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The optimal design of periodic structures under the macro scale and that of periodic materials under the micro scale are treated differently by the current topology optimization techniques. Nevertheless, a material point in theory could be considered as a unit cell in a periodic structure if the number of unit cells approaches to infinity. In this work, we investigate the equivalence between optimal solutions of periodic structures obtained from the macro scale approach on the structure level and those of material microstructures obtained from the micro scale approach using the homogenization techniques. The minimization of the mean compliance of the macrostructure with a volume constraint is taken as the optimization problem for both structural and material designs. On the macro scale, we solve the optimization problem by gradually increasing the number of unit cells until the solution converges, in terms of both the topology and the objective function. On the micro scale, the optimal microstructure of the material is obtained for the macrostructure under prescribed loading and support conditions. The microstructure of the material compares very well with the corresponding optimal topology from the periodic macrostructural design. This work reveals the equivalence of the solutions from the macro and micro approaches, and proves that an optimal finite periodic solution remains valid through cell refinement to infinite periodicity.

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... Interestingly, few of the above-mentioned approaches consider an actual full-scale analysis to identify whether a single-scale approximation of the multi-scale design is useful in engineering practice. To the authors' knowledge only the works of Zhang and Sun (2006), Pantz and Trabelsi (2008), Zuo et al. (2013), Liu et al. (2017), Groen and Sigmund (2018), Garner et al. (2019), Groen et al. (2019a), Kumar and Suresh (2019), Wu et al. (2019), Groen et al. (2020), Xue et al. (2020), andJiang et al. (2020) actually do such an important full-scale verification. This can be seen as a missed opportunity, especially since many works claim to do topology optimization to allow for additive manufacturing (AM). ...

... Notable works in this aspect are the works by Hollister and Kikuchi (1992) and Pecullan et al. (1999) in 2D and by Coelho et al. (2016) in 3D. Besides the effect on the elasticity tensor, the works of Zhang and Sun (2006) and Zuo et al. (2013) identify the difference between a 2D multi-scale model with uniform microstructures and a single-scale approximation for different amount of microstructure repetitions. Here we extend these studies to 3D using the cantilever subject to a downward corner load (Fig. 5a). ...

... Furthermore, it is interesting to see that the compliance values converge from below towards the compliance of the multi-scale design. This is in line with numerical observations by Zhang and Sun (2006) and Zuo et al. (2013). Furthermore, to make sure that the results are not biased by the microstructure location, we have shifted the infill for the designs 15% of the domain length in ydirection. ...

In this article, we demonstrate the state-of-the-art of multi-scale topology optimization for 3D structural design. Many structures designed for additive manufacturing consist of a solid shell surrounding repeated microstructures, so-called infill material. We demonstrate the performance of different types of infill microstructures, such as isotropic truss or plate lattice structures and show that the best results can be obtained using spatially varying and oriented orthotropic microstructures. Furthermore, we demonstrate how implicit geometry modeling using nTop platform can help to interpret these multi-scale designs as single-scale manufacturable designs (de-homogenization). More importantly, we demonstrate the small difference in performance between these multi-scale and single-scale designs through extensive numerical testing. The presented method is at least 3 orders of magnitude more efficient compared to standard density-based topology optimization, allowing for high-resolution 3D structures to be obtained on a standard workstation PC.

... For a single micro-structure repeated throughout the computational domain, it was demonstrated that an optimized periodic structure with sufficient finite geometric periodicity converges to optimized material unit cells with an infinite geometric periodicity obtained using homogenization [32,33]. However, it must be noted that this is true only for certain design problems, particularly for cases where localization due to boundary condition effects is not an issue and where the homogenized moduli to be optimized is dominant and known a priori. ...

... The results are in good agreement with Refs. [33,37] although the optimization methods slightly differ. As observed in these two references, the compliance increases with an increase in the number of unit cells. ...

... It is observed that the microstructures possess a vertical bar on the right edge to support the distributed load on the right beam edge. When decoupling the scales of the design problem, the optimization process fails to capture accurately the loading applied at the macroscale level, requiring non-design subdomains to be enforced to circumvent this issue (see for instance Ref. [33]). In contrast, this issue is handled directly with the proposed method. ...

Advanced manufacturing processes such as additive manufacturing offer now the capability to control material placement at unprecedented length scales and thereby dramatically open up the design space. This includes the considerations of new component topologies as well as the architecture of material within a topology offering new paths to creating lighter and more efficient structures. Topology optimization is an ideal tool for navigating this multiscale design problem and leveraging the capabilities of advanced manufacturing technologies. However, the resulting design problem is computationally challenging as very fine discretizations are needed to capture all micro-structural details. In this paper, a method based on reduction techniques is proposed to perform efficiently topology optimization at multiple scales. This method solves the design problem without length scale separation, i.e., without iterating between the two scales. Ergo, connectivity between space-varying micro-structures is naturally ensured. Several design problems for various types of micro-structural periodicity are performed to illustrate the method, including applications to infill patterns in additive manufacturing.

... The convergence of periodic structures towards homogenised microstructures is investigated in [15,16]. These papers show that an optimised periodic structure, with finite geometric periodicity, converges to optimised material unit cells with an infinite geometric periodicity, obtained using homogenisation for a single microstructure repeated throughout the computational domain. ...

... This paper restricts itself to a single-scale optimisation approach, where the periodic microstructural details filling a given domain are optimised for the macroscale response. The complete macroscopic structure and its response is considered, while resolving all microstructural details as compared to the homogenisation approach common to all of the above papers, except [15,16]. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro-and macroscopic length scales. ...

... It is observed that the microstructure converges very fast and the overall topology does not qualitatively change for M x > 4, therefore only a select few have been shown out of the investigated set, M x ∈ {2, 4, 6, 8, 12, 16, 32, 64}. This trend was also observed in [16] for the same problem, although the optimised designs differ. This difference in topology is likely due to a difference in the effective length scale of the current approach and that of [16]; the former using density filtering combined with a projection at η = 0.5 and the latter using BESO. ...

This paper applies topology optimisation to the design of structures with periodic and layered microstructural details without length scale separation, i.e. considering the complete macroscopic structure and its response, while resolving all microstructural details, as compared to the often used homogenisation approach. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro- and macroscopic length scales. This results in microstructures tailored for specific applications rather than specific properties. Manufacturability is further ensured by the use of robust topology optimisation.
Dealing with the complete macroscopic structure and its response is computationally challenging as very fine discretisations are needed in order to resolve all microstructural details. Therefore, this paper shows the benefits of applying a contrast-independent spectral preconditioner based on the multiscale finite element method (MsFEM) to large structures with fully-resolved microstructural details. It is shown that a single preconditioner can be reused for many design iterations and used for several design realisations, in turn leading to massive savings in computational cost.
The density-based topology optimisation approach combined with a Heaviside projection filter and a stochastic robust formulation is used on various problems, with both periodic and layered microstructures. The presented approach is shown to allow for the topology optimisation of very large problems in MATLAB, specifically a problem with 26 million displacement degrees of freedom in 26 hours using a single computational thread.

... Huang and Xie et al. [23] presented a method for topology optimization of periodic structures using the bi-directional evolutionary structural optimization technique. Zuo et al. [24,25]studied the optimal design problem of periodic structures and established a direct relationship between the optimal design of material microstructures and the topology optimization of macro periodic structures. He et al. [26]presented a topology optimization algorithm for periodic structures with different geometries and irregular meshes. ...

... Based on SIMP method, the structural compliance C can be rewritten as: (25) The structural compliance C derives the relative density ...

s paper proposes a new optimization method to solve periodic layout optimization of a cyclic symmetric structure by means of the guide-weight method. The mathematical model for periodic layout optimization under multiple constraints is built on a single working condition. By constructing a virtual fan-shaped subdomain, periodic layout optimization of the cyclic symmetric structure is transformed into the conventional topology optimization of the virtual fan-shape subdomain. Starting from the Lagrange equation, the general iterative criterion of the virtual fan-shaped subdomain is deduced by the guide-weight method. Taking the problem of minimum compliance as an example, the iterative criterion between relative density and strain energy density of elements in the virtual fan-shaped subdomain is obtained under the condition of mass constraint. The physical significance of iterative criterion is explained and the optimization flow of periodic layout optimization is provided. Two examples of the cyclic symmetric structure show that the proposed method is effective and robust to solve the problem of periodic layout optimization.

... Response: This sentence and the last sentence could be changed as follow if appropriate. "Comparing with the homogenization-based topology optimization, it is shown that with the refinement of cells, the optimized periodic structure is gradually convergent to a similar structure designed by the homogenization method [32,35]. ...

... As a result, methods achieving microstructural connectivity in multiscale topology optimization are presented [29][30][31]. Second, the design objective such as compliance would be overestimated once a specific length scale is prescribed because the periodic structures are theoretically infinite small [32]. To this end, a scale-related higher-order homogenization method [33] Another method to design periodic structures is the macroscale topology optimization with periodicity constraints [ , ]. ...

Topology optimization of macroperiodic structures is traditionally realized by imposing periodic constraints on the global structure, which needs to solve a fully linear system. Therefore, it usually requires a huge computational cost and massive storage requirements with the mesh refinement. This paper presents an efficient topology optimization method for periodic structures with substructuring such that a condensed linear system is to be solved. The macrostructure is identically partitioned into a number of scale-related substructures represented by the zero contour of a level set function (LSF). Only a representative substructure is optimized for the global periodic structures. To accelerate the finite element analysis (FEA) procedure of the periodic structures, static condensation is adopted for repeated common substructures. The macrostructure with reduced number of degree of freedoms (DOFs) is obtained by assembling all the condensed substructures together. Solving a fully linear system is divided into solving a condensed linear system and parallel recovery of substructural displacement fields. The design efficiency is therefore significantly improved. With this proposed method, people can design scale-related periodic structures with a sufficiently large number of unit cells. The structural performance at a specified scale can also be calculated without any approximations. What's more, perfect connectivity between different optimized unit cells is guaranteed. Topology optimization of periodic, layerwise periodic, and graded layerwise periodic structures are investigated to verify the efficiency and effectiveness of the presented method.

... As this factor increases one may evaluate convergence of the apparent properties to the theoretical values from homogenization. This scale-size effects analysis has been carried out by other authors although restricted to material symmetries, mostly to two-dimensional microstructures and investigating mainly the convergence of the mean compliance, few elastic moduli and fundamental frequency [25][26][27][28][29][30][31][32]. The aim of the present work is to extend this to three-dimensional anisotropic material microstructures. ...

... Several studies on scale-size effects show the mean compliance (energy) convergence [25,26,[29][30][31]. For example, according to the theoretical discussion presented in [31], one expects the chain of inequalities in Eq. (14) to hold. ...

Periodic homogenization models are often used to compute the elastic properties of periodic composite materials based on the shape/periodicity of a given material unit-cell. Conversely, in material design, the unit-cell is not known a priori, and the goal is to design it to attain specific properties values – inverse homogenization problem. This problem is often solved by formulating it as an optimization problem. In this approach, the unit-cell is assumed infinitely small and with periodic boundary conditions. However, in practice, the composite material comprises a finite number of measurable unit-cells and the stress/strain fields are not periodic near the structure boundary. It is thus critical to investigate whether the obtained topologies are affected when applied in the context of real composites. This is done here by scaling the unit-cell an increasing number of times and accessing the apparent properties of the resulting composite by means of standard numerical experiments.

... With the explosive advancement of additive manufacturing (AM) techniques, the design space of sandwich structures is greatly enlarged, which makes it possible to fill the core layer with periodic porous material with complicated configurations [7][8][9]. However, there are still some AM constraints that need to be carefully considered during the design process [10][11][12][13]. ...

Graded lattice sandwich structures (GLSSs) enable superior structural performances due to the continuously-varying configurations and properties of lattices in space. This paper proposes a design method for GLSSs by multiscale topology optimization. In this method, the geometrical configuration of a prototype lattice cell (PLC) is described by an explicit topology description function (TDF). Based on some sample lattice cells, a Kriging metamodel is constructed to predict the effective property of each lattice cell. Based on the Kriging metamodel, the thickness optimization of two solid face-sheets and the distribution optimization of lattice cells in the core layer are respectively implemented. Driven by their equivalent densities, the configurations of lattice cells with similar topological features are generated by interpolating the shape of the PLC. Then, the graded lattice cells (GLCs) are generated. Using the proposed method, the computational burden involved in design of GLSSs can be reduced significantly. What's more, the manufacturability of GLSSs can be guaranteed by constructing a proper PLC with the help of TDF. Numerical examples in terms of compliance and natural frequency optimization of GLSSs are provided to verify the advantages of the proposed method. Also, bending tests are performed on the GLSSs fabricated by additive manufacturing (AM). The results reveal that GLSSs are stiffer and have larger natural frequencies than the uniform lattice sandwich structures.

... One such adaption is to implement periodic design in which a single unit-cell is optimized and then replicated to form a larger periodic structure (Huang and Xie 2008;Zuo et al. 2011). Typically, the unit-cell in periodic optimization is either considered to be infinitely small relative to the macrostructure (Cadman et al. 2013;Osanov and Guest 2016;Wu et al. 2021) or of finite size (Zhang and Sun 2006;Huang and Xie 2008;Chen et al. 2010;Xie et al. 2012;Zuo et al. 2013). The former is predominantly utilized for the design of microstructural materials while the latter, which is the focus of this study, is primarily utilized for large macrostructures. ...

Periodic topology optimization has been suggested as an effective means to design efficient structures which address a range of practical constraints, such as manufacturability, transportability, replaceability and ease of assembly. This study proposes a new approach for design of finite periodic structures by allowing variable orientation states of individual unit-cells. In some design instances of periodic structures, the unit-cell may exhibit certain geometric features allowing multiple possible assembly orientations (e.g. facing up or down). For the first time, this work incorporates such assembly flexibility within the periodic topology optimization, which enables to greatly expand the conventional periodic design space and take more advantage of structural periodicity. Given its broad applications, a methodology for the design of more efficient periodic structures while maintaining the same degree of periodic constraint may be of significant benefit to engineering practice. In this study, several numerical examples are presented to demonstrate the effectiveness of this new approach for both static and vibratory criteria. Brute force analysis is also utilized to compare all possible assembly configurations for several periodic structures with a small number of unit-cells. A heuristic approach is suggested for selecting more beneficially oriented configurations in periodic structures with a large number of unit-cells for which an exhaustive search may be computationally infeasible. It is found that in all the presented cases, the oriented periodic structures outperform the conventional non-oriented (or namely translational) periodic counterparts. Finally, an educational MATLAB code is provided for replication of the design results in this paper.

... The discrepancy of the stiffness performances computed by two methods has been investigated in detail. 64,65,66 In this appendix, we focus on the ...

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

... For this example, it seems reasonable that more materials distribute in the upper and lower boundary and the location near the loading point of the optimal designs. More in detail, the pattern of the optimized base cell of region 1 in those three situations is basically consistent, i.e., similar to the base cell of a square short beam in [51]. Since the relative thickness of the transition layer is set to be one-base-cell width, by increasing the number of base cells in homogenized regions, as shown in Fig. 11, the transition layer's relative thickness decreases. ...

With the rapid developments of modern fabrication techniques, architected structures are increasingly used in many application areas, e.g., lightweight structures, heat exchangers, energy absorption components, aircraft engines, etc. To systematically design optimized architected structures with favorable manufacturability in terms of exact sizes and good connectivity, in the present work, an enhanced multidomain topology optimization method is developed. The design domain is divided into several subdomains and boundary layers between them first. Periodic base cells with exact sizes are distributed in each subdomain, and analyzed and optimized approximately based on the homogenization method with coarse meshes to reduce the numerical efforts. Besides, gradient boundary layers discretized by fine meshes are optimized between subdomains and boundaries to connect adjacent base cells with different patterns and address the boundary effect. Quadtree technology is used to match the meshes of different sizes. Numerical examples verify both the effectiveness and efficiency of the proposed approach for designing manufacturable optimized architected structures.

... Periodic feature structures have attracted the attention of academic and industrial technicians with their unique structural form, excellent visual aesthetics, extensibility in array direction, physical versatility, and good manufacturing and design [20][21][22][23][24][25]. Many topology optimization methods have been presented to research periodic topology optimization, including the homogenization method, evolutionary structural optimization method [26][27][28][29], and independent continuous mapping method [30][31]. ...

The stacker crane is a long-and-thin structure with a large length-to-width ratio. It is difficult to obtain a topology configuration with good period properties using traditional optimization methods. While the mathematical model of periodic topology optimization–in which the elements’ relative densities are selected as design variables, and mean compliance as the objective function–is established. To find a topology configuration with a good period property, an additional constraint condition must be imported into the mathematical model. According to the optimization criteria method, the iterative formula of design variables is derived in the virtual sub domain. To verify the capability and availability of the proposed method, periodic topology optimization of a single-mast stacker crane is investigated in this paper. The results show that configurations with good periodicity can be obtained when the number of sub domains is varied. After considering mean compliance and complexity, the optimal configuration has eight periods. A preliminary lightweight design scheme is proposed based on this configuration of a stacker crane, which is a periodic feature structure.

... The structural compliances of the optimized SSGCCs for MBB beam versus the macro mesh sizes are plotted in Fig. 20. It is obvious that the structural compliance increases as the macro mesh size is refined, which matches well the numerical observations in the works [60,61]. On the one hand, this is caused by the fact that the periodicity of the sandwich layers is increased as the sequential refined macro mesh, resulting in the design freedom reduction, where the single layer is arranged periodically along its height direction to generate sandwich layers. ...

Exploring ultralight sandwich structures with superior load-bearing performance is one of the important topics in structural optimization. This paper proposes a novel multiscale topology optimization method to achieve the design of high-performance sandwich structures with graded cellular cores (SSGCCs). In this method, the thicknesses of two solid face-sheets, the graded distribution of cellular sandwich cores at a single layer and their configurations are optimized to well suit for loading conditions, where the single layer is arrayed periodically at its height direction to obtain sandwich layers. Specifically, at macroscale, the variable thickness sheet (VTS) method with the capacity of generating an overall free material distribution pattern, is applied to optimize the thicknesses of two solid face-sheets and achieve the graded distribution of cellular sandwich cores at a single layer. At microscale, a progressive optimization scheme is employed to topologically optimize multiple representative cellular cores (RCCs) at a single layer, so as to achieve their similar topological configurations. With a shape interpolation method, the configurations of graded cellular cores (GCCs) with essential interconnections can be obtained by interpolating the shapes of these RCCs with similar topological features. In order to reduce the computational burden on evaluating effective properties of GCCs by the homogenization method, a Kriging metamodel is constructed based on some key cellular cores as sample points, and adopted to predict the effective properties of all the GCCs. Both 2D and 3D numerical examples are provided to test the validity and advantages of the proposed method for designing SSGCCs.

... The dimensions of the unit cell range from large to small as compared with the dimensions of the whole structure to highlight the Chapter 1. Introduction 7 size effect. By assuming the material microstructures are infinitely small, the inverse homogenization designs for macroscopic structural performance were compared with the mono-scale topology optimization framework in [181,192]. ...

Mechanical and physical properties of complex heterogeneous materials are determined on one hand by the composition of their constituents, but can on the other hand be drastically modified by their microstructural geometrical shape. Topology optimization aims at defining the optimal structural or material geometry with regards to specific objectives under mechanical constraints like equilibrium and boundary conditions. Recently, the development of 3D printing techniques and other additive manufacturing processes have made possible to manufacture directly the designed materials from a numerical file, opening routes for totally new designs. The main objectives of this thesis are to develop modeling and numerical tools to design new materials using topology optimization. More specifically, the following aspects are investigated. First, topology optimization in mono-scale structures is developed. We primarily present a new evolutionary topology optimization method for design of continuum structures with smoothed boundary representation and high robustness. In addition, we propose two topology optimization frameworks in design of material microstructures for extreme effective elastic modulus or negative Poisson's ratio. Next, multiscale topology optimization of heterogeneous materials is investigated. We firstly present a concurrent topological design framework of 2D and 3D macroscopic structures and the underlying three or more phases material microstructures. Then, multiscale topology optimization procedures are conducted not only for heterogeneous materials but also for mesoscopic structures in the context of non-separated scales. A filter-based nonlocal homogenization framework is adopted to take into account strain gradient. Finally, we investigate the use of topology optimization in the context of fracture resistance of heterogeneous structures and materials. We propose a first attempt for the extension of the phase field method to viscoelastic materials. In addition, Phase field methods for fracture able to take into account initiation, propagation and interactions of complex both matrix and interfacial micro cracks networks are adopted to optimally design the microstructures to improve the fracture resistance

... Other studies have been devoted to topology optimization of structures in a context of non-separated scales (see e.g. [42][43][44][45]. However, to our best knowledge, the present work is the first to take into account the effects of strain gradient in the topology optimization through an appropriate homogenization scheme combined with the topology optimization strategy. ...

We present a topology optimization for lattice structures in the case of non-separated scales, i.e. when the characteristic dimensions of the periodic unit cells in the lattice are not much smaller than the dimensions of the whole structure. The present method uses a coarse mesh corresponding to a homogenized medium taking into strain gradient through a non-local numerical homogenization method. Then, the topological optimization procedure only uses the values at the nodes of the coarse mesh, reducing drastically the computational times. We show that taking into account the strain gradient within the topological optimization procedure brings significant increase in the resulting stiffness of the optimized lattice structure when scales are not separated, as compared to using a homogenized model based on the scale separation assumption.

... Huang et al. (2013) introduced a topology optimization algorithm for design of porous materials and composites so that the resulting macrostructure has the maximum stiffness. To reveal the equivalence of the solutions from the macro and micro approaches, Zuo et al. (2013) compared optimal material microstructures with optimal periodic structures. Xia and Breitkopf (2014b, 2014a, 2015b addressed concurrent design of material and structure within the FE 2 nonlinear multi-scale analysis framework. ...

Purpose
In pure material design, the previous research has indicated that lots of optimization factors such as used algorithm and parameters have influence on the optimal solution. In other words, there are multiple local minima for the topological design of materials for extreme properties. Therefore, the purpose of this study is to attempt different or more concise algorithms to find much wider possible solutions to material design. As for the design of material microstructures for macro-structural performance, the previous studies test algorithms on 2D porous or composite materials only, it should be demonstrated for 3D problems to reveal numerical and computational performance of the used algorithm.
Design/methodology/approach
The presented paper is an attempt to use the strain energy method and the bi-directional evolutionary structural optimization (BESO) algorithm to tailor material microstructures so as to find the optimal topology with the selected objective functions. The adoption of the strain energy-based approach instead of the homogenization method significantly simplifies the numerical implementation. The BESO approach is well suited to the optimal design of porous materials, and the generated topology structures are described clearly which makes manufacturing easy.
Findings
As a result, the presented method shows high stability during the optimization process and requires little iterations for convergence. A number of interesting and valid material microstructures are obtained which verify the effectiveness of the proposed optimization algorithm. The numerical examples adequately consider effects of initial guesses of the representative unit cell (RUC) and of the volume constraints of solid materials on the final design. The presented paper also reveals that the optimized microstructure obtained from pure material design is not the optimal solution any more when considering the specific macro-structural performance. The optimal result depends on various effects such as the initial guess of RUC and the size dimension of the macrostructure itself.
Originality/value
This paper presents a new topology optimization method for the optimal design of 2D and 3D porous materials for extreme elastic properties and macro-structural performance. Unlike previous studies, the presented paper tests the proposed optimization algorithm for not only 2D porous material design but also 3D topology optimization to reveal numerical and computational performance of the used algorithm. In addition, some new and interesting material microstructural topologies have been obtained to provide wider possible solutions to the material design.

... In other words, if the algorithm for designing periodic structures is adopted (section 5.4), the homogenization based material design result corresponds to the periodic design result when number of periodic cells goes to infinity. A comparison study with this aim was provided by Xie et al. [165] and Zuo et al. [192]. The optimization model for this particular problem can be formulated as following ...

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.

... One approach to circumvent this limitation, albeit at significant computational expense, is to maintain the unit cell as the design domain but then propagate the unit cell design to create a finitely periodic structure on which the nonlinear analysis and sensitivity analysis are performed. Of course, one must confirm that enough unit cells have been repeated to properly represent the periodic material (124,125). Elastic properties and symmetries, and their sensitivities, may remain estimated using unit cell-based homogenization. This finite periodicity approach was recently implemented to optimize energy absorption under a volume constraint (126), assuming an elastic perfectly plastic constitutive model with geometric nonlinearities considered (117,119). ...

Advanced manufacturing processes provide a tremendous opportunity to fabricate materials with precisely defined architectures. To fully leverage these capabilities, however, materials architectures must be optimally designed according to the target application, base material used, and specifics of the fabrication process. Computational topology optimization offers a systematic, mathematically driven framework for navigating this new design challenge. The design problem is posed and solved formally as an optimization problem with unit cell and upscaling mechanics embedded within this formulation. This article briefly reviews the key requirements to apply topology optimization to materials architecture design and discusses several fundamental findings related to optimization of elastic, thermal, and fluidic properties in periodic materials. Emerging areas related to topology optimization for manufacturability and manufacturing variations, nonlinear mechanics, and multiscale design are also discussed.

... Concerning the optimization involving periodic structures, the topology optimization has been applied to find optimal design of the periodic cells using the based density approach, [16,17] and more recently with the evolutionary technique, [18,19,20], considering static and dynamic problems. ...

This work applies the topology optimization technique to an acoustic-structure coupled system with periodic geometry constraint in order to obtain the optimal layout of the design domain for the minimization of the pressure frequency response in the acoustic fluid. The displacement-pressure formulation (u–p) is used for the finite element analysis of the coupled system and external harmonic excitations are applied in the system. The design domain of the coupled system is considered to be composed of identical unit cells. A periodic geometry constraint is applied in the design domain considering the fluid-structure interaction and the objective function. Appling the modified bi-directional evolutionary structural optimization (BESO) technique to the system, the design domain is evolving towards the optimal topology of the unit cells through removing/adding material accordingly to the sensitivity analysis. The influence of the total number of unit cells composing the periodic structure and the aspect ratio of the unit cells are investigated in the minimization of the objective function. In order to show the capability and efficiency of the proposed formulation, two acoustic-structure systems are optimized for several cell configurations, different aspect ratios of the periodic unit cells and excitation frequencies.

... More recently, Huang et al. (2012) used BESO to design multifunctional periodic composites having both extremal magnetic permeability and electrical permittivity and Yang et al. (2011) did the same, however for stiffness and thermal conductivity. There have been a number of recent publications in applying BESO methods to topology optimization of material microstructures Huang et al. 2013a;Zuo et al. 2013). ...

Topology optimization has evolved rapidly since the late 1980s. The optimization of the geometry and topology of structures has a great impact on its performance, and the last two decades have seen an exponential increase in publications on structural optimization. This has mainly been due to the success of material distribution methods, originating in 1988, for generating optimal topologies of structural elements. Previous methods suffered from mathematical complexity and a limited scope for applicability, however with the advent of increased computational power and new techniques topology optimization has grown into a design tool used by industry. There are two main fields in structural topology optimization, gradient based, where mathematical models are derived to calculate the sensitivities of the design variables, and non gradient based, where material is removed or included using a sensitivity function. Both fields have been researched in great detail over the last two decades, to the point where structural topology optimization has been applied to real world structures. It is the objective of this review paper to present an overview of the developments in non gradient based structural topology and shape optimization, with a focus on evolutionary algorithms, which began as a non gradient method, but have developed to incorporate gradient based techniques. Starting with the early work and development of the popular algorithms and focusing on the various applications. The sensitivity functions for various optimization tasks are presented and real world applications are analyzed. The article concludes with new applications of topology optimization and applications in various engineering fields.

... Recently, Huang et al. (2012) utilized BESO to design multifunctional periodic composites with both extremal magnetic permeability and electrical permittivity and Yang et al. (2011) did so for stiffness and thermal conductivity. A number recent works have investigated BESO methods for topology optimization of material microstructures Huang et al. 2013;Zuo et al. 2013). ...

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method’s focuses on new developments and novel applications.

Topology optimization is increasingly being used to design the architecture of porous cellular materials with extreme elastic properties. Herein, we look to extend the design problem to the nonlinear regime and aim to maximize the energy absorption capacity until failure of the base solid occurs locally. This results in a problem formulation where the nonlinear properties are estimated using a finitely periodic structure. An interesting base material choice for energy absorption are bulk metallic glasses for which we optimize the designs and fabricate them through a thermoplastic processing method. Testing to full densification reveals that the governing mechanisms for these topologically-optimized structures are combinations of buckling and yielding at the strut-level. As a consequence, they offer superior total energy absorption over the traditional honeycomb topologies. Investigations of the same topologies made of polyether ether ketone suggest future directions on how to improve the post-peak response of topology-optimized cellular materials.

Graded lattice sandwich structures (GLSSs) enable superior structural performances due to the continuously-varying configurations and properties of lattices in space. This paper proposes a design method for GLSSs by multiscale topology optimization. In this method, the geometrical configuration of a prototype lattice cell (PLC) is described by an explicit topology description function (TDF). Based on some sample lattice cells, a Kriging metamodel is constructed to predict the effective property of each lattice cell. Based on the Kriging metamodel, the thickness optimization of two solid face-sheets and the distribution optimization of lattice cells in core layer are respectively implemented. Driven by their equivalent densities, the configurations of lattice cells with similar topological features are generated by interpolating the shape of the PLC. Then, the graded lattice cells (GLCs) are generated. Using the proposed method, the computational burden involved in design of GLSSs can be reduced significantly. What is more, the manufacturability of GLSSs can be guaranteed by constructing a proper PLC with the help of TDF. Numerical examples in terms of compliance and natural frequency optimization of GLSSs are provided to verify the advantages of the proposed method. Also, bending tests are performed on the GLSSs fabricated by additive manufacturing (AM). The results reveal that GLSSs are stiffer and have larger natural frequencies than the uniform lattice sandwich structures.

To realize extraordinary wave phenomena, metamaterials need to attain unique effective material properties. In this work, we propose an inverse design strategy for metamaterials with specific anisotropic EMD (effective mass density). Although the conventional inverse homogenization technique has been extended to various fields, few works have been published to explore the inverse realization of an EMD tensor, each component of which is supposed to gain a given value at a target frequency. To this end, we propose a calculation scheme, in which the EMD tensor can be calculated in a much similar way to the homogenized static stiffness. Therefore, the scheme is quite convenient for sensitivity analysis. The coating layer interfacing the core and matrix is chosen as the design region because it directly determines the motion of the core. The matrix layout, which not only contributes to the stiffness of the metamaterial but also highly affects the core's local motion, is chosen carefully. The perfect transmodal Fabry–Pérot interference phenomenon is considered in this work, and through several numerical examples, the phenomenon is ideally realized. The proposed design strategy could be critically useful in designing locally resonant metamaterials with general anisotropy.

Scale separation is often assumed in most multiscale topology optimization frameworks. In this work, topology optimization of heterogeneous structures consisting of inseparable unit cells is studied. The cell morphology is given first and remains unchanged during the optimization process. A nonlocal numerical homogenization method is used to construct a mesoscopic constitutive relationship between the material and the structural scales. Topology optimization is performed on heterostructures at the higher mesoscale, and only based on a coarse grid for computational savings. Numerical studies show that the structural stiffness has been significantly improved compared to classical multiscale topology optimization using separation assumptions. However, there is still a size dependence of the optimal mesostructure related to the characteristic effect of the unit lattice.

The goal of this research is to optimize an object's macroscopic topology and localized gradient material properties (GMPs) subject to multiple loading conditions simultaneously. The gradient material of each macroscopic cell is modeled as an orthotropic material where the elastic moduli in two local orthogonal directions we call x and y can change. Furthermore, the direction of the local coordinate system can be rotated to align with the loading conditions on each cell. This orthotropic material is similar to a fiber-reinforced material where the number of fibers in the local x and y-directions can change for each cell, and the directions can as well be rotated. Repeating cellular unit cells, which form a mesostructure, can also achieve these customized orthotropic material properties. Homogenization theory allows calculating the macroscopic averaged bulk properties of these cellular materials. By combining topology optimization with gradient material optimization and fiber orientation optimization, the proposed algorithm significantly decreases the objective, which is to minimize the strain energy of the object subject to multiple loading conditions. Additive manufacturing (AM) techniques enable the fabrication of these designs by selectively placing reinforcing fibers or by printing different mesostructures in each region of the design. This work shows a comparison of simple topology optimization, topology optimization with isotropic gradient materials, and topology optimization with orthotropic gradient materials. Finally, a trade-off experiment shows how different optimization parameters, which affect the range of gradient materials used in the design, have an impact on the final objective value of the design. The algorithm presented in this paper offers new insight into how to best take advantage of new AM capabilities to print objects with gradient customizable material properties.

An application of the fast multipole boundary element method (FMBEM) and an artificial immune system (AIS) to the optimization of porous structure effective elastic properties is presented. The FMBEM allows one to model complex geometries with much lower number of degrees of freedom in comparison to the finite element method, that is usually applied in computational homogenization. Representative volume elements (RVEs) are modelled, with displacement boundary conditions corresponding to a given strain state in the macro scale. Effective elastic constants of the material are calculated by using the averaged strains and stresses. Design variables considered in the optimization problem describe the geometry. The minimized objective function involves a metric that allows one to calculate the distance between two elasticity tensors: a current solution and a reference tensor that defines the desired properties. A benchmark problem of porous structure with maximized effective bulk modulus is solved.

The aim of this Thesis is to present efficient methods for optimising high-resolution problems of a multiscale and multiphysics nature. The Thesis consists of two parts: one treating topology optimisation of microstructural details and the other treating topology optimisation of conjugate heat transfer problems.
Part I begins with an introduction to the concept of microstructural details in the context of topology optimisation. Relevant literature is briefly reviewed and problems with existing methodologies are identified. The proposed methodology and its strengths are summarised.
Details on the proposed methodology, for the design of structures with periodic and layered microstructural details, are given and the computational performance is investigated. It is shown that the used spectral coarse basis preconditioner, and its associated basis reutilisation scheme, significantly reduce the computational cost of treating structures with fully-resolved microstructural details.
The methodology is further applied to examples, where it is shown that it ensures connectivity of the microstructural details and that forced periodicity of the microstructural details can yield an implicit robustness to load position. An example of expansion control of a structure under compression is treated in detail, where it is shown that taking boundary effects into account is paramount.
Part II starts with an introduction to conjugate heat transfer and briefly reviews relevant literature. The governing equations used to describe heat transfer and fluid flow are outlined, describing both a commonly-used simplified convection model and the full natural convection model.
Topology optimisation using the simplified model is investigated as a means to reduce the computational time of optimising heat sinks. The model is shown to be useful in an industrial context to provide a first approximation in the design of heat sinks. However, serious flaws and drawbacks of combining the model with topology optimisation are identified.
In order to take full advantage of topology optimisation for providing insight into optimal design of heat sinks, a full conjugate heat transfer model is introduced. Optimised heat sinks are presented for both two- and three-dimensional natural convection problems, where similarities and differences are discussed. Generally, the observations are in line with classical heat sink design, but topology optimisation spawns designs exhibiting optimal characteristics without any prerequisite knowledge.Furthermore, it is shown that when using the full model, the local convection coefficients and surface fluxes are in direct disagreement with the assumptions of the simplified model.
The computational performance and scalability of the developed framework is presented and it is shown that it allows for efficient optimisation of problems with more than 300 million degrees of freedom and almost 30 million design variables. Finally, the framework is used to generate novel passive coolers for light-emitting diode (LED) lamps, where a 20 − 25% lower temperature of the LED package is achieved as compared to reference designs, using around 16% less material.

Based on topology optimization techniques, the influences of skin thickness, stiffener height and periodic grid number on the optimized design of periodic grid stiffened structure was investigated. Firstly, topology optimization of grid stiffened structures was carried out for different combinations of skin thickness and stiffener height. Comparison of the results shows that there is uniform grid configuration with optimal structural efficiency for different material volume fraction. Secondly, the effect of periodic grid number upon optimal grid configuration of one or two-ply periodic grid stiffened structure were discussed. The results show that the optimal structural performance is tends to limits and the optimal configuration is change for stiffness optimization when periodic grid number increase, furthermore, when periodic grid number is more than 7×7, the effect on optimized grid layout vanish for multi-objective optimization. The results provide guidance for design and choice of grid layout of grid stiffened structure.

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.

A numerical method for the topological design of periodic continuous domains under general loading is presented. Both the analysis and the design are defined over a single cell. Confining the analysis to the repetitive unit is obtained by the representative cell method which by means of the discrete Fourier transform reduces the original problem to a boundary value problem defined over one module, the representative cell. The repeating module is then meshed into a dense grid of finite elements and solved by finite element analysis. The technique is combined with topology optimization of infinite spatially periodic structures under arbitrary static loading. Minimum compliance structures under a constant volume of material are obtained by using the densities of material as design variables and by satisfying a classical optimality criterion which is generalized to encompass periodic structures. The method is illustrated with the design of an infinite strip possessing 1D translational symmetry and a cyclic structure under a tangential point force. A parametric study presents the evolution of the solution as a function of the aspect ratio of the representative cell.

This paper proposes a method for topology optimization of periodic structures on dynamic problems by using an improved bidirectional evolutionary structural optimization (BESO) technique. Frequency optimization and frequency-stiffness optimization are formulated for periodic continuum structures at the macroscopic level under arbitrary loadings and boundaries. Numerical instabilities that occur in common topological frequency optimization are dealt with by eliminating singular and single-hinged elements and removing alternative
element groups in case of sudden drops of the relevant frequency. Layout periodicity of the optimal design is guaranteed by creating a representative unit cell (RUC) on the basis of a user-defined cell mode and averaging the sensitivities from all unit cells into the RUC. The capability and effectiveness of the proposed approach are demonstrated by numerical experiments with various cell modes.

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

This paper presents a new approach to designing periodic microstructures of cellular materials. The method is based on the bidirectional evolutionary structural optimization (BESO) technique. The optimization problem is formulated as finding a micro-structural topology with the maximum bulk or shear modulus under a prescribed volume constraint. Using the homogenization theory and finite element analysis within a periodic base cell (PBC), elemental sensitivity numbers are established for gradually removing and adding elements in PBC. Numerical examples in 2D and 3D demonstrate the effectiveness of the proposed method for achieving convergent microstructures of cellular materials with maximum bulk or shear modulus. Some interesting topological patterns have been found for guiding the cellular material design.

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.

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

This paper demonstrates a simple finite element implementation of Lagrange multipliers to model the mechanical behaviour of an orthotropic composite material. The research shows the proper set of kinematic boundary conditions that must be applied in 2D plane stress elasticity to achieve the correct unit strain vectors that, upon interrogation of the associated Lagrange multipliers, give the stresses induced by these strain vectors. From these stresses the terms in the elasticity matrix can be evaluated. As well as demonstrating the correct kinematic conditions required, the paper presents the consequences of applying intuitive but incorrect conditions. © 1997 John Wiley & Sons, Ltd.

The numerical solution of homogenization equations by the finite element (FE) method is explained briefly. The issue of extracting boundary conditions from the periodicity assumption is addressed and a direct method utilizing symmetry is presented. Using this method, the computation of the elements of the constitutive matrix of a composite material is reduced to a very conventional boundary value problem with known forces and boundary conditions which can be carried out with any FE code. Two examples are presented.

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

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

This paper extends recent advances in the topology optimization of fluid flows to the design of periodic, porous material microstructures. Operating in a characteristic base cell of the material, the goal is to determine the layout of solid and fluid phases that will yield maximum permeability and prescribed flow symmetries in the bulk material. Darcy’s law governs flow through the macroscopic material while Stokes equations govern flow through the microscopic channels. Permeability is computed via numerical homogenization of the base cell using finite elements. Solutions to the proposed inverse homogenization design problem feature simply connected pore spaces that closely resemble minimal surfaces, such as the triply periodic Schwartz P minimal surface for 3 − d isotropic, maximum permeability materials.

This is the second part of a three-paper review of homogenization and topology optimization. In the first paper, we focused on the theory and derivation of the homogenization equations. In this paper, motives for using the homogenization theory for topological structural optimization are briefly explained. Different material models are described and the analytical solution of the homogenization equations for the so called “rank laminate composites” is presented. The finite element formulation is explained for the material model, based on a miscrostructure consisting of an isotropic material with rectangular voids. Using the periodicity assumption, the boundary conditions are derived and the homogenization equations are solved, and the results to be used in topology optimization are presented. The third paper deals with the use of homogenization for structural topology optimization by using optimality criteria methods.

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

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