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An orthotropic material is characterized by nine independent moduli. The ratios between the Young’s moduli in three directions are indicative of the level of orthotropy and the bulk modulus is indicative of the overall stiffness. In this paper we propose a method for designing the stiffest orthotropic material which has prescribed ratios for Young’s moduli. The material is modeled as a microstructure in a periodic unit cell. By using the homogenization method, the elasticity tensors are calculated and its compliance matrix is derived. A Lagrangian function is constructed to combine the objective and multiple equality constraints. To enable a bi-section search algorithm, the upper and lower bounds on those multipliers are derived by using a strain energy approach. The overall optimization is based on the bi-directional evolutionary structural optimization (BESO) method. Examples of various orthotropy ratios are investigated. The topology presents a constant pattern of material re-distributed along the strongest axis while the overall stiffness is maintained.

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... Further advances in the design of periodic microstructures of multi-functional materials for a range of physical properties have been published in the meantime which proves how active this research area has been. For instance, quite recent research papers present optimal material designs for poroelastic actuators [6], elastic stiffness [7][8][9][10][11], Poisson's ratio [2,9], natural frequencies [12], thermal conductivity [10], fluid permeability [11], negative or zero compressibility [13], viscoelastic behavior [14][15][16], bone scaffolds [17], piezocomposites [18,19]. ...

... Nevertheless, the six fundamental tests in Eq. (8) can provide the full characterization of the elastic tensor, as seen in Eq. (13), and one takes advantage of that to check the expected tensor symmetry and thus correctness of the applied BC's. ...

... The results in Figs. [6][7][8][9][10][11] show that the diagonal tensor coefficients estimates applying Dirichlet-type BC's converge always from above to the homogenization predictions as n increases. In contrast, the Neumann-type BC's produce diagonal estimates that converge from below. ...

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.

... However, the deep understanding the mechanisms of NLC must lead to potential applications as the development of efficient biological structures, nanofluidic actuators or compensators for undesirable moisture-induced swelling of concrete/clay-based engineering materials [313]. Since NLC has been found in systems having a fixed topology, there has been a large amount of work aimed to design NLC materials using topology optimization techniques [211,294,[329][330][331][332]. These techniques are also being used in order to design materials with prescribed bulk, shear or Young moduli, Poisson ratios and other properties [211,294,[329][330][331][332] and may be employed to develop prototypes of a wide series of mechanical devices as bone implants [333]. ...

... Since NLC has been found in systems having a fixed topology, there has been a large amount of work aimed to design NLC materials using topology optimization techniques [211,294,[329][330][331][332]. These techniques are also being used in order to design materials with prescribed bulk, shear or Young moduli, Poisson ratios and other properties [211,294,[329][330][331][332] and may be employed to develop prototypes of a wide series of mechanical devices as bone implants [333]. ...

The structural and mechanical properties of the deltic, squaric and croconic cyclic oxocarbon acids were obtained using theoretical solid-state methods based in Density Functional Theory employing very demanding calculation parameters in order to yield realistic theoretical descriptions of these materials. The computed lattice parameters, bond distances, angles, and X-ray powder diffraction patterns of these materials were in excellent agreement with their experimental counterparts. The crystal structures of these materials were found to be mechanically stable since the calculated stiffness tensors satisfy the Born mechanical stability conditions. Furthermore, the values of the bulk modulus and their pressure derivatives, shear and Young moduli, Poisson ratio, ductility and hardness indices, as well as mechanical anisotropy measures of these materials were reported. A complete review of the literature concerning the negative Poisson ratio and negative linear compressibility phenomena is given together with the theoretical study of the mechanical behavior of cyclic oxocarbon acid materials. The deltic, squaric, and croconic acids in the solid state are highly anisotropic materials characterized by low hardness and relatively low bulk moduli. The three materials display small negative Poisson ratios. The croconic acid displays the phenomenon of negative linear compressibility for applied pressures larger than ~0.4 GPa directed along the direction of minimum Poisson ratio and undergoes a pressure induced phase transition at applied pressures larger than ~1.0 GPa.

... Following basically the same design procedure of [109], Yang et al. [173] extended the BESO method for designing the stiffest orthotropic material with prescribed ratios for Young's moduli, where the ratios between the Young's moduli in three directions indicate the level of orthotropy and the bulk modulus indicates the overall stiffness. Fig. 39 gives several orthotropic material design results with 50% volume constraint for various modulus ratios a 13 = a 23 from 1.0 to 0.5, where a 13 is the ratio of the moduli along direction 1 and 3 and similarly for a 23 . ...

... Orthotropic material design results for various modulus ratios a 13 = a 23[173] ...

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.

... Analyses on natural bone show that the spatial distribution of Young's modulus is smooth without sharp increase or decrease in certain directions. However, most existing periodic bone implants focused on simply reducing the stiffness (Gurtner and Durand, 2014;Sallica-Leva et al., 2013;Murr et al., 2010;Ahmadi et al., 2014;Yang et al., 2013;Parthasarathy et al., 2010;Chantarapanich et al., 2012;Challis et al., 2010;Wieding et al., 2014;Heinl et al., 2008) with very few seeking for structures with isotropic or controlled anisotropic elasticity (Gurtner and Durand, 2014;Ahmadi et al., 2014;Yang et al., 2013;Challis et al., 2010). Besides, most designs of bone implants were not truss-like lattice and had complex internal structures since they were obtained through topological optimization techniques Challis et al., 2010;Xie et al., 2014). ...

... Analyses on natural bone show that the spatial distribution of Young's modulus is smooth without sharp increase or decrease in certain directions. However, most existing periodic bone implants focused on simply reducing the stiffness (Gurtner and Durand, 2014;Sallica-Leva et al., 2013;Murr et al., 2010;Ahmadi et al., 2014;Yang et al., 2013;Parthasarathy et al., 2010;Chantarapanich et al., 2012;Challis et al., 2010;Wieding et al., 2014;Heinl et al., 2008) with very few seeking for structures with isotropic or controlled anisotropic elasticity (Gurtner and Durand, 2014;Ahmadi et al., 2014;Yang et al., 2013;Challis et al., 2010). Besides, most designs of bone implants were not truss-like lattice and had complex internal structures since they were obtained through topological optimization techniques Challis et al., 2010;Xie et al., 2014). ...

Recent advances in additive manufacturing make it possible to fabricate periodic lattice structures with complex configurations. However, a proper design strategy to achieve lattice structures with controlled anisotropy is still unavailable. There is an urgent need to fill this knowledge gap in order to develop mechanical metamaterials with prescribed properties. Here we propose two different methodologies to design lattice structures with controlled anisotropy. As examples, we created two new families of lattice structures with isotropic elasticity and cubic symmetric geometry. The findings of this work provide simple and effective strategies for exploring lightweight metamaterials with desired mechanical properties.

... Such methods include Genetic Algorithms (GAs) Balamurugan et al. (2008Balamurugan et al. ( , 2011, Jain and Saxena (2010), Madeira et al. (2010), Wang and Tai (2005), Zhou (2010), Wang et al. (2006), Guest and Genut (2010), Bureerat and Limtragool (2006), Manan et al. (2010), Artificial Immune Algorithms (Luh and Chueh 2004), Ant Colonies (Kaveh et al. 2008;Luh and Lin 2009), Particle Swarms (Luh et al. 2011), Simulated Annealing (Shim and Manoochehri 1997), Harmony Search (Lee and Geem 2004), Differential Evolution Schemes (Wu and Tseng 2010), Bacterial Foraging (Georgiou et al. 2014) and many others. For further non gradient based algorithms the reader is referred to the book written by Yang (2010). The aforementioned methods have been directly applied to topology optimization, their binary nature intuitively lends itself to the determination of solid/void material, however they have not experienced significant acceptance. ...

... performed a similar analysis, however with the objective of maximizing the bulk or shear modulus of the macrostructure with a volume constraint. Yang et al. (2013) applied the BESO method to the design of threedimensional orthotropic materials with predefined ratios for effective Young's moduli. The recent developments of the ESO/BESO algorithm are to apply such methods to microstructural optimization, which have typically been the focus of gradient based methods such as: homogenization and SIMP. ...

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

... It is a powerful method for the design of complex structures with multi-scale features (Chen and Huang, 2019). Yang et al. (2013) proposed a topology optimization method of periodic hole unit structure and designed a porous scaffold with a required Young's modulus, as shown in Figure 4. Radman et al. (2012) specified the volume or shear modulus of units, and optimized the primary unit through the antihomogeneous two-way advanced optimization technology, and established functionally gradient porous structure by the proper connection between adjacent basic units. Xiao et al. (2012) rearranged the structure of the model under the constraint of volume fraction to achieve the ideal stiffness through the topology optimization method and obtain optimal three-dimensional structure of porous scaffolds (Xiao et al., 2013). ...

Design an implant similar to the human bone is one of the critical problems in bone tissue engineering. Metal porous scaffolds have good prospects in bone tissue replacement due to their matching elastic modulus, better strength, and biocompatibility. However, traditional processing methods are challenging to fabricate scaffolds with a porous structure, limiting the development of porous scaffolds. With the advancement of additive manufacturing (AM) and computer-aided technologies, the development of porous metal scaffolds also ushers in unprecedented opportunities. In recent years, many new metal materials and innovative design methods are used to fabricate porous scaffolds with excellent mechanical properties and biocompatibility. This article reviews the research progress of porous metal scaffolds, and introduces the AM technologies used in porous metal scaffolds. Then the applications of different metal materials in bone scaffolds are summarized, and the advantages and limitations of various scaffold design methods are discussed. Finally, we look forward to the development prospects of AM in porous metal scaffolds.

... These can be based on Platonic and Archimedean solids, or strut-based structures based on crystal packing, such as body-centred-cubic (BCC) or facecentred cubic (FCC) structures [9]. Other unit cells are based on triply-periodic minimal surfaces (TPMS) [10], or use topology optimisation to create unit cells with specific mechanical properties [11][12][13]. Uniaxial compression testing is used to evaluate the mechanical performance, often plotting the change in stiffness as a function of the relative J o u r n a l P r e -p r o o f density of the lattice. However, this testing is usually only conducted in one direction. ...

Additive manufacturing (AM) enables fine control over the architecture and mechanical properties of porous lattice structures. Typical periodic unit cells used in porous structures are inherently anisotropic, may not be suitable for multi-axial applications and cannot reliably create features or struts with low build angle. This study used laser powder bed fusion (PBF) to create isotropic stochastic porous structures in stainless steel (SS316L) and titanium alloy (Ti6Al4V), with modifications that aimed to overcome PBF manufacturing limitations of build angles. The structures were tested in uniaxial compression (n = 5) in 10 load orientations relative to the structure, including the three orthogonal axes. The testing verified that no hidden peaks in elastic modulus existed in the stochastic structure. Modification to the structure reduced the standard deviation of the 10 elastic modulus values from 249 MPa to 101 MPa when made in SS316L and from 95.9 MPa to 52.5 MPa for Ti6Al4V, indicating the structures were more isotropic. These modified stochastic structures have improved stiffness isotropy and could be used for lightweighting and biomaterial applications, reducing the dependence of performance on build orientation, and allowing more flexibility for component orientation on the build platform.
Data availability
The raw and processed data required to reproduce these findings are available to download from Mendeley Data.

... In such a space, the material's normal and shear resistance can be tuned independently from other primal material parameters, such as from the material's effective Poisson ratio value [34]. Up to now, different anisotropic unit-cell designs with ultra-soft or ultra-stiff effective normal material resistance have been contrived [35][36][37]. Their normal modulus can be controlled to substantially vary among the different material loading directions, exceeding the isotropic limits by orders of magnitude [38,39]. ...

In the current work, we demonstrate the potential of structures made of chiral artificial materials to balance bending loads through tensile loads, exploiting their inner normal to shear strain coupling. To that scope, we employ beam structures which we architecture with tetrachiral unit-cells. For the latter, we quantify their inherently coupled normal to shear strain behavior, making use of homogenization analysis techniques. We subsequently derive the equations that characterize the bending mechanics of beams with an inner bending to normal loading coupling, starting from first principles. Thereupon, we compute the normal forces required to equilibrate the effect of bending loads on beam structures, providing relevant closed-form parametric expressions. Using the derived analytical formulas, we carry out both numerical simulations and experiments for the case of cantilever beams. Results suggest that the coupling of normal and shear deformations can be used as a primal load-balancing mechanism, providing new possibilities in the control of the artificial structure's kinematics and overall mechanics.

... The design of lattice structures by topology optimization is beneficial in order to obtain optimized light-weight structures with the desired design, structure and structural properties for a specific application. Within biomedical field, the optimization procedure for designing lattice structures with desired pore size, porosity/density percentage [17], as well as a stiffness close to the host bone [18,19] is required for implant applications. ...

Topology optimization approach was used for the design of Ti6Al4V ELI lattice structures with stiffness and density close to the human bone for implant applications. Three lattice designs with volume densities of 35 %, 40 % and 45 % and corresponding elastic modulus of 18.6 GPa, 23.1 GPa 27.4 GPa close to the human bone were generated. Laser powder bed fusion (LPBF) technique was used for the manufacturing of the specimens. Physical measurements and mechanical characterization of specimens were assessed by microCT analyses and compression test, perpendicular and parallel to the building direction of the specimens.
LPBF Ti6Al4V ELI manufactured lattice structures showed deviations in wall thickness in comparison with the generated designs, leading to an increase in relative porosity but also a decrease in elastic modulus in comparison with the original designs. Horizontal walls of the lattice structures showed higher wall thickness in comparison with the vertical walls, leading to anisotropic behaviour of the lattice structures. Higher elastic modulus and compression strength were obtained when thicker walls were oriented along the loading direction of the compression test, showing a complete failure by dividing the specimens into two neighbouring halves. All specimens showed 45° diagonal shear fracture along the structure. On the other hand, higher energy absorption at first maximum compression strength peak was observed when samples were tested parallel to the building direction (when thinner walls were oriented along the loading compression direction). Results showed that designed lattice structures can possess the levels of human bones’ stiffness and therefore can reduce/avoid stress shielding on implant applications.

... Topology optimization has received designers' extensive attention too. It has been widely used for designing structures and materials for desirable mechanical performance and physical properties (Sturm et al., 2010;Guest and Prévost, 2007;Yang et al., 2013). A ground structure-based topology optimization procedure is used by Namasivayam and Seepersad (2011) to decrease the deviation from intending surface profile of the unmanned aerial vehicle wing based on the ground structure method. ...

... Comprehensive reviews on this topic can be found in, e.g., Schaedler and Carter (2016) and Osanov and Guest (2016). Material microstructures have been designed to achieve (a) extremal properties, e.g., maximal bulk or shear moduli (Huang et al. 2011;Neves et al. 2000), negative Poisson's ratio (Sigmund 2000), extremal thermal expansion (Sigmund and Torquato 1997); (b) targeted properties, e.g., effective Young's moduli (Yang et al. 2013), Poisson's ratio (Sigmund 1995); (c) multifunctionality, e.g., simultaneous transportation of heat and electricity (Torquato et al. 2002), and stiffness and conductivity (Challis et al. 2008). ...

Functional performance benefits brought by graded materials or a variation of microstructures have been established. However, the challenge of connecting the adjacent microstructures remains open. Without connected microstructures, the design is only theoretical and cannot be realized in practice. This paper presents a shape metamorphosis approach to address this challenge. The approach formulates an optimization problem to minimize the difference between two shapes, where the shapes generated during the optimization iterations form the graded microstructures. A stiffness-based constraint is proposed to eliminate potential breakage or shrinkages in generating the intermediate shapes, which are used to build a transition zone for the two shapes to be connected. This also ensures that the optimal functionality is not compromised significantly. A series of numerical studies demonstrate that the proposed approach is capable of producing smoothly connected microstructures for multiscale optimization.

... The additional design degrees of freedom provide a mechanism to adjust mechanical attributes over a four-parameter, wider design space. During the last decades, a wide range of anisotropic inner material architectures has been invented that yield material designs with ultra-soft or ultra-stiff effective normal material behaviors [26][27][28]. More specifically, material architectures with normal elastic stiffness ratios that differ up to three orders of magnitude between their different loading directions have been reported (matrix composites, graphene spongy-shaped structures) [29,30]. ...

In the current work we analyze the bulk and normal to shear properties of a wide range of two-dimensional
artificial materials under small deformations. To that scope, we employ reentrant, chiral, triangular and rectangular-shaped
based unit-cell arrangements. We characterize their mechanical response, using a homogenization
analysis method. We identify inner material architectures of high and low relative resistance to
pressure and to shear loads. We classify the metastructures’ mechanical behavior with respect to the one expected
for common isotropic engineering materials. We observe considerable differences for most of the analyzed
structures for which we provide insights making use of symmetry analysis notions. What is more, we
identify lightweight inner material designs for certain bulk and normal to shear behaviors to be obtained, while
we unravel the role of the Poisson’s ratio value on the retrieved effective attributes. We quantify the impact of
alterations in the metamaterials’ inner design parameters, such as in the slenderness ratio of its inner elements,
reporting unit-cell designs that yield macroscopic relative mechanical properties (bulk and normal to shear
attributes) that are insensitive to inner element slenderness changes.

... Radman et al. [2013] utilized the bidirectional evolutionary structural optimization (BESO) technique [Querin et al., 1998;Yang et al., 1999] to design 2D and 3D unit cells of cellular isotropic materials with maximum bulk and shear moduli. This technique was also used by Yang et al. [2013] for designing 3D orthotropic materials with prescribed Young's moduli ratios. For the design of 3D elastic porous microstructures,Özdemir [2014] made use of the topological derivatives based on the associated mathematical framework. ...

Optimal design of porous and periodic microstructures through topology identification of the associated periodic unit cell (PUC) constitutes the topic of this work. Here, the attention is confined to two-phase heterogeneous materials in which the topology identification of manufacturable 3D-PUC is conducted by means of a topology optimization technique. The associated objective function is coupled with 3D numerical homogenization approach that connects the elastic properties of the 3D-PUC to the target product. The topology optimization methodology that is adopted in this study is the combination of solid isotropic material with penalization (SIMP) method and optimality criteria algorithm (OCA), referred to as SIMP-OCA methodology. The fairly simple SIMP-OCA is then generalized to handle the topology design of 3D manufacturable microstructures of cubic and orthotropic symmetry. The performance of the presented methodology is experimentally validated by fabricating real prototypes of extremal elastic constants using additive manufacturing. Experimental evaluation is performed on two designed microstructures: an orthotropic sample with Young’s moduli ratios E2/E1=2.5, E3/E1=2 and a cubic sample with negative Poisson’s ratio of −0.19. In all practical examples studied, laboratory measurements are in reasonable agreement with the prescribed values; thus, corroborating the applicability of the proposed methodology.

... Topology optimization has received designers' extensive attention too. It has been widely used for designing structures and materials for desirable mechanical performance and physical properties (Sturm et al., 2010;Guest and Prévost, 2007;Yang et al., 2013). A ground structure-based topology optimization procedure is used by Namasivayam and Seepersad (2011) to decrease the deviation from intending surface profile of the unmanned aerial vehicle wing based on the ground structure method. ...

Purpose
This paper aims to present a new topology method in designing the lightweight and complex structures for 3D printing.
Design/methodology/approach
Computer-aided design (CAD) and topology design are the two main approaches for 3D truss lattices designing in 3D printing. Though these two ways have their own advantages and have been used by the researchers in different engineering situations, these two methods seem to be incompatible. A novel topology method is presented in this paper which can combine the merits of both CAD and topology design. It is generally based on adding materials to insufficient parts in a given structure so the resulting topology evolves toward an optimum.
Findings
By using the topology method, an optimized-Kagome structure is designed and both 3D original-Kagome structure and 3D optimized-Kagome structure are manufactured by fused deposition modeling (FDM) 3D printer with ABS and the compression tests results show that the 3D optimized-Kagome has a higher specific stiffness and strength than the original one.
Originality/value
The presented topology method is the first work that using the original structure-based topology algorithm other than a boundary condition-based topology algorithm for 3D printing lattice and it can be considered as general way to optimize a commonly used light-weight lattice structure in strength and stiffness.

... Certain honeycomb and hybrid accordion cellular solids have been shown to satisfy the previously described stiffness characteristics [13,14]. Creating metamaterials of controlled anisotropy tuned to be ultra-soft or ultra-stiff and lightweight has become increasingly important not only in morphing, but also in biomechanical, civil and mechanical engineering applications [15][16][17][18][19]. Three dimensional unit cells have been designed so as to yield a priori specified material stiffness ratios, using evolutionary structural optimization methods [20]. Moreover, different cubic-shaped lattices and origami lattices have been employed to achieve optimal bulk and shear moduli and controlled Young's modulus values [21,22]. ...

In the current work, we elaborate two-dimensional metamaterials of controlled anisotropy. To that scope, we employ diamond and octagon-shaped planar lattices with and without inner links. Using a dedicated homogenization technique, we derive closed-form expressions for the lattice's effective mechanical properties. We analyse the effect of the lattice's configuration on the metamaterial's effective static properties, identifying configurations with mechanical attributes desirable for morphing, biomedical and mechanical engineering applications. We thereafter compute the lattice's wave propagation characteristics, deriving a link between the metamaterials' static and dynamic properties. In particular, we analyse the longitudinal and shear wave phase velocity dependence on the lattice's geometric configuration. Thereupon, we identify architectural arrangements for which the phase velocity vanishes in certain propagation directions , exhibiting wave propagation isolation characteristics. We demonstrate that the detected isolation features can systematically arise for lattice architectural designs that yield highly anisotropic static properties (thus high material moduli ratios) and anti-auxetic material behaviours (thus non-negative Poisson's ratio values).

... The BESO method has many advantages over ESO in terms of computational efficiency, robustness of the method and manufacturability of the final topology (Shobeiri 2016;Huang, Xie, and Burry 2007). This kind of optimization has been successfully applied to a large variety of optimization problems, including compliance minimization (Shobeiri 2015(Shobeiri , 2016, frequency maximization (Huang, Zuo, and Xie 2010), displacement constraint Zuo, Xie, and Huang 2012) and material moduli maximization (Yang et al. 2013). ...

... The BESO method has many advantages over the ESO method in terms of computational efficiency, robustness of the method and manufacturability of the final topology [26,28]. This kind of optimization has been successfully applied to a large variety of optimization problems including compliance minimization [26][27], frequency maximization [31], displacement constraint [32,33] and material moduli maximization [34]. This paper extends the BESO method for topology optimization of continuum structures to the strut-and-tie modelling of reinforced concrete structures. ...

This article presents a method for the automatic generation of optimal
strut-and-tie models in reinforced concrete structures using a bi-directional evolutionary structural optimization method. The methodology presented is developed for compliance minimization relying on the Abaqus finite element software package. The proposed approach deals with the generation of truss-like designs in a three-dimensional environment, addressing the design of corbels and joints as well as bridge piers and pile caps. Several three-dimensional examples are provided to show the capabilities of the proposed framework in finding optimal strut-and-tie models in reinforced
concrete structures and verifying its efficiency to cope with torsional
actions. Several issues relating to the use of the topology optimization for strut-and-tie modelling of structural concrete, such as chequerboard patterns, mesh-dependency and multiple load cases, are studied. In the last example, a design procedure for detailing and dimensioning of the strut-and-tie models is given according to the American Concrete Institute (ACI) 318-08 provisions.

... Since then, the material design concept based on topology optimization has been developed in various material systems including elastic [16][17], thermoelastic [18][19], viscoelastic [20][21] and piezoelectric [22][23] materials, also including materials with fluid permeability [24] and negative Poisson's ratio (NPR) [25][26]. Other topology optimization methods were also developed in material design, such as level set method [27][28][29] and ESO/BESO methods [30][31][32][33][34][35][36]. A systematic review on microstructural design including nonlinearities through topology optimization was given in the literature [37][38]. ...

Recent studies have shown that the stiffness of composites in one or more directions could increase dramatically when the Poisson’s ratios of constituent phases approach the thermodynamic limits. In this paper, we establish a computational framework for the topology design of the microstructure of a composite material whose constituent phases have distinct Poisson’s ratios. In this framework, the composite is assumed to be composed of periodic microstructures and the effective mechanical properties are determined through the numerical homogenization method. Topology optimization for maximizing the effective Young’s modulus is performed to find the optimal distribution of material phases, subject to constraints on the volume fractions of the constituent phases. Four 3D numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. Various microstructures of optimized composites have been obtained for different objective functions and for different parameters.

... Shen et al. (2012) aplicaram otimização topológica em materiais esponjosos tendo como critério de otimização a energia de absorção dos sistemas multi-escalas formados por seções de esponjas vegetais. Um estudo sobre a otimização de micro-estruturas 3D compostas de materiais ortotrópicos é apresentado por Yang et al. (2013). O método evolucionário pode ser encontrado ainda no trabalho de Huang et al. (2013) para otimização sistemas compostos por micro-estruturas, materiais celular, e macro-estruturas, materiais compósitos, sob o critério de rigidez. ...

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.

... In a recent BESO study [13], a general orthotropic design problem was considered where there was no information on the absolute target value for each modulus. Instead, the ratio between the moduli was specified as a constraint, for example, a constant ratio between the effective Young's moduli in the three principal directions. ...

Novel and efficient structural and material designs can be realized by topology optimization that is capable of maximizing the performance of structural systems under given constraints. The bi-directional evolutionary structural optimization (BESO) method has been developed into an effective tool for topology optimization of load-bearing structures and materials. The latest advances of BESO are aimed at expanding its practical applications to a wider range of structural systems on both macro and micro scales. This paper presents recent developments of BESO for optimal design problems of a variety of structural systems ranging from buildings of large scales to materials of micro scales. Selected applications are introduced to demonstrate the capability of BESO. Examples presented in this paper are based on research and industrial projects of the Centre for Innovative Structures and Materials (http://www.rmit.edu.au/research/cism) at RMIT University.

... Through years of development, the current form of BESO has become a gradient-based mathematical optimization method with convergent and mesh-independent algorithms [7]. Recent applications of BESO have addressed a range of macro structural optimization problems including compliance minimization [9], frequency maximization [14], displacement constraints [12,27], as well as inverse homogenization [18] problems such as material moduli maximization [24,17] and concurrent designs [13,26]. ...

This paper presents a 100-line Python code for general 3D topology optimization. The code adopts the Abaqus Scripting Interface that provides convenient access to advanced finite element analysis (FEA). It is developed for the compliance minimization with a volume constraint using the Bi-directional Evolutionary Structural Optimization (BESO) method. The source code is composed of a main program controlling the iterative procedure and five independent functions realising input model preparation, FEA, mesh-independent filter and BESO algorithm. The code reads the initial design from a model database (.cae file) that can be of arbitrary 3D geometries generated in Abaqus/CAE or converted from various widely used CAD modelling packages. This well-structured code can be conveniently extended to various other topology optimization problems. As examples of easy modifications to the code, extensions to multiple load cases and nonlinearities are presented. This code is intended for educational purposes and would be useful for researchers and students in the topology optimization field. With further extensions, the code could solve sophisticated 3D conceptual design problems in structural engineering, mechanical engineering and architecture practice. The complete code is given in the appendix section and can also be downloaded from the website: www.rmit.edu.au/research/cism/.

... BESO has also been successfully applied to a wide range of material design problems, e.g. maximum bulk or shear modulus (Huang et al., 2011), tailored stiffness orthotropy (Yang et al., 2013), functionally graded materials (Radman et al., 2013) and multi-scale design of composite materials and structures (Zuo et al., 2013). These materials can be constructed as arrays of microstructures and then fabricated using advanced manufacturing technologies such as additive manufacturing (Challis et al., 2010). ...

There has been considerable interest in materials exhibiting negative or zero compressibility. Such
materials are desirable for various applications. A number of models or mechanisms have been proposed
to characterize the unusual phenomena of negative linear compressibility (NLC) and negative area
compressibility (NAC) in natural or synthetic systems. In this paper we propose a general design technique
for finding metamaterials with negative or zero compressibility by using a topology optimization
approach. Based on the bi-directional evolutionary structural optimization (BESO) method, we establish a
systematic computational procedure and present a series of designs of orthotropic materials with various
magnitudes of negative compressibility, or with zero compressibility, in one or two directions. A physical
prototype of one of such metamaterials is fabricated using a 3D printer and tested in the laboratory under
either unidirectional loading or triaxial compression. The experimental results compare well with the
numerical predictions. This research has demonstrated the feasibility of designing and fabricating
metamaterials with negative or zero compressibility and paved the way towards their practical
applications.

The engineering of tendon and ligament tissue biocompatible restoration materials constitutes a long-standing engineering challenge, from the chemical, biological and mechanical compatibility analysis and design perspective. Their mechanics are inherently anisotropic, exceeding the potential limits of common, non-architected engineering materials. In the current contribution, the design of advanced material or “metamaterial” architectures that can emulate the mechanical properties observed in native tendon and ligament tissues is analytically, experimentally, and numerically investigated. To that scope, anisotropic metamaterial designs that are based on rectangular cuboid architectures with and without inner body-centered strengthening cores are considered. Thereupon, the metamaterial design specifications required for the approximation of the highly anisotropic tissue performance, namely of the characteristic elastic, shear, and Poisson's ratio attributes are studied. It is shown that certain strengthened, anisotropic body-centered cuboid lattice architectures allow for substantial effective metamaterial stiffness along the primal tissue loading direction, upon a rather low shear loading resistance. The previous mechanical attributes come along with Poisson's ratio values well above unity and moderate relative density values, furnishing a combination of material characteristics that is highly desirable in restoration praxis. The analytically and numerically guided anisotropic metamaterial performance is experimentally reproduced both for the case of uniaxial and shear loads, using a microfabrication stereolithography additive manufacturing technique. The obtained scanning electron microscopy images highlight the fabrication feasibility of the identified metamaterial architectures, in scales that are directly comparable with the ones reported for the natural tissues, having feature sizes in the range of some 10ths of micrometers and elastic attributes within the range of clinical observation.

Background
Dense titanium (Ti) fusion cages have been commonly used in transforaminal lumbar interbody fusion. However, the stiffness mismatch between cages and adjacent bone endplates increases the risk of stress shielding and cage subsidence.Methods
The current study presents a multiscale optimization approach for porous Ti fusion cage development, including microscale topology optimization based on homogenization theory that obtains a unit cell with prescribed mechanical properties, and macroscale topology optimization that determines the layout of framework structure over the porous cage while maintaining the desired stiffness. The biomechanical performance of the designed porous cage is assessed using numerical simulations of fusion surgery. Selective laser melting is employed to assists with fabricating the designed porous structure and porous cage.ResultsThe simulations demonstrate that the designed porous cage increases the strain energy density of bone grafts and decreases the peak stress on bone endplates. The mechanical and morphological discrepancies between the as-designed and fabricated porous structures are also described.Conclusion
From the perspective of biomechanics, it is demonstrated that the designed porous cage contributes to reducing the risk of stress shielding and cage subsidence. The optimization of processing parameters and post-treatments are required to fabricate the designed porous cage. The present multiscale optimization approach can be extended to the development of cages with other shapes or materials and further types of orthopedic implants.Graphic abstract

In this study, a level set topology optimization method is introduced and used to design optimized periodic architected materials for the maximum macrostructural stiffness considering thermoelasticity. The design variables are defined at the microscopic scale and updated by minimizing the total structural compliance induced by mechanical and thermal expansion loads at the macroscopic scale. The two scales are coupled by the effective elasticity tensor calculated through the homogenization theory. A decomposition method is constructed to formulate several sub-problems from the original optimization problem thereby this otherwise computationally expensive problem can be efficiently solved, especially when the number of material subdomains is large. The proposed method is demonstrated through several numerical examples. It is shown that a macrostructural geometry and boundary conditions have a significant impact on the optimized material de-signs when thermoelastic effects are considered. Porous material with a well-designed microstructure is preferred over solid material when a thermal load is non-zero. Moreover, when a larger number of material microstructures are allowed in optimization, the overall performance is improved due to the expanded design space.

In this paper, we proposed a new material design method by microstructure topology optimization. Novelty of the proposed method is to target the whole nonlinear volume-averaged effective stress-strain curve of microstructure representative volume element (RVE) rather than aiming specific values such as strength, stiffness or Poisson ratio. J2 plasticity model with a linear isotropic hardening model was chosen for local residuals. Global residuals are computed within nonlinear finite element framework for the topology optimization. Sensitivities of the objective function augmented with the residuals and adjoint response vectors with respect to design variables are derived with details and their numerical computational procedures were also presented. Microstructure topologies showing two different targeted stress-strain curves under uniaxial and biaxial loadings were obtained by using the method of moving asymptotes (MMA) optimization algorithm. Accuracy of the sensitivity computations was verified and numerical examples demonstrated a potential of the proposed method in applications to multiscale topology optimization.

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

The utilization of “anisotropy” is becoming an important direction for realizing highly-functionalized products overcoming the current functional limits for materials with isotropic properties. In particular for the use in aerospace and biomedical industries, some products are exposed to anisotropic stress fields. Therefore, the products should be anisotropic along the functional axis. Additive manufacturing (AM) technology was originally specialized to fabricate the complicated structures in an arbitrary way. However, when it comes to metal AM, the control of material anisotropy has received an increasing attention in recent years. In this review article, the latest findings regarding the control of mechanical anisotropy through the modifications of macroscopic porous structure, microstructure, and crystallographic orientation by powder-based metal AM technologies are introduced. The powder-based metal AM is able to control a wide range of anisotropy from crystallographic texture to pore structure, however, their simultaneous control is yet challenging. Attainment of this heightens the value of metal AM technology and enhances the AM products’ functions.

One of the critical issues in orthopaedic regenerative medicine is the design of bone scaffolds and implants that replicate the biomechanical properties of the host bones. Porous metals have found themselves to be suitable candidates for repairing or replacing the damaged bones since their stiffness and porosity can be adjusted on demands. Another advantage of porous metals lies in their open space for the in-growth of bone tissue, hence accelerating the osseointegration process. The fabrication of porous metals has been extensively explored over decades, however only limited controls over the internal architecture can be achieved by the conventional processes. Recent advances in additive manufacturing have provided unprecedented opportunities for producing complex structures to meet the increasing demands for implants with customized mechanical performance. At the same time, topology optimization techniques have been developed to enable the internal architecture of porous metals to be designed to achieve specified mechanical properties at will. Thus implants designed via the topology optimization approach and produced by additive manufacturing are of great interest. This paper reviews the state-of-the-art of topological design and manufacturing processes of various types of porous metals, in particular for titanium alloys, biodegradable metals and shape memory alloys. This review also identifies the limitations of current techniques and addresses the directions for future investigations.

One of the critical issues in orthopaedic regenerative medicine is the design of bone scaffolds and implants that replicate the biomechanical properties of the host bones. Porous metals have found themselves to be suitable candidates for repairing or replacing the damaged bones since their stiffness and porosity can be adjusted on demands. Another advantage of porous metals lies in their open space for the in-growth of bone tissue, hence accelerating the osseointegration process. The fabrication of porous metals has been extensively explored over decades, however only limited controls over the internal architecture can be achieved by the conventional processes. Recent advances in additive manufacturing have provided unprecedented opportunities for producing complex structures to meet the increasing demands for implants with customized mechanical performance. At the same time, topology optimization techniques have been developed to enable the internal architecture of porous metals to be designed to achieve specified mechanical properties at will. Thus implants designed via the topology optimization approach and produced by additive manufacturing are of great interest. This paper reviews the state-of-the-art of topological design and manufacturing processes of various types of porous metals, in particular for titanium alloys, biodegradable metals and shape memory alloys. This review also identifies the limitations of current techniques and addresses the directions for future investigations.

Chronic pain is a major health issue and most patients suffer from spontaneous pain. Previous studies suggest that Huperzine A (Hup A), an alkaloid isolated from the Chinese herb
Huperzia serrata
, is a potent analgesic with few side effects. However, whether it alleviates spontaneous pain is unclear. We evaluated the effects of Hup A on spontaneous pain in mice using the conditioned place preference (CPP) behavioral assay and found that application of Hup A attenuated the mechanical allodynia induced by peripheral nerve injury or inflammation. This effect was blocked by atropine. However, clonidine but not Hup A induced preference for the drug-paired chamber in CPP. The same effects occurred when Hup A was infused into the anterior cingulate cortex. Furthermore, ambenonium chloride, a competitive inhibitor of acetylcholinesterase, also increased the paw-withdrawal threshold but failed to induce place preference in CPP. Therefore, our data suggest that acetylcholinesterase in both the peripheral and central nervous systems is involved in the regulation of mechanical allodynia but not the spontaneous pain.

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

This paper proposes a fully coupled multi-scale finite element model for the mechanical description of a new composite material inspired in wood cell-walls. The constitutive response of the composite is described by means of a representative volume element (RVE) in which the fibre is represented as a periodic alternation of rigid and soft portions of material. Furthermore, at a lower scale the overall constitutive behavior of the fibre is modelled as a single material defined by a second RVE. Numerical tests demonstrate substantial gains in terms of resistance to failure, toughness and in the control of the overall flexibility/stiffness balance in the material. © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11

We consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion. The elastic properties of orthotropic composites are characterized by nine independent moduli. We provide bounds for six of these, namely the three Youngs moduli and three in-plane shear moduli. The bounds are optimal and correlate the six moduli.

Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.

The behaviour of a periodic composite material depends not only on the properties of the constituent phases but also strongly on the microstructural texture of those phases such as spheres, lamellae and needles. This paper shows how to design the microstructure for a specific extremal bulk (effective) thermal conductivity in a three-phase composite medium. An inverse homogenization technique that is driven by the computational topology optimization algorithm is presented. Apart from benchmarking examples such as the Vigdergauz-type and sandwich-like architectures, a series of new single length-scale designs of microstructures are generated from this procedure. The topological design results are validated by comparing their conductivities against the empirical formulae in the two-phase composites. This study interestingly finds that the phase interfaces yielded from the topology optimization highly resemble the constant mean curvature surfaces. A comparison of their respective attainability with the Milton–Kohn physical bounds is made and the equivalence of these two sets of topologies is consequently justified.

Evolutionary Structural Optimization (ESO), is a numerical method of structural optimization that is integrated with finite element analysis (FEA). Bi-directional ESO (BESO) is an extension to this method and can begin with minimal amount of material (only that necessary to support the load and support cases) in contrast to ESO which uses an initially oversized structure. Using BESO the structure is then allowed to grow into the optimum design or shape by both adding elements where the stresses are the highest and taking elements away where stresses are the lowest. In conducting this research, a methodology was developed (and integrated into the ESO program EVOLVE) which produced the optimal 3D finite element models of a structure in a more reliable way than the traditional ESO method. Additionally, the BESO method was successfully extended to multiple load cases for both 2D and 3D. Two different algorithms were used to find the best structure experiencing more than one load case and the results of each are included.

Evolutionary structural optimization (ESO) method was originally developed based on the idea that by system-atically removing the inefficient material, the residual shape of the structure evolves toward an optimum. This paper presents an extension of the method called bidirectional ESO (BESO) for topology optimization subject to stiffness and displacement constraints. BESO allows for the material to be added as well as to be removed to modify the structural topology. Basic concepts of BESO including the sensitivity number and displacement extrapolation are proposed and optimization procedures are presented. Integrated with the finite element analysis technique, BESO is applied to several two-dimensional plane stress problems. Its effectiveness and efficiency are examined in comparison with the results obtained by ESO. It is found that BESO is more reliable and computationally more efficient than ESO in most cases. Its capability and limitation are discussed. Nomenclature C = mean compliance E = Young's modulus K* = element stiffness matrix / = thickness of plate Uj = displacement at the constrained location M* = limit of the displacement constraint ii' = element displacement vector due to real load u ij -element displacement vector due to unit virtual load acting at the location of the displacement constraint W = weight of current structure W^ = weight of maximum structure V^ b j = objective weight WSbT (1) = first local minimum of the objective weight \y|) T b 1 j l(2) = second local minimum of the objective weight W 0?t -weight of optimal topology WQ = weight of structure of the full design area W* = target weight a = sensitivity number A = increment v = Poisson's ratio

This paper presents the bidirectional evolutionary structural optimization (BESO) method for the design of two-phase composite materials with optimal properties of stiffness and thermal conductivities. The composite material is modeled by microstructures in a periodical base cell (PBC). The inverse homogenization method is used to derive the properties of effective bulk modules and thermal conductivities. BESO procedures are presented to optimize the two individual properties and their various combinations. Two numerical examples are studied. The effectiveness of BESO has been demonstrated by good comparisons to the benchmark microstructures and the Hashin-Shtrikman bounds.

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

Elastic energy is the measure for the overall stiffness/flexibility of a solid. For orthotropic materials, a combined finite element/optimization procedure is described, by which the orientation field of the most stiff/flexible solid can be determined. A study of these bounds for possible materials is performed and more general insight obtained. Recent analytical results from sensitivity analysis are applied and the paper also adds to theoretical aspects, such as proving coinciding directions for principal strains and principal stresses, even with different (but optimal) principal directions of the material.

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.

The plant cell wall is treated as a two phase fibre composite material in which the fibres are dispersed, in an isotropic matrix, in the plane of the cell wall with an angular distribution f(). If f () can be represented by a gaussian it is shown that the elastic stiffness constants of the cell wall can be easily evaluated. The theory is applied to a model of the earlywood of Pinus radiata and the theoretical variation of the longitudinal Young's Modulus with mean fibrilar direction is compared with that determined experimentally.Die pflanzliche Zellwand wird im allgemeinen behandelt wie ein im wesentlichen aus zwei Phasen bestehendes Fasermaterial, bei dem die Fasern in einer isotropen Matrixsubstanz gelst und in der Zellwandebene mit einem Winkel f() verteilt sind. Sofern f() normal verteilt ist, knnen die elastischen Konstanten der Zellwand verhltnismig einfach berechnet werden. Diese Theorie wird auf das Modell von Kiefern-Frhholz (Pinus radiata) angewendet, und die theoretisch ermittelte nderung des longitudinalen Youngs Moduls mit der mittleren Faserrichtung wird mit experimentell bestimmten Werten verglichen.

The balance between stiffness and strength design is considered in the present paper. For materials with different levels of orthotropy (including isotropy), we optimize the density distribution as well as the orientational distribution for a short cantilever problem, and discuss the tendencies in design and response (energy distributions and stress directions). For a hole in a biaxial stress field, the shape design of the boundary hole is also incorporated.The resulting tapered density distributions may be difficult to manufacture, for example, in micro-mechanics production. For such problems a penalization approach to obtain black and white designs, i.e. uniform material or holes, is often applied in optimal design. A specific example is studied to show the effect of the penalization, but is restricted here to an isotropic material.When the total amount of material is not specified, a conflict between optimal design for stiffness and optimal design for strength appears. The computational results of such a case study are shown.

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.

The paper presents a new class of two-phase isotropic composites with extremal bulk modulus. The new class consists of micro geometrics for which exact solutions can be proven and their bulk moduli are shown to coincide with the Hashin–Shtrikman bounds. The results hold for two and three dimensions and for both well- and non-well-ordered isotropic constituent phases. The new class of composites constitutes an alternative to the three previously known extremal composite classes: finite rank laminates, composite sphere assemblages and Vigdergauz microstructures. An isotropic honeycomb-like hexagonal microstructure belonging to the new class of composites has maximum bulk modulus and lower shear modulus than any previously known composite.Inspiration for the new composite class comes from a numerical topology design procedure which solves the inverse homogenization problem of distributing two isotropic material phases in a periodic isotropic material structure such that the effective properties are extremized.

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

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.

Tissue scaffolds are typically designed and fabricated to match native bone properties. However, it is unclear if this would lead to the best tissue ingrowth outcome within the scaffold as neo-tissue keeps changing the stiffness of entire construct. This paper presents a numerical method to address this issue for design optimization and assessment of tissue scaffolds. The elasticity tensors of two different types of bones are weighted by different multipliers before being used as the targets in scaffold design. A cost function regarding the difference between the effective elasticity tensor, calculated by the homogenization technique, and the target tensor, is minimized by using topology optimization procedure. It is found that different stiffnesses can lead to different remodeling results. The comparison confirms that bone remodeling is at its best when the scaffold elastic tensor matches or is slightly higher than the elastic properties of the host bone.

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