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Topological design of microstructures of cellular materials for maximum bulk or shear modulus

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Abstract

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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... For a sandwich structure, because the cell size of the core layer is much smaller than the whole structure, the effect of the micro-structural properties of the core layer on the macroscopic structural performance can be determined by homogenization theory [29][30][31]. However, a large amount of computing resources will be required to calculate the effect of the micro-structure on the performance of the macrostructure directly by using the homogenization method [16]. ...
... In the present work, based on the mechanical characteristics of the core layer, which withstands the shear stress and deformation of the sandwich beam, the BESO [29] method was employed to optimize a core composed of a periodic base cell with the maximum shear modulus under a prescribed volume constraint. The dynamic response characteristics and energy absorption ability of the sandwich beam with the maximum shear stiffness core layer under blast impact loading were analyzed by the finite element method. ...
... The macroscopic properties of a cellular material can be estimated by the homogenization theory [29] when it is composed of a periodic base cell (PBC) repeatedly and the PBC is smaller than the structure size. The effective elasticity tensor of a periodic material can be computed as ...
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Based on the mechanical characteristics of the core layer, which withstands the shear stress and deformation of a sandwich beam, a topology optimization framework based on the bi-directional evolutionary structural optimization method is proposed to optimize the core layer composed of a periodic base cell with extreme shear stiffness. The effects of the volume fraction, filter radius, and initial periodic base cell (PBC) aspect ratio on the micro-topology of the core and the dynamic response process, core compression, and energy absorption capacity of the sandwich beams under blast impact loading were analyzed by the finite element method. The results demonstrated that the over-pressure action stage was coupled with the core compression stage. Under the same loading and mass per unit area, the sandwich beam with a 20% volume fraction core layer had the best blast resistance. The filter radius has a slight effect on the shear stiffness and blast resistances of the sandwich beams, but increasing the filter radius could slightly improve the bending stiffness. Upon changing the initial PBC aspect ratio, there are three methods for PBC evolution: the first is to change the angle between the adjacent bars, the second is to further form holes in the bars, and the third is to combine the first two methods. However, not all three methods can improve the energy absorption capacity of the structure. Changing the aspect ratio of the PBC arbitrarily may lead to worse results. More detailed studies are necessary if further optimization is to be achieved.
... Zhou et al. [49] developed a systematic method to design the multi-phase microstructural composites with tailored thermal conductivity on Milton-Kohn bounds. Huang et al. [50] have applied the bi-directional evolutionary structural optimization (BESO) method and AH approach to design the microstructure of cellular materials for maximum bulk and shear modulus. Zhang et al. [24] proposed the alternative strain energy method to get the same accuracy as the AH method and used it to design the microstructure with specific properties. ...
... The BESO technique is mathematically simple and easy to implement. And the local strain and stress values are the only input data for the BESO technique to conduct the topology optimization [50]. There are no intermediate densities in BESO, which contributes to generating binary structures. ...
... As shown in Fig. 3(a-d), the optimal microstructures exhibit the square pattern with different details. The optimal values of the objective function in this paper approach the results of Huang et al. [50], but their optimal microstructures are different due to the non-unique solution of topology optimization. Fig. 3(f) shows the evolutional histories of bulk modulus and volume fraction with respect to iteration when the volume fraction constraint is 30%. ...
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Besides constituting components, the properties of composites are highly relevant to their microstructures. The work proposed a fast Fourier transform (FFT)-based inverse homogenization method implemented by the bi-directional evolutionary structural optimization (BESO) technique to explore the vast potential of cellular materials. The periodic boundary condition of self-repeated representative volume elements can be naturally satisfied in the FFT-based homogenization scheme. The objective function of the optimization problem is the specific moduli or the quadratic difference between the effective value and the target, which are obtained in terms of mutual strain energies. Its sensitivity to the design variable, namely the elemental density, is derived from the adjoint variable method and used as the criterion to remove or add material in local elements. Numerical examples show that the proposed method generates a series of architected cellular materials with maximum modulus, negative Poisson's ratio, and specific elasticity tensor. FFT-based homogenization in the method demands less memory usage but has high efficiency. Thus, it can achieve topology optimization of unit cell with one million hexahedral elements. This approach contributes to the extended application of FFT-based homogenization and can guide the microstructure design of mechanical metamaterials.
... When the volume ratio was 30%, the optimization problem converged after successively searching in 17 sub-domains with 2674 finite element calls, and the bulk modulus was 0.93. The optimized bulk modulus value was the same as in the results [52] but showed a different topological configuration. It was also clearly observed that the optimized configurations of the unit cell with 50% material usage, which were obtained by the proposed method, were consistent with the results [52]. ...
... The optimized bulk modulus value was the same as in the results [52] but showed a different topological configuration. It was also clearly observed that the optimized configurations of the unit cell with 50% material usage, which were obtained by the proposed method, were consistent with the results [52]. Moreover, the results of all the numerical examples were slightly better than the results [52]. ...
... It was also clearly observed that the optimized configurations of the unit cell with 50% material usage, which were obtained by the proposed method, were consistent with the results [52]. Moreover, the results of all the numerical examples were slightly better than the results [52]. Hence, the effectiveness of the proposed method for 3D microstructural material design was validated. ...
Article
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Powerful gradient-free topology optimization methods are needed for structural design concerning complex responses. In this paper, a novel gradient-free optimization method is proposed by integrating the material-field series expansion topological parameterization and the deep neural networks, providing two-fold advances: firstly, it generally reduces the massive topological design variables to fewer than 200, while keeps the capability to represent relative complex 3D topologies and clear boundaries; secondly, by constructing a sequential neural network surrogate model, it sufficiently explores the reduced design space and is capable of handling multi-peak and discontinuous optimization problems. The effectiveness of this method is illustrated via several design problems, among which the optimized material effective bulk modulus achieves 98% of the H-S bound and the highly-nonlinear peak weld stress in a phone dropping process is decreased by 16.59%. This method reduces the computational time by 1~4 orders of magnitude compared with the coarse-mesh-based gradient-free methods, and it is the first time to successfully conduct gradient-free 3D topology optimization with thousands of finite elements. The method’s ease of implementation and compatibility with various simulation software, brings topology optimization into complex industrial applications and proves that gradient-free technology represents an effective optimization benchmark for improving structural performance.
... A multitude of architected truss (meta-)materials has been introduced theoretically and experimentally with as-designed properties. Classical examples include extreme properties such as designs with high-strength-and/or stiffness-to-weight ratios [10], further unconventional properties such as auxeticity [11,12] or near-infinite bulk-to-shear modulus ratios [12,13,14,15]. Advancing from linear to nonlinear and dynamic material behavior, properties of interest have included energy absorption [16], acoustic wave tuning [17,18], controllable nonlinear stress-strain behavior and shape recoverability under large deformations [19,6], insensitivity to imperfections [20,21,22], and damage tolerance [23]. ...
... based on the primitive Bravais lattice vectors a i . In 2D, the reciprocal lattice vectors k 1 and k 2 follow from (14) with a 3 = (0, 0, 1). ...
... That is, we ensure that ∇φ k is locally tangential to the (local) k-vector defined through (14) at every point on the macroscale. ...
Preprint
Full-text available
We introduce a computational framework for the topology optimization of cellular structures with spatially varying architecture, which is applied to functionally graded truss lattices under quasistatic loading. We make use of a first-order homogenization approach, which replaces the discrete truss by an effective continuum description to be treated by finite elements in a macroscale boundary value problem. By defining the local truss architecture through a set of Bravais vectors, we formulate the optimization problem with regards to the spatially varying basis vectors and demonstrate its feasibility and performance through a series of benchmark problems in 2D (though the method is sufficiently general to also apply in 3D, as discussed). Both the displacement field and the topology are continuously varying unknown fields on the macroscale, and a regularization is included for well-posedness. We argue that prior solutions obtained from aligning trusses along the directions of principal stresses are included as a special case. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).
... Zhang et al. [16] designed microcomposites with auxetic behaviours over a large strain range by combining a density-based topology optimisation with a mixed stress/deformation driven nonlinear homogenisation method. The second is the functional micro-composites with the maximum effective bulk modulus (EBM) which has drawn much attention [21][22][23][24], promoting increasing applications in load-bearing structures [25][26][27][28]. Huang et al. [22] presented a design approach based on the bidirectional evolutionary structural optimisation for the periodic micro-composites with maximum EBM using the [14], (b) functional microstructures under deformation and failure [26], and (c) multi-material micro-composites [27,28]. ...
... The second is the functional micro-composites with the maximum effective bulk modulus (EBM) which has drawn much attention [21][22][23][24], promoting increasing applications in load-bearing structures [25][26][27][28]. Huang et al. [22] presented a design approach based on the bidirectional evolutionary structural optimisation for the periodic micro-composites with maximum EBM using the [14], (b) functional microstructures under deformation and failure [26], and (c) multi-material micro-composites [27,28]. homogenisation theory and finite element analysis within a periodic base cell. ...
... In this study, except for the FSE index, we also extended a decisionmaking approach using Eq. (22) to evaluate the solutions for micro-composites under contradictive objectives for obtaining the optimal solution. By doing so, the evaluation result was also utilised to confirm that obtained by the FSE index. ...
Article
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This study develops a new multi-material topology optimisation framework for design of periodic micro-composites with optimal functional performance and reduced stress concentration. First, multi-material topology optimisation is developed based on the alternating active phase algorithm and inverse homogenisation method with the sensitivity analysis derived for specific property objective i.e., negative Poisson's ratio (NPR) or maximum effective bulk modulus (EBM) and (p-norm macroscopic) stress objective. Then, the effects of initial material distribution and weight ratio (w1, w2 assigned to the property and stress objectives, respectively) are investigated, and the evaluation indices are also developed to obtain the optimal solution. Further, two cases related to the design of micro-composites for maximised either NPR or EBM with reduced maximum stress are performed. The results show that when designing the multi-material NPR micro-composites, the decrease of w1/w2 contributes to a general decease of both NPR and maximum stress. While in designing the maximum EBM, decreasing w1/w2 leads to the reduced maximum stress and simultaneously reduced EBM; hereby, a decision-making method as well as the proposed evaluation index are both applied and compared for acquiring the optimal result. This study provides new methods and solutions to multi-material micro-composites design for future industrial applications.
... Gao et al. [15] proposed an efficient topological design method of 3D microstructural materials to obtain extreme mechanical properties by integrating a 3D energy-based homogenization method into a parametric level set method. Huang et al. [37] adopted the Bi-directional Evolutionary Structural Optimization (BESO) method to design the material microstructures with maximum bulk and shear modulus. Liu et al. [11] developed a gradient-free design method for cellular material microstructures by combining the material-field series-expansion (MFSE) and sequential Kriging-based optimization algorithm [38] , and various high-performance microstructural configurations approaching the H-S upper bound could be obtained. ...
... It can be observed that although the extra material with E 4 =100 is added into the case, the optimized topology with smooth microstructural boundaries and clear material interfaces can still be obtained by the proposed method. The combination layouts of three materials exhibit the well-known prismatic-like pattern, as given in the references [11,37]. Particularly, the hard material (green, E 4 =100) is placed at the dominantly interval frame, while the weakest material (blue, E 2 =10) is mainly located at external frame and the moderate material (red, E 3 =50) mainly plays the role of connection. ...
Article
Micro-architectured materials with periodically configured microstructures are widely used in various engineering fields due to their superior mechanical properties. Compared with conventional topology design with one single material, topological design with multiple materials can provide a better solution as the expanded design freedom. This paper presents a level set-based multi-material topological design method for micro-architectured materials involving three phases or more. In this method, a difference-set-based multi-material level set model is employed to represent the topology of each phase, in which N+1 phases can be precisely represented by the sequential difference-set of N level set functions without any overlaps and redundant regions. By using an alternating active-phase algorithm, the multi-material topology design problem with N+1 phases is splitted into N(N+1)/2 binary-phase topological design sub-problems, where each sub-problem involves fewer design variables and volume constraints. The effective properties of multiphase micro-architectured materials are evaluated by a numerical homogenization approach, and their topological evolutions of two active phases in each sub-problem can be readily achieved by updating a single level set function, which greatly facilitate the extension to topological design problems with more phases. Both numerical examples and additive-manufactured specimens demonstrate the effectiveness of the proposed method for designing multiphase micro-architectured materials.
... According to classical conclusions by Ashby [5] and a detailed review by Bhate et al. [6], porous structures can be classified as beam-based cellular structures, i.e. lattices, and surface-based cellular structures, i.e. foams. Both classes of porous structures were studied by Huang et al. [7] using a new simulation approach called bidirectional evolutionary structural optimization (BESO). Optimized cell topologies and densities for different load directions (compression, shear, tension) were found. ...
... One of the oldest known TPMS is the Schwarz p (primitive) surface which can be imagined as hollow spheres in simple cubic stacking with open cells. Thus, it is plausible that Huang et al. [7] used this configuration for their comparison of foam versus lattice topologies. Afshar et al. [11] extended their investigation to a large variety of crystallographic groups of TPMS both simulating, printing and testing the deformation and the collapse behavior of these structures. ...
Article
Full-text available
Infill structures of additive manufactured components give not only stability, but can be used as a design feature regarding lightweight and mechanical properties. In this study, an infill structure was investigated which consists of hollow spheres in a hexagonal closed packing. An additional characteristic is, that the contact between the spheres is modelled as an intersection. A script-based design approach was used for generating the structure within a cylinder. Thus, the sphere diameter, the sphere wall thickness and the outer shell thickness can be varied. The deformation behavior was studied using the Finite Element Method (FEM). Compression tests on additive manufactured cylindrical specimens verify the FEM results: (1) appropriate modelling of the contact between the spheres enhances the stability to the structure. (2) Increasing the relative density increases the mechanical properties. (3) Lower values for the relative sphere wall thickness result in an improved ductility.
... In the literature, different multi-scale TO methods for designing ACMs are available: they are based on (a) the homogenisation method [72,73], (b) the LSM [57,58,[74][75][76][77], (c) the SIMP approach [61,[78][79][80][81] or (d) the BESO method [82]. These strategies are often applied at the scale of the ACM RVE to find the optimal topology satisfying the requirements of the problem at hand. ...
... -Current multi-scale TO approaches for ACMs suffer from three main limitations. The first one is related to the problem formulation, which often includes only requirements on prescribed values of the macroscopic elastic tensor components of the ACM as done in [35,58,61,63,64,74,75,77,[79][80][81][82][83]103]. However, in real-world engineering applications the RVE topology at the lower scale must be optimised in order to satisfy design requirements on macroscopic structural responses, like compliance, mass, strength, etc. ...
Thesis
Cette thèse porte sur l’intégration des spécificités des problèmes multi-échelle et des procédés de fabrication additive (FE), dans l’algorithme d'optimisation topologique (OT) basé sur le champ de pseudo-densité (utilisé en tant que descripteur topologique) et sur les hypersurfaces NURBS (de l'anglais non-uniform rational basis spline) développé au laboratoire I2M de Bordeaux. L’objectif est de faciliter le travail du concepteur lors des différentes étapes de la chaîne numérique AM, en réduisant le temps consacré à chaque étape. Pour ce faire, cette thèse aborde deux grands défis. Tout d’abord, le développement d’une stratégie de reconstruction de surface semi-automatique pour reconstruire et intégrer dans un environnement CAO la frontière de la topologieoptimisée en minimisant les ressources informatiques (temps, mémoire, etc.) dédiées à cette tâche.Deuxièmement, l’intégration des spécificités des problèmes multi-échelle dans le processus d’OT. Concernant cet aspect, un cadre théorique/numérique permettant l’optimisation simultanée des descripteurs topologiques définis à plusieurs échelles a été développé. Dans ce contexte, des exigences de conception de nature différente, telles que la contrainte de fabrication sur l'épaisseur minimale imprimable, la condition de séparation d'échelle, la légèreté, la souplesse généralisée (en présence de conditions aux limites mixtes de Neumann-Dirichlet non nulles), ont été incluses dans la formulation du problème, en exploitant les propriétés du formalisme NURBS. L’efficacité de la méthode d’OT multi-échelles proposée a été testée sur des problèmes 2D et 3D tirés de la littérature et validée par les résultats d’essais.
... Guest and Prévost [6] dealt with the topology optimization of the fluid flows in the design of porous periodic materials. Challis et al. [7], Amstutz et al. [8] and Gao et al. [9] proposed level set-based approaches for the design of microstructures, and Huang et al. in [10,11] presented a Bi-directional Evolutionary Structural Optimization (BESO)-method-based approach for the optimal design of periodic microstructures. A detailed review of the different methodologies in the optimal design of materials together with a description of the variety of the approaches presented so far to deal with the topology optimization of the macro design concurrently with the micro design can be seen in [12]. ...
... The three approaches implemented are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis following the combined approximations approach. In most MOR approaches, the approximation of the displacement field is taken into account in the calculation of the sensitivities through the expression of Equation (10). ...
Article
Full-text available
The main part of the computational cost required for solving the problem of optimal material design with extreme properties using a topology optimization formulation is devoted to solving the equilibrium system of equations derived through the implementation of the finite element method (FEM). To reduce this computational cost, among other methodologies, various model order reduction (MOR) approaches can be utilized. In this work, a simple Matlab code for solving the topology optimization for the design of materials combined with three different model order reduction approaches is presented. The three MOR approaches presented in the code implementation are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis (AR) following the combined approximations approach. The complete code, containing all participating functions (including the changes made to the original ones), is provided.
... A multitude of architected truss (meta-)materials has been introduced theoretically and experimentally with as-designed properties. Classical examples include extreme properties such as designs with high strength-and/or stiffness-to-weight ratios [10], further unconventional properties such as auxeticity [11,12], or near-infinite bulk-to-shear modulus ratios [12][13][14][15]. Advancing from linear to nonlinear and dynamic material behavior, properties of interest have included energy absorption [16], acoustic wave tuning [17,18], controllable nonlinear stress-strain behavior and shape recoverability under large deformations [6,19], insensitivity to imperfections [20][21][22], and damage tolerance [23]. ...
... That is, we ensure that ∇ϕ k is locally tangential to the (local) k-vector defined through (14) at every point on the macroscale. Finding the field ϕ : Ω R d , in principle, requires the solution of an over-determined system. ...
Article
We introduce a computational framework for the topology optimization of cellular structures with spatially varying architecture, which is applied to functionally graded truss lattices under quasistatic loading. We make use of a first-order homogenization approach, which replaces the discrete truss by an effective continuum description to be treated by finite elements in a macroscale boundary value problem. By defining the local truss architecture through a set of Bravais vectors, we formulate the optimization problem with regards to the spatially varying basis vectors and demonstrate its feasibility and performance through a series of benchmark problems in 2D (though the method is sufficiently general to also apply in 3D, as discussed). Both the displacement field and the topology are continuously varying unknown fields on the macroscale, and a regularization is included for well-posedness. We argue that prior solutions obtained from aligning trusses along the directions of principal stresses are included as a special case. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).
... As far as the strategies dedicated to the multi-scale design of LSs and of metamaterials are concerned, different multi-scale TO methods are available in the literature. They are based on (a) the homogenisation method [23,24], (b) the LSM [25][26][27][28][29][30], (c) the SIMP approach [31][32][33][34][35] or (d) the BESO method [36]. These strategies are often applied at the scale of the LS RVE, in order to find the optimal topology satisfying the requirements of the problem at hand. ...
... As it can be inferred from this non-exhaustive literature survey, current multi-scale TO approaches for LSs suffer from three main limitations. The first one is related to the problem formulation, which often includes only requirements on prescribed values of the macroscopic elastic tensor components of the LS as done in [25][26][27]30,31,[33][34][35][36][37][38][39][40]46]. However, in real-world engineering applications the RVE topology at the lower scale must be optimised in order to satisfy design requirements on macroscopic structural responses, like compliance, mass, strength, etc. ...
Article
Full-text available
This work discusses three aspects of topology optimisation (TO) problems dealing with structural stiffness maximisation of anisotropic continua under mixed inhomogeneous Neumann–Dirichlet boundary conditions (BCs). Firstly, the total potential energy (TPE) is introduced as intuitive measure of the structural stiffness, instead of the work of applied forces and displacements (WAFD). Secondly, it is proven that the WAFD under mixed BCs is not a self-adjoint functional, while the one related to the TPE is always a self-adjoint functional, regardless of the BCs nature. Thirdly, the influence of the anisotropy, of the applied BCs and of the design requirement on the volume fraction on the optimised topology is investigated: depending on these features, the optimal solutions of the two problem formulations, i.e., minimisation of the functional involving the TPE or minimisation of the WAFD subject to a constraint on the volume fraction, can coincide. The problem is formulated in the context of a special density-based TO approach wherein a Non-Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor, i.e., the pseudo-density field. The properties of NURBS entities are exploited to derive the gradient of the physical responses involved in the problem formulation and to easily satisfy the minimum length scale requirement (related to manufacturing needs). The differences between TPE-based and WAFD-based formulations and the effectiveness of the proposed method are shown on 2D and 3D problems.
... In this context, a single-or a multi-objective topology optimization at the microscale can drive the design of new unit cells matching target properties at the macroscale, potentially in a multi-physics framework. For instance, the optimization of homogenized elastic properties is tackled in [29][30][31] with the aim of maximizing the bulk (or shear) modulus. To this aim, the authors control specific components of the homogenized elastic tensor or resort to the minimization of the compliance of a given structural part. ...
... Inverse homogenization is the procedure that allows us to design microstructures with prescribed properties at the macroscale. The required features are mathematically commuted into a goal functional J and into suitable constraints driving a topology optimization process to be solved in the unit cell Y ⊂ R 2 whose periodic repetition yields the cellular material [26,27,29,38,43]. According to a density-based approach, a standard way to perform such an optimization leads us to define an auxiliary scalar field, ρ, that models the relative material density at the microscale. ...
Article
Full-text available
We present a new algorithm to design lightweight cellular materials with required properties in a multi-physics context. In particular, we focus on a thermo-elastic setting by promoting the design of unit cells characterized both by an isotropic and an anisotropic behavior with respect to mechanical and thermal requirements. The proposed procedure generalizes the microSIMPATY algorithm to a thermo-elastic framework by preserving all the good properties of the reference design methodology. The resulting layouts exhibit non-standard topologies and are characterized by very sharp contours, thus limiting the post-processing before manufacturing. The new cellular materials are compared with the state-of-art in engineering practice in terms of thermo-elastic properties, thus highlighting the good performance of the new layouts which, in some cases, outperform the consolidated choices.
... Topology optimization for inverse homogenization problems (IHPs) (Sigmund, 1994) is a powerful and effective method to find optimal microstructures. Many methods have been developed for the microstructure topology optimization, such as density-based method (Aage et al., 2015;Groen and Sigmund, 2018), isogemetric topology optimization (Gao et al., 2019(Gao et al., , 2020, bidirectional evolutionary structural optimization (Huang et al., 2012(Huang et al., , 2011, and level set method (Vogiatzis et al., 2017;Li et al., 2018). ...
Preprint
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We propose a high-performance GPU solver for inverse homogenization problems to design high-resolution 3D microstructures. Central to our solver is a favorable combination of data structures and algorithms, making full use of the parallel computation power of today's GPUs through a software-level design space exploration. This solver is demonstrated to optimize homogenized stiffness tensors, such as bulk modulus, shear modulus, and Poisson's ratio, under the constraint of bounded material volume. Practical high-resolution examples with 512^3(134.2 million) finite elements run in less than 32 seconds per iteration with a peak memory of 21 GB. Besides, our GPU implementation is equipped with an easy-to-use framework with less than 20 lines of code to support various objective functions defined by the homogenized stiffness tensors. Our open-source high-performance implementation is publicly accessible at https://github.com/lavenklau/homo3d.
... Different kinds of new constraints have been imposed during the optimization process in recent years, such that further structural design problems can be addressed effectively and practically [27][28][29][30][31][32][33][34][35][36]. In addition, topology optimization has been applied in transdisciplinary research such as biomechanical morphogenesis [37][38][39] and metamaterial designs [40][41][42][43]. ...
Article
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Ribbed slabs are widely used in the building industry. Designing ribbed slabs through conventional engineering techniques leads to limited structural forms, low structural performance and high material waste. Topology optimization is a powerful tool for generating free-form and highly efficient structures. In this research, we develop a mapping constraint optimization approach to designing ribbed slabs and shells. Compared with conventional ones, the presented approach is able to produce designs with higher performance and without isolated ribs. The approach is integrated into three optimization methods and used to design both flat slabs and curved shells. Several numerical examples are used to demonstrate the effectiveness of the new approach. The findings of this study have potential applications in the design of aesthetically pleasing and structurally efficient ribbed slabs and shells.
... The BESO method has become a widely used design technique in both academic research (e.g., thermal conduction [15], biomechanics [16,17], acoustics [18], microstructural materials [19,20], and nano-photonic designs [21]) and industrial applications (e.g., architecture [22], automotive [23], aircraft [24] and railway vehicles [25]). ...
Article
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The bi-directional evolutionary structural optimisation (BESO) has attracted much interest in recent decades. However, the high computational cost of the topology optimisation method hinders its applications in large-scale industrial designs. In this study, a parallel BESO method is developed to solve high-resolution topology optimisation problems. An open-source computing platform, FEniCS, is used to parallelise the finite element analysis (FEA) and optimisation steps. Significant improvements in efficiency have been made to the FEA and the filtering process. An iterative solver, a reanalysis approach and a hard-kill option in BESO have been developed to reduce the computational cost of the FEA. An isotropic filter scheme is used to eliminate the time-consuming elemental adjacency search process. The efficiency and effectiveness of the developed method are demonstrated by a series of numerical examples in both 2D and 3D. It is shown that the parallel BESO can efficiently solve problems with more than 100 million tetrahedron elements on a 14-core CPU server. This work holds great potential for high-resolution design problems in engineering and architecture. Keywords: Topology optimisation, Bi-directional evolutionary structural optimisation, FEniCS, Parallel computing, High-resolution
... Even though the BESO can add or remove elements, it is common to start with a design full of Material 1 to avoid a topology predefinition and ensure that each of the elements has the same probability of being part of the final topology. The four elements filled with Material 2 at the center of the base cell are introduced to avoid the uniform sensitivity distribution and allow the algorithm to initialize the design process (Huang et al. 2011Picelli et al. 2015). Two different cases are presented for the NTE and the PTE problems to compare the final topology of metamaterials with different values of negative thermal expansion and mesh dependency. ...
Article
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The use of computational evolutionary strategies in the design of metamaterials with desired thermal expansion coefficients is uncommon due to the discrete nature of the design variables. This work presents a Bi-directional Evolutionary Structural Optimization (BESO) based methodology for designing orthotropic metamaterials with a specific thermal expansion coefficient using an objective function considering only the thermal expansion coefficients, with no constraints on geometry or stiffness. Topologies of the metamaterials, composed of two material phases and a void, are obtained using a material interpolation between neighboring material phases and three easy-to-implement numerical strategies to stabilize the evolutionary process. Two are on the sensitivity calculation and one is on the addition ratio’s value. The strategies applied to the sensitivity numbers are proposed to avoid the positive and negative values of the elemental sensitivity numbers and the element change between no neighboring materials. Additionally, the addition ratio’s value reduction strategy assures the convergence of the thermal expansion properties to the desired value. The homogenization method is used to obtain the equivalent thermal expansion properties of the designed materials. Some numerical examples are presented to show the potential and effectiveness of the proposed methodology.
... In addition to typical dead and live loads, dynamic seismic loads were also considered. A numerical elastoplastic analysis based on deflections and cracking patterns validated the results of evolutionary design (Huang, Radman, & Xie, 2011). ...
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Today's architects must be proficient in the majority of tools associated with construction, building, and design creation. Due to the increasing rapidness and complexity of software development, architects must innovate and modify their working methodologies. In addition, architecture students must possess software skills, thus they are taught actual software whose primary function is to facilitate the creation of a design by applying their knowledge of design with structure. Innovative structural design with topology optimization is a cutting-edge method for utilizing digital architecture. In many instances, there is a disparity between the vision of the architect and the sensibility of the engineer, as well as between the aesthetic or appearance of the structure and its corresponding skeleton. Form is the domain of the architect, while function is the responsibility of the engineer. However, both architects and engineers are typically concerned with "function," even if in very different ways. Topology optimization is a technique that maximizes the structural performance of a system by optimizing material placement within a specific design space, a specific set of loads, boundary conditions, and constraints. If the topology optimization method is taught alongside conventional methods during structural design studies at the beginning of architectural education, a model of education that is more in line with contemporary technological approaches can be provided. In this study, parametric design and topology optimization methods were taught in the 2nd term architectural design studio training course's staircase design study for the development of students' structural comprehension. This technique, which is relatively difficult for a second-semester student, has given them the ability to apply a variety of tools. MİMARİ TASARIMA GİRİŞ DERSİ İÇİN TOPOLOJİ OPTİMİZASYONU YÖNTEMİ ÖZET Bugünün mimarları inşaat, bina ve tasarım oluşturma ile ilgili araçların çoğunda yetkin olmalıdır. Yazılım geliştirmenin artan hızı ve karmaşıklığı nedeniyle, mimarlar çalışma metodolojilerini yenilemeli ve değiştirmelidir. Ek olarak, mimarlık öğrencilerinin yazılım becerilerine sahip olmaları gerekir, bu nedenle onlara, birincil işlevi tasarım bilgilerini yapı ile uygulayarak bir tasarımın oluşturulmasını kolaylaştırmak olan yazılımlar öğretilmelidir. Topoloji optimizasyonu, mimari yapıtları üretmek için son teknolojiyi kullanan yenilikçi bir yapısal tasarım yöntemdir. Pek çok durumda, mimarın vizyonu ile mühendisin duyarlılığı arasında ve yapının estetiği veya görünümü ile ona karşılık gelen iskelet arasında bir uyumsuzluk vardır. Form mimarın, işlev ise mühendisin sorumluluğundadır. Bununla birlikte, hem mimarlar hem de mühendisler, çok farklı şekillerde olsa bile tipik olarak "işlev" ile ilgilenirler. Topoloji optimizasyonu, belirli bir tasarım alanı, belirli bir yük seti, sınır koşulları ve kısıtlamalar içindeki malzeme yerleşimini optimize ederek bir sistemin yapısal performansını en üst düzeye çıkaran bir tekniktir. Mimarlık eğitiminin başlangıcında yapısal tasarım çalışmalarında geleneksel yöntemlerin yanı sıra topoloji optimizasyon yöntemi öğretilirse, çağdaş teknolojik yaklaşımlara daha uygun bir eğitim modeli sağlanabilir. Bu çalışmada, 2. dönem mimari tasarım stüdyosu eğitimi dersi kapsamında, merdiven tasarımı çalışmasında öğrencilerin yapısal kavrayışlarının geliştirilmesine yönelik parametrik tasarım ve topoloji optimizasyon yöntemleri öğretilmiştir. İkinci dönem öğrencisi için nispeten zor olan bu teknik, onlara çeşitli araçları uygulama becerisi kazandırmıştır.
... The numerical homogenization methods have been developed as a basis for multiscale design applications, i.e., topology optimization [21][22][23][24][25], and functional structural design [26][27][28][29], in which the input object is first partitioned into a voxel grid as a coarse scale, then the periodic microstructures are filled into the grid with homogeneous material properties. However, such high-frequency calling of the numerical homogenization method makes it highly time-consuming for simulation and optimization in those applications. ...
Article
Microstructures are attracting academic and industrial interest because of the rapid development of additive manufacturing. The numerical homogenization method has been well studied for analyzing mechanical behaviors of microstructures; however, it is too time-consuming to be applied to online computing or applications requiring high-frequency calling, e.g., topology optimization. Data-driven homogenization methods are considered a more efficient choice but the microstructures are limited to cubic shapes, therefore are unsuitable for periodic microstructures with a more general shape, e.g., parallelepipeds. This paper introduces a fine-designed 3D convolutional neural network (CNN) for fast homogenization of parallelepiped microstructures, named PH-Net. Superior to existing data-driven methods, PH-Net predicts the local displacements of microstructures under specified macroscopic strains instead of direct homogeneous material, empowering us to present a label-free loss function based on minimal potential energy. For dataset construction, we introduce a shape–material transformation and voxel-material tensor to encode microstructure type, base material and boundary shape together as the input of PH-Net, such that it is CNN-friendly and enhances PH-Net on generalization in terms of microstructure type, base material, and boundary shape. PH-Net predicts homogenized properties hundreds of times faster than numerical homogenization and even supports online computing. Moreover, it does not require a labeled dataset and thus the training process is much faster than current deep learning methods. Because it can predict local displacement, PH-Net provides both homogeneous material properties and microscopic mechanical properties, e.g., strain and stress distribution, and yield strength. We also designed a set of physical experiments using 3D printed materials to verify the prediction accuracy of PH-Net.
... It was suggested to build a big structure on the shore with a combination of commercial, public, and recreational activities, offering 75,000 m2 of attractive useable space. In addition to organically formed columns, this design features a 3D Extended ESO roof (Huang, Radman, & Xie, 2011). ...
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When it comes to the development of forms, TO software offers limitless options. Each new boundary condition, the shape of the design space, or the constraints, will result in the generation of a new form that is tailored to the user’s specifications. Using parametric design to implement the form-forming principles of topological optimization enables designers and architects to search for architectural form of structures and ideas in a streamlined and straightforward manner. The use of numerical methods as a basis for generating forms has led to the creation of intriguing, intricate structures with increased value. It is not constructed of prefabricated architectural solutions, but rather a unique response to a specific situation. Geometric shapes in architecture are never the optimal shapes obtained from a form-finding process driven solely by structural optimization; rather, they embody and integrate multiple criteria. It could be assumed that there is a correlation between these natural processes and the design techniques given in this book. This can also be seen as an evolutionary technique that is not constrained by the availability of calculation and analytic techniques. Each particular structure must be completely specified and modeled in order to be evaluated in digital processes. Cognitive form, through its tectonics and space, alters the established design tenets. The complete integration of structural engineering into the architectural design process does not guarantee good architecture or innovative space and forms, but it makes their existence possible. Now, more than ever before, engineers are embracing the natural world and poetically utilizing its logic to achieve architecture’s potential. Future studies will primarily concentrate on enhancing optimization techniques for skeletal structures, such as space trusses and frames. This is because it is possible to directly express computational findings as constructible configurations in the real world. There are numerous methodological obstacles, such as buckle considerations and production limits. This leaves a lot of room for ongoing research and development aimed at providing architects and engineers with consistent, efficient, and dependable computational design tools.
... The quest for optimal stiff and strong materials depends on many aspects, including the loading conditions and constituent base materials. Stiffness-optimal materials meeting theoretical upper bounds have been obtained via systematic design approaches [6,7,8,9,10]. It has been shown that plate microstructures reach the Hashin-Shtrikman bounds [11] in the low volume fraction limit and remain within 10% of the theoretical upper bounds at moderate volume fractions. ...
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This paper proposes a methodology for architecting microstructures with extremal stiffness, yield, and buckling strength using topology optimization. The optimized microstructures reveal an interesting transition from simple lattice like structures for yield-dominated situations to hierarchical lattice structures for buckling-dominated situations. The transition from simple to hierarchical is governed by the relative yield strength of the constituent base material as well as the volume fraction. The overall performances of the proposed microstructures indicate that optimal strength is determined by the buckling strength at low volume fractions and yield strength at high volume fractions, regardless of the base material's relative yield strength.
... Moreover, evolutionary structural optimization (ESO) methods have been developed, based on the concept that low stress material parts can be removed or deleted and thereafter recovered in the bi-directional formulation, known as BESO [13]. Given relative density constraints, the BESO method has been used to identify periodic patterns that optimally sustain bending loads [14], or maximize the shear and bulk resistance of a cellular material [15]. Moreover, minimum compliance based techniques have been elaborated, identifying optimal truss-like microstructural patterns [16]. ...
Article
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In the current work, a numerical method for the inverse engineering of metamaterials is elaborated. The method is based on the combination of asymptotic homogenization schemes with genetic algorithms and it makes use of the complete set of parameters contained in the target compliance tensor. As such, it can be used to compute lattice unit-cell patterns that meet target macroscale elastic, shear, Poisson’s ratio and normal to shear strain coupling performances for the first time. The elaborated formulation applies to both constant and variable target relative density metamaterial designs, identifying metamaterial architectures within and beyond orthotropy. Different relevant case-study examples are provided, highlighting the potential of the formulation to capture a wide range of effective metamaterial behaviors. The accuracy of the results is additionally verified through commercial code, dedicated Abaqus finite element models, as well as through experimental testing of 3D-printed, periodic metamaterial samples. The scheme has a substantially low computational cost, so that a wide range of inverse engineering tasks can be performed within a computing time of a few minutes, using regular power, personal computing machines.
... Because of the problem of the optimization dependency to the initial problem design variables, the researches for using binarization methods was established (i.e., the design domain is either zero or 1 from the initialization phase, updating till final design). Therefore, binarized topology optimization such as ESO, which has gained popularity [32,33], due to binarization of the initial design domain and the common doctrine of zeroing the central element value of the periodic structure to work as a trigger of inverse homogenization [34,35]. Performing ESO with this prescribed setting will lead to uniformity of design as well as limiting the chances to fall into local minima. ...
Article
Lightweight and high heat conductive solid structures are playing important role in various fields of engineering. To maximize the design performance for such structures, we investigated multiscale topology optimization for excessive lightweight heat-conductive porous structures and introduced a mathematical optimization model formulation for concurrently optimizing the structures (macrostructure) and the constitutive pores (microstructure). The microscale is considered a representative volume element and designed using the asymptotic homogenization method. For each iteration, the effective heat conductivity tensor of the microstructure is evaluated during the optimization process and used as the heat conductivity of the macrostructure. Sensitivity analysis on this concurrent optimization scheme was derived to address the macro and microstructure coupling. To broaden the scope of the research applicability, three topology optimization methods, i.e., SIMP, level set and ESO are investigated, and the results are compared and discussed. The suggested formulations showed a successful application of the concurrent multiscale optimization formulations and good coupling on the macro and microscale. Also, the formulations demonstrated a strong influence between the macro and the microscale of the design problem for the topology optimization methods. Increasing the design freedom by introducing various microstructures for the macro design domain showed superior performance associated with attaining high weight reduction. In addition, the concurrent optimization scheme has enabled the microstructures to attain a good spatial layout of materials while taking into account the weight reduction constraint. The spatial arrangements of the designed microstructures have achieved conducting heat in a shorter path toward the heatsink zone of macrostructure design. This allows attaining good performance with high weight reduction. Furthermore, numerical examples of different mesh numbers were used to study mesh dependency of multiscale topology optimization. The scope of the study was broadened by the inclusion of 3D case studies. Implementing Isosurface technique to achieve high detail model was also used to attain high detailed concurrently optimized design with minimal mesh number to minimize the computational cost. The 3D optimized case was investigated experimentally.
... For example, Sigmund (2000), Zhang et al. (2007) and Amstutz et al. (2010) designed microstructures with maximized bulk moduli. Also, the maximum shear modulus (Neves et al. 2000;Huang et al. 2011;Xia and Breitkopf 2015) and the minimization of negative Poisson's ratio (Wang et al. 2014;Yang et al. 2019) are also considered as the objective of microstructure optimization. Collet et al. (2018) and Coelho et al. (2019) designed strength-oriented microstructures by introducing stress constraints. ...
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Natural structures and some researches about artificial three-level structures have demonstrated that these structures have good performance in many aspects. For obtaining such structures, the traditional two-scale design method needs very finer meshes, which lead to expensive computational costs. Therefore, our purpose is to propose an efficient topology optimization method for designing the three-level structure with excellent performance. The proposed method that is also called the three-scale design method in this paper divides the design domains into three scales, which are connected by the homogenization method. At each scale, the optimal material layout can be found by using the SIMP (Solid Isotropic Material with Penalization) method. Then, the proposed three-scale method is integrated into the topology optimization, two optimization strategies are provided to design the three-level structure. The first design strategy considers structural compliance as an optimization objective, which is usually common in multi-scale design. The decoupled sensitivity analysis method is used to improve the computational efficiency of this algorithm. Another effective strategy is to take buckling performance as the optimization objective, it can build a direct link between the good structural performance of the multi-level structure and optimization formula. Several numerical examples are provided to verify the effectiveness of the two design strategies. Meanwhile, the results of performance analysis show that adding a third scale does improve the performance of the structure in some aspects, such as buckling performance, robustness and ultra-light.
... On the other hand, non-uniform lattice structures, i.e., with variable elementary cells, can provide a more tailored distribution of elements aimed at achieving improved mechanical performance. Generally, such problems are addressed by means of homogenization-based approaches [28][29][30][31] which are based on homogenization theory [32,33]. Chen et al. [34] presented a parametrization modeling approach based on pre-generated meshes that is applicable to large structures. ...
Article
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Metal lattice structures produced by means of additive techniques are attracting increasing attention thanks to the high structural efficiency they can offer. In order to achieve the maximum structural performance, numerical design techniques are used almost exclusively, thus based on CAE-FEM codes. Nevertheless, the current manufacturing facilities do not yet guarantee defect-free components, and, therefore, such imperfections need to be introduced in the numerical models too. The present work aims to describe a FE modelling technique for lattice structures based on the use of beam and shell elements, and therefore with a very reduced computational cost. The main structural parameters, such as weight and stiffness and strength, are used to drive the model calibration. Simple mathematical relationships, based on Experimental-CAD-FEM comparisons, are provided to estimate the error related to the numerical model in a simple and fast way. The validation was performed by three-point bending test on flat specimen with regular octet-truss microstructure both with and without external skin. The test articles were produced in Ti6Al4V and by means of the electron beam melting (EBM) technology. The results obtained are in excellent agreement with the experimental ones, indeed the maximum error is about 3%. All this indicates these methodologies as possible tools for evaluating the performance of such kinds of high-tech structures.
... This method is a bridge that connects the optimization of micro and macro scales. Based on this design method, many algorithms have been developed to optimize the geometry of the material unit to obtain the desired or extreme effective performance, such as negative Poisson's ratios [20,21], limit thermal expansion [22], limit shear modulus or bulk modulus [23,24], functional graded properties [25,26] and others. However, most of the above methods mainly focus on the design of microstructures, but lack the simultaneous optimization of macrostructure. ...
Article
Cellular structures are often used to improve the stiffness, fatigue strength, damage tolerance, and other superior functioning properties of structures in engineering. This paper presents a novel multiscale concurrent topology optimization method to optimize structures which are periodically filled with multiple microstructures and connected by solid interfaces. This method generates better structural performance and does not need to preprocess the initial design domain. It enables to save the computational resources, and improves the freedom of structural optimization design to a certain extent. Solid interface layers are built to connect different microstructure blocks, and hence the full degrees of design freedom of microstructures can be further explored. At the macroscale, a novel piecewise projection and a series of gradient-based filtering operations are proposed to distinguish the microstructure blocks and interface layers, respectively. Then, an improved ordered solid isotropic material with penalization (SIMP) method is proposed to optimize the spatial distribution of different microstructures with an affordable computational cost. At the microscale, the microstructures are generated by the numerical homogenization method. Microstructures with different volume constraints are treated as different materials, and the volume fraction limit value of the microstructure corresponds to the design variable of the improved ordered SIMP method. Finally, the compliance minimization problem under the constraint of material volume fractions is investigated, and sensitivity analysis is derived. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed method.
... homogenized scale. Some of the methodologies adopted to perform topology optimization at the macroscale have been employed to design the periodic internal structure of cellular materials (see, for instance, Sigmund 1994;Huang et al. 2011;Noël and Duysinx 2017). The high versatility of direct and inverse homogenization justifies the adoption of these techniques for diverse applications (see, e.g., Ivarsson et al. 2020;Bruggi and Corigliano 2019). ...
Article
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A flexible problem-specific multiscale topology optimization is introduced to associate different areas of the design domain with diverse microstructures extracted from a dictionary of optimized unit cells. The generation of the dictionary is carried out by exploiting micro-SIMP with AnisoTropic mesh adaptivitY (microSIMPATY) algorithm, which promotes the design of free-form layouts. The proposed methodology is particularized in a proof-of-concept setting for the design of orthotic devices for the treatment of foot diseases. Different patient-specific settings drive the prototyping of customized insoles, which are numerically verified and successively validated in terms of mechanical performances and manufacturability.
... The boundary conditions are commonly considered with periodic displacement and anti-periodic traction. Predicting the homogenized physical properties has been considered as a basis of many applications, i.e., topological optimization [3,4,5,6], and functional structural design [7,8,9,10]. The most popular computational architecture is the voxel-based two-scale microstructure framework, in which the object is first partitioned into a voxel grid as a coarse scope, then the periodic microstructures are filled into the grid with pre-homogenized material properties for simulation and optimization. ...
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With the rapid development of additive manufacturing, microstructures are attracting both academic and industrial interests. As an efficient way of analyzing the mechanical behaviors of microstructures, the homogenization method has been well studied in the literature. However, the classic homogenization method still faces challenges. Its computational cost is high for topological optimization that requires highly repeated calculation. The computation is more expensive when the microstructure is deformed from a regular cubic, causing changes for the virtual homogeneous material properties. To conquer this problem, we introduce a fine-designed 3D convolutional neural network (CNN), named DH-Net, to predict the homogenized properties of deformed microstructures. The novelty of DH-Net is that it predicts the local displacement rather than the homogenized properties. The macroscopic strains are considered as a constant in the loss function based on minimum potential energy. Thus DH-Net is label-free and more computation efficient than existing deep learning methods with the mean square loss function. We apply the shape-material transformation that a deformed microstructure with isotropic material can be bi-transformed into a regular structure with a transformed base material, such that the input with a CNN-friendly form feeds in DH-Net. DH-Net predicts homogenized properties with hundreds of acceleration compared to the standard homogenization method and even supports online computing. Moreover, it does not require a labeled dataset and thus can be much faster than current deep learning methods in training processing. DH-Net can predict both homogenized material properties and micro-mechanical properties, which is unavailable for existing DL methods. The generalization of DH-Net for different base materials and different types of microstructures is also taken into account.
... In this context, a single-or a multi-objective topology optimization at the microscale can drive the design of new unit cells matching target properties at the macroscale, potentially in a multi-physics framework. For instance, the optimization of homogenized elastic properties is tackled in [29][30][31] with the aim of maximizing the bulk (or shear) modulus. To this aim, the authors control specific components of the homogenized elastic tensor or resort to the minimization of the compliance of a given structural part. ...
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We present a new algorithm to design lightweight cellular materials with required properties in a multi-physics context. In particular, we focus on a thermo-mechanical setting, by promoting the design of unit cells characterized both by an isotropic and an anisotropic behaviour with respect to mechanical and thermal requirements. The proposed procedure generalizes microSIMPATY algorithm to a multi-physics framework, by preserving all the good properties of the reference design methodology. The resulting layouts exhibit non-standard topologies and are characterized by very sharp contours, thus limiting the post-processing before manufacturing. The new cellular materials are compared with the state-of-art in engineering practice in terms of thermo-mechanical properties, thus highlighting the good performance of the new layouts which, in some cases, outperform the consolidated choices.
... Material design with desired bulk and shear moduli is certainly one of the active topics. Existing works include, but are not limited to, the design of materials with maximum bulk/shear modulus or with a prescribed shear modulus [46,47,48,49]. Topology optimization techniques were used in these designs. ...
Article
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In recent years, machine learning methods have been applied increasingly in material design and discovery. While these approaches have demonstrated promising performance and produced novel materials that are unachievable previously, most existing machine learning methods are supervised methods and need millions of labeled training data. The construction of the training data requires massive computational resource and is extremely time-consuming, forming a major bottleneck of machine learning approaches. In this work, an efficient artificial neural network-based inverse design method is developed for the design of architectured composite materials with novel properties. By adopting an adaptive learning and optimization strategy, the design space can be effectively explored, thereby greatly reducing the number of required labeled training data. In addition, a generative adversarial network is used to generate design candidates which drastically reduces the number of design variables and speeds up the optimization process. The excellent performance of the method is demonstrated on the design of several novel composite materials such as materials with high toughness, high stiffness near theoretical upper bound. Compared with some existing machine learning based methods, a two-order-magnitude reduction in the number of labeled training data has been achieved while maintaining the same level of design performance.
... Replicating the osteons and canals of cortical bone and the plates and struts of trabecular bone are exceptional mathematical challenges. Researchers have attempted to define bone's unit cells with fundamental architectural shapes, triply periodic minimal surfaces (TPMS) and topology optimization [30][31][32]. Fundamental shapes include cube, diamond and tetrahedral unit cells and are straightforward to 3D print. TPMS have a repeating 3D structure whose area between any given boundaries is as small as possible, the simplest example being soap film. ...
Article
3D-printing innovations are being explored as a uniting framework for the future of individualized joint replacement. The ability to convert 2D medical images to adjustable 3D models means a patient’s own anatomy can serve as the foundation for implant design. There are three biomimetic design considerations to understand the research on these new implants. First, optimizing the unit cell of 3D models can give researchers the essential building block necessary to 3D-print reliable artificial joints. Second, adequate porosity when designing a 3D-printed biomimetic joint is a balance between strength and the need for osseointegration. Third, functionally graded material as a design principle connects unit cell and porosity to create a 3D-printed product with complex properties along different spacial axes. 3D printing offers the opportunity to incorporate biomimetic design principles that were previously unobtainable with traditional manufacturing methods.
... The homogenization-based scheme builds the cross-scale connection by bridging the microscopic structures and the macroscopic equivalent material properties (Zhang et al. 2021a, b). At the early stage, the inverse homogenization method was developed to design unit microstructures exhibiting maximum shear and bulk moduli (Huang et al. 2011;Sigmund 2000), outstanding buckling strength (Wang and Sigmund 2020), negative Poisson's ratio (Andreassen et al. 2014), etc. Moreover, the unit microstructures were also tailored to employ excellent physical features, including magnetic and electrical permittivity (Huang et al. 2012), fluid permeability (Guest and Prévost 2006), acoustic wave propagation (Zhang et al. 2021a, b) and others. ...
Article
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Porous infill, rather than the solids, can provide high stiffness-to-weight ratio, energy absorption, thermal insulation, and many other outstanding properties. However, porous structure design to date have been majorly performed with topology optimization under small deformation assumption. The effect of porosity control under large deformation is not explored yet. Hence, this paper exploits the topological design method of porous infill structures under large deformational configuration. Specifically, the neo-Hookean hyperelasticity model is adopted to simulate the large structural deformation, and the adjoint sensitivity analysis is performed accordingly with the governing equation and constraint. The maximum local volume fractions before and after deformation are concurrently constrained and especially for the latter, the representative volume points (RVPs) are modeled and tracked for evaluating the local volume fractions subject to the distorted mesh configuration. The local volume constraints are then aggregated with the P-norm method for a global expression. Iterative corrections are made to the P-norm function to rigorously restrict the upper bound of the maximum local volume. Finally, several benchmark cases are investigated, which validate the effectiveness of the proposed method.
... 5,6 With the advantages of a simple algorithm and easy connection with the finite element analysis program, it has been widely used in structural topology optimization. 14,15 The level set method was proposed in the topology optimization field by Wang et al. and Allaire et al. 8,9 It describes the boundary of the entity by the contour plane of a one-dimension higher function, and the evolution process of the boundary is determined by solving the Hamilton-Jacobi equation. By further developing, the level set method, a parametric way with radial basis functions is used, 16 which can avoid solving the Hamilton-Jacobi equation and improve calculation efficiency. ...
Article
Based on the variable density method, this paper proposes a boundary density evolutionary topology optimization method. The method uses a material interpolation model without penalization. Combined with the density grading filtering method, an optimal topology with only 0/1 cells can be obtained. Compared with the solid isotropic microstructures with penalization method (SIMP), no penalty factor is required in the material interpolation model; compared with the evolutionary structural optimization method (ESO), intermediate‐density elements are allowed in the optimization process, but the concept of gradually removing the low‐utilization materials near the boundary in the ESO method is retained. After the optimal result is obtained, the structural boundary element is processed by the level set of nodal strain energy, and the optimization result with smooth boundaries similar to the level set method (LSM) can be obtained. The proposed method has the superiority of the variable density method, and it also combines the advantages of the evolutionary method and the level set method, so which is named as boundary density evolution (BDE) method. The four static and one dynamic optimization examples illustrate the stability and efficiency of the proposed method.
... Guest and Prévost (2006) optimized multifunctional materials for maximized stiffness and fluid permeability. Metamaterials with negative Poisson's ratio (Wang et al., 2014;Alomarah et al., 2020), functionally graded properties (Radman et al., 2013), maximum bulk or shear modulus (Huang et al., 2011), and so forth have been thoroughly investigated. ...
Article
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This research presents a lattice structure topology optimization (LSTO) method that significantly expands the design space by creating a novel candidate lattice that assesses an extremely large range of effective material properties. About the details, topology optimization is employed to design lattices with extreme directional tensile or shear properties subject to different volume fraction limits and the optimized lattices are categorized into groups according to their dominating properties. The novel candidate lattice is developed by combining the optimized elementary lattices, by picking up one from each group, and then parametrized with the elementary lattice relative densities. In this way, the LSTO design space is greatly expanded for the ever increased accessible material property range. Moreover, the effective material constitutive model of the candidate lattice subject to different elementary lattice combinations is pre-established so as to eliminate the tedious in-process repetitive homogenization. Finally, a few numerical examples and experiments are explored to validate the effectiveness of the proposed method. The superiority of the proposed method is proved through comparing with a few existing LSTO methods. The options of concurrent structural topology and lattice optimization are also explored for further enhancement of the mechanical performance.
... Recently, these materials are receiving increasing attention as structural and functional materials [15,16]. The latter occur widely in nature [17]. The mechanical behaviour depends mainly on the type of base material, relative density, morphology and topology [18]Erreur ! ...
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This study aims to modify and characterize an open-cell polyurethane foam with a density of ∼28.7 kg/m³ to study the mechanical behavior in static compression and cyclic fatigue. A technique is presented for transforming this type of foam from a conventional state with a positive Poisson's ratio to an auxetic transformed state with a negative Poisson's ratio (density of ∼119.59 kg/m³) via a specially designed device. The results obtained show the degree of influence of the transformation on the mechanical behavior of this type of foam. The cyclic fatigue tests of auxetic foam were carried out under control-displacement for different loading (r) levels (0.725, 0.75, 0.80, 0.85, 0.90 and 0.95). The monitoring of cyclic fatigue damage of auxetic samples allowed tracing the evolution of the stiffness degradation (F/F0) as a function of the number of cycles N taking place in two different phases. The hysteresis loops of the different r levels were identified as a function of the number of cycles. The evaluation of the dissipated energy (Ed) and damping (η) as a function of N was also carried out. The results show that the maximum forces involved between 28 N and 580 N are considerably higher than those previously found in the literature. This leads to a significant dependence of the hysteresis loops, Ed and η of the auxetic foam as a function of the number of cycles and the loading rate, with a remarkable stiffness performance compared to other auxetic foams.
... The properties are obtained generally by homogenization and can also be optimized. These optimized properties are of many kinds and include for instance terms of the thermo-elasticity tensors (Sigmund 1994) like bulk or shear modulus (Huang et al. 2011), Poisson's ratio , or thermal expansion (Sigmund and Torquato 1997). This is usually done using periodic boundary conditions (Xia and Breitkopf 2015a). ...
Article
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Multi-scale topology optimization (a.k.a. micro-structural topology optimization, MTO) consists in optimizing macro-scale and micro-scale topology simultaneously. MTO could improve structural performance of products significantly. However, a few issues related to connectivity between micro-structures and high computational cost have to be addressed, without resulting in loss of performance. In this paper, a new efficient multi-scale topology optimization (EMTO) framework has been developed for this purpose. Connectivity is addressed through adaptive transmission zones which limit loss of performance. A pre-computed database of micro-structures is used to speed up the computing. Design variables have also been chosen carefully and include the orientation of the micro-structures to enhance performance. EMTO has been successfully tested on two-dimensional compliance optimization problems. The results show significant improvements compared to mono-scale methods (compliance value lower by up to 20% on a simplistic case or 4% on a more realistic case), and also demonstrate the versatility of EMTO.
Article
An intelligent microstructural design method based on deep learning is proposed considering performance indicators that contains boundary information and homogenized elastic modules. Microstructure dataset is established by random boundary method and homogenization method. Random boundary method is proposed to design microstructures under given boundary information, and homogenization method is utilized to acquire homogenized elastic modules. A generative and adversarial network with gradient penalty is developed to establish the high-dimensional mapping between performance indicators and microstructure. The Wasserstein distance is imported to overcome mode collapse. Numerical simulation shows that the pre-trained network successfully achieved corresponding microstructure design by given performance indicators.
Article
This work deals with the multi-scale topology optimisation (TO) of multi-material lattice structures. The proposed approach is based on: non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the representative volume element (RVE), an improved multiphase material interpolation (MMI) scheme to penalise the element stiffness tensor of the multi-material RVE, the strain energy-based homogenisation method (SEHM) to carry out the scale transition. In this context, the design requirements are defined at different scales and their gradient is evaluated by exploiting the properties of the NURBS entities and of the SEHM. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the topological descriptors of the material phases composing the RVE. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, a sensitivity analysis of the optimised configuration of the RVE to the parameters tuning the shape of the NURBS entity is conducted. Finally, the influence of the starting point and of the macroscopic loads on the optimal solution is investigated.
Article
Microstructures, i.e., architected materials, are designed today, typically, by maximizing an objective, such as bulk modulus, subject to a volume constraint. However, in many applications, it is often more appropriate to impose constraints on other physical quantities of interest. In this paper, we consider such generalized microstructural optimization problems where any of the microstructural quantities, namely, bulk, shear, Poisson ratio, or volume, can serve as the objective, while the remaining can serve as constraints. In particular, we propose here a neural-network (NN) framework to solve such problems. The framework relies on the classic density formulation of microstructural optimization, but the density field is represented through the NN’s weights and biases. The main characteristics of the proposed NN framework are: (1) it supports automatic differentiation, eliminating the need for manual sensitivity derivations, (2) smoothing filters are not required due to implicit filtering, (3) the framework can be easily extended to multiple-materials, and (4) a high-resolution microstructural topology can be recovered through a simple post-processing step. The framework is illustrated through a variety of microstructural optimization problems.
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Designing composite materials according to the need of applications is fundamentally a challenging and time-consuming task. A deep neural network-based computational framework is developed in this work to solve the forward (predictive) and the inverse (generative) composite design problem. The predictor model is based on the popular convolution neural network architecture and trained with the help of finite element simulations. Conventionally, a large amount of training data is required for accurate prediction from neural network models. A novel data augmentation strategy is proposed in this study which significantly saves computational resources in the training phase. It shown that the data augmentation approach is general and can be used in any setting involving periodic microstructures. We next use, the property predictor model as a feedback mechanism in the neural network-based generator model. The proposed predictive-generative model is used to obtain the composite microstructure for various requirements such as maximization of elastic properties, specified elastic constants, etc. The efficacy of the proposed predictive-generative model is demonstrated by solving certain class of problems. It is envisaged that the developed model coupled with data augmentation strategy will significantly reduce the cost and time associated with the composite material designing process for varying application requirements.
Article
This paper presents the microstructural topology optimization for the prescribed relaxation moduli of viscoelastic composites. The microstructures of the viscoelastic composites are assumed to be composed of periodic unit cells. A numerical homogenization method is systematically developed to obtain the effective relaxation modulus of the unit cell. The density-based topology optimization is adopted to find optimal microstructures of the viscoelastic composites. The design objective is to minimize the difference between the prescribed relaxation moduli and effective relaxation moduli at specified time instants. An adjoint sensitivity analysis is developed to compute the derivatives of the effective relaxation moduli with respect to the design variable. Several numerical examples demonstrate the effectiveness of the proposed approach. Various optimized microstructures of viscoelastic composites are presented and discussed based on the volume fraction of each constituent material, relaxation time, and stiffness.
Article
The porous scaffold – a substitute for artificial bone – is vital to bone defect repair and bone tissue regeneration in tissue engineering. In consideration of the limitations of few existing design methods, the significant difference between the internal pore structure of porous scaffolds and natural bone microstructure, and the low controllability of design methods. Based on Voronoi-Tessellation (VT) and random function theory, this paper proposed a design method with controllable pore density, adjustable pore shape and size, and heterogeneous pore distribution. The stress change law of porous scaffolds under compression was analyzed using finite element simulation experiments; the relationship between the porosity and the structural strength of porous scaffolds was obtained. Furthermore, the anisotropy of porous scaffolds was studied, and the change law of the macroscopic equivalent elastic modulus with the change in porosity of porous scaffolds was acquired. Under the same porosity, the mechanical properties of the porous scaffold with a uniform pore distribution and the porous scaffold designed in this paper were analyzed and compared; the damage extension laws of two different porous scaffolds were obtained. The results demonstrate that the design method proposed in this paper has high controllability, and the parametric modeling process is both random and controllable. The internal pore structures of porous scaffolds generated by the design method are closer to the microstructure of natural bone tissue – the structural strength satisfies the requirements of artificial implants. Graphical abstract [Formula: see text]
Article
This paper develops a multi-scale topology optimization method that realizes optimized structural stiffness design while achieves inter-connectivity among the heterogeneous unit cells. Specifically, about the technical details, lattice structure topology optimization (LSTO) is conducted by optimizing the parameter field of the specially-designed multi-variable lattices, through which the optimized lattice parameters reflect the density and stress states of the associated macro-element. Then, the macro-elements with close lattice parameters are gathered into clusters, providing the initial guess for the next-step freeform optimization. Finally, multiscale topology optimization (MTO) through the inverse homogenization approach is performed to further design the unit cell structures. The unit cell structures for each cluster are forced to be identical to save homogenization-related computational resources and the interconnectivity is ensured due to the optimized and perfectly connected initial guess from LSTO. Using the proposed method, three classical numerical examples are studied that prove the effects of improved mechanical performance, ensured micro-structure inter-connectivity, and the affordable computing scale. Finally, mechanical tests are conducted to verify the design performance benefits of the proposed method.
Article
Topology optimization (TO) of large-scale structures is a computationally demanding process that challenges its widespread adoption in industrial design. We present a new partition based TO framework applicable to the design of large-scale problems and apply it to mechanical stiffness optimization problems. The method employs, first, a data-driven physical partitioning of strain energy contours to partition the design domain, and then a partition based TO. In contrast to conventional topology optimization in which the number of design variables is typically equal to the number of elements in the discretized domain, the proposed method assigns density design variables to each spatial partition leading to significant computational cost reduction and convergence acceleration. The constitutive matrix required for finite element analysis is iteratively determined according to the Hashin-Shtirkman upper bounds based on partition densities. Once the optimized partition densities are achieved, the manufacturable binary structure is realized by mapping a set of generated high-performance isotropic microstructure cells onto partition elements. To validate the capability, effectiveness, and efficiency of the proposed method, several numerical examples are provided. The optimized structures using macrostructural analysis exhibit comparable performance to the conventional SIMP method for small-size problems. Besides, the TO results for the large-scale problems suggest significant computational cost efficiency.
Article
Bio-inspired engineering design has drawn increased attention in recent years for the excellent structural and mechanical properties exhibited by biological systems. In this study, mechanical properties (bulk and shear moduli) of a lattice structure inspired by morphological characteristics of young balsa wood were investigated. The effect of design parameters on the mentioned properties was investigated through an energy-based homogenization technique. A genetic algorithm was used to determine the optimal lattice topology under two different cases (maximization of bulk modulus/Maximization of combined moduli (bulk and shear modulus)). Results demonstrate the effectiveness of the bio-inspired lattice structure. An interesting pattern has been found to guide the cellular material design.
Article
Metamaterials are synthetic materials designed to have unique properties like negative Poisson ratio (NPR). NPR metamaterials, also known as auxetics, offer significant value in applications that require high energy absorption, e.g., packing materials, medical knee pads, footwear. However, material uncertainty arising out of manufacturing tolerance, inhomogeneity of material properties, and others could lead to significant variations in the response of the metamaterials. Thus, a SIMP based robust topology optimization (RTO) design for the NPR metamaterials under material uncertainty is investigated. The weighted mean and variance of the deterministic objective function is utilized to form a robust objective function. The variation in effective Poisson’s ratio with respect to the lower bound goes from 15.40% to 105% with deterministic topology optimization. In contrast, RTO produces more stable designs and shows the variation of only 1.72% to 2.54%. Several parametric studies are used to demonstrate the feasibility of the proposed RTO methodology.
Article
In this research, a new procedure is presented for two-scale construction of three-dimensional microstructures. Based on two-point correlation functions and phases recovery algorithm the fine and coarse scale microstructures are constructed separately for arbitrary volume fractions. All required correlation functions are calculated using an exponentially decreasing sinusoidal function that makes all constructed microstructures to be bicontinuous for a wide range of volume faction. Using the Boolean subtraction of fine scale microstructure from the coarse scale one, the intended multiscale microstructure is constructed. Connectivity evaluation of constructed microstructure reveals that even for low volume fractions, the connectivity of different phase regions will be established throughout the three-dimensional volume element that is crucial if conduction of heat, ion, electron, etc. is required. Another important property of the constructed microstructure is that, due to the periodic assumption, the final sections of the volume element gradually become similar to the initial ones in three dimensions and as a result it is possible to use the volume element as a unit cell to construct an arbitrary large structure with complete continuity, smoothness and isotropy.
Article
Here we consider the possible bulk and shear moduli of planar polycrystals built from a single crystal in various orientations. Previous work gave a complete characterization for crystals with orthotropic symmetry. Specifically, bounds were derived separately on the effective bulk and shear moduli, thus confining the effective moduli to lie within a rectangle in the (bulk, shear) plane. It was established that every point in this rectangle could be realized by an appropriate hierarchical laminate microgeometry, with the crystal taking different orientations in the layers, and the layers themselves being in different orientations. The bounds are easily extended to crystals with no special symmetry, but the path to constructing microgeometries that achieve every point in the rectangle defined by the bounds is considerably more difficult. We show that the two corners of the box having minimum bulk modulus are always attained by hierarchical laminates. For the other two corners we present algorithms for generating hierarchical laminates that attain them. Numerical evidence strongly suggests that the corner having maximum bulk and maximum shear modulus is always attained. For the remaining corner, with maximum bulk modulus and minimum shear modulus, it is not yet clear whether the algorithm always succeeds, and hence whether all points in the rectangle are always attained. The microstructures we use are hierarchical laminate geometries that at their core have a self-similar microstructure, in the sense that the microstructure on one length scale is a rotation and rescaling of that on a smaller length scale.
Article
This paper presents the Bézier extraction based isogeometric approach to multi-objective topology optimization (TO) of periodic microstructures. The approach utilizes the B-Splines based Bézier elements as the finite element (FE) representation at both macro and micro levels. The equivalent elastic properties of the representative volume element (RVE) are computed by the numerical homogenization method with the periodic boundary conditions (PBCs) on the control points (CPs) of the B-Spline mesh. The multi-objective optimization problems including the material bulk modulus maximization, negative Poisson's ratio (NPR) and concurrent topology optimization (CTO) of the composite structures are formulated by using the Bézier elements based isogeometric analysis (IGA). The distance-based density distribution on the CPs of the B-Spline mesh is proposed as the initialization for designing the RVE, which is more robust than the uniform density distribution towards the optimal results. Several numerical examples are presented to illustrate the effectiveness of the proposed approach, and a variety of isotropic and anisotropic RVEs and composite structures are obtained. Meanwhile, various influences on the optimal design of the RVE and macrostructure are also discussed.
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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.
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Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.
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Being one of new generation of composites, functionally graded materials (FGMs) possess gradually changed physical properties due to their compositional and/or microstructural gradients. In literature, exhaustive studies have been carried out in compositional modeling and design, while limited reports are available for microstructural optimization. This article presents an inverse homogenization method for the design of two-phase (solid/void) FGM microstructures, whose periodic base cells (PBCs) vary in a direction parallel to the property gradient but periodically repeat themselves in the perpendicular direction. The effective elasticity tensor at each PBC is estimated in terms of the homogenization theory. The overall difference between the effective tensor and their target is minimized by seeking for an optimal PBC material topology. To preserve the connectivity between adjacent PBCs, three methods, namely connective constraint, pseudo load, and unified formulation with nonlinear diffusion are proposed herein. A number of two-dimensional examples possessing graded volume fraction and Young’s modulus but constant positive or negative Poisson’s ratios are presented to demonstrate this computational design procedure.
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This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.
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Achieving negative permittivity and negative permeability signifies a key topic of research in the design of metamaterials. This paper introduces a level-set based topology optimization method, in which the interface between the vacuum and metal phases is implicitly expressed by the zero-level contour of a higher dimensional level-set function. Following a sensitivity analysis, the optimization maximizes the objective based on the normal direction of the level-set function and induced current flow, thereby generating the desirable patterns of current flow on metal surface. As a benchmark example, the U-shaped structure and its variations are obtained from the level-set topology optimization. Numerical examples demonstrate that both negative permittivity and negative permeability can be attained.
Article
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.
Article
This paper presents two computational models to design the periodic microstructure of cellular materials for optimal elastic properties. The material equivalent mechanical properties are obtained through a homogenization model. The two formulations address the problem of finding the optimal representative microstructural element for periodic media that maximizes either the weighted sum of the equivalent strain energy density for specified multiple macroscopic strain fields, or a linear combination of the equivalent mechanical properties. Constraints on material volume fraction and material symmetries are considered. The computational models are established using finite elements and mathematical programming techniques and tested in several numerical examples.
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Cellular solids include engineering honeycombs and foams (which can now be made from polymers, metals, ceramics, and composites) as well as natural materials, such as wood, cork, and cancellous bone. This new edition of a classic work details current understanding of the structure and mechanical behavior of cellular materials, and the ways in which they can be exploited in engineering design. Gibson and Ashby have brought the book completely up to date, including new work on processing of metallic and ceramic foams and on the mechanical, electrical and acoustic properties of cellular solids. Data for commercially available foams are presented on material property charts; two new case studies show how the charts are used for selection of foams in engineering design. Over 150 references appearing in the literature since the publication of the first edition are cited. It will be of interest to graduate students and researchers in materials science and engineering. © Lorna J. Gibson and Michael F. Ashby, 1988 and Lorna J. Gibson and Michael F. Ashby, 1997.
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There are several well-established techniques for the generation of solid-void optimal topologies such as solid isotropic material with penalization (SIMP) method and evolutionary structural optimization (ESO) and its later version bi-directional ESO (BESO) methods. Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed in this paper. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures with one or multiple materials. The results show that the optimal designs from the present BESO method are independent on the degree of penalization. The resulted optimal topologies and values of the objective function compare well with those of SIMP method.
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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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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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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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In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level-set method as a region representation with a moving boundary model. As a boundary optimization problem, the structural boundary description is implicitly embedded in a scalar function as its iso-surfaces. Such level-set models are flexible in handling complex topological changes and are concise in describing the material regions of the structure. Furthermore, by using a simple Hamilton–Jacobi convection equation, the movement of the implicit moving boundaries of the structure is driven by a transformation of the objective and the constraints into a speed function that defines the level-set propagation. The result is a 3D structural optimization technique that demonstrates outstanding flexibility in handling topological changes, the fidelity of boundary representation, and the degree of automation, comparing favorably with other methods in the literature based on explicit boundary variation or homogenization. We present two numerical techniques of conjugate mapping and variational regularization for further enhancement of the level-set computation, in addition to the use of efficient up-wind schemes. The method is tested with several examples of a linear elastic structure that are widely reported in the topology optimization literature.
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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.
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Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural optimization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summarizes latest developments in BESO for stiffness optimization problems and compares BESO with other well-established optimization methods. Through a series of numerical examples, this paper provides answers to those critical comments and shows the validity and effectiveness of the evolutionary structural optimization method. KeywordsEvolutionary Structural Optimization (ESO)-Bi-directional ESO (BESO)-Local optimum-Optimal design-Displacement constraint
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
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
We utilize two different approaches, homogenization theory and discrete network analyses, to study the mechanical and transport properties of two-dimensional cellular solids (honeycombs) consisting of either hexagonal, triangular, square or Voronoi cells. We exploit results from homogenization theory for porous solids (in the low-density limit) to establish rigorous bounds on the effective thermal conductivity of honeycombs in terms of the elastic moduli and vice versa. It is shown that for hexagonal, triangular or square honeycombs, the cross-property bound relating the bulk modulus to the thermal conductivity turns out to be an exact and optimal result. The same is true for the cross-property bound linking the shear or Young's modulus of the triangular honeycomb to its conductivity. For low-density honeycombs, we observe that all of the elastic moduli do not depend on the Poisson's ratio of the solid phase. The elastic-viscoelastic correspondence principle enables us to conclude that all of the viscoelastic moduli of honeycombs in the low-density limit are proportional to the complex Young's modulus of the solid phase. Such structures have real Poisson's ratios and the loss tangent is the same for any load.
Article
This is the second part of a three-paper review of homogenization and topology optimization. In the first paper, we focused on the theory and derivation of the homogenization equations. In this paper, motives for using the homogenization theory for topological structural optimization are briefly explained. Different material models are described and the analytical solution of the homogenization equations for the so called “rank laminate composites” is presented. The finite element formulation is explained for the material model, based on a miscrostructure consisting of an isotropic material with rectangular voids. Using the periodicity assumption, the boundary conditions are derived and the homogenization equations are solved, and the results to be used in topology optimization are presented. The third paper deals with the use of homogenization for structural topology optimization by using optimality criteria methods.
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.
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