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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.

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... Topology optimization is highly promising for achieving the material-structure-performance integrated design, the related approaches [1][2][3][4][5][6][7][8][9][10] have been extensively explored in the past few decades since the pioneering work of Bends and Kikuchi [11]. The advances in multiscale concurrent topology optimization methods [12][13][14][15][16][17][18] lead to new opportunity to simultaneously optimize the microstructure and macroscopic material distribution with help of the homogenizationbased methods [11,19], which have gained more interest in the engineering fields [20][21][22][23][24][25][26]. Till now, the relatively mature multiscale topology optimization methods basically only consider one kind of material microstructure, and the corresponding macroscopic structure is assumed to be periodically configured by a series of periodic unit cells (PUCs) with the same topology, as schematically marked with the red point in Fig. 1. ...

... In the past decades, the studies about topological design of concurrent topology optimization has been mainly focused on the stiffness maximization problems under multiscale volume constraints [22][23]25,36,[38][39]. The safety and reliability of traditional concurrent design will be greatly challenged when the design is subjected to dynamic loads. ...

... However, it is worth noting that the derivation in Eqs. (21)(22) will lead to extremely excessive burden of computational cost because the ...

There are many requirements in engineering fields to improve the structural performance. Thus, the hierarchical structures have attracted lots of attention because of the appealing material efficiency and functionality, e.g., the bearing capacity and interfacial strength from large aerospace structures to small device facilities. However, the dynamic topology optimization design of these structures composed of multiple microstructures remains challenge by the connectivity issue between different microstructures and the extensive computational cost when handling the interval-frequency problem during multiscale concurrent design. In this work, the connectivity model is proposed for improving the manufacturability of hierarchical structure, which includes the connectable layer scheme between different microstructures and the enlarged filter domain. To break the bottleneck of computation cost induced by the simultaneous optimization of macro-microscopic design variables, the combined method of modal superposition and model order reduction is incorporated to efficiently compute the frequency response. Two response vectors are presupposed before decoupling sensitivity analysis. In addition, in order to implicitly ensure the bearing capacity of the structure, the stiffness maximization design is integrated into the response minimization problem when the frequency interval range is high. The layer-wise graded cantilever beam is used as an example to demonstrate the effectiveness and accuracy of proposed method in handling dynamic concurrent topology optimization problem. The manufacturability and systematic performance of hierarchical structures are significantly improved when they simultaneously experience the interval-frequency harmonic and static loads.

... Xu and Xie [14] proposed a concurrent multiscale topology optimization method for two-dimensional macro-and microstructures to minimize the displacement response mean square. Xu et al. [15] presented a concurrent multiscale topology optimization method aiming to maximize the macrostructural stiffness with multiphase material. Yan et al. [16] proposed a concurrent topology optimization method to get the topology of a macrostructure, a microstructure, and orientation of microstructures. ...

... (14)) and the adjoint equation (Eq. (15)) in the macrostructure, the state equation (Eq. (16)) and the adjoint equation (Eq. ...

... (14) and (16), the adjoint Eqs. (15) and (17), and the constraint Eq. (18) are all satisfied, the perturbation expansion of the Lagrange function is expressed as Eq. (25). ...

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

... This addition or removal depends on the sensitivity analysis. Sensitivity analysis is the estimation of the response of the structure to the modification of design variables and is dependent on the calculation of derivatives [70][71][72][73][74][75][76][77][78][79][80]. 13 A New BEM for Modeling and Optimization of 3T Fractional Nonlinear Generalized Magneto… DOI: http://dx.doi.org ...

... The second step is to substitute for _ U respectively, in Eq. (85) and solve the resulted equation for the threetemperature fields. The third step is to correct the displacement using the computed three-temperature fields for the Eq.(80). The fourth step is to compute _ U ...

... For composite material design and selection, understanding and quantifying the links between material structure at the nano/mesoscale/microscale and their macroscopic effects is, therefore, essential and requires the integration of several models that have overlapping scales (polymer chemistry, matrix-fibre interface, fibre properties and topologies, etc.) [3,4]. This implies the need for development and integration of models to describe the behaviour of composite materials at different scales as well as material-processing-property relationships [5]. ...

... Epoxy resins are reasonably stable to chemical attacks and are excellent adherents having slow shrinkage during curing and no emission of volatile gases. Four (4) formulations with well-known ingredients (molecular structure) and well know properties will be used: is mainly used to cure the epoxy resin, which causes a chemical reaction without changing its own composition. The curing time mainly depends on the hardener and epoxy mixing ratio. ...

This paper shares and contributes to a ground-breaking vision developed and being implemented which consists in the integration of materials modelling methodologies and knowledge-based systems with business process for decision making. The proposed concept moves towards a new paradigm of material and process selection and design by developing and implementing an integrated multi-disciplinary, multi-model and multi-field approach together with its software tool implementation for an accurate, reliable, efficient and cost effective prediction, design, fabrication, Life Cycle Engineering (LCE), cost analysis and decision making. This new paradigm of integrated material design is indeed endowed with a great potential by providing further insights that will promote further innovations on a broad scale.

... This methods are being utilized by few authors to minimized the mean compliance of thermoelasctic structures considering, temperature's distribution independent of the thermoelastic design (Rodrigues and Fernandes, 1995), material volume constraints (Xia and Wang, 2008), thermoelastic stress loads (Gao and Zhang, 2010), a multi-scale and multimaterial analysis (Xu et al., 2016). Moreover, the BESO method has been used to minimized the compliance of structures under design-dependent loads (Picelli et al., 2014(Picelli et al., , 2017. ...

... The sensitive number can be found by the gradient of the objective function (C) with respect to the design variable x ij (Xu et al., 2016). Assuming that E 1 > E 2 ... > E n , the sensitive numbers are calculated as follows: ...

... With the multiple material phase interpolation scheme at hand (section 4.1), it is straightforward to extend the above mentioned simultaneous design frameworks for the design of multi-phase two-scale structures. Xu et al. [170] proposed to design thermoelastic composite structures, where the macroscale structure is assumed to be made a composite material with three constituent phases. 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. ...

... Fig. 58(a) gives one representative stiffness maximization design of a thermoelastic composite structure. The underlying composite material is assumed to be made of one void phase and two solid phases with different Young's mod- [170] uli and thermal expansion coefficients. Fig. 58(b) and (c) are the design results corresponding to cases of ∆T = 0 • C and ∆T = 500 • C, respectively. ...

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.

... For example, due to the fact that the fuzzy interfaces of traditional multi-material designs will lead to the local optimum issue, some advanced multi-material methods are developed in Refs. [12][13][14][15][16][17][18][19][20][21][22][23][24] for providing the clear and smooth boundaries for multi-material structures. To decouple the nonlinearity problem between total mass and variables, several efficient optimization strategies [25][26][27][28][29][30][31] are presented for avoiding the variable-dependent derivative of the total mass w.r.t design variables in the traditional multi-material model. ...

Multi-material structures offer the superior performance and broader designable freedom, and have gained increasing interest benefitting from the advances in additive manufacturing (AM) technique. However, the effect of anisotropic yield strength of these AM-fabricated parts on to-pology optimization of multi-material structures has so far received little attention. In the current work, an anisotropic strength-based topology optimization method for multi-material structures is developed for maximizing strength performance under total mass constraint, which allows for the consideration of directional dependence and tension-compression asymmetry of yield strengths based on the Tsai-Wu yield criterion. Another challenge for multi-material structures design is that the possible permutations of numerous candidate materials are unlimited, it is impossible that all combinations of multiple candidate materials are tested for the optimal strength performance of multi-material structures. To solve this problem, we propose a prese-lection method for the inclusion or exclusion of candidate materials based on their elastic properties, yield strengths and densities before optimization. Afterwards, our topology optimization method can automatically find the optimal combination of volume fractions of different candidate materials. Several 2D and 3D numerical examples are investigated for minimizing the maximum of failure indexes of strength for fixed total mass. The advantages of multi-material structures are verified by comparing to single-material designs. The effectiveness of the prese-lection method of candidate materials is also verified.

... These suggested mechanisms are relying on the hygral expansion of a material to achieve motion or deformation. In addition, the design of porous moisture displacement inverter can achieve a high level of sensitivity while also maintaining a low weight, resulting in improving durability and attaining phenomenal light weighing (Fujioka et al. 2022;Kato et al. 2018;Xu et al. 2016;Zhang et al. 2019). ...

This study explores the application of concurrent multiscale topology optimization in the design of lightweight hygral-activated porous compliant mechanisms. The proposed approach utilizes two distinct representations of the design problem, namely macro and microscale domains, to achieve an optimized design. By implementing a concurrent multiscale topology optimization framework, the effective properties of the microscale, including elastic and hygral diffusivity tensors and the hygral expansion coefficient, are calculated and used as the hygro-elastic modeling effective properties of the macroscale. Furthermore, this study considers hygral transport in solids and hygral evaporation, thereby enabling simultaneous consideration of hygral transfer physics. A sensitivity analysis of the proposed concurrent optimization scheme was performed to address both macro and microstructure coupling, as well as hygro-elastic physics coupling. Numerical simulations were conducted for various single and multiple microstructure systems to investigate their performance. A study was also carried out to examine the impact of incorporating multiple microstructures into a single macro design domain on the macrostructure's dependency. The results showed that the use of multiple microstructures improved the design freedom and performance-to-weight ratio of the macrostructure.

... The multiscale concurrent design problem is gaining more attention due to its capability to simultaneously design the lattice materials' macroscale distribution and their corresponding microstructural topology on the microscale [4]. That means that besides the structural design, the material properties can also be designed according to the design requirements and may further open up the design space [5]. ...

The concurrent design of different lattice material microstructures and their corresponding macro-scale distributions has great potential in achieving both lightweight and desired multiphysical performances. In such design problems, the lattice microstructures are usually separately optimized on the basis of the homogenization method, and the possibly poor connectivity between them is a key factor that severely hinders the fabrication and application of optimized two-scale structures. To handle the microstructure connectivity issue, this paper proposes a novel microstructure connectable strategy to bridge the gap between the two-scale and the full-scale model using the material-field series expansion (MFSE) method. Assuming different lattice material types in several pre-defined macro-scale regions, describe all types of microstructural topology with different portions of one material field function and update them simultaneously during the optimization process. Benefits from the material field definition with spatial correlation, the microstructures are well-connected without requiring additional constraints in the topology optimization model. The energy-based homogenization method is utilized for bridging the two-scale with different microstructures, while a decoupled sensitivities analysis for the microscale is employed to enhance the computation efficiency. Additionally, the proposed method significantly reduces the dimension of design variables, resulting in lower optimizer spending. The effectiveness and efficiency of the proposed method are demonstrated by several benchmark two-scale problems. Compared to density-based connectable methods, the proposed framework is easy to implement and reduces computational time by an order of magnitude in the 2D case.

... compared with the other two strategies. Currently, these three MTO strategies have been employed for design of structures under different scenarios, such as dynamics [42][43][44][45][46][47], thermoelastic [48][49][50] and thermal insulation problems [51,52]. ...

... Liu et al. [6] proposed a concurrent topology optimization method to simultaneously obtain the optimal macrostructure and uniformly distributed microstructure, where the Porous Anisotropic Material with Penalization (PAMP) and Solid Isotropic Material with Penalization (SIMP) models were applied to interpolate the material stiffness at the macro and micro scales, respectively. The proposed method has been extended to various other problems, such as frequency [7,8], thermoelastic structure [9,10], and vibration acoustic designs [11,12]. Based on the Bi-directional Evolutionary Structural Optimization (BESO) method, Yan et al. [13] presented a concurrent topology optimization method, where the material orientations were also optimized. ...

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

... 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.

... With the advancement of addictive manufacturing techniques, the complex structures of various materials achieved by structural topology optimization can be realized [12], which extends the researches on the deterministic topology optimization of various inhomogeneous materials [13][14][15]. Xu et al. [16] investigated a novel two-phase multi-material topology optimization scheme that considered the thermal expansion based on the bi-directional evolutionary structural optimization method. Li et al. [17] investigated the topology optimization for the functionally graded cellular composites with metamaterials by level sets. ...

An efficient scheme for the robust topology optimization considering hybrid bounded uncertainties (RTOHBU) is proposed for the graphene platelets (GPLs) reinforced functionally graded materials (FGMs). By introducing the concept of the layer-wise FGMs, the properties of the GPLs reinforced FGMs are calculated based on the Halpin-Tsai micromechanics model. The practical boundedness of probabilistic variables is naturally ensured by utilizing a generalized Beta distribution in constructing the robust topology optimization model. To address the issue of lacking the information of critical loads in existing topology optimization approaches considering hybrid uncertainties, a gradient-attributed search is carried out at first based on the hypothesis of linear elasticity to determine the critical loads leading to the worst structural performance. Subsequently, the statistical characteristics of the objective structural performance under such critical loads are efficiently evaluated by integrating the univariate dimension reduction method and the Gauss-Laguerre quadrature, the accuracy of which is verified by the comparison analysis utilizing the results of Monte Carlo simulation as references. Furthermore, a novel realization vector set is constructed for the bounded probabilistic uncertainties to parallelize the sensitivity analysis and accelerate the optimization process. All the proposed innovations are integrated into the robust topology optimization scheme, the effectiveness and efficiency of which are verified by three illustrative examples.

... . 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 benefit of MMA algorithm is that it replaces the original nonlinear, non-convex optimization problem by a sequence of approximating convex subproblems which are much easier to solve. The implemented MMA is based on the bi-directional evolutionary structural optimization (BESO), which is the evolutionary topology optimization approach that allows modification of the structure by either adding efficient material or removing inefficient material to or from the structure design [88][89][90][91][92][93][94][95][96]. This addition or removal depends upon the sensitivity analysis. ...

The main purpose of this chapter is to propose a new boundary element formulation for the modeling and optimization of three-temperature nonlinear generalized magneto-thermoelastic functionally graded anisotropic (FGA) composite microstructures’ problems, which is the gap of this study. Numerical results show that anisotropy and the functionally graded material have great influences on the nonlinear displacement sensitivities and nonlinear thermal stress sensitivities of composite microstructure optimization problem. Since, there are no available data for comparison, except for the problems with one-temperature heat conduction model, we considered the special case of our general study based on replacing three-temperature radiative heat conductions with one-temperature heat conduction. In the considered special case, numerical results demonstrate the validity and accuracy of the proposed technique. In order to solve the optimization problem, the method of moving asymptotes (MMA) based on the bi-evolutionary structural optimization method (BESO) has been implemented. A new class of composite microstructures problems with holes or inclusions was studied. The two-phase magneto-thermoelastic composite microstructure which is studied in this chapter consists of two different FGA materials. Through this chapter, we investigated that the optimal material distribution of the composite microstructures depends strongly on the heat conduction model, functionally graded parameter, and shapes of holes or inclusions.

... Among the existed researches associated with the multi-material topology optimization, two-scale concurrent topology optimization methods have been used for the multi-phase infill structure design [52][53][54]. To name some recent researches: Xu et al. [55] proposed a multiscale optimization model for composite thermoelastic macrostructure and microstructure with multi-phase material accounting for maximum stiffness. Li et al. [56] presented a new multiscale topology optimization method for the concurrent design of multi-phase composite structures under a certain range of excitation frequencies based on a parametric color level set method. ...

Porous infill structures have been widely studied and used in additive manufacturing because of their lightweight and excellent mechanical properties. However, without considering graded multi-material infill pattern, existed infill design methods have not yet fully tapped the potential of multiple materials in structural design. In this paper, a systematic multi-phase infill design method is proposed to generate graded multi-material infill structures. The method builds upon a unified multi-material density-based topology optimization framework, in which a modified multi-phase material interpolation formulation is presented to represent the relationship between the stiffness matrix and design variables. To generate spatial-varying and multi-phase structures distributed in the interior of a design domain, the maximum local material volume constraints are imposed on each phase material in the neighborhood of each element in the design domain. A series of relaxations is introduced into the framework to facilitate the implementation of the gradient-based optimization algorithm. The whole design process is performed on a full-size finite element analysis, so it can avoid the separation of scales and naturally guarantee the optimized infills to be smoothly connected. The applicability and effectiveness of the proposed multi-phase infill generation model are then demonstrated by several typical numerical examples with the objective of minimum compliance.

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

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

... Pedersen and Pedersen (2010) demonstrated that minimum compliance designs do not lead to maximum strength designs in the presence of thermoelastic loading for 2D and 3D structures and suggest an alternative problem based on obtaining uniform energy density. The challenges of material interpolation in topology optimization for thermoelasticity is one of the motivations for researchers to create and explore other methods based on clear {0, 1} designs, such as the bi-directional evolutionary structural optimization (BESO) and the level set topology optimization (LSTO) methods (Li et al. 2001;Xu et al. 2016;Xia and Wang 2008;Takallozadeh and Yoon 2017;Giusti et al. 2017;Deng and Suresh 2017). Another reason for developing binary design methods is the clear definition of structural boundaries. ...

The future perspective of using topology optimization to solve challenging design-dependent physics problems motivates the creation of methods with clear structural boundaries and well-defined volume. This paper develops the topology optimization of binary structures (TOBS) method to include design-dependent fluid pressure and constant thermal expansion loads. Topology design in thermoelastic and fluid pressure problems have been only handled separately up to date. To the authors’ best knowledge, this is the first work to consider both type of loads simultaneously within a structural topology optimization framework. The TOBS method uses discrete design variables, sensitivity filtering, and formal mathematical programming (integer linear optimization) to achieve convergent and mesh-independent solutions. The discrete nature of the method presents attractive features when dealing with design-dependent body and surface loads. In this paper, we use the structural mean compliance and volume as functions for optimization. The sensitivity analysis is carried out using the adjoint and semi-analytical methods. Numerous examples are shown to design novel structural designs which perform well under the applied fluid pressure and thermal loads. The observed computational times signify the practicability of integer programming for structural optimization problems.

... The concurrent two-scale topology optimization approaches have attracted intense attentions in the recent years. These approaches have been applied to many fields like mechanical designs (Xia and Breitkopf 2014;Xu and Cheng 2018;Yan et al. 2016;Yan et al. 2014;Yan et al. 2013), thermoelastic structure designs (Deng et al. 2013;Xu et al. 2016), and some multi-physics problems (Liang and Du 2015;Liang and Du 2019;Vicente et al. 2016;. Nonetheless, to the best knowledge of the authors, so far the concurrent two-scale topology optimization methods with respect to sound radiation power have not been developed/ applied systematically to the design of PnCs materials/structures. Thus, it is necessary and valuable to carry out the work of the concurrent topology optimization to the PnCs design problem in the present paper. ...

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.

... Homogenised behaviour of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenisation is the basis for computational topology optimisation [3,10,24,41,86,116,184,205,208,210,211] which will give rise to the next generation of architectured materials as it can already be seen in the works of [8,48,71,83,96,119,120,128,132,149,171,204,207]. ...

... This type of the concurrent topology optimization with a kind of material microstructures has the simple formulation, and no connectively issue is occurred in the optimization. Recently, this kind of designs has been applied to many problems, such as the multi-phase (Da et al. 2017;Long et al. 2018), the dynamic (Niu et al. 2009;Zuo et al. 2013;Yan et al. 2015;Vicente et al. 2016), the thermal design (Yan et al. 2015;Xu et al. 2016), the robust (Guo et al. 2015;Zheng et al. 2019), the manufacturing (Yan et al. 2017), and the nonlinear analysis (Kato et al. 2018). ...

This paper presents the compact and efficient Matlab codes for the concurrent topology optimization of multiscale composite structures not only in 2D scenario, but also considering 3D cases. A modified SIMP approach (Sigmund 2007) is employed to implement the concurrent topological design, with an energy-based homogenization method (EBHM) to evaluate the macroscopic effective properties of the microstructure. The 2D and 3D Matlab codes in the paper are developed, using the 88-line 2D SIMP code (Struct Multidisc Optim 43(1): 1-16, 2011) and the 169-line 3D topology optimization code (Struct Multidisc Optim 50(6): 1175-1196, 2014), respectively. This paper mainly contributes to the following four aspects: (1) the code architecture for the topology optimization of cellular composite structures (ConTop2D.m and ConTop3D.m); (2) the code to compute the 3D isoparametric element stiffness matrix (elementMatVec3D.m); (3) the EBHM to predict the macroscopic effective properties of 2D and 3D material microstructures (EBHM2D.m and EBHM3D.m); and (4) the code to calculate the sensitivities of the objective function with respect to the design variables at two scales. Several numerical examples are tested to demonstrate the effectiveness of the Matlab codes, which are attached in the Appendix, also offering an entry point for new comers in designing cellular composites using topology optimization.

... Homogenised behaviour of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenisation is the basis for computational topology optimisation [3,10,24,41,86,116,184,205,208,210,211] which will give rise to the next generation of architectured materials as it can already be seen in the works of [8,48,71,83,96,119,120,128,132,149,171,204,207]. ...

Architectured materials involve geometrically engineered distributions of microstructural phases at a scale comparable to the scale of the component, thus calling for new models in order to determine the effective properties of materials. The present chapter aims at providing such models, in the case of mechanical properties. As a matter of fact, one engineering challenge is to predict the effective properties of such materials; computational homogenization using finite element analysis is a powerful tool to do so. Homogenized behavior of architectured materials can thus be used in large structural computations, hence enabling the dissemination of architectured materials in the industry. Furthermore, computational homogenization is the basis for computational topology optimization which will give rise to the next generation of architectured materials. This chapter covers the computational homogenization of periodic architectured materials in elasticity and plasticity, as well as the homogenization and representativity of random architectured materials.

... [22][23][24] The determinant of stiffness matrix cannot distinguish specific stiffness values in a certain direction. 25,26 The minimum and maximum stiffness [27][28][29] can reflect variation range of the stiffness value of the mechanism, and the corresponding eigenvector directions of the minimum and maximum stiffness of mechanism represent the minimum and maximum stiffness direction, respectively. ...

In this article, the most contribution is to propose a novel general stiffness model to analyze the stiffness of a wall-climbing hexapod robot. First, we propose a new general stiffness model of serial mechanism, which includes the linear and nonlinear stiffness models. By comparison, the nonlinear stiffness model is a variable stiffness model which introduces the external load force as a variable, obtaining that the nonlinear stiffness model can greatly improve the accuracy of stiffness model than linear stiffness model. Then, the stiffness model of one leg of the robot and the overall stiffness model of the robot are derived based on the general stiffness model. Next, to improve the stiffness of the robot, a new minimum and maximum stiffness are introduced, which provide with effective reference for the selection and optimization of the structural parameters of the robot. Finally, we develop a new wall-climbing hexapod robot based on selection and optimization of the structural parameters, then the experiments are used to show that the selection of structure parameters of the robot effectively improve the stiffness of the robot.

... The method has been effectively applied to a wide range of problems like nonlinear structures [28], natural frequency maximization [29,30], material opti- 50 mization and multiscale problems [31, 32,33,34,35,36], multiphysics problems [37,38], etc. Another discrete topology optimization method was proposed by Svanberg and Werme [39] where the authors effectively proposed a sequential integer linear programming approach, where one starts with a coarse mesh to solve an optimization problem and uses the final solution of this problem as the 55 initial solution for optimization on a refined mesh and so on. ...

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement.

... In the last decade, researchers have been pushing the methods of topology optimization to new limits, specially in multiphysics design [49,67,17,48,52,63,54]. This has been pointed out by Deaton and Grandhi [16] as one of the trending topics in topology optimization. ...

This work presents an extended bi-directional evolutionary structural optimization (BESO) method applied to static structural design problems considering the interaction between viscous fluid flows and linearly elastic structures. The fluid flow is governed by incompressible and steady-state Navier-Stokes equations. Both domains are solved with the finite element method and simplifying conditions are assumed for the fluid-structure coupling, such as small structural displacements and deformations in a staggered method. The presented BESO method aims to minimize structural compliance in a so called “wet” optimization problem, in which the fluid loads location, direction and magnitude depend on the structural layout. In this type of design-dependent loading problem, density-based topology optimization methods require extra numerical techniques (usually mixed models with overlapping domains) in order to model the interaction of different governing equations during the optimization procedures. In this work, the discrete nature of the evolutionary topology optimization approach allows the fluid-structure boundaries to be modelled and modified straightforwardly by switching the discrete design variables between fluid and structural finite elements. Therefore, separate domains are used in this approach. Numerical results show that the BESO-based methods can be applied to this kind of multiphysics problem effectively and efficiently.

... 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]. Vicente et al. [30] presented concurrent topology optimization models for minimizing the frequency responses. ...

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.

In this paper, a new multiscale concurrent topology optimization method for thermoelastic structures considering the iterative variation of temperature field is proposed for the first time, which breaks the limitation that previous multiscale concurrent topology optimization studies being compliable merely to uniform temperature field. In this method, the iterative variations of macroscopic structural heat transfer, structural temperature, structural force transfer, structural displacement, design-dependent thermal stress load, microscopic effective thermal conductivity, effective elasticity and effective thermal expansion coefficient are all taken into consideration. In order to establish a compact hierarchical thermoelastic coupling equation on the above iterative factors, firstly, a thermoelastic coupling matrix with a distinct physical meaning is proposed to address the issues on accuracy of thermal stress loads and solution of adjoint sensitivity multipliers caused by design-dependent varying temperature field, and this matrix can be used as a new manner to solve homogenized effective thermal stress coefficient. Secondly, the compact coupling equation is derived using multiscale adjoint sensitivity analysis and its effectiveness is illustrated by comparative cases. Finally, the generality and stability of proposed method are illustrated through diverse scenarios involving compliance optimization, multimaterial concurrent design, maximum displacement control, multicellular structure design, asymmetric boundary conditions and three-dimensional structures. It is obvious that this pioneering approach has a broad potential in advanced integrated structures and materials design of thermoelastic structures.

An intelligent microstructural design method based on deep learning is proposed considering performance indicators that contains boundary information and homogenized elastic modules. Microstructure dataset is established by random boundary method and homogenization method. Random boundary method is proposed to design microstructures under given boundary information, and homogenization method is utilized to acquire homogenized elastic modules. A generative and adversarial network with gradient penalty is developed to establish the high-dimensional mapping between performance indicators and microstructure. The Wasserstein distance is imported to overcome mode collapse. Numerical simulation shows that the pre-trained network successfully achieved corresponding microstructure design by given performance indicators.

The realisation of sophisticated hierarchically patterned multiphase steels has the potential to enable unprecedented properties in engineering components. The present work explores the controlled creation of patterned multiphase steels in which the patterns are defined by two different crystal structures: face centre cubic or fcc (austenite) and body centre cubic or bcc (martensite). These austenite/martensite mesostructures are generated by solid-solid phase transformations during the application of localised laser heat treatments in a Fe-Ni-C alloy. In particular, four patterned configurations are analysed in this work consisting of one or two horizontal austenite line structures imprinted in a base of as-quenched or tempered martensite. Digital image correlation analysis during tensile testing of the developed materials showed that both the strength of the base martensite and the mesostructure at the gauge have a strong effect on the resulting properties. Clear differences were observed among the configurations in strain partitioning, hardening of the different constituents and failure. The uniform elongation and tensile strength are increased with respect to that of the reference martensite and austenite, respectively. Concepts explored in this work can be extended to more complex patterns and other base microstructures, opening novel strategies to engineer properties in steel and other alloys.

Thermal‐mechanical coupling environments extensively exist for engineering structures. How to effectively reduce thermoelastic deformation is distinctly important. Cellular structures have superior performances in stiffness and multi‐functionality with lightweight, and an excellent design can be expected by reasonably designing the microstructural topology and their distributions. However, existing studies are limited to designing homogenous cellular structures for thermoelastic response problems, although heterogeneous cellular structures have larger design spaces. This paper intends to investigate the topology optimization problem for designing a special type of heterogeneous cellular structure, i.e., quasi‐periodic cellular structure, for thermoelastic responses. The quasi‐periodic cellular structure is a composition of microstructures with similar but not the same topology across the macro domain, which ensures a good compromise between manufacturability and performance. By introducing the erode‐dilate operators to describe quasi‐periodic cellular structural topology, a simple single‐loop topology optimization model for minimizing the deformation variance of specific lines with multi‐material and multi‐constraint is established. Three types of design variables are included, and their sensitivity analyses are derived for using the gradient‐based optimization solver to obtain updates simultaneously. Designs with shape‐preserving and required stiffness under thermoelastic loadings can be obtained. The examples presented show the capabilities of the proposed procedure and validate the effectiveness of the quasi‐periodic cellular structures in improving structural performance for thermoelastic problems.

This paper presents a robust topology optimization (RTO) framework for thermoelastic hierarchical structures with hybrid uncertainty. Firstly, the thermoelastic concurrent optimization model is established and the uncertainties with interval random parameters are integrated into the thermoelastic hierarchical structure. Then, a reliable and cost-effective hybrid uncertainty perturbation analysis method (HUPAM) is derived for a quick estimate of the robust objective function subject to the mechanical and thermal loads. Finally, by calculating the design variables sensitivities of macroscale and microscale, the robust topological design can be generated efficiently. To obtain clear and optimal topologies for both macro- and micro- structures, the bi-directional evolutionary structural optimization (BESO) method is adopted. Some 2D and 3D numerical examples are presented to demonstrate the influences of the hybrid uncertainties on the final designs. The results also show that the proposed method can effectively improve the thermoelastic structural performance when it comes to uncertainties.

A concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties is discussed in this work. A robust topology optimization method is proposed for structures composed of periodic microstructures under thermal and mechanical coupled loads. The robust objective function is defined as a linear combination of the mean and standard variance under the worst case for the robust optimization model. An efficient hybrid orthogonal polynomial expansion (HOPE) method is developed to evaluate the robust objective function. The sensitivities for the robust topology optimization are then calculated based on the uncertainty analysis. Three numerical examples are provided to verify the effectiveness of the proposed method, and the Monte-Carlo-Scanning (MCS) test is used to validate the numerical accuracy of our proposed method. For comparison purpose, the topology optimizations under deterministic assumptions are also provided for these examples to show the importance of considering hybrid uncertainties.

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.

Two-dimensional (2-D) micro-architectured mechanical metamaterials are designed using a topology optimization approach that integrates a parametric level set method (PLSM) with a meshfree method based on compactly supported radial basis functions (CS-RBF). The PLSM is employed as the optimization algorithm to achieve desired microstructures with targeted material properties. The effective elastic properties, including the bulk modulus, shear modulus and Poisson’s ratio, are predicted using a strain energy-based homogenization method and the CS-RBF meshfree algorithm. Two sets of optimizations are implemented: one is for the case of a single solid material, and the other is for the case of two solid materials, each involving an additional void phase. Three numerical examples are provided for each case with the same optimization objectives: maximizing the effective bulk modulus, maximizing the effective shear modulus, and minimizing the effective Poisson’s ratio under given volume fraction constraints. The numerical results reveal that the newly proposed approach can generate smooth topological boundaries and optimal microstructures. In particular, the current method can topologically optimize auxetic metamaterials with a negative Poisson’s ratio. It can also be extended to design other periodic metamaterials, including those with a negative coefficient of thermal expansion or frequency bandgaps.

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

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

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.

The classical material-by-design approach has been extensively perfected by materials scientists, while engineers have been optimising structures geometrically for centuries. The purpose of architectured materials is to build bridges across the microscale of materials and the macroscale of engineering structures, to put some geometry in the microstructure. This is a paradigm shift. Materials cannot be considered monolithic anymore. Any set of materials functions, even antagonistic ones, can be envisaged in the future. In this paper, we intend to demonstrate the pertinence of computation for developing architectured materials, and the not-so-incidental outcome which led us to developing large-scale additive manufacturing for architectural applications.

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.

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.

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 method for design of optimized poroelastic materials which under internal pressurization turn into actuators for application in, for example, linear motors. The actuators are modeled in a two-scale fluid–structure interaction approach. The fluid saturated material microstructure is optimized using topology optimization in order to achieve a better macroscopic performance quantified by vertical or torsional deflections. Constraints are introduced to ensure a certain deflection/extension ratio of the actuator.

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.

In this paper, optimum stress distribution for hollow plates composed of linear cellular materials (LCMs), a kind of truss-like
material, is investigated. To reduce the computational cost, we model the material as micropolar continua representation.
Two classes of design variables, relative density, and cell-size distribution of truss-like materials are to be determined
by optimization under given total material volume constraint. And the concurrent designs of material and structure are obtained
for three different optimization formulations. For the first formulation, we aim at the minimization of the maximum stress
that appears at the initial uniform design; for the second formulation, we minimize the highest stress within the specified
point set. As the yield strength of truss-like material is dependent on the relative material density, we minimize the ratio
of stress over the corresponding yield strength along the hole boundary in our third formulation, which maximizes the strength
reserve and seems more rational. The numerical results for the three objectives validate the concurrent optimization method
proposed in this paper. And the influence of ply angle (angle between the principle direction of material and the axes of
the system’s coordinates) on the optimum result is discussed. The dependence of optimum design on finite element meshes is
also investigated. An approximate discrete model is established to verify the method proposed in this paper, and the stress
concentration near a hole is reduced significantly.

This paper presents a method for topology optimization of periodic structures using the bi-directional evolutionary structural
optimization (BESO) technique. To satisfy the periodic constraint, the designable domain is divided into a certain number
of identical unit cells. The optimal topology of the unit cell is determined by gradually removing and adding material based
on a sensitivity analysis. Sensitivity numbers that consider the periodic constraint for the repetitive elements are developed.
To demonstrate the capability and effectiveness of the proposed approach, topology design problems of 2D and 3D periodic structures
are investigated. The results indicate that the optimal topology depends, to a great extent, on the defined unit cells and
on the relative strength of other non-designable part, such as the skins of sandwich structures.

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

The solid isotropic material with penalization (SIMP) method is used in topology optimization to solve problems where the variables are 0 or 1. The theoretical convergence properties have not been exhaustively studied. In this paper a theorem on convergence under weaker assumptions than those previously assumed is given.

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.

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.

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

This paper presents a level set-based topology optimization method for the design of negative permeability dielectric metamaterials. Metamaterials are artificial materials that display extraordinary physical properties that are unavailable with natural materials. The aim of the formulated optimization problem is to find optimized layouts of a dielectric material that achieve negative permeability. The presence of grayscale areas in the optimized configurations critically affects the performance of metamaterials, positively as well as negatively, but configurations that contain grayscale areas are highly impractical from an engineering and manufacturing point of view. Therefore, a topology optimization method that can obtain clear optimized configurations is desirable. Here, a level set-based topology optimization method incorporating a fictitious interface energy is applied to a negative permeability dielectric metamaterial design problem. The optimization algorithm uses the Finite Element Method (FEM) for solving the equilibrium and adjoint equations, and design problems are formulated for both two- and three-dimensional cases. First, the level set-based topology optimization method is explained, and the optimization problems for the design of metamaterials are then discussed. Several optimum design examples for the design of dielectric metamaterials that demonstrate negative effective permeability at prescribed frequencies are provided to confirm the utility and validity of the presented method.

Viscoelastic damping material attached on the surface of a structure is widely used to suppress the resonance vibration in aerospace, automobiles, and various other applications. A full treatment of damping material is not an effective method because the damping effect is not significantly increased compared to that obtained by an effective partial damping treatment. In addition, the weight of the structure is increased significantly, which can cause poor system performance. Topology optimization is recently implemented in order to find an effective optimal damping treatment. The objective function is maximization of the damping effect (i.e. the modal loss factor) and the constraint is a maximum allowable volume of damping material. In this paper we compare the modal loss factors obtained by topology optimization to the ones obtained by other approaches, in order to determine which approach provides a better damping treatment (i.e. higher value of the modal loss factor). As a result, topology optimization provides about up to 61.14 per cent higher modal loss factor, as confirmed by numerical example. The numerical model for finite element analysis and topology optimization is also experimentally validated by comparing the numerical results to the experimental modal loss factors.

This paper deals with the sensitivity analysis of structural acoustic performance in presence of non-proportional damping and optimal layout design of the damping layer of vibrating shell structures under harmonic excitations. The structural system with a partially-covered damping layer has a non-proportional global damping matrix. Therefore, the method of complex mode superposition in the state space is employed in the dynamic response analysis. The sound pressure is calculated with the structural response solution by using the boundary element method. In this context, an adjoint variable scheme for the design sensitivity analysis of sound pressure is developed. In the optimal design problem, the design objective is to minimize the structural vibration-induced sound pressure at a specified point in the acoustic medium by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is employed, and the relative densities of the damping material are considered as design variables. Numerical examples are given to illustrate the validity and efficiency of this approach. The influences of the excitation frequency, the damping coefficients and the locations of the reference point on the optimal topologies are also discussed.

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

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.

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

The bidirectional evolutionary structural optimization (BESO) method that allows the material removed and added to the structure is proposed for maximizing stiffness of nonlinear structures. The performance of the structures were gradually improved by removing and adding elements and the optimization process was stopped when both the objective volume and the prescribed. The BESO method provides a more robust and more efficient algorithm for searching for the optimal solution. The BESO method gains computational efficiency because the total elements become less. The improvements can be significant for problems involving local buckling. The numerical results show that the stiffness of structures optimized using combined geometrically and materially nonlinear modeling is higher than that using linear analysis.

The present paper is concerned with the layout optimization of resonating actuators using topology optimization techniques. The goal of the optimization is a maximization of the magnitude of steady-state vibrations for a given excitation frequency. The problem formulation includes an external viscous damper at the output port which models a working load on the structure. Copyright © 2002 John Wiley & Sons, Ltd.

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.

We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly
elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint
on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the
material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material
used.
We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such
that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q.
This increases the probability to obtain a zero-one solution of the relaxed problem.

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.

This paper is devoted to minimum stress design in structural optimization. The homogenization method is extended to such a framework and yields an efficient numerical algorithm for topology optimization. The main idea is to use a partial relaxation of the problem obtained by introducing special microstructures which are sequential laminated composites. Indeed, the so-called corrector terms of such microgeometries are explicitly known, which allows us to compute the relaxed objective function. These correctors can be interpreted as stress amplification factors, caused by the underlying microstructure.

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

Shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by Bendsoe and Kikuchi together with various examples which may justify validity and strength of the present approach for plane structures.

Electromagnetic structures that incorporate certain structural periodicities are known to display special behavior when subjected to electromagnetic waves, and can be designed to have specific functions such as inhibiting the intrusion of electromagnetic waves of certain frequencies into the periodic structure.This paper proposes a novel topology optimization method for periodic microstructures of electromagnetic materials using the concept of propagation behavior to implement designs that inhibit electromagnetic wave propagation. First, a way to apply topology optimization to the design of electromagnetic structures is briefly discussed. Next, the design specifications are clarified, and a new objective function is proposed to satisfy these specification. The optimization algorithm is developed using sequential linear programming (SLP) and the adjoint variable method (AVM). Several numerical examples are provided to confirm that the proposed method is capable of automatically generating physically reasonable periodic structures that have desired specified functions without relying on a predefined basic lattice.