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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.

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... Huang and Xie [2] divided the macrostructures into prescribed cells and studied the scale effect of cells considering periodic constraint. Xie et al. [3] investigated the scale effect further and concluded that the optimized structure would converge to a certain repetitive pattern with various length scales of the cells. Also, Almeida et al. [4] introduced SPR constraints into an FGM-SIMP (Functionally Graded Material -Solid Isotropic Material with Penalization) model for optimization of functionally graded materials. ...

... Solutions with different divisions of cell domain, in comparison, verify the conclusion by Xie et al. [3] that the topological configuration converges to a certain pattern under the same working condition as the division of cell changes, and one can expect the component pattern to remain stable as the cell division becomes denser. The component patterns shown in Table 1 exhibit consistency in topology except for some minor variations in the roundness of boundaries, which are caused by the poorer ability of geometry description of coarser finite element mesh as the cell scale reduces since the same mesh is adopted here for different Table 1 Solutions for MIC with different cell division. ...

Modular integrated construction (MIC) can realize modular production in factories and has the advantages of increasing material utilization, speeding up construction, and improving construction environment, which make it more and more widely accepted in engineering practice. But a critical issue for MIC is how to design standard structural modular to maximize its construction productivity. In this paper, a structural topology optimization method based on a parameterized level set is adopted to realize the modular design for stiffness maximum problem of structures. By introducing the symmetry and pattern repetition (SPR) constraints in the optimization model and combining the finite element-based numerical analysis method, the modular design of a structure can be obtained through topology optimization. As a result, the structure can be assembled with repeated modular of the same topology to realize the MIC. Several two-dimensional (2D) and three-dimensional (3D) numerical examples are studied to illustrate the effectiveness of the proposed method in building designs following the MIC concept.

... Computational homogenization techniques are typically used to design topologically optimized repeated unit cells [4,7,8,10,11,12]. Under homogenization-based methods, the scales of macro-and micro-structures are assumed to differ to enable the application of perturbation theory. ...

... Thus, the layouts obtained in this exercise agree closely with the existing solutions found in [18] and are reasonable in the sense that the optimized structures have enhanced shear stiffness for sustaining the given loads. The results also demonstrate that the microstructures and their compliance values converge asymptotically in high periodicity, which agrees with [14,12]. Although it is not shown herein, the objective values of the optimal layout without assumed periodicity is found to be a lower bound of optimized cellular structures since the latter restricts the material distribution in the design domain. ...

Owing to their tailorable physical properties, periodic cellular structures are considered promising materials for use in various engineering applications. To fully leverage the potential of such structures, it will be necessary to develop an optimized design method capable of producing intricate material layouts without sacrificing manufacturability. This paper presents a topology optimization framework for designing manufacturable, multi-material cellular structures that are to be subjected to temperature change. Under this framework, multi-material layouts within designable unit cells are represented using level-set functions and corresponding Boolean operations; by assuming a common length scale between these unit cells and the macrostructure, the manufacturability of optimized designs is guaranteed. Increases in computational cost and storage requirement are minimized by applying the Guyan reduction method, in which the secondary degree of freedom is condensed out to reduce the size of the discretized model. The design capabilities of the proposed method were investigated using several numerical models, with the results demonstrating that the method achieves overall improvements in performance as a result of its expanded design space.

... For some problems considering the overall structural performance, e.g., the structural compliance or the natural frequencies of the structure, luckily, the answer is affirmative. To be specific, the numerical examples of Moses et al. [39] and Zuo et al. [40] suggested that the objective function value would converge by increasing the number of repeated base cells of the composites for topology optimization; Xie et al. [41] further found that the optimal topologies of the base cells converge rapidly to specific patterns as the increasement of repetitive base cells. Those results suggest that the homogenization method can still be used to predict the effective property of architecture structures with finite (e.g., no less than 10) repetitive base cells for the sake of both accuracy and efficiency. ...

... Figure 14 shows the corresponding optimized designs. For the design domain containing only one homogenized region, the pattern (Fig. 14a) is similar to Fig. 2 of [41], where the flanges in the base cell are to resist bending deformation, and the cross is to strengthen the shear-bearing capacity. In the second case, as shown in Fig. 14b, the patterns in two homogenized regions are reflectional symmetric. ...

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.

... An appealing way to deal with such an engineering issue is to use multiple identical sub-components which are small relative to the design domain, and thus more easily manufactured and transported. Topology optimization of such repetitive sub-components, known as periodic optimization, enables compromise between system complexity and costs (Zhang and Sun 2006;Huang and Xie 2008;Zuo 2009;Chen et al. 2010;Xie et al. 2012), which is particularly attractive in mass-production and simplicity of assembly. ...

... An example of 2D translational periodicity is the subdivision of a rectangular design domain into an m by n set of cells (a 1 by 1 representing typical single component design) with each unit cell exhibiting an identical design. In spite of its importance, there has been limited work available on finite periodic topology optimization to date, with most focusing on stiffness criteria (Zhang and Sun 2006;Huang and Xie 2008;Xie et al. 2012) and some extending to frequency and conductivity criteria (Zuo 2009;Chen et al. 2010). ...

Design of engineering structures may benefit from reduction in assembly complexity through use of periodic components, in which uniform sub-structures combine to form a relatively simple topology. The benefits of periodic structures include lower manufacture costs as well as ease of assembly. Recent developments in periodic topology optimization have shown its efficacy for addressing a range of design objectives. However, constraints such as assembly conditions and the connection configuration of periodic sub-components present limiting factors in the application of periodic optimization to real-world engineering problems. This study addresses the current knowledge gap in periodic optimization assembly through inclusion of common interfacing connections between periodic components, such as screws, welds, or rivets, thus accounting for real assembly conditions. A bi-directional evolutionary structural optimization (BESO) method and solid isotropic material with penalization (SIMP) method are presented, for stiffness and frequency criteria, which simultaneously optimizes the topology of the periodic components and the joint configuration connecting components. Elemental sensitivities are derived and utilized to drive the design of both the periodic component and the connection layouts. Iterative updating of the topological design, guided by elemental sensitivities, allows for optimization of the periodic topology for given objective functions. To demonstrate the effectiveness of the proposed method, optimized structures are explored through different periodicities. Application of the methodology presented in this study will assist in providing new design capabilities to reduce the costs of manufacturing, transport, and assembly through optimized periodic components.

... Even though the authors tried to reduce the computational resource requirements, the computation cost is substantially higher than the homogenization-based approaches. It has been shown that the optimal solution of a unit cell converges rapidly to that obtained by inverse homogenization as the number of repetitive cells increases (beyond five or six in the case of mechanical properties) [19,20]. In addition, for boundary effect, as shown in Ref. [21], the thickness of boundary layer has the same scale of the unit cell. ...

... Results in Refs. [19][20][21] suggested that the homogenization method can efficiently offer a reasonable approximation for a large number of repeated unit cells, when some global measures, e.g., mean compliance, are taken into consideration. Liu et al. [22] divided the structural domain into several subdomains and boundary layers. ...

With the rapid developments of advanced manufacturing and its ability to manufacture microscale features, architected materials are receiving ever increasing attention in many physics fields. Such a design problem can be treated in topology optimization as architected material with repeated unit cells using the homogenization theory with the periodic boundary condition. When multiple architected materials with spatial variations in a structure are considered, a challenge arises in topological solutions, which may not be connected between adjacent material architecture. This paper introduces a new measure, connectivity index (CI), to quantify the topological connectivity, and adds it as a constraint in multiscale topology optimization to achieve connected architected materials. Numerical investigations reveal that the additional constraints lead to microstructural topologies, which are well connected and do not substantially compromise their optimalities.

... Even though the authors tried to reduce the computational resource requirements, the computation cost is substantially higher than the homogenization-based approaches. It has been shown that the optimal solution of a unit cell converges rapidly to that obtained by inverse homogenization as the number of repetitive cells increases (beyond five or six in the case of mechanical properties) [19,20]. In addition, for boundary effect, as shown in Ref. [21], the thickness of boundary layer has the same scale of the unit cell. ...

... Results in Refs. [19][20][21] suggested that the homogenization method can efficiently offer a reasonable approximation for a large number of repeated unit cells, when some global measures, e.g., mean compliance, are taken into consideration. Liu et al. [22] divided the structural domain into several subdomains and boundary layers. ...

... This evidences the recent uses of the BESO method on multiphysics problems (Picelli et al., 2015c). The discrete nature of these methods are also recently explored in multiscale problems (Xie et al., 2012;Zuo et al., 2013;Xia and Breitkopf, 2014;Huang et al., 2015;Xia and Breitkopf, 2015). Similar approaches can also be adopted by other topology optimization methods with explicit boundaries definition, such as level set based methods (Luo et al., 2012;Shu et al., 2014;. ...

... Through the last 10 years, the methods of topology optimization have been under a considerable scientific effort to be extended to different physical phenomena problems. One may cite aerolastic structures (Maute and Allen, 2004), acoustics design (Duhring et al., 2008;Silva and Pavanello, 2010;Yoon, 2013), thermo-elastic stresses (Gao et al., 2008), fluid flows (Aage et al., 2008) and fluid-structure interaction (Kreissl et al., 2010;Yoon, 2010;Andreasen and Sigmund, 2013), acoustic-structure responses Shu et al., 2014;Vicente et al., 2015), multiscale analysis (Xie et al., 2012;Xia and Breitkopf, 2014) and others. ...

The aim of this thesis is the development of a computational tool for the design of structures considering fluid-structure interaction using topology optimization. A methodology of structural topology optimization is proposed in association with finite element formulations of fluid-structure coupled problems. In this type of problems, the structure undergoes fluid loading, i.e., pressure and/or viscous loads. The difficulties in designing fluid loaded structures arise due to the variation of location, direction and magnitude of the loads when the structural shape and topology change along the optimization procedure. This turns out to be an additional difficulty for the traditional density-based topology optimization methods. In density-based methods, the pressure loaded surfaces are not explicitly defined due to the existence of intermediate density elements. In this thesis, it is suggested an alternative methodology to handle this type of design-dependent loads. With an extended bi-directional evolutionary structural optimization (BESO) method associated with different fluid-structure formulations, pressures and viscous loads can be modelled straightforwardly for any structural topology due to the discrete nature of the method. Thus, the problem is solved without any need for pressure load surfaces parametrization. The BESO methodology is extended considering the procedures of switching fluid-structure-void elements, new sensitivity analyses and constraints. Steady state problems are considered, including linear elasticity for the structural analysis and Laplace, Helmholtz and incompressible Navier-tokes flow equations for the fluid analysis. Constant and non constant loads are modelled. Several examples and applications are explored with the proposed methodology.

... 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.

... Currently, size effect has become a hot topic in the mechanical analysis of lattice materials (e.g., Refs. [11][12][13][14][15][16]). Yan et al. [13] discussed the nature of the size effect by using the RVE method. ...

... Zhang et al. [14,15] studied the integrated design of porous materials and structures with scale-coupled effects. Xie et al. [16] found that the topologies of the unit cells converged rapidly to certain patterns by increasing the number of unit cells. ...

The size effects of microstructure of lattice materials on structural analysis and minimum weight design are studied with extented multiscale finite element method (EMsFEM) in the paper. With the same volume of base material and configuration, the structural displacement and maximum axial stress of micro-rod of lattice structures with different sizes of microstructure are analyzed and compared. It is pointed out that different from the traditional mathematical homogenization method, EMsFEM is suitable for analyzing the structures which is constituted with lattice materials and composed of quantities of finite-sized micro-rods. The minimum weight design of structures composed of lattice material is studied with downscaling calculation of EMs-FEM under stress constraints of micro-rods. The optimal design results show that the weight of the structure increases with the decrease of the size of basic sub-unit cells. The paper presents a new approach for analysis and optimization of lattice materials in complex engineering constructions.

... For the design of microstructural materials, the unit cell, or the microstructure, is infinitely small compared to the macrostructure [27][28][29][30]. On the other hand, the focus of this study lies in unit cells of finite size, and is primarily used for the design of large macrostructures [31][32][33]. ...

In recent years, topology optimization of periodic structures has become an effective approach to generating efficient designs that meet a variety of practical considerations, including manufacturability, transportability, replaceability, and ease of assembly. Traditional periodic structural optimization typically restricts designs to a uniform assembly configuration utilizing only one type of unit cell. This study proposed a novel clustering-based approach for periodic structural optimization, which allows variable orientations of individual unit cells. A dynamic k-means clustering strategy is introduced to categorize all unit cells into distinct groups and gradually eliminate less efficient unit cells from the optimized design. Meanwhile, a novel technique is introduced to identify and select more efficient orientations of unit cells during the optimization process. Several numerical examples are presented to demonstrate the effectiveness of the proposed approach. The results show that periodic structures with clustered oriented unit cells can significantly outperform their traditional periodic counterparts. This study not only incorporates assembly flexibility into periodic topology optimization but also utilizes multiple types of unit cells in a design, thereby further enhancing its structural performance.

... BESO is an extension of the Evolutionary Structural Optimisation (ESO) method that enables the addition of material to the structure in areas where it is most required while simultaneously removing inefficient material [21]. The BESO approach has been widely employed to enhance the microscopic stiffness [21][22][23] and nonlinear behaviour [17] of lattice materials. Jia et al. [16] employed a nongradient topology optimisation method to enhance the crashworthiness of a solid beam under impact loads. ...

This study evaluates topologically optimized lattice structures for high strain rate loading, crucial for impact resistance. Using the BESO (Bidirectional Evolution Structural Optimisation) topology optimisation algorithm, CompIED and ShRIED topologies are developed for enhanced energy absorption and impact resistance. Micromechanical simulations reveal CompIED surpasses theoretical elasticity limits for isotropic cellular materials, while the hybrid design ShRComp achieves theoretical maximum across all relative densities. Compared to TPMS, truss, and plate lattices, the proposed structures exhibit higher uniaxial modulus. Manufactured via fused deposition modeling with ABS thermoplastic, their energy absorption capabilities are assessed through compression tests and impact simulations. The ShRComp lattice demonstrates superior energy absorption under compression compared to CompIED. Impact analyses of CompIED and ShRComp sandwich structures at varying velocities show exceptional resistance to perforation and higher impact absorption efficiency, outperforming other classes of sandwich structures at similar densities. These findings position these new and novel topologies as promising candidates for impact absorption applications.

... For example, topology optimization can be combined with lattices, TPMS or other cellular structures. One way to do so is that a unit cell is optimized through topology optimization and the part is substituted by repetition of this optimized unit cell [25][26][27]. Another way is that topology optimization defines the envelop of the final part and the remaining part is entirely substituted by a periodic repetition of the chosen unit cell [28][29][30][31]. ...

Mass reduction of mechanical systems is a recurrent objective in engineering, which is often reached by removing material from its mechanical parts. However, this material removal leads to a decrease of mechanical performances for the parts, which must be minimized and controlled to avoid a potential system failure. To find a middle-ground between material removing and mechanical performances), material must be kept only in areas where it is necessary, for example using stress-driven material removal methods. These methods use the stress field to define the local material removal based on two local parameters: the local volume fraction vf and the structural anisotropy orientation β. These methods may be based on different types of cellular structure patterns: lattice-based or bio-inspired. The long-term objective of this study is to improve the performance of stress-driven methods by using the most efficient pattern. For this purpose, this study investigates the influence of vf and β on the mechanical stiffness of three planar cellular structures called Periodic Stress-Driven Material Removal (PSDMR) structures. The first, taken from the literature, is bio-inspired from bone and based on a square pattern. The second, developed in this study, is also bio-inspired from bone but based on a rectangular pattern. The third is a strut-based lattice pattern well documented in the literature for its isotropic behavior. These three patterns are compared in this study in terms of relative longitudinal stiffness, obtained through linear elastic compressive tests by finite element analysis. It is highlighted that each PSDMR pattern has a specific domain in which it performs better than the two others. In future works, these domains could be used in stress-driven material removal methods to select the most adequate pattern or a mix of them to improve the performances of parts.

... Hence, the advantage of using category-reduced planarization has disappeared. To overcome this setback, periodic constraints can be incorporated into topology optimization, so that panels within the same category will have the same topology (Xie et al., 2012). For example, a simple periodic constraint can be achieved by averaging the sensitivities of panels in the same category during the optimization process. ...

Free-form architectural design has gained significant interest in modern architectural practice. Due to their visually appealing nature and inherent structural efficiency, free-form shells have become increasingly popular in architectural applications. Recently, topology optimization has been extended to shell structures, aiming to generate shell designs with ultimate structural efficiency. However, despite the huge potential of topology optimization to facilitate new design for shells, its architectural applications remain limited due to complexity and lack of clear procedures. This paper presents four design strategies for optimizing free-form shells targeting architectural applications. First, we propose a topology-optimized ribbed shell system to generate free-form rib layouts possessing improved structure performance. A reusable and recyclable formwork system is developed for their effective and sustainable fabrication. Second, we demonstrate that topology optimization can be combined with funicular form-finding techniques to generate a rich variety of elegant designs, offering new design possibilities. Third, we offer cost-effective design solutions using modular components for free-form shells by combining surface planarization and periodic constraint. Finally, we integrate topology optimization with user-defined patterns on free-form shells to facilitate aesthetic expression, exemplified by the Voronoi pattern. The presented strategies can facilitate the usage of topology optimization in shell designs to achieve high-performance and innovative solutions for architectural applications.

... This makes it possible to analyze a single PBC of the material instead of the entire structure by FEA. In the context of micro-scale material optimization, PBCs have no specific physical size, and therefore, the optimized solution depends heavily on the scale effect of the unit cell (Xie et al. 2011). This becomes problematic when the optimized designs are to be manufactured. ...

Assembly complexity and manufacturing costs of engineering structures can be significantly reduced by using periodic mechanical components, which are defined by combining multiple identical unit cells into a global topology. Additionally, the superior energy-absorbing properties of lattice-based periodic structures can potentially enhance the overall performance in crash-related applications. Recent research developments in periodic topology optimization (PTO) have shown its efficacy for tackling new design problems and finding advanced novel structures. However, most of these methods rely on gradient information in the optimization process, which poses difficulties for crash problems where analytical sensitivities are usually not directly applicable. In this paper, we present an effective periodic evolutionary level set method (P-EA-LSM) for the optimization of periodic structures. P-EA-LSM uses a low-dimensional level-set representation based on moving morphable components to parametrize a single unit cell, which is replicated in the design domain according to a predefined pattern. The unit cell is optimized using an evolutionary algorithm and the structural responses are calculated for the entire system. We initially assess the performance of P-EA-LSM using three 2D minimum compliance test cases with varying periodicities. Our results demonstrate that our approach produces solutions comparable to other state-of-the-art methods for PTO while keeping a low dimensionality of the optimization problem. Subsequently, we effectively evaluate the capabilities of P-EA-LSM in a crashworthiness scenario. This particular application highlights the significant potential of the method, which does not rely on analytical sensitivities.

... The stiffness maximization may possibly be attained either by taking the maximum deflection or the mean compliance as goal functionals to be minimized. Notwithstanding an idea on the stiffness is more likely given by the former approach, many good results in the literature have been attained by considering the latter (see, e.g., [81,82,83,84,85]). In [86], in fact, the former objective functional has been considered for cantilever beams and it has been shown that the free end deflection decreases by an extra 10% in comparison with the one obtained by minimizing the mean compliance. ...

This dissertation is focused on the utility of variational principles and the vast possibilities they offer as powerful tools for a suggestive use to solve optimization problems in structural mechanics.
To this purpose, an introduction to the analytical approach to continuous dynamic optimization problems and the development of a dedicated computational method are addressed in the first part of the dissertation.
For the sake of establishing a level of practical effectiveness and clarifying their vitality, a few concrete applications ranging from shape optimization problems for thin-walled axisymmetric pressure vessels and straight and curved beams to material optimization problems for functionally graded elastic bodies are addressed.
The corresponding decision variables are the meridian shape, the cross sectional area distribution and the mechanical properties distributions along specific directions throughout the body, respectively.
Potential performance criteria destined for optimization and possible structural constraints consist of reasonable combinations of lightweightness, storage capacity, compliance, resistance to buckling and load-bearing capacity.
These problems are formulated in the second part of the dissertation, solved and thoroughly discussed and, when possible, compared to literature.
In some cases, optimal solutions are derived analytically and are accompanied by prompt design charts, otherwise, in case of a cumbersome analytical tractability, they are obtained numerically by means of the computational method developed in the first part.

... 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.

... However, this method was not used in engineering until the maturity of finite element analysis in the 1980s. In the last few decades, several techniques like the homogenization method [9], the solid isotropic microstructure with penalization (SIMP) method [10,11], the evolutionary structural optimization (ESO) method [2,12], the bidirectional evolutionary structural optimization (BESO) method [13,14], the level set method [15][16][17], and the phase-field method [18][19][20] have been developed and widely used in aerospace [21,22], bio-chemical [23,24], mechanical [25,26], and structural engineering [27,28]. ...

The level set method can express smooth boundaries in structural topology optimization with the level set function's zero-level contour. However, most applications still use rectangular/hexahedral mesh in finite element analysis, which results in zigzag interfaces between the void and solid phases. We propose a reaction diffusion-based level set method using the adaptive triangular/tetrahedral mesh for structural topology optimization in this work. Besides genuinely expressing smooth boundaries, such a body-fitted mesh can increase finite element analysis accuracy. Unlike the traditional upwind algorithm, the proposed method breaks through the constraint of Courant-Friedrichs-Lewy stability condition with an updating scheme based on finite element analysis. Numerical examples for minimum mean compliance and maximum output displacement at specified positions, in both 2D and 3D, converge within dozens of iterations and present elegant structures.

... In consequence, various optimised solutions for the cellular configurations may be generated according to the different numbers of elements. In regard to such issue, Xie et al. (2012) investigated the scale effect and come to the conclusion that the cellular microstructures would convergence to a stable solution with increasing the amount of elements. ...

... Lightweight structures are the long-time pursuit of human society for more sustainable lives (Schaedler et al. 2011;Wang et al. 2016;Yin et al. 2017). With the development of understanding of material structure-property, designing materials from a variety of length scales, from nano-scale (Baughman et al. 2002), microscale (Evans et al. 2001;Xu et al. 2018;Xu et al. 2019) to macro-scale (Xie et al. 2012) and multi-scales (Yao et al. 2011;Zhang et al. 2019), is regarded as one of the most promising methodologies for improving and optimization of mechanical performance of structures (Lakes 1993;Zhang et al. 2011). ...

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.

... Basically, for certain constraints, topology optimization aims at finding the optimal material distribution and maximizing the performance of the resulting structure. Up to present, the most widely used topology optimization methods include: the density-based methods such as the Solid Isotropic Material with Penalization (SIMP) method [1,2], the homogenization method [3,4], and the evolutionary method [5]; the boundary evolution methods such as the phase field method [6,7], the level set method [8][9][10][11][12], the explicit interface represent method [13]; the methods with combinations of discrete components [14,15]. ...

In the framework of the parameterized level set method, the structural analysis and topology representation can be implemented in a decoupling way. A parameterized level set function, typically, using radial basis functions (RBFs), is a linear combination of a set of prescribed RBFs and coefficients. Once the coefficients are determined, the theoretical level set function is determined. Exploiting this inherent property, we propose a multi-discretization method based on the parameterized level set method. In this approach, a coarse discretization is applied to do the structural analysis whereas another dense discretization is employed to represent the structure topology. As a result, both efficient analysis and high-resolution topological design are available. Note that the dense discretization only accounts for a more precise and smooth description of the theoretical level set function rather than introduce extra design freedom or incur interference to structural analysis or the optimization process. In other words, this decoupling way will not add to the computational burden of structural analysis or result in non-uniqueness of converged results for a particular analysis setting. Numerical examples in both two-dimension and three-dimension show effectiveness and applicability of the proposed method.

... 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. ...

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.

... Various attempts have been made to link micro-scale infill topology to the density values obtained through optimization. Multi-scale optimization has been addressed in Yan et al. (2016), Sivapuram et al. (2016), and Yi et al. (2012). Recently, Dapogny et al. (2019) performed a 2D topology optimization considering specific infill patterns with anisotropic behavior. ...

Additive manufacturing (AM) has enabled the fabrication of artifacts with unprecedented geometric and material complexity. The focus of this paper is on the build the optimization of short fiber reinforced polymers (SFRP) AM components. Specifically, we consider optimization of the build direction, topology, and fiber orientation of SFRP components. All three factors have a significant impact on the functional performance of the printed part. While significant progress has been made on optimizing these independently, the objective of this paper is to consider all three factors simultaneously and explore their interdependency, within the context of thermal applications. Towards this end, the underlying design parameters are identified, appropriate sensitivity equations are derived, and a formal optimization problem is posed as an extension to the popular Solid Isotropic Material with Penalization (SIMP). Results from several numerical experiments are presented, highlighting the impact of build direction, topology, and fiber orientation on the performance of SFRP components.

... 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

... Otherwise, the simulation result will be poor in accuracy because of the scale effect [3]. Then, it is found in [3,39] that, consistent multi-scale design results could be derived when the material unit size falls below some threshold value, i.e., further reducing the unit cell size would not change the optimization result. This observation is also validated through experiments [40,41]. ...

This paper performs a combined numerical and experimental study to explore the role of minimum length scale constraints in multi-scale topology optimisation. Multi-scale topology optimisation is generally performed without considering the actual unit cell size, while an arbitrary value considerably smaller than the part is selected afterwards. However, this procedure would be problematic if including geometric constraints, e.g. minimum length scale constraints, since geometric constraints cannot be applied without knowing the unit cell dimensions. To address this issue, unit cell size should be defined beforehand, and guidelines will be provided in this work through a thorough numerical exploration, i.e. compliance minimisation multi-scale topology optimisation with different unit cell sizes and a consistent minimum length scale limit will be performed. The numerical results indicate that selecting the unit cell size considerably smaller than the part and larger than the length scale limit would be recommended. Then, experiments are conducted to explore the effect of minimum length scale limit on the stiffness and strength of the multi-scale design. It is observed that increasing the minimum length scale limit would reduce the structural mechanical performance in both aspects.

... 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]. ...

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.

... maximum effective stiffness [26], negative Poisson's ratio [27,28], extreme thermal expansion [29] and multifunctionality [30]. In the design of materials characterized by periodic unit cells, the sizes of the microstructures are usually assumed to be much smaller than that of the macrostructure, otherwise the optimal solution may exhibit certain scale effects [31,32]. On the other hand, the material microstructures can also be inhomogeneous. ...

This paper studies concurrent two-scale design optimization of composite structures filled with multiple microstructural unit cells. The task of the design problem is to simultaneously optimize microstructural configurations of the unit cells and their spatial distribution in the macroscale. To this end, a new topology optimization framework based on combined topology representation of the density model and the level set model is proposed. The homogenization method is used to link the material microstructural design and the macroscale design by evaluating the effective properties of the microstructures. In the microscale, topology optimization of multiple microstructural unit cells is performed with the density-based method. In the macroscale design, the distribution of multiple microstructural unit cells is optimized by the velocity field level set method, which inherits advantages of the implicit geometrical representation of the conventional level set model (relatively clear and smooth material boundaries/interfaces, more natural description of topological evolution). Moreover, the velocity field level set method maps the variational boundary shape optimization problem into a finite-dimensional design space, thus making it relatively easy and efficient to employ general mathematical programming algorithms to handle the multiple constraints and two types of design variables in the concurrent two-scale design problem. Numerical examples show that the present concurrent two-scale design method can generate meaningful designs of hierarchical cellular structures with well-defined boundaries and material interfaces.

... As a result, deviation arises in stress calculation when the present AABH method is used. However, since it has been shown both theoretically ( Dumontet, 1985 ) and numerically ( Xie et al., 2012 ) that such deviation is notable only within a thin boundary layer whose thickness spans several cell spacings, we add a non-designable thin region around the specimen boundary, to temporarily address the issue of boundary layer. Further studies concerning improving the accuracy of the proposed formulation may become one future research direction. ...

Graded microstructures have demonstrated their values in various engineering fields, and their production becomes increasingly feasible with the development of modern fabrication techniques, such as additive manufacturing. With the use of asymptotic analysis, we propose in this article a homogenisation framework to underpin the fast design of devices filled with quasi-periodic microstructures. With the introduction of a mapping function which transforms an infill graded microstructure to a spatially-periodic configuration, the originally complicated cross-scale problem can be asymptotically decoupled into a macroscale problem within a homogenised media and a microscale problem within a representative unit cell. For a given graded microstructure, the stress field and overall compliance computed by the proposed method are shown, both theoretically and numerically, to be consistent with the underlying fine-scale results. Upon linearisation, the computational cost associated with the proposed formulation is found to be as low as that in existing asymptotic-analysis-based homogenisation approaches, where only spatially periodic microstructures are considered. The present framework also exhibits interesting features in several other aspects. Firstly, smooth connectivity within graded microstructures is automatically guaranteed. Secondly, the configuration obtained here is naturally characterised by a finite length scale associated with the resolution of fabrication. The proposed approach effectively reproduces the optimal microstructure for the case of uniaxial loading where explicit solutions are available, and other numerical results are further provided.

... 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.

... As a result, deviation arises in stress calculation when the present AABH method is used. However, since it has been shown both theoretically ( Dumontet, 1985 ) and numerically ( Xie et al., 2012 ) that such deviation is notable only within a thin boundary layer whose thickness spans several cell spacings, we add a non-designable thin region around the specimen boundary, to temporarily address the issue of boundary layer. Further studies concerning improving the accuracy of the proposed formulation may become one future research direction. ...

Graded microstructures have demonstrated their values in various engineering fields, and their production becomes increasingly feasible with the development of modern fabrication techniques, such as additive manufacturing. In this article, we propose a novel homogenisation-aided topology optimisation (HATO plus) framework so as to underpin the fast design of devices decorated with spatially-varying microstructures. With the introduction of a mapping function which transforms a graded microstruc-ture to a spatially-periodic configuration, the originally complicated multiscale problem can be asymptotically decoupled into a macroscale problem within a homogenised media and a microscale problem within a unit cell. Under this framework, the data set needed for characterising a graded microstructure is significantly compressed, while the computational cost associated with the proposed linearised formulation is shown to be as low as that in traditional homogenisation-based topology optimisation approaches, where only periodic structures are considered. Compared to traditional methods, the present framework shows outstanding features in several aspects. Firstly, the variety of describable microstructures is significantly enhanced. Secondly, smooth connectivity within graded microstructures is automatically achieved under the proposed framework. Thirdly, the configuration obtained here is naturally associated with a finite characteristic length scale associated with the resolution of fabrication. The proposed approach is validated by reproducing optimal microstructures where explicit solutions are available, and more simulation results are further provid-1 ed to demonstrate the effectiveness of the proposed homogenisation-aided topology optimisation framework.

... The level set method for topology optimization has complicated algorithm with low efficiency in solving the Hamilton-Jacobi equation. One of the most popular used optimization methods is the evolutionary structural optimization (ESO) [19], [20], which has high calculate efficiency and can achieve the target easily. Comparing with other topology methods, ESO has attractive features in algorithm which can be easily implemented into commercial FEA software packages, and in numerical results with no "grey" area, which can be easily produced. ...

Abstract—In order to interact with human flexibly, the robots
need lightweight structure to adjust their configuration
conveniently and further save operation energy. It is a challenge
in design when the robots are proceeding tasks with a huge and
heavy body. This paper presents an improved framework of the
humanoid robot optimized by the evolutionary structural
optimization (ESO) method for lightweight design. By
analyzing the force of the structure using the finite element
software, the location with maximum stresses much smaller
than allowable stresses of the materials was found and then
removed. By comparison, the weight of the optimized
framework achieved 50.15% less than the original one without
changing the stiffness and vibration performance, improving
the material utilization and extending the service time of the
battery.

... Since a finite number of microstructures were actually used for solving the macroscopic displacement field, typically each element corresponding to a single microstructure, different amounts of elements would lead to different optimized microstructures. Zhang and Sun [21] and Xie et al. [22] investigated this scale-dependent effect and found that the optimized microstructure would converge to a certain configuration by decreasing the length scale ratios. An integrated structure-material design method was developed by Liu et al. [23], where both the topology of the macrostructure and its constituent microstructure were simultaneously optimized. ...

This paper proposes a novel multiscale concurrent design method to provide insight into the optimal structural design based on micro-architectures, where both the spatially-varying microstructural configurations and their macroscopic distribution are optimized in an integrated manner. A shape metamorphosis technology is developed to interpolate a prototype microstructure (PM) to generate a family of graded microstructures (GMs) that are connectable to each other in a natural way since they present similar topological features and material distribution patterns at their edges. The concurrent design optimizes configuration of the PM at the microscopic level and coordinates the compatible generated GMs at the macroscopic level in a double-loop manner, which ensures a sufficiently large design space. Numerical examples demonstrate that, compared to the one-scale design strategy, i.e. the macrostructure topology optimization and the homogenous microstructure design method, the proposed approach is able to produce remarkably improved optimized solutions. Furthermore, the obtained structures show good manufacturability to which the additive manufacturing technology is applicable without extensive post-modification.

... 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.

... The level set method for topology optimization has complicated algorithm with low efficiency in solving the Hamilton-Jacobi equation. One of the most popular used optimization methods is the evolutionary structural optimization (ESO) [19], [20], which has high calculate efficiency and can achieve the target easily. Comparing with other topology methods, ESO has attractive features in algorithm which can be easily implemented into commercial FEA software packages, and in numerical results with no "grey" area, which can be easily produced. ...

In order to interact with human flexibly, the robots need lightweight structure to adjust their configuration conveniently and further save operation energy. It is a challenge in design when the robots are proceeding tasks with a huge and heavy body. This paper presents an improved framework of the humanoid robot optimized by the evolutionary structural optimization (ESO) method for lightweight design. By analyzing the force of the structure using the finite element software, the location with maximum stresses much smaller than allowable stresses of the materials was found and then removed. By comparison, the weight of the optimized framework achieved 50.15% less than the original one without changing the stiffness and vibration performance, improving the material utilization and extending the service time of the battery.

... Through the last 10 years, the methods of topology optimization have been under a considerable scientific effort to be extended to different physical phenomenon problems. One may cite aerolastic structures [3], acoustics design [4][5][6], thermo-elastic stresses [7], fluid flows [8] and fluid-structure interaction [9][10][11], acousticstructure responses [12][13][14], multiscale analysis [15,16] and others. ...

... Recently, Andreasen and Sigmund [10] suggested a multi-scale topology optimization method to design poroelastic actuators. We also refer the readers to [11][12][13][14] for more successful applications of multi-scale optimization method in different fields. ...

... Huang et al. [17] maximized the stiffness of the macrostructure by optimizing the cellular materials and composites with periodic microstructures. Xie et al. [18] studied the convergence of topological patterns of optimal periodic structures. Radman et al. [19] designed the isotropic periodic microstructures of cellular materials based on the BESO technique. ...

... Tekoglu and Onck (2005) pointed out that the mechanical properties of a structure composed of porous materials are strongly dependent on the ratio of macrostructure to the size of the unit cell. Xie et al. (2012) found that the topologies of the unit cells converged rapidly to certain patterns by increasing the number of unit cells. Lipperman et al. (2008) implemented the maximum strength design for complex ground structure that is composed of a lattice material with actual size under periodic boundary conditions in uniaxial stress field. ...

This paper presents a concurrent optimization technique for structures composed of ultralight lattice materials. The optimization aims at obtaining the minimum structural compliance by optimizing the structural configuration in macroscale and the size of microcomponents of lattice materials concurrently with the specified base material volume. The microstructure of the lattice materials is assumed to be homogeneous to meet the manufacture practice. Optimization in two scales is integrated into one system with the extended multiscale finite element method. In addition, the influences from the finite size of the material microstructures on the optimal results are studied. The superiority of the concurrent optimization relative to the single-scale design of microstructures is indicated. Numerical experiments under linear and periodic boundary conditions validate the proposed optimization model and algorithm.

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.

The stiffness maximization of elastic straight Euler-Bernoulli beams under the action of linearly distributed loads is addressed. The goal is achieved by minimizing the average compliance, which is given by the value of internal elastic energy distributed over the length of the beam. Studies in the literature suggest considering this approach since it provides, unlike the minimization of the maximum deflection, a constant bending stress behavior along the beam axis. An isoperimetric constraint on the material volume is also considered and optimal solutions are derived by means of calculus of variations. Three types of boundary conditions are discussed, namely cantilever, simply supported and guided-simply supported beams. Introducing a well known relation between the cross sectional area and moment of inertia, closed-form solutions for several cross sections commonly used in engineering are derived. Finally, a sensitivity analysis with respect to the load parameters is addressed within a numerical example.

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.

This paper studies concurrent two-scale design optimization of composite structures filled with multiple microstructural unit cells. The task of the design problem is to simultaneously optimize microstructural configurations of the unit cells and their spatial distribution in the macroscale. To this end, a new topology optimization framework based on combined topology representation of the density model and the level set model is proposed. The homogenization method is used to link the material microstructural design and the macroscale design by evaluating the effective properties of the microstructures. In the microscale, topology optimization of multiple microstructural unit cells is performed with the density-based method. In the macroscale design, the distribution of multiple microstructural unit cells is optimized by the velocity field level set method, which inherits advantages of the implicit geometrical representation of the conventional level set model (relatively clear and smooth material boundaries/interfaces, more natural description of topological evolution). Moreover, the velocity field level set method maps the variational boundary shape optimization problem into a finite-dimensional design space, thus making it relatively easy and efficient to employ general mathematical programming algorithms to handle the multiple constraints and two types of design variables in the concurrent two-scale design problem. Numerical examples show that the present concurrent two-scale design method can generate meaningful designs of hierarchical cellular structures with well-defined boundaries and material interfaces.

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.

The application of structural optimization to fluid-structure multiphysics systems has gotten huge attention of the researches in the last years. However, the evolutionary approach of the optimization methods has not been investigated in this class of problems. The present work aims to propose, implement, and validate an evolutionary topology optimization for elasto-acoustic systems. In this work, a finite element analysis of the proposed systems is carried out using the u/p mixed formulation. The structural domain is governed by the linear equation of elasticity and described in terms of the displacements, u, and the fluid domain is featured by the Helmholtz equation via the primary variable of pressure, p. The BEFSO (Bi-directional Evolutionary Fluid-structural Optimization) method, here proposed, follows the procedure of the evolutionary methods in which the material removal/addition in the system occurs in the discrete way. It means that the material density, the variable project, can be 1 or 0 for solid or void elements, respectively. As part of the proposed methodology, it is developed a procedure to remove/add solid materials in the system in order to keep the interface between the domains well defined during the optimization process. Examples of optimization for 2D and 3D elasto-acoustic systems are presented, through which can be verified the efficiency of the optimization procedure developed and implemented in this work, as well the feasibility for engineering problems solution.

Extended Multiscale Finite Element Method (EMFEM) is employed to deal with the minimum compliance design of structures composed of periodic lattice materials with the sectional area of micro components as design variables under volume constraints. For unit cells with finite size, an optimization model which adapts to the structure with complex geometry and loading conditions has been established. The size effect of micro-structure of the lattice materials is discussed. Cantilever and L-shaped beams under linear boundary condition and periodic boundary condition are optimized with sequential quadratic programming algorithm. The reliability of the optimization model and algorithm are verified by the numerical examples. The paper presents a new approach for optimization of lattice materials to compose engineering constructions.

To obtain the optimal topological configuration for thermal conductive microstructures of composite materials, a topological optimization model of the periodic structure is established by the Independent Continuous Mapping method. In this model, minimized weight is taken as the objective and thermal compliance is the constraint condition. The thermal compliance constraint is approximately formulated using a first-order Taylor expansion. The image-filtering method is implemented by a partial differential equation to eliminate checkerboard patterns and mesh-dependence problems. To satisfy the periodic constraint, the designable domain is divided into a certain number of identical subdomains and the contribution coefficients of thermal compliance are redistributed. Optimal topological configurations with different periodic numbers and different loading conditions are compared and analyzed. Numerical results verify the validity of the proposed topology optimization method in designing the microstructures of composite materials for thermal conduction.

In order to obtain periodic material microstructure under macroscopic thermal conduction condition, the optimal topological model of periodic structure was built by solid isotropic material with penalization (SIMP) method. The volume fraction was referred as constraint and minimized thermal compliance was taken as optimization objective in this model. To satisfy the periodic constraint, the designable domain was divided into a certain number of identical unit cells and the thermal compliance was reallocated. The filtered variable implicitly as a solution of a partial differential equation (PDE) was applied to eliminate the checkerboard patterns and mesh-dependence problems efficiently. The optimal topological configurations were analyzed and compared with different numbers of unit cells and different load cases. The numerical results indicate that proposed periodic model is valid in design of periodic material microstructure with macroscopic steady state thermal conduction condition. Microstructure configurations are different when number of unit cells changes and it reflects the influences of size effect to periodic material design. With an increasing number of unit cells, the optimal results gradually converge to the results using homogenization method. ©, 2015, Beijing University of Aeronautics and Astronautics (BUAA). All right reserved.

Full Article
Multiphysics systems including dynamic fluid-structure interaction problems have hardly been studied in several fields of mechanical engineering. Among others, we can cite the researches in vibrations of submerged structures and the design of poroacoustic absorbing systems. Structural Topology Optimization can be applied in this class of problems in order to obtain new materials and structures. In this paper, it is presented the topology optimization based on volume constraints and natural frequency maximization of fluid-structure interaction problems. The method used in this work is the Bi-directional Evolutionary Structural Optimization (BESO), which consists in a successive elimination and replacement of elements in the design domain. This domain is defined initially and through a sensitivity analysis of the structure’s eigenvalue solution, the evolutionary algorithm remove or add solid elements. The aim of this work is to propose a new version of the BESO method applied to fluid-structure interaction systems. We consider the case of free vibration of structures attached to a fixed fluid domain. Numerical results show that the BESO method can be applied to this kind of multiphysics problem.

IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References

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

In this paper we seek to summarize the current knowledge about numerical instabilities such as checkerboards, mesh-dependence and local minima occurring in applications of the topology optimization method. The checkerboard problem refers to the formation of regions of alternating solid and void elements ordered in a checkerboard-like fashion. The mesh-dependence problem refers to obtaining qualitatively different solutions for different mesh-sizes or discretizations. Local minima refers to the problem of obtaining different solutions to the same discretized problem when choosing different algorithmic parameters. We review the current knowledge on why and when these problems appear, and we list the methods with which they can be avoided and discuss their advantages and disadvantages.

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.

To ensure manufacturability and mesh independence in density-based topology optimization schemes, it is imperative to use
restriction methods. This paper introduces a new class of morphology-based restriction schemes that work as density filters;
that is, the physical stiffness of an element is based on a function of the design variables of the neighboring elements.
The new filters have the advantage that they eliminate grey scale transitions between solid and void regions. Using different
test examples, it is shown that the schemes, in general, provide black and white designs with minimum length-scale constraints
on either or both minimum hole sizes and minimum structural feature sizes. The new schemes are compared with methods and modified
methods found in the literature.

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 presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

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.

This paper presents a concurrent topology optimization method to simultaneously achieve the optimum structure and material microstructure for minimum system compliance. Microstructure is assumed to be uniform in macro-scale to meet manufacturing requirements. Design variables for structure and material microstructure are independently defined and then integrated into one system with the help of homogenization theory. Penalization approaches are adopted in both scales to ensure clear topologies, i.e. SIMP (Solid Isotropic Material with Penalization) in micro-scale and PAMP (Porous Anisotropic Material with Penalization) in macro-scale. Numerical experiments for two examples validate the proposed method and also demonstrate the superiority of truss-like materials. (c) 2008 Elsevier Ltd. All rights reserved.

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

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

It is of great importance for the development of new products to find the best possible topology or layout for given design objectives and constraints at a very early stage of the design process (the conceptual and project definition phase). Thus, over the last decade, substantial efforts of fundamental research have been devoted to the development of efficient and reliable procedures for solution of such problems. During this period, the researchers have been mainly occupied with two different kinds of topology design processes; the Material or Microstructure Technique and the Geometrical or Macrostructure Technique. It is the objective of this review paper to present an overview of the developments within these two types of techniques with special emphasis on optimum topology and layout design of linearly elastic 2D and 3D continuum structures. Starting from the mathematical-physical concepts of topology and layout optimization, several methods are presented and the applicability is illustrated by a number of examples. New areas of application of topology optimization are discussed at the end of the article. This review article includes 425 references.

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.

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that
have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive
debate on this important topic.

A highly efficient new method for the sizing optimization of large structural systems is introduced in this paper. The proposed technique uses new rigorous optimality criteria derived on the basis of the general methodology of the analytical school of structural optimization. The results represent a breakthrough in structural optimization in so far as the capability of OC and dual methods is increased by several orders of magnitude. This is because the Lagrange multipliers associated with the stress constraints are evaluated explicitly at the element level, and therefore, the size of the dual-type problem is determined only by the number of active displacement constraints which is usually small. The new optimaliy criteria method, termed DCOC, will be discussed in two parts. Part I gives the derivation of the relevant optimality criteria, the validity and efficiency of which are verified by simple test examples. A detailed description of the computational algorithm for structures subject to multiple displacement and stress constraints as well as several loading conditions is presented in Part II.

Two types of solutions may be considered in generalized shape optimization. Absolute minimum weight solutions, which are rather unpractical, consist of solid, empty and porous regions. In more practical solutions of shape optimization, porous regions are suppressed and only solid and empty regions remain. This note discusses this second class of problems and shows that a solid, isotropic microstructure with an adjustable penalty for intermediate densities is efficient in generating optimal topologies.

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 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.

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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Topological design of modu-lar structures under arbitrary loading Optimal design of peri-odic linear elastic microstructures A critical review of established methods of struc-tural topology optimisation

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A critical review of established methods of structural topology optimisation

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doi:210.1007/s00158-00007-00217-00150

Optimal shape design as a material distribution problem

Bendsøe MP (1989) Optimal shape design as a material distribution
problem. Struct Optim 1:193–202