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Maximizing the effective Young’s modulus of a composite material by exploiting the Poisson effect

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Abstract

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.

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... Relying on separate interpolation of Young's modulus and Poisson's ratio in topology optimization, Strek et al. (2014) succeeded to disclose the Poisson's ratio effect in sandwich-structured composites. Furthermore, this model was extended to establish a computational framework to optimize composite microstructures and concurrent structures in consideration of the hybrid of PPR and NPR materials (Long et al., 2016;Long et al., 2017a;Long et al., 2017b). With void in micro-scale, Jia et al. (Jia et al., 2018) proposed two concurrent models to discuss the Poisson's ratio effect. ...
... Until now, studies based on topology optimization (Long et al., 2016;Long et al., 2017aLong et al., , 2017bJia et al., 2018) are primarily focused on improving the stiffness of NPR materials in micro-scale through mixing with the PPR materials. Recently, the auxetic cellular structure design demonstrates the potential in large-scale structural systems (Cabras and Brun, 2016;Subramani et al., 2016;Magalhaes et al., 2016). ...
... E (2) . Elemental Young's modulus and Poisson's ratio are penalized by the SIMP interpolation (Strek et al., 2014;Long et al., 2016;Jia et al., 2018): ...
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Purpose Negative Poisson’s ratio (NPR) material has huge potential applications in various industrial fields. However, lower Young’s modulus due to the porous form limits its further applications. Based on the topology optimization technique, this paper aims to optimize the structure consisting two isotropic porous materials with positive Poisson’s ratio (PPR) and NPR and void. Design/methodology/approach Under prescribed dual-volume fraction constraints, the structural compliance is taken as the objective. Young’s modulus and Poisson’s ratio are, respectively, interpolated and expressed with Lamé’s parameters for easier programming. Accordingly, the sensitivities can be derived through the chain rule. Several two- and three-dimensional illustrative examples are presented to demonstrate the capability and effectiveness of the proposed approach. The influences of Poisson’s ratios, volume fractions and Young’s moduli on the optimized results are investigated. Findings For NPR materials having unique load responses, the resulting topologies of PPR and NPR materials have distinct material distributions in comparison of the results from two PPR materials. Furthermore, it is observed that higher structural stiffness can be achieved from the hybrid of PPR and NPR materials than that obtained from the structures made of individual constituent materials. Originality/value A topology optimization methodology is proposed to design structures composed of PPR and NPR materials.
... An in-depth investigation was conducted to nd the in uence of effective Young's moduli when Poisson effect was considered. Long et al. [44] established a topology optimization framework for maximizing the effective Young's modulus of composite containing multiple constituent phases with distinct Poisson's ratios. However, no reports can be found in literature regarding the design of composites with extremal bulk or shear modulus made of constituent phases with signi cantly different Poisson's ratios. ...
... 5 shows a parabola curve of the resulting effective Young's modulus for various Poisson's ratios of phase 2. Parts of the results exceed the Voigt estimation of 1.75. It can be observed that the enhancement effects of the effective Young's modulus are dependent on the difference between the Poisson's ratio of two constituent materials, which agrees well with the phenomena in literature [43,44]. The optimized topologies in Fig. 4 are different with those seeking for maximizing the effective Young's modulus [44]. ...
... It can be observed that the enhancement effects of the effective Young's modulus are dependent on the difference between the Poisson's ratio of two constituent materials, which agrees well with the phenomena in literature [43,44]. The optimized topologies in Fig. 4 are different with those seeking for maximizing the effective Young's modulus [44]. It can be inferred that the optimized topologies highly in uence the mechanical properties of the resulting composites. ...
Article
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A methodology for achieving the maximum bulk or shear modulus in an elastic composite composed of two isotropic phases with distinct Poisson’s ratios is proposed. A topology optimization algorithm is developed which is capable of finding microstructures with extreme properties very close to theoretical upper bounds. The effective mechanical properties of the designed composite are determined by a numerical homogenization technique. The sensitivities with respect to design variables are derived by simultaneously interpolating Young’s modulus and Poisson’s ratio using different parameters. The so-called solid isotropic material with penalization method is developed to establish the optimization formulation. Maximum bulk or shear modulus is considered as the objective function, and the volume fraction of constituent phases is taken as constraints. The method of moving asymptotes is applied to update the design variables. Several 3D numerical examples are presented to demonstrate the effectiveness of the proposed structural optimization method. The effects of key parameters such as Poisson’s ratios and volume fractions of constituent phase on the final designs are investigated. A series of novel microstructures are obtained from the proposed approach. It is found that the optimized bulk and shear moduli of all the studied composites are very close to the Hashin-Shtrikman-Walpole bounds.
... Similar increase in stiffness was found in laminate composites with alternating layers of materials with negative Poisson's ratio (NPR) and positive Poisson's ratio (PPR) [37][38][39]. Long et al. [40] proposed a topology optimization algorithm to acquire the maximum effective Young's modulus. Other similar research also proposed the methodology for designing high-stiffness composites by considering Poisson effect [41]. ...
... The above studies [36][37][38][39][40][41] focused on the design of highstiffness composite constructed by constituent phases of distinct Poisson's ratios. However, topology optimization of macrostructure composed of such composites might not bring optimal solutions when considering complex boundary conditions. ...
Article
This paper introduces a two-scale concurrent topology optimization method for maximizing the frequency of composite macrostructure that are composed of periodic composite units (PCUs) consisting of two isotropic materials with distinct Poisson’s ratios. Interpolation of Poisson’s ratios of different constituent phases is used in PCU to exploit the Poisson effect. The effective properties of the composite are computed by numerical homogenization and integrated into the frequency analysis. The sensitivities of the eigenvalue of macro- and micro-scale density are derived. The design variables on both the macro- and micro-scales are efficiently updated by the well-established optimality criteria methods. Several 2D and 3D illustrative examples are presented to demonstrate the capability and effectiveness of the proposed approach. The effect of the micro-scale volume fraction and Poisson’s ratio of the constituent phases on the optimal topology are investigated. It is observed that higher frequency can be achieved at specific range of micro-scale level volume fraction for optimal composites than that obtained from structures made of individual base materials.
... Young's modulus and Poisson's ratio are respectively interpolated for employing topology optimization technique to exploit Poisson's ratio effect in sandwich structured composites Strek et al. 2014 . Besides maximizing the effective Young's modulus Long et al. 2016, Long et al. Long, Han, and Gu 2017 extended the above model to reveal the Poisson's ratio effect in concurrent structure under volume fraction constraint and mass constraint. ...
... Superscript numbers 1 and 2 characterize the PPR and NPR base materials, respectively. The penalization factors have the values of 4 a= and 1 b = for better convergence and clearer topologies Long et al. 2016 . ...
Article
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Negative Poisson’s ratio (NPR) material attracts a lot of attentions for its unique mechanical properties. However, achieving NPR is at the expense of reducing Young’s modulus. It has been observed that the composite stiffness can be enhanced when blending positive Poisson’s ratio (PPR) material into NPR material. Based on the respective interpolation of Young’s modulus and Poisson’s ratio, two concurrent topology optimization problems with different types of constraints, called Problem A and B, are respectively discussed to explore the Poisson’s ratio effect in porous microstructure. In Problem A, the volume constraints are respectively imposed on macro and micro structures; in Problem B, besides setting an upper bound on the total available base materials, the micro thermal insulation capability is considered as well. Besides considering the influence of micro thermal insulation capability on the optimized results in Problem B, the similar and dissimilar influences of Poisson’s ratios, volume fractions in Problem A and B are also investigated through several 2D and 3D numerical examples. It is observed that the concurrent structural stiffness resulting from the mixture of PPR and NPR base materials can exceed the concurrent structural stiffness composed of any individual base material.
... Successive investigations included the study of the influence of the Poisson's ratio of constituent phases and topologies of NPR inclusions in hybrids (Strek et al. 2014;Shufrin, Pasternak, and Dyskin 2015;Zuo and Xie 2014). Long et al. (2016) applied topology optimization to maximize the effective Young's modulus with NPR and positive Poisson's ratio (PPR) material. ...
... As explained in Example 4, the system performance mainly depends on the stiffness of the composite. Similarly to the composite design (Shufrin, Pasternak, and Dyskin 2015;Long et al. 2016), the increase in the effective Young's modulus can be achieved when the value of the NPR inclusion approaches −1, with the other phase being fixed. The tendency of the mean compliance of the concurrent design fits well with the stiffness in the composite design. ...
Article
Most studies on composites assume that the constituent phases have different values of stiffness. Little attention has been paid to the effect of constituent phases having distinct Poisson’s ratios. This research focuses on a concurrent optimization method for simultaneously designing composite structures and materials with distinct Poisson’s ratios. The proposed method aims to minimize the mean compliance of the macrostructure with a given mass of base materials. In contrast to the traditional interpolation of the stiffness matrix through numerical results, an interpolation scheme of the Young’s modulus and Poisson’s ratio using different parameters is adopted. The numerical results demonstrate that the Poisson effect plays a key role in reducing the mean compliance of the final design. An important contribution of the present study is that the proposed concurrent optimization method can automatically distribute base materials with distinct Poisson’s ratios between the macrostructural and microstructural levels under a single constraint of the total mass.
... In the microstructural topology optimization design, the asymptotic homogenization [13,24], which actually plays a role of connection between the macro and micro quantiles, is utilized to predict the effective properties of the periodic microstructures. Later on, various novel materials with either extreme proprieties or prescribed properties [25][26][27][28][29][30][31] have been developed by incorporating the topology optimization methods with the numerical homogenization theory. The inverse homogenization approach explores novel material microstructures without any preliminary knowledge on their optimal topologies. ...
... And then the sensitivities of the objective functions can be easily calculated by substituting (27) into (28) or (29). In addition, the sensitivities of the volume constraints can be derived by: the material phase boundaries represented by standard four-node Gaussian integration strategy ( Fig. 2(b)) will deviate away from the actual material interfaces (Fig. 2(a)). ...
Article
This paper presents a systematic optimization design method for the multiphase auxetic metamaterials with different deformation mechanisms in both 2D and 3D scenarios. In this method, the parametric color level set (PCLS) is developed to describe different material phases in the microstructures, in which at most 2L materials phases can be precisely represented by only L description functions without any overlaps. Furthermore, clear and smooth material interfaces can be guaranteed in the design, and multiple material usage constraints can be efficiently handled by the well-established gradient-based algorithm. The shape derivative theory is introduced to analyze the design sensitivities for the optimization problem. The effective elasticity properties of the multiphase composites are evaluated by the numerical homogenization method under periodic boundary conditions. Various symmetric conditions are defined and enforced to induce the re-entrant and chiral patterns in both 2D and 3D metamaterials. Several numerical examples are provided to demonstrate the features of the proposed method in tailoring different types of multiphase auxetics. The multiphase metamaterials with two and more material phases are discussed. It is shown that the presented design method can be used to devise both the re-entrant and chiral auxetic metamaterials with excellent auxetic and stiffness properties.
... Czarnecki et al. presented models of auxetic materials resulting from optimal distribution of Young's modulus within the composite material volume [32,33]. Long et al. maximized the effective Young's modulus of a composite with auxetic inclusions [34]. Pozniak et al. [35] investigated composites with elliptic inclusions using the finite element method and showed that one could tailor a material of practically arbitrary elastic parameters. ...
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This paper presents a study of new two-dimensional composite structures with respect to their thermomechanical properties. The investigated structures are based on very well-known auxetic geometries—i.e., the anti-tetrachiral and re-entrant honeycomb—modified by additional linking elements, material which is highly sensitive to changes of temperature. The study shows that temperature can be used as a control parameter to tune the value of the effective Poisson’s ratio, which allows, in turn, changing its value from positive to negative, according to the temperature applied. The study shows that such thermoauxetic behavior applies both to composites with voids and those completely filled with material.
... Since the homogenization [Guedes and Kikuchi (1990)] has been established, it is rapidly becoming popular to combine with topology optimization to develop an inverse design procedure for seeking 4 the best topologies of PUCs with superior material effective properties [Guest and Prévost (2006); Huang et al. (2011);Long et al. (2016); Sigmund (1994); Torquato et al. (2003)]. The homogenization is utilized to evaluate material effective properties which work as the objective function, and topology optimization is applied to evolve the topology of the material cell until the expected effective property is gained. ...
Preprint
Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson's ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.
... Since the homogenization [Guedes and Kikuchi (1990)] has been established, it is rapidly becoming popular to combine with topology optimization to develop an inverse design procedure for seeking the best topologies of PUCs with superior material effective properties [Guest and Prévost (2006); Huang et al. (2011);Long et al. (2016); Sigmund (1994); Torquato et al. (2003)]. The homogenization is utilized to evaluate material effective properties which work as the objective function, and topology optimization is applied to evolve the topology of the material cell until the expected effective property is gained. ...
Article
Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson’s ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.
... Structural topology optimization, which aims to determine the best material distribution in a predefined domain, is established as a powerful intelligent means in the conceptual design stage of engineering products [1][2][3][4], especially in the field of aeronautics and astronautics [5][6][7][8]. Since the pioneer research by Bendsøe and Kikuchi [9] two decades ago, the relative works are extensively expounded in both academic and industrial fields. ...
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This study presents a novel non-probabilistic reliability-based topology optimization (NRBTO) framework to determine optimal material configurations for continuum structures under local stiffness and strength limits. Uncertainty quantification (UQ) analysis under unknown-but-bounded (UBB) inputs is conducted to determine the feasible bounds of structural responses by combining a material interpolation model with stress aggregation function and interval mathematics. For safety reasons, improved interval reliability indexes that correspond to displacement and stress constraints are applied in topological optimization issues. Meanwhile, an adjoint-vector based sensitivity analysis is further discussed from which the gradient features between reliability measures and design variables are mathematically deduced, and the computational difficulties in large-scale variable updating can be effectively overcome. Numerical examples are eventually given to demonstrate the validity of the developed NRBTO methodology.
... In this context, a nanocomposite can be defined as a combination of two or more constituent materials or phases, showing high thermal stability, high strength, chemical resistance, improved catalytic properties and so on. [18][19][20][21] Despite many available nanocomposites, the combination of perovskite based nanocomposites is scarce in the literature. La 2 CuO 4 and LaCoO 3 based-perovskite nanocomposites can be synthesized by different physical and chemical routes such as, solid state reaction, 6,13,22,23 sol-gel, 8,24 combustion, 25,26 hydrothermal 27,28 and high energy ball milling. ...
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Cobalt substituted La2CuO4/LaCoO3 perovskite nanocomposites were prepared using microwave combustion method. X-ray diffraction analysis showed that Co2+ substitution in La2CuO4 induced the formation of secondary LaCoO3 phase with rhombohedral structure along with the existing orthorhombic (La2CuO4) structure. The orthorhombic/rhombohedral structured nanocomposites possessed an average crystallite size in the range 36 - 46 nm and 21 - 49 nm respectively. The Rietveld analysis confirmed the formation of these two phases. The appearance of FT-IR bands around 682 and 516 cm-1 were correlated to the orthorhombic stretching modes while 580 and 445 cm-1wereassociated to rhombohedral stretching modes thereby confirming the two-phase perovskites system. Optical study revealed, two linear regions depicting two band gaps related to La2CuO4 and LaCoO3 phases. The band gap was foundto increase with the increase in Co2+ doping fraction. Morphological observations using scanning electron microscopy showed intragranular pores and fused grains with distinct grain boundaries. The modification in magnetic properties was associated most probably withthe exchange of A and B sites within La2CuO4 host lattice because of Co2+ doping, as well as the important phase composition of La2CuO4/LaCoO3 occurrence from ferromagnetic-to-paramagnetic transition. The BET surface area of pure and doped nanocomposites was found to vary considerably within the range 12.2 - 43.7 m2/g. The as-fabricated La2CuO4/LaCoO3 perovskite nanocomposites were evaluated for the conversion of glycerol to formic acid in liquid phase batch reactor at atmospheric conditions. This La3+ based nanocomposite behaves as an efficient bifunctional catalyst with high conversion efficiency and selectivity around 96.5% and 95.1% respectively.
... Although the proposed size optimization methodology can improve the STL of PLSSHT to a certain extent, it may not still able to meet the requirement of the noise reduction of actual applications. To this end, continuum topology optimization technique [34][35][36][37][38][39][40] can be a promising candidate with strong capacities for significantly improving the structural functionalities. Considering this, future work will focus on maximizing the STL of PLSSHT by introducing holes to the trusses, making up the unit cells, with simultaneously reducing the weigh via topology optimization methods. ...
... Meanwhile, since the homogenization theory [18] is developed to evaluate macroscopic effective properties based on the topologies of periodical material microstructures, it has become popular to combine the homogenization theory with topology optimization to formulate an inverse design procedure of PUCs for gaining the specific effective properties [19]. Many researchers have devoted considerable efforts to optimize or tailor material effective properties, so that microstructured materials with various novel effective properties have been presented, such as the extreme mechanical properties [20][21][22][23], maximum stiffness and fluid permeability [24,25], exotic thermomechanical properties [26], negative Poisson's ratio (NPR) [27][28][29][30] and extreme thermal properties [31]. Micro-structured materials design has become one of the most promising applications of topology optimization [7,32]. ...
Article
This paper proposes an effective method for the design of 3D micro-structured materials to attain extreme mechanical properties, which integrates the firstly developed 3D energy-based homogenization method (EBHM) with the parametric level set method (PLSM). In the 3D EBHM, a reasonable classification of nodes in periodic material microstructures is introduced to develop the 3D periodic boundary formulation consisting of 3D periodic boundary conditions, 3D boundary constraint equations and the reduced linearly elastic equilibrium equation. Then, the effective elasticity properties of material microstructures are evaluated by the average stress and strain theorems rather than the asymptotic theory. Meanwhile, the PLSM is applied to optimize microstructural shape and topology because of its positive characteristics, like the perfect demonstration of geometrical features and high optimization efficiency. Numerical examples are provided to demonstrate the advantages of the proposed design method. Results indicate that the optimized 3D material microstructures with expected effective properties are featured with smooth structural boundaries and clear interfaces.
... Another widely adopted design approach is to use numerical structural optimization techniques, mostly topology optimization, e.g. the studies in [24,25,26,27,28,29,30,31,32,33,34,35,36], shape and size optimization, e.g. the studies in [37,38,39,40,41,42,43,44,45], as well as deep-learning-based approaches, e.g., in [46,47]. Among these works, the recently developed isogeometric analysis (IGA) based on non-uniform rational basis spline (NURBS) demonstrates a great potential in designing auxetics with refined details on curved geometry features. ...
Article
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Taking advantage of the powerful design potentials of isogeometric analysis, an integrated shape and size optimization framework for designing tetra chiral and anti-chiral auxetics is proposed to be capable of obtaining excellent designs without complicated implementation efforts. The framework utilizes a non-uniform rational basis spline (NURBS) based parametrization method that describes the chiral and anti-chiral structures with a small number of size and shape parameters. With this effective framework, systematic design studies considering both plane strain and stress conditions are performed to provide bounding graphs for the best achievable auxeticities under different stiffness requirements. Designs with tunable effective properties are also provided to demonstrate the capabilities of the proposed framework. The potential for electronic-skin applications is illustrated.
... Nowadays, composite materials offer several advantages, because of the significant enhancement of their properties over their individual constituents as well as to conventional materials. Generally, a nanocomposite contains at least two materials/phases at the nanoscale regime, and is preferred due to its high thermal stability, chemical resistance, high strength, enhanced catalytic properties and so on [9][10][11]. Although many nanocomposites can be found in the literature, perovskite/perovskite nanocomposites are scarce reported. ...
... Structural topology optimization, which aims to determine the best material distribution in a predefined domain, is established as a powerful intelligent means in the conceptual design stage of engineering products [1][2][3][4], especially in the field of aeronautics and astronautics [5][6][7][8]. Since the pioneer research by Bendsøe and Kikuchi [9] two decades ago, the relative works are extensively expounded in both academic and industrial fields. ...
Article
This paper presents the interval reliability-based topology optimization (IRBTO) framework and an effective solution procedure for interval parametric structures to achieve optimal material configurations under consideration of local stiffness and strength failure. Firstly, ε-relaxed stress criterion and global stress aggregation approach are involved to circumvent the stress singularity and multi-constrained problems. Combined the orthogonal polynomial expansion with the set allocation theorem, an interval dimension-by-dimension method (IDDM) is proposed to determine feasible bounds of structural responses under unknown-but-bounded load and material uncertainties. The interval reliability (IR) is then applied to handle the limited reliability constraints of concerned displacements and the global stress measure. Meanwhile, the adjoint-vector based sensitivity analysis of presented IR indexes to design variables is further discussed to avoid expensive computational cost from the large-scale nature of variable updating. The usage, rationality, and superiority of the developed methodology are demonstrated by several case applications.
... In this branch, the Level Set Method (LSM) [15][16][17], the phase field method [18,19], the recently proposed Moving Morphable Components/Voids (MMC/V) method [20][21][22][23] and the Bubble method [24,25] have been obtained considerable discussions. These developed TO methods have been also applied to address several different numerical problems, like the dynamic optimization [26][27][28], compliant mechanisms [29,30], stress problems [31][32][33], robust designs [34][35][36], materials design [37][38][39][40][41], concurrent topology optimization [42][43][44][45][46][47][48]. ...
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Topology Optimization (TO) is a powerful numerical technique to determine the optimal material layout in a design domain, which has accepted considerable developments in recent years. The classic Finite Element Method (FEM) is applied to compute the unknown structural responses in TO. However, several numerical deficiencies of the FEM significantly influence the effectiveness and efficiency of TO. In order to eliminate the negative influence of the FEM on TO, IsoGeometric Analysis (IGA) has become a promising alternative due to its unique feature that the Computer-Aided Design (CAD) model and Computer-Aided Engineering (CAE) model can be unified into a same mathematical model. In the paper, the main intention is to provide a comprehensive overview for the developments of Isogeometric Topology Optimization (ITO) in methods and applications. Finally, some prospects for the developments of ITO in the future are also presented.
... Ye et al. (2020) proposed a method for designing stiffer mechanical metamaterials based on NPR properties. Long et al. (2016) performed topology optimisation to maximise the effective Young's modulus and determine the optimal distribution of 3D material microstructures, whose continuous phases exhibit different Poisson's ratios. Wang et al. (2017) obtained lattice materials using the multiscale isogeometric topology optimisation technique. ...
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Isotropic materials are widely used because they retain their properties regardless of the measurement directions. However, their mechanical properties tend to deteriorate owing to disturbances in the base material parameters. This study, for the first time, systematically investigates multiple isotropic materials by considering material uncertainties with respect to the elastic modulus and Poisson's ratio. A smooth bi-directional evolutionary topology optimisation method, is employed to identify optimal topological configurations for material microstructures, macroscopic equivalent properties of the microstructures are evaluated via the comprehensible energy-based homogenisation method. A non-intrusive polynomial chaos expansion model is employed to implicitly quantify uncertainties in the base material subject to Gaussian distributions. Additionally, both the expectations and standard variations in extreme properties are considered as the objective function, and a measurement index is defined to determine whether the designed microstructures are isotropic. Lastly, the deterministic and uncertain cases for numerical examples are compared to demonstrate the efficiency of the proposed method.
... Similarly, for each composite material, a variation in Young's modulus is observed as a function of the thickness of the sample. This is in full agreement with the results of the work on topological optimization methodology for maximizing the stiffness of composites [38]. ...
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... Clausen et al. 18 designed and fabricated 3D auxetic material micro-structures undergoing large deformations. Long et al. 19 performed topology optimization to maximize the effective Young's modulus, so as to obtain the optimal distribution of 3D material micro-structures whose constituent phases consist of non-identical Poisson's ratios. A novel TO method was presented based on the independent point-wise density interpolation to obtain a bimaterial chiral metamaterial 20 . ...
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We demonstrate that the consideration of material uncertainty can dramatically impact the optimal topological micro‑structural configuration of mechanical metamaterials. The robust optimization problem is formulated in such a way that it facilitates the emergence of extreme mechanical properties of metamaterials. The algorithm is based on the bi‑directional evolutionary topology optimization and energy‑based homogenization approach. To simulate additive manufacturing uncertainty, combinations of spatial variation of the elastic modulus and/or, parametric variation of the Poisson’s ratio at the unit cell level are considered. Computationally parallel Monte Carlo simulations are performed to quantify the effect of input material uncertainty to the mechanical properties of interest. Results are shown for four configurations of extreme mechanical properties: (1) maximum bulk modulus (2) maximum shear modulus (3) minimum negative Poisson’s ratio (auxetic metamaterial) and (4) maximum equivalent elastic modulus. The study illustrates the importance of considering uncertainty for topology optimization of metamaterials with extreme mechanical performance. The results reveal that robust design leads to improvement in terms of (1) optimal mean performance (2) least sensitive design, and (3) elastic properties of the metamaterials compared to the corresponding deterministic design. Many interesting topological patterns have been obtained for guiding the extreme material robust design.
... In terms of investigating the mechanical property of the composite bi-material and the influences of the interface and volume fraction, the effective mechanical property was illustrated [46]. Effective mechanical properties are defined here as the overall mechanical properties of combined specimen that contains both rock and backfill. ...
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Purpose – Proposes a methodology for dealing with the problem of designing a material microstructure the best suitable for a given goal. Design/methodology/approach – The chosen model problem for the design is a two-phase material, with one phase related to plasticity and another to damage. The design problem is set in terms of shape optimization of the interface between two phases. The solution procedure proposed herein is compatible with the multi-scale interpretation of the inelastic mechanisms characterizing the chosen two-phase material and it is thus capable of providing the optimal form of the material microstructure. The original approach based upon a simultaneous/sequential solution procedure for the coupled mechanics-optimization problem is proposed. Findings – Several numerical examples show a very satisfying performance of the proposed methodology. The latter can easily be adapted to other choices of design variables. Originality/value – Confirms that one can thus achieve the optimal design of the nonlinear behavior of a given two-phase material with respect to the goal specified by a cost function, by computing the optimal form of the shape interface between the phases.
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Abstract Numerical homogenization is an efficient way to determine effective macroscopic properties, such as the elasticity tensor, of a periodic composite material. In this paper an educational description of the method is provided based on a short, self-contained Matlab implementation. It is shown how the basic code, which computes the effective elasticity tensor of a two material composite, where one material could be void, is easily extended to include more materials. Furthermore, extensions to homogenization of conductivity, thermal expansion, and fluid permeability are described in detail. The unit cell of the periodic material can take the shape of a square, rectangle, or parallelogram, allowing for all kinds of 2D periodicities.
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The evolutionary structural optimization (ESO) and basic ESO (BESO) processes and given various illustrative examples are described. Such processes are based on the concept of slowly removing inefficient materials from a structures so that the residual structure evolves towards the optimum. It is shown that the simple ESO and BESO algorithms are capable of solving a wide range of shape and topology optimization problems.
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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.
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A sandwich-structured composite is a special class of composite materials that is fabricated by attaching two thin but stiff layers to a lightweight but thick core. Composites analyzed in this paper consist of two different materials: auxetic and structural steel. The optimization criterion is minimum compliance for the load case where the frame's top boundary is downward loaded. Outer layers are made of steel while the middle layer is two-phase solid material composite. Only the middle layer is optimized by means of minimization of the objective function defined as the internal strain energy. In the first part of this paper we study the application of the solid isotropic material with penalization (SIMP) model to find the optimal distribution of a given amount of materials in sandwich-structured composite. In the second part we propose a multilayered composite structure in which internal layers surfaces are wavy. In both cases the total energy strain is analyzed.
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We systematically design materials using topology optimization to achieve prescribed nonlinear properties under finite deformation. Instead of a formal homogenization procedure, a numerical experiment is proposed to evaluate the material performance in longitudinal and transverse tensile tests under finite deformation, i.e. stress-strain relations and Poisson's ratio. By minimizing errors between actual and prescribed properties, materials are tailored to achieve the target. Both two dimensional (2D) truss-based and continuum materials are designed with various prescribed nonlinear properties. The numerical examples illustrate optimized materials with rubber-like behavior and also optimized materials with extreme strain-independent Poisson's ratio for axial strain intervals of εi∈[0.00,0.30].
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The aim of this study was to design isotropic periodic microstructures of cellular materials using the bidirectional evolutionary structural optimization (BESO) technique. The goal was to determine the optimal distribution of material phase within the periodic base cell. Maximizing bulk modulus or shear modulus was selected as the objective of the material design subject to an isotropy constraint and a volume constraint. The effective properties of the material were found using the homogenization method based on finite element analyses of the base cell. The proposed BESO procedure utilizes the gradient-based sensitivity method to impose the isotropy constraint and gradually evolve the microstructures of cellular materials to an optimum. Numerical examples show the computational efficiency of the approach. A series of new and interesting microstructures of isotropic cellular materials that maximize the bulk or shear modulus have been found and presented. The methodology can be extended to incorporate other material properties of interest such as designing isotropic cellular materials with negative Poisson's ratio.
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Materials with good vibration damping properties and high stiffness are of great industrial interest. In this paper the bounds for viscoelastic composites are investigated and material microstructures that realize the upper bound are obtained by topology optimization. These viscoelastic composites can be realized by additive manufacturing technologies followed by an infiltration process. Viscoelastic composites consisting of a relatively stiff elastic phase, e.g. steel, and a relatively lossy viscoelastic phase, e.g. silicone rubber, have non-connected stiff regions when optimized for maximum damping. In order to ensure manufacturability of such composites the connectivity of the matrix is ensured by imposing a conductivity constraint and the influence on the bounds is discussed.
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We present a method to design manufacturable extremal elastic materials. Extremal materials can possess interesting properties such as a negative Poisson’s ratio. The effective properties of the obtained microstructures are shown to be close to the theoretical limit given by mathematical bounds, and the deviations are due to the imposed manufacturing constraints. The designs are generated using topology optimization. Due to high resolution and the imposed robustness requirement they are manufacturable without any need for post-processing. This has been validated by the manufacturing of an isotropic material with a Poisson’s ratio of ν=-0.5ν=-0.5 and a bulk modulus of 0.2% times the solid base material’s bulk modulus.
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The design of periodic microstructural composite materials to achieve specific properties has been a major area of interest in material research. Tailoring different physical properties by modifying the microstructural architecture in unit cells is one of the main concerns in exploring and developing novel multi-functional cellular composites and has led to the development of a large variety of mathematical models, theories and methodologies for improving the performance of such materials. This paper provides a critical review on the state-of-the-art advances in the design of periodic microstructures of multi-functional materials for a range of physical properties, such as elastic stiffness, Poisson’s ratio, thermal expansion coefficient, conductivity, fluidic permeability, particle diffusivity, electrical permittivity and magnetic permeability, etc.
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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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This paper presents a new approach to designing periodic microstructures of cellular materials. The method is based on the bidirectional evolutionary structural optimization (BESO) technique. The optimization problem is formulated as finding a micro-structural topology with the maximum bulk or shear modulus under a prescribed volume constraint. Using the homogenization theory and finite element analysis within a periodic base cell (PBC), elemental sensitivity numbers are established for gradually removing and adding elements in PBC. Numerical examples in 2D and 3D demonstrate the effectiveness of the proposed method for achieving convergent microstructures of cellular materials with maximum bulk or shear modulus. Some interesting topological patterns have been found for guiding the cellular material design.
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The Voigt estimation or the rule of mixture has been believed to be the upper bound of the effective Young’s modulus of composites. However, this is only true in the situations where the Poisson effect is not significant. In this paper, we accurately derived the effective compliance matrix for two-phase layered composites by accounting for the Poisson effect. It is interesting to find that the effective Young’s modulus in both transverse and longitudinal direction can exceed not only the Voigt estimation, but also the Young’s modulus of the stiffest constituent phase. Moreover, the longitudinal (or parallel connection) Young’s modulus is not always larger than the transverse (or serial connection) one. For isotropic composites, it has also been demonstrated that the Voigt estimation is not the upper bound for the effective Young’s modulus. Therefore, one should be careful in applying the well known bound estimations on the effective Young’s modulus of composites if one of the phases is near its incompressibility limit.
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Analytical models and complementary finite element (FEM) calculations are developed to evaluate the influence of Poisson ratio on the mechanical properties of biocomposites with staggered mineral platelets in parallel alignment in a soft protein matrix. As the Poisson ratio of the soft matrix is increased towards the incompressibility limit 0.5, the staggered biocomposites are significantly stiffened in both longitudinal and transverse directions, with the transverse stiffness increasing by more than two orders of magnitude. Since the Poisson ratio of most soft biological tissues is very close to 0.5, it is concluded that the Poisson ratio can play a crucial role in the mechanical properties of biocomposites. Based on the theoretical analysis, some discussions are made on several mechanisms that are responsible for the superior properties of biocomposites.
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We present a variational method to address the topology optimization problem the phase transition method. A phase-field model is employed based on the phase-transition theory in the fields of mechanics and material sciences. The topology optimization is formulated as a continuous problem with the phase-field as design variables within a fixed reference domain. All regions are described in terms of the phase field which makes no distinction between the solid, void and their interface. The van der Waals-Cahn-Hilliard theory is applied to define the variational topology optimization as a dynamic process of phase transition. The Γ-convergence theory is then adapted for an approximate solution to this free-discontinuity problem. As a result, a two-step, alternating numerical procedure is developed which treats the whole design domain simultaneously without any explicit tracking of the interface. Within this variational framework, we show that a regularization theory can be incorporated to lead to a well-posed formulation. We also show that the phase-field model has a close relationship with the general Mumford-Shah model of image segmentation in computer vision. The proposed variational method is illustrated with several two-dimensional examples that have been extensively used in the recent literature on topology optimization, especially in the homogenization-based methods. Extension of the proposed method to the general problems of multiple material phases other than just solid and void is discussed, and it is further suggested that such a variational approach may represent a promising alternative to the widely-used material distribution model for the future development in topology optimization.
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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.
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A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.
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The theory of elasticity predicts a variety of phenomena associated with solids that possess a negative Poisson's ratio. The fabrication of metamaterials with a 'designed' microstructure that exhibit a Poisson's ratio approaching the thermodynamic limits of 1/2 and -1 increases the likelihood of realising these phenomena for applications. In this work, we investigate the properties of a layered composite, with alternating layers of materials with negative and positive Poisson's ratio approaching the thermodynamic limits. Using the finite element method to simulate uniaxial loading and indentation of a free standing composite, we observed an increase in the resistance to mechanical deformation above the average value of the two materials. Even though the greatest increase in stiffness is gained as the thermodynamic limits are approached, a significant amount of added stiffness can be attained, provided that the Young's modulus of the negative Poisson's ratio material is not less than that of the positive Poisson's ratio material. (C) 2008 Elsevier B.V. All rights reserved.
Article
This paper presents a systematic investigation into the computational design of multi-phase microstructural composites with tailored isotropic and anisotropic thermal conductivities. The composites are assumed to be periodically ranked by base cells (representative vol-ume elements) whose best possible geometric configurations make the composite's bulk or effective thermal conductivity attaining to the target Milton–Kohn bounds. To avoid checkerboard patterns and generate edge-preserving results in topology optimization, a nonlinear diffusion technique is exploited by introducing the generalized interface energy into the objective function. The adjoint variable method is used to formulate the sensitivity of the objective functions with respect to multi-phase design variables (''relative density"), which guides the method of moving asymptotes to converge along the steepest direction. Unlike the typical density-based method (e.g. SIMP), the penalty factor is no longer needed in this present method after the local conductivity is interpolated by the Hashin–Shtrikman bound other than commonly-used arithmetic bound. In addition to the conventional Vigdergauz-like structures, three new classes of single-length-scale microstructures are generated to closely approach the isotropic Hashin–Strikman bounds in three-phase and two-dimen-sional cases. This paper also generated sandwich-like microstructures attaining to the anisotropic Milton–Kohn bounds.
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Topology optimization is used to systematically design periodic materials that are optimized for multiple properties and prescribed symmetries. In particular, mechanical stiffness and fluid transport are considered. The base cell of the periodic material serves as the design domain and the goal is to determine the optimal distribution of material phases within this domain. Effective properties of the material are computed from finite element analyses of the base cell using numerical homog-enization techniques. The elasticity and fluid flow inverse homogenization design problems are formulated and existing tech-niques for overcoming associated numerical instabilities and difficulties are discussed. These modules are then combined and solved to maximize bulk modulus and permeability in periodic materials with cubic elastic and isotropic flow symmetries. The multiphysics problem is formulated such that the final design is dependent on the relative importance, or weights, assigned by the designer to the competing stiffness and flow terms in the objective function. This allows the designer to tailor the micro-structure according to the materials' future application, a feature clearly demonstrated by the presented results. The meth-odology can be extended to incorporate other material properties of interest as well as the design of composite materials.
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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.
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Variational principles in the linear theory of elasticity, involving the elastic polarization tensor, have been applied to the derivation of upper and lower bounds for the effective elastic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. When the ratios between the different phase moduli are not too large the bounds derived are close enough to provide a good estimate for the effective moduli. Comparison of theoretical and experimental results for a two-phase alloy showed good agreement.
Article
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.
Article
Application of piezoelectric materials, such as hydrophones and naval sonars, requires an improvement in their performance characteristics. This improvement can be obtained by designing new types of piezocomposite materials that possess a richer class of properties. The effective properties of a composite material depend on the topology of its unit cell (or microstructure) and the properties of its constituents. By changing the unit cell topology, better performance characteristics can be obtained in the piezocomposite. In this work, we have extended the optimal design method of piezocomposite microstructures proposed in the previous work [1] to three dimensional (3D) topologies, considering static applications such as hydrophones. This method uses topology optimization techniques and homogenization theory, and consists of finding the distribution of the material and void phases in a periodic unit cell that optimizes the performance characteristics of the piezocomposite. The optimization problem is subjected to constraints such as property symmetry and stiffness. An additional constraint was added in order to penalize the amount of intermediate densities generated in the final design. The optimized solution is obtained using Sequential Linear Programming (SLP). In order to calculate the effective properties of a unit cell with complex topology, a general homogenization method applied to piezoelectricity was implemented using the finite element method (FEM). This homogenization method has no limitations regarding volume fraction or shape of the composite constituents. The main assumption is the periodicity of the unit cell. Microstructures obtained show a large improvement in performance characteristics compared to pure piezoelectric material or simple designs of piezocomposite unit cells. Finally, a hydrophone made of one layer of the unit cells obtained by using microstructure design is suggested. An FEM analysis is done to evaluate the performance improvement of such transducer.
Article
This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).
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Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming.To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds.The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.
Article
The evolutionary structural optimisation (ESO) method has been under continuous development since 1992. Originally the method was conceived from the engineering perspective that the topology and shape of structures were naturally conservative for safety reasons and therefore contained an excess of material. To move from the conservative design to a more optimum design would therefore involve the removal of material. The ESO algorithm caters for topology optimisation by allowing the removal of material from all parts of the design space. With appropriate chequer-board controls and controls on the number of cavities formed, the method can reproduce traditional fully stressed topologies. If the algorithm was restricted to the removal of surface-only material, then a shape optimisation problem (along the lines of the Min–Max type problem) is solved. Recent research by the authors has presented and benchmarked an additive evolutionary structural optimisation (AESO) algorithm that, with appropriate decision making, starts the evolutionary optimisation procedure from a minimal kernel structure that connects the loading points to the mechanical constraints. Naturally this is unevenly and overly stressed, and material is subsequently added to the surface to reduce localised high stress regions. AESO only adds material to the surface, the present work describes the combining of basic ESO with the AESO to produce bi-directional ESO (BESO) whereby material can be added and removed. This paper shows that this method provides the same results as the traditional ESO. This has two benefits, it validates the ESO concept, and as the examples demonstrate, BESO can arrive at an optimum faster than ESO. This is especially true for 3D structures, since the structure grows from a small initial one rather than contracting from a, sometimes, huge initial one where around 90% of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Both 2D and 3D structures are examined and multiple load cases are applied.
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After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.
Article
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.
Article
We develop and test an algorithmic approach to the boundary design of elastic structures. The goal of our approach is two-fold: first, to develop a method which allows one to rapidly solve the two-dimensional Lamé equations in arbitrary domains and compute, for example, the stresses, and second, to develop a systematic way of modifying the design to optimize chosen properties. At the core, our approach relies on two distinct steps. Given a design, we first apply an explicit jump immersed interface method to compute the stresses for a given design shape. We then use a narrow band level set method to perturb this shape and progress towards an improved design. The equations of 2D linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers. This effectively reduces the dimensionality of the problem by one. Once the stresses are found, the level set method, which represents the design structure through an embedded implicit function, is used in the second step to alter the shape, with velocities depending on the stresses in the current design. Criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated.
Article
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.
Article
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.
Optimal shape design as a material distribution problem
  • M P Bendsoe
Bendsoe MP. Optimal shape design as a material distribution problem. Struct Opt 1989;1(4):193-202.