrate the expected behaviour.

In this paper we address the problem of structural shape and topology optimization in a multi-material domain. A level-set method is employed as an alternative approach to the popular homogenization-based methods of rule of mixtures for multi-material modeling. A multi-phase level-set model is adapted for material and topology representation. This model eliminates the need for a material interpolation or phase mixing scheme. It only requires m level-set functions to represent a structure of n=2m different material phases, in a principle similar to combining colors from the three primary colors. Therefore, this multi-phase model may be referred to as a “color” level-set representation which has its unique benefits: it is flexible to handle complex topologies; it substantially reduces the number of model functions when n>3; it automatically avoids the problem of overlap between material phases of a conventional partitioning approach. We describe numerical techniques for efficient and robust implementation of the method, by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain. This would separate the issues of accuracy in numerical calculations of the physical equation and in the level-set model propagation. A gradient projection method is described for incorporating multiple constraints in the problem. Finally, the benefits and the advantages of the developed method are illustrated with several 2D examples of mean compliance minimization of multi-material structures.

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.

To bring down noise levels in human surroundings is an important issue and a method to reduce noise by means of topology optimization is presented here. The acoustic field is modeled by Helmholtz equation and the topology optimization method is based on continuous material interpolation functions in the density and bulk modulus. The objective function is the squared sound pressure amplitude. First, room acoustic problems are considered and it is shown that the sound level can be reduced in a certain part of the room by an optimized distribution of reflecting material in a design domain along the ceiling or by distribution of absorbing and reflecting material along the walls. We obtain well defined optimized designs for a single frequency or a frequency interval for both 2D and 3D problems when considering low frequencies. Second, it is shown that the method can be applied to design outdoor sound barriers in order to reduce the sound level in the shadow zone behind the barrier. A reduction of up to 10 dB for a single barrier and almost 30 dB when using two barriers are achieved compared to utilizing conventional sound barriers.

Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming.To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds.The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

The design of interior cutouts in laminated composite panels is of great importance in aerospace, automobile and structural engineering. Based on the Tsai–Hill failure criterion of the first ply, this paper presents a newly developed Fixed (FG) Grid Evolutionary Structural Optimization (ESO) method to explore shape optimization of multiple cutouts in composite structures. Different design cases with varying number of cutouts, ply orientations and lay-up configurations are taken into account in this study. The examples demonstrate that the optimal boundaries produced by FG ESO are much smoother than those by traditional ESO. The results show the remarkable effects of different opening numbers and various lay-up configurations on resulting optimal shapes. The paper also provides an in-depth observation in the interactive influence of the adjacent cutouts on the optimal shapes.

The optimal control problem of minimizing the dynamic response of anisotropic symmetric or antisymmetric composite laminated rectangular plates with various boundary conditions is presented using various plate theories. The objective of the present control problem is to minimize the dynamic response of the plate with minimum possible expenditure of force. The dynamic response of the structure comprises a weight sum of the control objective (the total vibrational energy) and a penalty functional of the control force. In addition to the active control, the layer thickness and the orientation angle of the material fibers are taken as optimization design variables. The explicit solutions for the optimal force and controlled deflections are obtained in forms of double series using the Liapunov–Bellman theory. The effectiveness of the proposed control and the behavior of the controlled structure are investigated. Various numerical results including the effect of boundary conditions, number of layers, anisotropy ratio, aspect ratio, and side-to-thickness ratio on the control process for symmetric and antisymmetric laminates are presented.

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.

A multiobjective optimal design methodology is developed for lightweight, low-cost composite structures of improved dynamic performance. The design objectives may include minimization of damped resonance amplitudes (or maximization of modal damping), weight, and material cost. The design vector includes micromechanics, laminate, and structural shape parameters. Constraints are imposed on static displacements, static and dynamic ply stresses, dynamic amplitudes, and natural frequencies. The effects of composite damping tailoring on the dynamics of the composite structure are incorporated. Applications on a cantilever composite beam and plate illustrate that only the proposed multiobjective formulation, as opposed to single-objective functions, may simultaneously improve the objectives. The significance of composite damping in the design of advanced composite structures is also demonstrated, and the results indicate that the minimum-weight design or design methods based on undamped dynamics may fail to improve the dynamic performance near resonances.

Design and fabrication of biphasic cellular materials with transport properties -a modified bidirectional evolutionary structural optimization procedure and MATLAB program

- Zhou Shiwei
- Cadman Joseph
- Chen Yuhang
- Li Wei
- Yi Xie
- Huang Min
- Xiaodong

Zhou Shiwei, Cadman Joseph, Chen Yuhang, Li Wei, Xie Yi Min, Huang
Xiaodong, et al. Design and fabrication of biphasic cellular materials with
transport properties -a modified bidirectional evolutionary structural
optimization procedure and MATLAB program. Int J Heat Mass Transfer
2012;55(25-26):8149-62.