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... Later, the Bi-directional Evolutionary Structural Optimization (BESO) method, proposed by Yang et al. [3], made possible not only to remove but also add elements to the structure during the optimization process. Shortly thereafter, the so-called new BESO algorithm [4] has been used in many works, since also provides solutions for some important numerical problems, such as checkerboard pattern and mesh-dependence. All these improvements made the method even more popular and highly used in many engineering applications [5,6]. ...

... (1), and knowing that ∇p · n = 0 for the rigid wall case, the weak form can then be written as Eqn. (4). ...

... This section presents the Bi-directional evolutionary structural optimization (BESO) method related with acoustic problems [4,5,13,15]. The main steps of the methodology are given as follows: ...

In the past few years, acoustic-mechanical devices have become widely used, which increased the demand for noise control solutions. One of the approaches to solve such
problems consists in designing noise barriers. However, finding the best topology for these barriers can be a complex task. In this work it is proposed a methodology to design periodic noise barriers, composed of rigid materials, using the bi-directional evolutionary structural optimization (BESO) method. The acoustic problem is modeled using the Helmholtz equation and solved by the finite element procedure, while a material interpolation scheme is used for switching acoustic and rigid elements. The optimization problem is defined as the minimization of the average square pressure amplitude in a specific region of the acoustic domain, while the volume of the barrier is reduced. The sensitivity analysis was carried out by the gradient of the objective function with respect to the design variable. Two cases are presented in order to show the capabilities of the proposed approach. In the first one, periodic conditions are imposed in the entire system, while in the second non-periodic conditions are considered. The results showed that, although the barrier volume was reduced by 35% in both cases, the objective function decreased at least 68.80%.

... Several techniques based on topology optimization have been used by researchers in recent decades, such as the Solid Isotropic Material with Penalization (SIMP) (Bendsøe & Sigmund, 2003), Level-Set Method (LSM) (Sethian and Wiegmann, 2000), and Bidirectional Evolutionary Structural Optimization (BESO) (Huang & Xie, 2010). Among the researches that address the subject of optimization in structural analysis with multi-material, it is possible to mention Wang and Wang (2004) for employing a multi-phase level-set model in a multi-material criteria domain. ...

... In standard approaches, topology optimization is used to adjust some design parameters to achieve some objectives, such as minimum volume, without violating certain constraints that usually is found in engineering problems. According to Huang and Xie (2010), for the stiffness optimization of structures with multiple materials, it is assumed that the elasticity moduli of different materials ( ) are ranked 1 , 2 , ⋯ , , ⋯ , ...

... As for the addition or removal of the elemental material by the ESO method, this occurs through numerical analysis of the sensitivity number -Equation (3). Thus, for the case of a standard ESO multi-material, the sensitivity number for stiffness optimization is expressed as a function of the elemental compliance, an inverse measure of the importance of the element to the overall stiffness of the member, as described in Huang and Xie, 2010. ...

... Although topology optimization has good prospects, it is recognized as the most difficult task in the field of structural design. Specifically, topology optimization finds optimal models by determining the best locations and geometry of cavities in the [1]. Topology optimization is performed using the results of a finite element analysis (FEA). ...

... A topology optimization problem aims to minimize compliance, while meeting various constraints, such as a given amount of material, weight, manufacturing requirements, costs, etc., Huang and Xie, [2010] [1]. The most comprehensive goal in topological optimization is to minimize compliance, equation (1). ...

... A topology optimization problem aims to minimize compliance, while meeting various constraints, such as a given amount of material, weight, manufacturing requirements, costs, etc., Huang and Xie, [2010] [1]. The most comprehensive goal in topological optimization is to minimize compliance, equation (1). ...

Because of the rapid development key industries like aerospace, energy, rail transportation, engineering machinery and other industries there exist a growing demand for high speed and precise manufacturing of large and heavy parts. CNC Machine Tools with kinematical feed chains with great distance between slides, Gantry type, are the main tool to machine large and heavy parts. In such type of machines, the mobile element (the mechanical coupling between the two kinematical feed chains) has a significant effect upon the static and dynamic characteristics of the machine. Deformation of the mobile element has a significant impact upon machining accuracy. A study of the static proprieties and the deformation that effect the mobile element is extremely useful for improving the machining accuracy and efficiency of the machine. The weight of the mobile element it is extremely important and effects the overall dimension and weight of the machine. Using FEA analysis on a 3D CAD model this papers objectives are mesh optimization and topological optimization of the mobile element. This study will show that the topological optimization preformed on the mobile element has reduce the inertial forces, 10 to 30 %, while maintaining rigidity.

... Topology optimisation originated in the work of Michell who proposed criteria for the generation of optimal frame structures [21]. More recently, the field has advanced with the work of Bendsøe & Kikuchi in 1988 with the introduction of the "homogenisation method" of topology optimisation [22]: solid isotopic material with penalisation (SIMP) [23], evolutionary structural optimisation (ESO) [24], bi-directional evolutionary structural optimisation (BESO) [24], level-set method [25], and ground structure method [26] (Figure 2). Commonly, topology optimisation (TO) requires some form of design domain discretisation. ...

... Topology optimisation originated in the work of Michell who proposed criteria for the generation of optimal frame structures [21]. More recently, the field has advanced with the work of Bendsøe & Kikuchi in 1988 with the introduction of the "homogenisation method" of topology optimisation [22]: solid isotopic material with penalisation (SIMP) [23], evolutionary structural optimisation (ESO) [24], bi-directional evolutionary structural optimisation (BESO) [24], level-set method [25], and ground structure method [26] (Figure 2). Commonly, topology optimisation (TO) requires some form of design domain discretisation. ...

... This research applies the BESO method as a basis for the design of truss structures; however, any TO methods which are applied in a similar manner as BESO can be used with the proposed method. The underlying BESO MATLAB script used in this work can be found in [24]. Discussion of the theory behind topology optimisation or the implementation of the BESO method is beyond the scope of this work; however, interested readers can find more information in [24]. ...

Generative design refers to the automated design of components through the use of computer-aided engineering (CAE) tools. This is an enabling technology which allows reduced lead times in component design, particularly for custom and unique parts; improved certification of components; efficient exploration of the design space; and results in optimised design outcomes. Topology optimisation (TO) is a systematic tool for defining efficient material distributions for given boundary conditions, whereas shape optimisation (SO) is useful for refining identified part geometries with respect to design constraints. These qualities make TO and SO suitable for automating the embodiment and detailed design phases respectively for use in generative design. However, integration of these tools is hindered by challenges in the parameterisation of TO solutions, which are typically manually resolved. This manual resolution imposes a limit on the rate in which data can be generated as well as the scope of the solution space. In response to this challenge, a novel parameterisation method is proposed to directly integrate topology and shape optimisation to achieve an automated design process. The benefits of the proposed method are reduced computational cost and improved automation, which is suitable for investigating a large number of potential solutions and selecting and refining an optimal design for a given loading condition. The proposed method has been used within this paper for the design of 2D trusses as a proof of concept.

... The goal of TO is to determine the optimal distribution of the material within a given design domain under prescribed requirements. Since its introduction in the late 1980s [9], numerous aproaches have been proposed in the literature [10,11], as the homogenization method [9,12], Solid Isotropic Material with Penalization (SIMP) method [10,13], Level Set Method (LSM) [14,15], Moving Morphable Components (MMC) method [16], Evolutionary Structural Optimization (ESO) method [17], and its improved version, i.e., the Bi-directional ESO [18,19] method. ...

... The computation of the derivatives of both objective and constraint functions with respect to the design variables is needed to solve problem (19) through a deterministic algorithm. This task is achieved by exploiting the local support property of Equation (9). ...

... Moreover, the Globally-Convergent Method of Moving Asymptotes (GC-MMA) algorithm [42] has been used to perform the solution search for the constrained non-linear programming problem (CNLPP) of Equation (19). The parameters governing the behavior of the the GC-MMA algorithm are given in Table 1. ...

This work deals with heat conduction problems formulation in the framework of a CAD-compatible topology optimization method based on a pseudo-density field as a topology descriptor. In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field and, on the other hand, on the well-known Solid Isotropic Material with Penalization (SIMP) approach. The resulting method is then referred to as NURBS-based SIMP method. In this background, heat conduction problems have been reformulated by taking advantage of the properties of the NURBS entities. The influence of the integer parameters, involved in the definition of the NURBS hyper-surface, on the optimized topology is investigated. Furthermore, symmetry constraints, as well as a manufacturing requirement related to the minimum allowable size, are also integrated into the problem formulation without introducing explicit constraint functions, thanks to the NURBS blending functions properties. Finally, since the topological variable is represented by means of a NURBS entity, the geometrical representation of the boundary of the topology is available at each iteration of the optimization process and its reconstruction becomes a straightforward task. The effectiveness of the NURBS-based SIMP method is shown on 2D and 3D benchmark problems taken from the literature.

... To improve the performance and robustness of the ESO method, the bi-directional evolutionary structural optimization (BESO) method allows void elements to be readmitted during the form-finding process. [17][18][19]. The solid isotropic material with penalization (SIMP) method gradually changes the elemental properties between solid and void with material interpolation schemes [20,21]. ...

... where the compliance C is equal to the work of external force, U the global displacement vector, K the global stiffness matrix, F the force vector, i v the elemental volume, f v the target volume fraction of rib material, and min ρ a non-zero design variable to avoid singularity [17,20]. The sensitivity of each element is calculated as (no negative sign for BESO): ...

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.

... Further, to optimize the elastic tensor of scaffolds, the SIMP method is applied (Sturm et al., 2010). Besides, through the use of ESO-based topology optimization, a series of unit cells exhibiting maximum shear and bulk modulus with predefined stiffness ratios and functionally graded structure were obtained (Huang and Xie, 2010). ...

... The rutile and anatase phases present in the TiO 2 layer showed the capability of formation of HA deposit, an ECM component of natural bone that promotes bone regrowth and improved osseointegration (de Jonge et al., 2008). An evenly distributed and well separated pore with a maximum of 1 μm in diameter was observed on the porous TiO 2 layer when anodization was carried out in the presence of H 2 SO 4 and H 3 PO 4 (Xie et al., 2010). On the other hand, anodization in the presence of acetic acid (CH 3 COOH) led to the formation of a much more heterogeneous surface, with larger craters and grooves roughly 200 μm in size. ...

... Therefore, the BESO strategy differs from the ESO method, in terms of the criterion to update the stiffness tensor of each element: instead of using the stress state as indicator of rejection, the displacement field is used. The method has been reformulated in [21,22], by adding features to obtain mesh-independent results, without checker-board pattern and by introducing a sensitivity number averaging method to speed up convergence. The works on ESO and BESO methods mentioned above make use of the socalled hard-kill technique to remove/add elements. ...

... To this purpose, the STL file has been elaborated in CATIA ® environment and it has been manually split in 4 patches for an overall number of N = 4151 TPs, as indicated in Tab. 7. 1. The results of the SPM to get the surface parametrisation (for each patch) are illustrated in Fig. 7.21, while the TPs cloud and the optimal NURBS surfaces at the end of the MEP and of the DOP are shown in Fig. 7. 22. ...

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.

... In this section, a topological optimization is presented for maximizing the fracture resistance of elastoplastic anisotropic material described in Section 2. The method is based on the BESO method [71][72][73]. In the proposed procedure, the fracture resistance is defined as the cumulative external work of the structure for a complete loading, from the initiation of cracks to failure, under the constraint of material volume fraction. ...

... containing the discrete values for all elements of the mesh is defined. Following the well-known BESO formulation [71], the design variables take the values ∈ [0; 1], = 1 for solid material, whereas = 0 corresponds to voids. For stability considerations, displacement-controlled loading is adopted. ...

3D-printed structures may be characterized by anisotropic fracture behavior because of their layered nature. Depending on the orientation of the sample during the layer deposition, a completely different mechanical response can be obtained, ranging from quasi-brittle to elastoplastic, and with large variations in the maximum stress to failure. In this study, an optimization framework is proposed for 3D-printed samples to maximize their resistance to fracture with respect to both the orientation of the deposited layers during the process and the topology of the sample. To achieve this, a phase-field anisotropic elastoplastic fracture model is combined with a Bidirectional Evolutionary Structural Optimization topology optimization. The model makes it possible to predict the response of the structure until failure with respect to the orientation of the deposited layers in the 3D-printing process and then optimize this orientation to maximize the mechanical response. A large increase in fracture resistance can be obtained by optimizing the orientation, and a significant increase in fracture resistance can be achieved using the present nonlinear anisotropic topology optimization compared with the use of linear topology optimization.

... It should be noted that the BESO method performs iterative optimization through an important parameter called the evolutionary rate ER whose value is normally set to 2% [8]. For the SGA-BESO method, the main control parameters are the minimum and maximum values of P c and P m (i.e., P c_min , P c_max , and P m_min and P m_max ). ...

... where τ is the allowable convergence limit (set to 0.1% in this paper). Equation (8) indicates that the change in structural compliance over the last 10 iterations is sufficiently small. Table 1 shows all the final results obtained from the three methods. ...

To accelerate the convergence rate and obtain robust optimal results with clear profiles of structural topologies, this paper proposes a hybrid multi-population genetic algorithm (MPGA) and bi-directional evolutionary structural optimization (BESO) method for structural topology optimization. Each element in the design domain is treated as an individual and the elemental sensitivity is taken as the fitness function of one individual. Based on these treatments, MPGA operators, including crossover, mutation, migration and selection, are modified to adapt to compliance minimization problems. Additionally, some key parameters are controlled to guarantee a convergent solution and to solve the structural unconnectivity problem. A case is used to verify the effectiveness and efficiency of the proposed method. The numerical results show that the proposed method is efficient, and compared with the BESO method and combined simple genetic algorithm and BESO method, the proposed method provides a powerful ability in searching for better robust solutions and improving convergence speed.

... To normalize the stiffness distribution and smooth the jagged surfaces, we introduce the filter scheme into the shape optimization. The filter scheme is based on the digital image processing technique, which has been generally used in topology optimization to suppress the phenomenon of checkerboard and mesh dependency [32,33]. In shape optimization, only double filter schemes are effective to normalize the strongly nonlinear behavior of beam elements. ...

... (10), otherwise continue step 2 until the convergence criterion is satisfied. Herein, the convergence criterion referring to the literature [32] is defined: = (16) where change1 and change 2 denote the differences in compliance and volume. M=10 is used to ensure the stability of the compliance at least in 20 iterations. 1 and 2 represent the allowable convergence error, which is used to measure if the convergency is reached. ...

Evolutionary node shifting is an effective approach to the shape optimization of spatial structures for excellent mechanical performance. In this paper, a novel shape optimization algorithm is proposed, which is applicable to complex gridshell structures of irregular geometries and non-uniform grids. With the objective of maximizing the structural stiffness, the nodal coordinates are iteratively updated according to the sensitivity information. The perturbation displacements referring to the inverse hanging method are applied to the initial flat model. By analysis, we found that the irregular grids give rise to non-smooth gradient fields, which results in jagged surfaces. To normalize the non-smooth gradient fields and to prevent the optimization process from falling into local optima, a double filter scheme is introduced in the process of optimization. A variety of examples are presented to demonstrate that the proposed algorithm can effectively solve shape optimization problems of general gridshell structures.

... A method using a graphic diversity measure was proposed to generate diverse solutions for topology optimization based on the solid isotropic material with penalization (SIMP) method [3,33]. Some simple and effective strategies for achieving diverse and competitive structure designs were proposed based on the bidirectional evolutionary structural optimization (BESO) method [17,16,34]. ...

... 11b and 11c. If there is no penalty for the length of any member, the optimized structure is shown in Fig. 11d, and its volume is 16 show that by penalizing a bar, the number of nodes and bars of the optimized structure can also be reduced without affecting the volume to an unacceptable level. Therefore, a diversified design can be obtained by penalizing one certain bar. ...

Structural topology optimization plays an important role in obtaining conceptual designs in the preliminary design stage. However, traditional structural optimization methods can only generate one optimized design for the material distribution under certain constraints. The optimized structure could have some disadvantages, such as an unattractive appearance, difficulty in manufacturing, or high construction cost. Therefore, it is more practical to produce multiple designs that not only have high structural performance but also have substantially different forms from which the designer can choose. Two strategies were explored in this study for generating diverse truss structures, namely, the penalizing length method (PLM) and the modifying ground structure method (MGSM). Using the proposed PLM, it is possible to delete unneeded bars in the optimized structure, such as very slender bars, and the cross-sectional areas of the remaining bars will be automatically redistributed to ensure structural nodal stability. In addition, by generating overlapping potential bars in the ground structure, the structural instability problem caused by pin joints can be overcome. Two-dimensional and three-dimensional numerical examples were provided to indicate the effectiveness of the proposed methods. The numerical results showed that the proposed methodologies can generate diverse structures while maintaining structural performance.

... This smoothing approach is somehow similar with the one used in BESO in order to smooth the sensitivity numbers [30] to perform the removal/adding of elements. ...

... In order to obtain conclusions about the quality of structure configuration, these are to be compared with structures obtained with a BESO [30,33,34] approach on structural optimization. ...

Weight reduction is one of the main concerns when designing any component as it reduces material cost and green house gas emissions, among other aspects. Several numerical approaches exist in the literature with the objective of having any component with known mechanical loading become optimized in terms of mass minimization and stiffness maximization. Thus, the objective of this work is the development of optimized structures maintaining the same geometry by means of cellular materials, namely the gyroid infill, and generating functionally graded cellular structures with higher stiffness-to-weight ratio. Remodelling algorithms based on biological phenomena, namely bone growth, as well as Bi-evolutionary structural optimization (BESO) were employed to obtain the density map allowing the material functional gradient distribution. Smoothing functions were tested as a possibility of enhancing stiffness as abrupt density changes are avoided. The gyroid infill was characterized in order to create a phenomenological law based on bone remodelling laws. The gyroid law was implemented on the analysis FEMAS (opens-source, academic and educational FEM and meshless method software) software which presented the density map as an output. Each gradient consisted on areas at a similar density being concatenated into one solid. The different solids, at different density levels, are assembled thus creating the material functional gradient. Lastly, simulations consisted on three distinct and benchmark flexural load cases. Specimens were printed using FFF technology in PLA (E = 3145 MPa, ν = 0.3) having then been tested experimentally according to the appropriate load case. Numerical results correlated with the experimental results in terms of accuracy between theoretical and experimental stiffness where there was a greater accuracy for the specimens subject to a Four-Point bending load case, where only a 16% gap was verified between numerical and experimental flexural stiffness.

... Further, to optimize the elastic tensor of scaffolds, the SIMP method is applied (Sturm et al., 2010). Besides, through the use of ESO-based topology optimization, a series of unit cells exhibiting maximum shear and bulk modulus with predefined stiffness ratios and functionally graded structure were obtained (Huang and Xie, 2010). ...

... The rutile and anatase phases present in the TiO 2 layer showed the capability of formation of HA deposit, an ECM component of natural bone that promotes bone regrowth and improved osseointegration (de Jonge et al., 2008). An evenly distributed and well separated pore with a maximum of 1 μm in diameter was observed on the porous TiO 2 layer when anodization was carried out in the presence of H 2 SO 4 and H 3 PO 4 (Xie et al., 2010). On the other hand, anodization in the presence of acetic acid (CH 3 COOH) led to the formation of a much more heterogeneous surface, with larger craters and grooves roughly 200 μm in size. ...

Friction stir welding is solid-state joining technique widely used in automobile, aerospace, and ship building sectors for welding similar and dissimilar metals and alloys without expending huge amount of energy. The welded joints have relatively good strength, corrosion/wear characteristics as well as fine microstructure leading to limited defects. Friction stir welding was basically started for joining soft metals and alloys. Aluminum and its alloys have wide use in industrial sectors due to their light weight and good strength. Friction stir welding is also intended for joining of aluminum with low melting point metals like copper, zinc, and magnesium sheets. Unlike fusion welding schemes, here the frictional heat between the tool and weld materials does not create the temperatures to melting points; rather, it plasticizes the region temporarily and solidifies the mixture at the joint interface. This work deals with the sustainable friction stir welding of aluminum alloys of variable compositions. The influence of cutting parameters and tool geometry on the mechanical characteristics of joint is discussed. Tensile strength and impact resistance of welded portions are obtained. Theoretical energy required in each case along with other data is estimated for all experimental cases.

... BESO enables the previously deleted elements to be recovered [28,32]. Both ESO and BESO are easily interfaced with commercial FE codes and can resolve problems from many disciplines, e.g., civil, mechanical and aerospace [33]. The typical statement of problem using BESO is stated as: ...

... However, the raw elemental sensitivity numbers of void elements are set to zero in [23]. Though Xia et al. [23] used the so-called soft-kill BESO method [33], the stress values of void elements were not taken into account. Thus, it is deduced that these two factors are responsible for the poor convergence of the method proposed in [23] in the framework of the conventional BESO method. ...

... Built upon the ESO, the bi-directional evolutionary structural optimization (BESO) method is an improvement of the ESO, which not only gradually removes inefficient elements but also allows efficient elements to be added to the structure [13][14][15]. Since there are only two states of solid or void for each clement in the ESO/BESO method, the structural boundary obtained is clear and unambiguous [16]. ...

... In this way, these two variables can describe each element's structural layout and material properties. According to relevant research [16], when p > 3, there is little difference between the results obtained by the soft-kill BESO method (with soft elements) and the hard-kill BESO method (with no soft element). However, the purpose of introducing soft material as void elements is to realize the bidirectional change of elements, including the change from void to solid elements and the change from solid to void elements. ...

Structural optimization techniques for single-material continuum have been well developed and widely applied. However, topology optimization techniques with multiple materials, especially those with distinctly different mechanical properties in tension and compression, require significant improvement. Built on the bi-directional evolutionary structural optimization (BESO) technique for a single material, this paper proposes a method which takes material utilization as the criterion for determining the importance of the materials to the whole structure and assigns a suitable material for each element according to the sum of the principal stresses to obtain an efficient multi-material distribution. Numerical examples are presented to demonstrate the effectiveness of the proposed method in material saving and safety enhancement. The application of the method to the topology optimization of bridges with different support conditions and multi-span continuous bridges shows that the new technique has significant practical value for the conceptual design of multi-material structures.

... A otimização topológica de estruturas não lineares requer mais tempo computacional do que o requerido para uma análise linear. A eficiência computacional, para tais situações, é considerada um caso crítico, especialmente para análises tridimensionais (Huang and Xie, 2010). ...

Resumo. O projeto de estruturas otimizadas está se tornando cada vez mais importante devido à escassez de recursos, competições tecnológicas e questões de proteção ambiental. Ao considerar vários casos de carga e materiais, a otimização estrutural busca produzir um projeto que seja econômico e seguro. A otimização topológica tornou-se bastante difundida entre as engenharias e atualmente é aplicada em muitos campos de pesquisa. A maioria dos estudos de otimização topológica são realizados sob as premissas do comportamento elástico linear, que idealiza o material e impõe o equilíbrio em relação à configuração de referência (não deformada) da estrutura. No entanto, as suposições de linearidade são muito restritivas para problemas avançados. Neste contexto, este trabalho irá verificar um algoritmo baseado no método de otimização topológica com não linearidade geométrica para problemas de estado plano de tensão. O código proposto foi desenvolvido no software MATLAB. A verificação do código com não linearidade geométrica ocorre a partir da replicação do caso de uma viga em balanço carregada na extremidade. O processo de otimização com não linearidade, ao contrário da otimização topológica linear, deve apresentar dependência da magnitude da carga. Os resultados obtidos demonstram uma boa estabilidade durante o processo iterativo e topologias assimétricas são geradas. Palavras-chave: Não linearidade geométrica, Topologia assimétrica, Otimização estrutural, Otimização topológica.

... Afterwards, they were considered as the discrete version of the standard SIMP scheme (Sigmund and Maute, 2013). Nowadays evolutionary approaches are still being improved (Huang and Xie, 2010) and are part of several commercial software due to their simplicity to be utilized with finite element software (Deaton and Grandhi, 2014 ...

Dynamic balancing is an important field of study in high-speed robotics and spatial robots. Taking into account robot dynamic balancing performance for robot design leads to low base vibrations, high precision and short cycle times. With the aim to develop a comprehensive robot design for dynamic balancing, structural topology optimization is studied in this research work as a tool for designing dynamically balanced robots, also called reactionless robots. The suitability of the proposed methodology is confirmed by accomplishing an optimized design of a reactionless four-bar linkage and the partial dynamic balancing of five-bar robotic mechanism. The significance of the dynamically balanced four-bar linkage is related to the possibility to exploit this optimized linkage as a special leg for building reactionless robots. Besides, the five-bar robot is very important due to its industrial applications, where it is typically used in pick-and-place operations.

... The feasibility of this method has been proved in many scientific and industrial fields [55][56][57][58][59][60][61]. PSO requires a solution space conventionally achieved by discrete methods by optimizing the algorithms. ...

Optimizing heat conduction layout is essential during engineering design, especially for sensible thermal products. However, when the optimization algorithm iteratively evaluates different loading cases, the traditional numerical simulation methods usually lead to a substantial computational cost. To effectively reduce the computational effort, data-driven approaches are used to train a surrogate model as a mapping between the prescribed external loads and various geometry. However, the existing model is trained by data-driven methods, which require intensive training samples from numerical simulations and do not effectively solve the problem. Choosing the steady heat conduction problems as examples, this paper proposes a physics-driven convolutional neural networks (PD-CNNs) method to infer the physical field solutions for randomly varied loading cases. After that, the particle swarm optimization (PSO) algorithm is used to optimize the sizes, and the positions of the hole masks in the prescribed design domain and the average temperature value of the entire heat conduction field is minimized. The goal of reducing heat transfer is achieved. Compared with the existing data-driven approaches, the proposed PD-CNN optimization framework predicts field solutions that are highly consistent with conventional simulation results. However, the proposed method generates the solution space without pre-obtained training data. We obtained thermal intensity results for holes 1, hole 2, hole 3, and hole 4 with 0.3948, 0.007, 0.0044, and 0.3939, respectively, by optimization PD-CNN model.

... As a material layout design method, topology optimization allows the full potential of materials to be exploited (Bendsøe and Kikuchi 1988), meanwhile maintaining a rigorous mathematical expression. It has been widely employed in engineering structure design (Huang and Xie 2010), especially in the design of aircraft and aerospace structures (Zhu et al. 2016). As of today, the approaches of topology optimization have been developed into two main branches. ...

Radar cross section (RCS) reduction for cavities is essential in flight vehicle design. As a conventional method to reduce the RCS, coating radar absorbing materials has been widely employed in engineering. Nevertheless, radar absorbing coatings (RAC) will additionally increase the structural weight. In this paper, a topology optimization approach is introduced for the layout design of RAC on the inner cavity walls. The objective of this problem is to minimize the mean value of RCS under the prescribed incident angles. A SIMP-like model is employed to represent the relative impedance of areas of intermediate density. The design variable is iteratively updated during the optimization process using a gradient-based algorithm. The RCS of the cavity is computed by the iterative physical optics method, which is utilized for the subsequent analytical gradient derivation. The validity of the proposed method is demonstrated by optimizing the RAC layout of two different shaped cavities. In both numerical examples, when optimizing the RCS in both planes with a weight of 1:1 and a volume fraction of 50%, the highest RCS loss rate in both horizontal and pitch planes is 18.02% and the lowest is only 6.89%. The optimization results indicate that the proposed method can be employed as a design procedure to consider both weight cost and cavity RCS reduction when coating the absorbing materials, instead of the classical experience-based RAC distributions.

... We briefly reviewed the development of BESO from the most primitive ESO to the current generally popular version. One may refer to the monograph of BESO written by Huang and Xie [151] for more details of this approach, and refer to a comprehensive review raised by Xia et al. [404] for recent advances of the approach. We can list a series of literature in which the BESO approach has always been redefined for solving various problems, such as stiffness and natural frequency optimization [152,156], periodic structure [153], multiple materials [155,143], multiple constraints [154], to name a few. ...

The objective of this thesis is to develop density based-topology optimization methods for several challenging dynamic structural problems. First, we propose a normalization strategy for elastodynamics to obtain optimized material distributions of the structures that reduces frequency response and improves the numerical stabilities of the bi-directional evolutionary structural optimization (BESO). Then, to take into account uncertainties in practical engineering problems, a hybrid interval uncertainty model is employed to efficiently model uncertainties in dynamic structural optimization. A perturbation method is developed to implement an uncertainty-insensitive robust dynamic topology optimization in a form that greatly reduces computational costs. In addition, we introduce a model of interval field uncertainty into dynamic topology optimization. The approach is applied to single material, composites and multi-scale structures topology optimization. Finally, we develop a topology optimization for dynamic brittle fracture structural resistance, by combining topology optimization with dynamic phase field fracture simulations. This framework is extended to design impact-resistant structures. In contrast to stress-based approaches, the whole crack propagation is taken into account into the optimization process.

... In 1999, Yang et al. [2] modified the ESO technique by allowing not only material removal, but also addition to the design domain. After a series of modifications that included sensitivity filters [3] and material interpolation schemes [4], Huang and Xie [5] proposed the new BESO approach, being extensively used ever since. ...

... On est ainsi dans un problème dit combinatoire, ou le nombre de possibilités à explorer est donné ici par 2 MˆN . La problématique est alors de mettre en place une stratégie permettant d'explorer cet espace, ce qui n'a pas à ce jour été étudié sauf avec des approches heuristiques [69][70][71][72]. ...

Aujourd'hui, le développement de systèmes micromécatroniques nécessite d’intégrer un ensemble de fonctionnalités notamment l'actionnement et la mesure dans un volume de plus en plus réduit. La conception de tels systèmes demeurent ainsi une démarche délicate de par leur complexité et la prise en considération des effets de couplage entre la structure et ses mécanismes d’actionnement et de mesure. Plusieurs solutions ont été proposées dans la littérature en particulier les méthodes de conception optimale. Elles permettent entre autres le placement optimal du couple actionneur/capteur au sein d’une structure porteuse, l’optimisation simultanée ou séparée des dimensions des actionneurs, la conception avec performances garanties, la conception à partir de blocs élémentaires passifs et actifs, etc. Cependant ces approches se limitent à un ensemble de paramètres discrets, réduisant le champ de conception des structures étudiées en imposant des hypothèses sur la forme initiale du système. Pour aboutir à des systèmes micromécatroniques performants avec une forte densité d’intégration, nous proposons dans cette thèse de développer un outil de conception à base d'optimisation topologique. La thèse a donc trait à démontrer l’intérêt de cette méthode dans le cadre de la conception de structures piézoélectriques. Tout d'abord, la physique du matériau piézoélectrique a été intégrée à l'approche d'optimisation topologique SIMP (Solid Isotropic with Material Penalization). Ensuite, l'approche étendue a été implémentée sous la forme d'un code informatique MATLAB permettant de simplifier sa manipulation par des utilisateurs non-experts. Enfin, deux structures, un actionneur 1D et un positionneur 2D, ont été conçues au travers de cet outil puis réalisées. Les simulations numériques et les validations expérimentales ont montré un bon accord entre les structures optimisées et les prototypes réalisés démontrant ainsi l’intérêt et le potentiel de cet approch e d'optimisation.

... There exist numerous pedagogical TO MATLAB codes online in Sigmund 2001;Suresh 2010;Challis 2010;Huang and Xie 2010;Andreassen et al. 2011;Saxena 2011;Talischi et al. 2012b;Wei et al. 2018;Ferrari and Sigmund 2020;Picelli et al. 2020 that can help a user to learn and explore various optimization techniques. In addition, one may refer to the articles by Han et al. (2021a); Wang et al. (2021) for a comprehensive list of TO educational codes. ...

This paper provides a simple, compact and efficient 90-line pedagogical MATLAB code for topology optimization using hexagonal elements (honeycomb tessellation). Hexagonal elements provide nonsingular connectivity between two juxtaposed elements and, thus, subdue checkerboard patterns and point connections inherently from the optimized designs. A novel approach to generate honeycomb tessellation is proposed. The element connectivity matrix and corresponding nodal coordinates array are determined in 5 (7) and 4 (6) lines, respectively. Two additional lines for the meshgrid generation are required for an even number of elements in the vertical direction. The code takes a fraction of a second to generate meshgrid information for the millions of hexagonal elements. Wachspress shape functions are employed for the finite element analysis, and compliance minimization is performed using the optimality criteria method. The provided MATLAB code and its extensions are explained in detail. Options to run the optimization with and without filtering techniques are provided. Steps to include different boundary conditions, multiple load cases, active and passive regions, and a Heaviside projection filter are also discussed. The code is provided in Appendix A, and it can also be downloaded along with supplementary materials from https://github.com/PrabhatIn/HoneyTop90.

... An extension of the ESO method is the wellknown Bi-directional Evolutionary Structural Optimisation (BESO) [14]. Later, the BESO approach has been reformulated in [15,16], by adding features to obtain mesh-independent results, without checker-board pattern and by introducing a sensitivity number averaging method to speed up convergence. Recently, new evolutionarybased procedures for TO have been developed in the framework of the level-set method (LSM) to obtain smooth topology boundary [17]. ...

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.

... TO basically aims to achieve the best material distribution to maximize the stiffness-to-weight ratio by minimizing the total strain energy (i.e., compliance) while considering predefined design constraints such as stress and/or volume fraction constraints. Various TO methods have been developed such as solid isotropic material with penalization (SIMP) [2], level-set methods [3,4], the evolutionary structural optimization (ESO) [5], and the bidirectional evolutionary structural optimization (BESO) [6,7]. The design variables are set only binary values of 0 or 1 in the ESO and BESO method, and these values correspond to void and solid materials respectively. ...

In this study, the lightweight final topologies of marine structural elements are studied using three-dimensional non-local topology optimization algorithm based on peridynamics. Peridynamics is the new nonlocal formulation of classical continuum mechanics that can easily model complex singularities such as cracks or voids. For optimization algorithm, the minimization of strain energy is chosen as the objective function while satisfying the pre-defined volume constraint. PD is coupled with a continuous density-based topology algorithm method, namely Optimality Criteria (OC). In this method, optimization results are examined by considering the definition of continuous design variables as each PD particle has its relative density value between 0 and 1. As a numerical case study, two ship mid-section geometries with the same cross-sections but different thicknesses are selected to perform PD-based 3D optimization. Obtained results are extensively presented and compared with respect to varying volume fraction and thickness of the mid-ship section to investigate the robustness and efficiency of the PD-TO. Overall, it is demonstrated that PD-TO methodology can be effectively utilized to generate stiff and lightweight design of the marine structural elements.

... Conventional topology optimization methods are aimed at achieving the solution that maximizes the structural performance under a certain volume fraction constraint, and its basic problem can be described mathematically as below (Huang and Xie, 2010b): ...

With the ability to generate forms with high efficiency and elegant geometry, to-pology optimization has been increasingly used in architectural and structural designs. However , the conventional topology optimization techniques aim at achieving the structurally most efficient solution without any potential for architects or designers to control the design details. This paper introduces three strategies based on Bi-directional Evolutionary Structural Optimization (BESO) method to artificially pre-design the topological optimized structures. These strategies have been successfully applied in the computational morphogenesis of various structures for solving practical design problems. The results demonstrate that the developed methodology can provide the designer with structurally efficient and topologically different solutions according to their proposed designs with multi-filter radii, multi-volume fractions, and multi-weighting coefficients. This work establishes a general approach to integrating objective topology optimization methods with subjective human design preferences, which has great potential for practical applications in architecture and engineering industry.

... Generally in each TO approach, the design reaches the optimum design in some steps. In each step, the finite element method calculates the objective function and evaluates improvements of the performance (Bendsoe and Sigmund, 2013;Huang and Xie, 2010;Bendsøe and Sigmund, 1999). The optimizer can use gradient-based methods like Optimality criteria (OC) algorithm and Method of Moving Asymptotes (MMA). ...

In this paper, a new non-gradient-based topology optimization (TO) method proposed. Simulated annealing (SA) with crystallization factor used to generate new solutions. During this process, the newly generated solutions evaluated based on the SA concept. A density filter also applied to remove the discontinuity of shapes. The innovation of this method is applying the history of accepted or rejected solutions by the crystallization factor. Results of compliance minimization of cantilever and MBB-beams from the proposed method compared with the results of gradient-based methods. The main advantage of the proposed method is the improvement of convergence of the results as well as no need for the derivative of the objective function.

... There exist numerous pedagogical TO MATLAB codes online in Sigmund 2001;Suresh 2010;Challis 2010;Huang and Xie 2010;Andreassen et al. 2011;Saxena 2011;Talischi et al. 2012b;Wei et al. 2018;Ferrari and Sigmund 2020;Picelli et al. 2020 that can help a user to learn and explore various optimization techniques. In addition, one may refer to the articles by Han et al. (2021a); Wang et al. (2021) for a comprehensive list of TO educational codes. ...

This paper provides a simple, compact and efficient 90-line pedagogical MATLAB code for topology optimization using hexagonal elements (honeycomb tessellation). Hexagonal elements provide nonsingular connectivity between two juxtaposed elements and, thus, subdue checkerboard patterns and point connections inherently from the optimized designs. A novel approach to generate honeycomb tessellation is proposed. The element connectivity matrix and corresponding nodal coordinates array are determined in 5 (7) and 4 (6) lines, respectively. Two additional lines for the meshgrid generation are required for an even number of elements in the vertical direction. The code takes a fraction of a second to generate meshgrid information for the millions of hexagonal elements. Wachspress shape functions are employed for the finite element analysis, and compliance minimization is performed using the optimality criteria method. The provided Matlab code and its extensions are explained in detail. Options to run the optimization with and without filtering techniques are provided. Steps to include different boundary conditions, multiple load cases, active and passive regions, and a Heaviside projection filter are also discussed.

... The BESO method, a further extension of ESO, was taken up by Mezzina et al. (2012) to compare ST models for the design of concrete bridge decks subjected to seismic in-plane actions with models found by the use of the load path method by Schlaich et al. (1987). Hardjasaputra (2015) and Palmisano et al. (2014) also investigate the finding of ST models for reinforced concrete structures, consulting the software BESO2D or rather BESO3D by RMIT University, Australia (no further details which version) by Huang and Xie (2010), with the second publication investigating several nonlinear constitutive laws. ...

Structural optimization within concrete construction has been increasingly taken up in research within the last two decades. Possible drivers are the need for material-reduced and thus resource-efficient structures as well as recent advancements in automated concrete construction. However, structural concrete is characterized by nonlinear material behavior. Consequently, the merge of structural concrete design and topology optimization is not trivial. This paper reviews and assesses the topic of topology optimization within concrete construction, carrying out an extensive quantitative as well as qualitative review on practical and numerical applications. The following research areas are identified: Multimaterial modeling, stress constraints, concrete damage modeling, strut and tie modeling, combined truss-continuum topology optimization, the consideration of multiple load cases, a focus on construction techniques and alternative approaches. Although the number of research papers dealing with the topic of topology optimization in concrete construction is numerous, there are only few that actually realized topology optimized concrete structures. In addition, only a little number of experiments was performed for an objective evaluation of the found geometries so far. Concluding this review, a list of future challenges, like the incorporation of sustainability measurements within the optimization process, is given and thus serves as a guidance for subsequent research.

... Topology optimization aims at finding optimal material distribution within the prescribed design domain and achieving the best performance of the structure. Since the seminar paper of Bendsøe et al. [1] in 1988, several topology optimization methods have been developed, including the homogenization method [2,3], the solid isotropic material with penalization (SIMP) method [4,5], the level-set method (LSM) [6][7][8], the bi-directional evolutionary structural optimization (BESO) [9][10][11][12]. With the recent development of additive manufacturing, multi-material structures can be comfortably fabricated and play an important role in practical engineering applications. ...

... Starting from landmark papers published in the late 1980s [4], a substantial literature has emerged in the field of topology optimization. To date, structural topology optimization has been developed in many different directions, including density-based methods [5][6][7], level set-based methods [8][9][10][11][12][13][14], evolutionary methods [15][16][17], phase field methods [18,19], moving morphable components methods [20][21][22][23] and several others. A detailed comparative review of the above methods is beyond the scope of this paper. ...

Standard moving morphable component (MMC)-based topology optimization methods use free components with explicitly geometrical parameters as design units to obtain the optimal structural topology by moving, deforming and covering such components. In this study, we intend to present a method for geometrically nonlinear explicit topology optimization using moving wide-Bézier components with constrained ends. Not only can the method efficiently avoid the convergence issues associated with nonlinear structural response analysis, but it can also alleviate the component disconnection issues associated with the standard MMC-based topology optimization methods. The numerical investigations proposed in this work indicate that the proposed method allows us to obtain results in accordance with the current literature with a more stable optimization process. In addition, the proposed method can easily achieve minimum length scale control without adding constraints.

... The TO algorithm includes generating new solutions and evaluates them based on defined criteria [5]. A variety of methods have been proposed to generate new solutions and evaluate them [2]. ...

Topology optimization (TO) of engineering products is an important design task to maximize performance and efficiency, which can be divided into two main categories of gradient-based and non-gradient-based methods. In recent years, significant attention has been brought to the non-gradient-based methods, mainly because they do not demand access to the derivatives of the objective functions. This property makes them well compatible to the structure of knowledge in the digital design and simulation domains, particularly in Computer Aided Design and Engineering (CAD/CAE) environments. These methods allow for the generation and evaluation of new evolutionary solutions without using the sensitivity information. In this work, a new non-gradient TO methodology using a variation of Simulated Annealing (SA) is presented. This methodology adaptively adjusts newly-generated candidates based on the history of the current solutions and uses the crystallization heuristic to smartly control the convergence of the TO problem. If the changes in the previous solutions of an element and its neighborhood improve the results, the crystallization factor increases the changes in the newly random generated solutions. Otherwise, it decreases the value of changes in the recently generated solutions. This methodology wisely improves the random exploration and convergence of the solutions in TO. In order to study the role of the various parameters in the algorithm, a variety of experiments are conducted and results are analyzed. In multiple case studies, it is shown that the final results are well comparable to the results obtained from the classic gradient-based methods. As an additional feature, a density filter is added to the algorithm to remove discontinuities and gray areas in the final solution resulting in robust outcomes in adjustable resolutions.

... The structural shape gradually changes in the direction of increasing the minimum internal force and improving the force transmission efficiency of components. This process is similar to the idea of evolutionary structural optimization [25], fully-stressed design, and the optimality criteria approach. ...

Branching structures are mechanically efficient in supporting large-span structures, such as free-form roofs. To support a roof with a specified geometry, we present a novel shape and topology optimization method to find the optimal branching structure in this paper. In the proposed method, the branching structure is modelled as a cable-net, while the reaction forces in z-direction from the roof are converted to the upward external loads at the branching structure. The force densities of the members are the design variables. The optimal branching structure can be obtained by minimizing one of the several types of objective functions, e.g., the sum of the strain energy, the sum of the absolute value of the internal axial forces, and the sum of the absolute value of the force densities, etc. The shape of the branching structure represented by the nodal coordinates is determined by solving the linear equilibrium equations in terms of force densities. The topology is optimized by removing the members with small axial forces and incorporating the closely spaced nodes. If the allowable stress is specified for the members, their cross-sectional areas can be directly calculated from the force densities and member lengths. Hence, it is very convenient to use force densities for simultaneously optimizing the cross-section, shape, and topology of a branching structure. Numerical examples show that this method can be easily applied to a 2D problem; however, for a 3D problem, the tolerances of the constraints on the reaction forces should be relaxed. It is also shown that considering the roof supports as variables is effective for finding an optimal shape of 3D branching structure supporting a free-form surface.

... The topology optimization, which aims to determine the best layout of material in a predefined domain, has become a powerful tool in engineering structures since the pioneering research by Bendsøe and Kikuchi (Bendsøe and Kikuchi 1988). Over the past few decades, many typical approaches, including the solid isotropic material with penalization (SIMP) model (Bendsøe 1989;Rozvany et al. 1992), evolutionary structural optimization (ESO) method (Huang and Xie 2010;Xia and Breitkopf 2014), the level set method (Mei and Wang 2004;Wei and Wang 2009), and moving morphable components (MMC) strategy (Zhang et al. 2016;Zhang et al. 2017), have been proposed and a large number of studies (Cheng and Jiang 1992;Wang et al. 2019a;Qiu et al. 2019) have been published. It should be noted that most of the developments have concentrated on static topology optimization. ...

This study investigates a novel reliability-based topology optimization framework to determine optimal material configurations for freely vibrating continuum structures with unknown-but-bounded uncertainties. Firstly, the optimization formulation for freely vibrating continuum structures with fundamental eigenfrequency and eigenfrequency gap constraints is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the natural eigenfrequencies under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization model and the performance measure approach is proposed to overcome the convergence difficulties. Meanwhile, the sensitivity analysis is further discussed based on the chain rule, in which the sensitivities of the target performance measure to the bounds of eigenfrequencies are calculated and the sensitivities of the eigenfrequencies with respect to the design variables are deduced. Numerical examples are eventually given to demonstrate the validity of the developed reliability-based topology optimization methodology.

... Klarbring et al. [12] adopted the solid isotropic material penalization (SIMP) model [13,14] to minimize energy release rate of material cracking based upon the virtual crack extension technique. Shobeiri [15] investigated the topology optimization for cracked structures by the bi-directional evolutionary structural optimization method (BESO) [16], in which the element free Galerkin (EFG) method was used to model fracture. Kang et al. [17] proposed to integrate the linear elastic fracture mechanics (LEFM) technique into the topology optimization for making the structures insensitive to initial cracks. ...

In this paper, a recently-developed phase-field damage model is incorporated into the topology optimization framework to take into account crack initiation and propagation in a path-dependent fashion. The proposed topological design can enhance fracture resistance of structures made of brittle materials such as advanced ceramics. For the first time, a path-dependent shape derivative is developed in a step-wise manner during the nonlinear fracture analysis, which enables to drive the topology optimization properly. To measure the fracture resistance of structure, a p-norm function is formulated to aggregate the phase-field variables into a single constraint. To demonstrate the effectiveness of the presented topology optimization procedure, three 2D benchmark examples with single-phasic material and one 3D biomedical example with biphasic materials are studied here. The comparison with the topological designs based upon conventional linear elastic finite element analysis without the damage model indicates that the proposed method can significantly improve the fracture resistance of structures with more efficient use of materials. The proposed method is anticipated to provide an effective approach for sophisticated path-dependent topological design of structures reducing severe stress concentration and high risks of fracture failure.

The aim of this paper is to propose a novel
computational technique of applying reliability-based
design to thermoelastic structural topology optimization. Therefore, the optimization of thermoelastic
structures’ topology based on reliability-based design
is considered by utilizing geometrical nonlinearity
analysis. For purposes of introducing reliability-based
optimization, the volume fraction parameter is viewed
as a random variable with a normal distribution having a mean value and standard deviation. The Monte
Carlo simulation approach for probabilistic designs is
used to calculate the reliability index, which is used
as a constraint related to the volume fraction constraint of the deterministic problem. A new bi-directional evolutionary structural optimization scheme is
developed, in which a geometrically nonlinear thermoelastic model is applied in the sensitivity analysis.
The impact of changing the constraint of a defned
volume of the required design in deterministic problems is examined. Additionally, the impact of altering
the reliability index in probabilistic problems is investigated. The efectiveness of the suggested approach
is shown using a benchmark problem. Additionally,
this research takes into account probabilistic thermoelastic topology optimization for a 2D L-shaped beam
problem.

A multi‐objective optimization design method for the cross‐brace structure of a computer numerical control gantry machine tool is proposed to improve mechanical performance. Dynamic and static performance indexes of the cross brace are analyzed. On basis of sensitivity analysis, size parameters which have great influence on mass and dynamic and static performance indexes of the cross brace are selected as design variables for optimization. Objective values under design variables are obtained by orthogonal experimental method, and an objective function is fitted by response surface method. After verifying the fitting effect of the objective function, entropy weight method is used to calculate weight coefficient of each optimization objective, and a comprehensive performance optimization model of the cross brace is built. Accelerated particle swarm optimization (APSO) intelligent algorithm is used to solve the comprehensive performance optimization model. Comparison results before and after the optimization show that the mass of optimized cross brace is reduced by 8.377%, the maximum stress is reduced by 13.252%, and the first‐order natural frequency is increased by 2.183%. The proposed optimization method provides a new idea to improve the dynamic and static performance of the cross brace while reducing its mass. On basis of sensitivity analysis, size parameters which have great influence on dynamic and static performance of the cross brace are selected as design variables for optimization, and a comprehensive performance optimization model of the cross brace is built by orthogonal experiment and response surface. Accelerated particle swarm optimization intelligent algorithm is used to solve the model.

Optimization of thermal compliance and linear elastostatic compliance is a common research task that can increase heat dissipation and structural stiffness. In particular, support structures in additive manufacturing (Kuo et al. in Struct. Multidiscip. Optim. 57:183–195, Jan. 2018) or the optimization of heat sinks (Zhou et al. in Struct. Multidiscip. Optim. 54:1045–1060, Oct. 2016) often require the consideration of both objectives. This article deals with the multi-objective topology optimization of frame structures using the weighted sum method for 2D and 3D wireframe meshes. Therefore, an element stiffness matrix is used, which couples both contributions. This automatically leads to holistic design proposals in terms of considering both objectives.KeywordsMulti-objectiveTopology OptimizationUnit CellGrid Structure

This work proposes a general formulation of the design requirement on the structural displacements evaluated, simultaneously, at different points of the structure in the framework of a density-based algorithm for topology optimization. The algorithm makes use of Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field describing the part topology and of the Solid Isotropic Material with Penalization approach. The proposed formulation takes advantage of the properties of the NURBS basis functions to ensure continuity between the pseudo-densities of adjacent elements, without the need of defining further filters. In particular, the multi-displacement requirement is formulated in the most general sense, by considering displacements on loaded and non-loaded regions. The gradient of structural displacements is evaluated in closed form by using the adjoint method and the properties of the NURBS blending functions. Moreover, a sensitivity analysis of the optimized topology to the integer parameters, involved in the definition of the NURBS hyper-surface, is carried out. The effectiveness of the proposed approach is proven through meaningful 2D and 3D benchmarks.

Topology optimization (TO) has been a popular design method among CAD designers in the last decades. This method optimizes the given design domain by minimizing/maximizing one or more objective functions, such as the structure's stiffness, and at the same time, respecting the given constraints like the volume or the weight reduction. For this reason, the companies providing the commercial CAD/FEM platforms have taken this design trend into account and, thus, have included TO in their products over the last years. However, it is not clear which features, algorithms, or, in other words, possibilities the CAD designers do have using these software platforms. A comparative study among the most applied topology optimization software was conducted for this research paper. First, the authors developed an online database of the identified TO software in the form of a table. Interested CAD designers can access and edit its content, contributing in this way to the creation of an updated library of the available TO software. In addition, a deeper comparison among three commercial software platforms-SolidWorks, ANSYS Mechanical, and ABAQUS-was implemented using three common case studies-(1) a bell crank lever, (2) a pillow bracket, and (3) a small bridge. These models were designed, optimized, and validated numerically, as well as compared for their strength. Finally, the above software was evaluated with respect to optimization time, optimized designs, and TO possibilities and features.

The emerging field of soft robotics presents a new paradigm for robot design in which “precision through rigidity” is replaced by “cognition through compliance.” Lightweight and flexible, soft robots have vast potential to interact with fragile objects and navigate unstructured environments. Like octopuses and worms in nature, soft robots’ flexible bodies conform to hard objects and reconfigure for different tasks, delegating the burden of control from brain to body through embodied cognition. However, because of the lack of efficient modeling and simulation tools, soft robots are primarily designed by hand. Typically, hard components from rigid robots or living creatures are heuristically substituted for comparable soft ones. Autonomous design and manufacturing methodologies are urgently required to produce bespoke, high‐performing robots. Currently, design methodologies exist between simple but realistic parametric optimizations, and evolutionary algorithms which simulate morphology and control coevolution. To find high‐performing designs, novel high‐fidelity simulators and high‐throughput manufacturing and testing processes are required to explore the complex soft material, morphology and control landscape, blending simulation, and experimental data. This article reviews the state of the art in autonomous soft robotic design. Existing manual and automated designs are surveyed and future directions to automate soft robot design and manufacturing are presented. By using soft and functional materials to deform around objects and adapt to new environments, soft robotics has the potential to revolutionize material handling and terrain navigation. But in the absence of accurate modeling tools, they are still laboriously designed manually. This article reviews progress toward autonomous modeling, simulation, and design.

This paper presents an explicit topology optimization approach using BEM‐based moving morphable void (MMV). Structural analysis in the conventional MMV approach is performed on a fixed mesh over the entire design domain. However, in the proposed method structural boundaries are described using B‐splines and discretized with boundary elements making explicit representation of the boundaries possible. In this regard, the proposed approach has following merits: a) the use of weak material, which is often adopted to mimic voids can be completely avoided; b) exact and explicit description of voids in MMV allows the capture of tiny structural features in a robust and effective way; c) numerical instabilities stemming from the use of fixed meshes can be naturally avoided. Several numerical examples are provided to demonstrate the effectiveness of the proposed approach.

This article aims to present a novel topological design approach, which is inspired by the famous density method and parametric level set method, to control the structural complexity in the final optimized design and to improve computational efficiency in structural topology optimization. In the proposed approach, the combination of radial basis function and the SIMP formula is introduced to describe the distribution of the fictitious density field in the design domain. By changing the radius and distribution of radial function, the structural complexity can be controlled. Meanwhile, it is found that the proposed method can naturally avoid checkerboard design. In order to improve the computational efficiency affected by the number of design variables, we propose to redefine the support points so that the number of support points is much smaller than that of the observation points of the radial function. It follows that the number of design variables can be reduced to a great extent. Several numerical examples are tested to show the feasibility and effectiveness of the presented method.

This paper first introduces the guide-weight criterion into the topology optimization problems for maximization of the fundamental eigenfrequency of vibrating continuum structures. The traditional solid isotropic material with penalization model is modified to eliminate the artificial localized modes. Based on this modified model, the iteration formula of the design variables is derived using the guide-weight criterion. An iterative mass control strategy is adopted to satisfy the equality constraint on the final mass and to stabilize the iteration process. Additionally, a mass preserving density filter based on Heaviside function is used to solve the gray transition problem. Several typical examples are used to validate the proposed method. Numerical results show that the proposed method is capable of achieving iterative convergence and clear profiles of topologies; meanwhile, the optimal results obtained by the proposed method agree well with those obtained by the commonly used bi-directional evolutionary structural optimization (BESO) method. In particular, the proposed method has a faster convergence rate than the BESO method.

Topology optimization is the procedure of determining the connectivity, shape and location of voids inside a given design domain. This chapter proposes a quantum‐inspired evolutionary algorithm (QEA) for structural topology optimization. It reviews the formulation of the continuum structural topology optimization model and the characteristics of quantum computing. The chapter also proposes the detailed implementation of a QEA‐based topology optimization. It discusses two issues: the density‐based continuum structural topology optimization formulation and the characteristics of quantum computing, which may be conducive to reducing the computational expense of topology optimization. The entire theoretical procedure of quantum‐inspired topology optimization, which mainly combines FEM analysis, design sensitivity solution and quantum‐inspired search, is accomplished for continuum structures. As the experimental implementation of quantum computation needs initialization, coherent manipulation, control and read of the fragile quantum system, practically building quantum computers has proved extremely difficult.

Structural topology optimization aims to design mechanical structures by seeking the optimal material layout within a given design space. Within this framework, this paper addresses the minimization of the structural mass under stress and buckling constraints, formulated as a nonlinear combinatorial optimization problem. An algorithm is proposed for such a problem, that follows a topological gradient-based approach. The adjoint method is applied to efficiently compute the constraint gradients. An iterative algorithm for buckling analysis, featuring low memory requirements, is also proposed. Numerical results, including a real application arising in the aeronautical field, illustrate the efficiency of the two proposed algorithms.

This work aims to perform the topology optimizationof frequency separation interval of continuous elastic bi-dimensional structures in the high-frequency domain. The studied structures are composed of two materials. The proposed algorithm is an adaptation of the Bidirectional Evolutionary Structural Optimization (BESO). As the modal density is high in this frequency domain, the objective function, based on the weighted natural frequency, is formulated to consider an important number of modes. To implement the algorithm, a mode tracking method is necessary to avoid problems stemming from mode-shifting and local modes. As the obtained results by using structural dynamics analysis present quasi-periodic topology, further calculations are done to compare the results with and without imposed periodicity. A dispersion analysis based on wave propagation theory is performed by using the unit cell previously obtained from the structural optimization to investigate the band gap phenomenon. The resulting band gaps from the dispersion analysis are compared with respect to the dynamic behavior of the structure. The topology optimization methodology and the wave propagation analysis are assessed for different boundary conditions and geometries. Comparison between both analyses shows that the influence of the boundary conditions on the frequency separation interval is small. However, the influence from the geometry is more pronounced. The optimization procedure does not present significant numerical instability. The obtained topologies are well-defined and easily manufacturable, and the obtained natural frequency separation intervals are satisfactory.

ESO-type topology optimization methods are re-examined and several improvements are suggested. Owing to objection to the term Evolutionary Sturctural Optimisation, the modified solution strategy will be termed SERA (Sequential Element Rejections and Admissions).

We examine both the evolutionary structural optimisation (ESO) and solid isotropic microstructure with penalisation (SIMP)
methodologies by investigating a cantilever tie–beam. Initially, both ESO and SIMP produce designs with higher objective function
values relative to a previously published ‘intuitive’ design. However, after careful investigation of the numerical parameters
such as the initial design domain and the mesh size, both methods obtain designs that have lower objective function values
relative to the intuitive design. Thus, a clearer understanding of the numerical parame- ters and their influence on optimisation
methods is achieved.

A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external
excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher
order, or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that
the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present
paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies
of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and
several illustrative results are presented.

In a finite element analysis (FEA), the contours of element von Mises stress and stiffness sensitivity number are found to be very similar. This paper shows there to be an equivalence between the von Mises stress criterion and the stiffness criterion for element elimination or addition in evolutionary structural optimization. The examples presented demonstrate the same resulting topologies during the evolving process using the two different criteria. The effect of numerical errors on topology is also investigated. It is concluded that the optimized topologies of a structure using the fully stressed criterion and the minimum compliance or maximum stiffness criterion are equivalent.

Checkerboard patterns are quite common in various fixed grid finite element based structural optimization methods. In the
evolutionary structural optimization procedure, such checkerboard patterns have been observed under various design criteria.
The presence of checkerboard patterns makes the interpretation of optimal material distribution and subsequent geometric extraction
for manufacturing difficult. To prevent checkerboarding, an effective smoothing algorithm in terms of the surrounding element’s
reference factors is proposed in this paper. The approach does not alter the mesh of the finite element model, nor increase
the degree of freedom of the structural system, therefore, it does not affect the computational efficiency. To demonstrate
the capabilities of this algorithm, a wide range of illustrative examples are presented in this paper.

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.

Most existing studies of 2D problems in structural topology optimization are based on a given (limit on the) volume fraction
or some equivalent formulation. The present note looks at simultaneous optimization with respect to both topology and volume
fraction, termed here “extended optimality”. It is shown that the optimal volume fraction in such problems — in extreme cases
— may be unity or may also tend to zero. The proposed concept is used for explaining certain “quasi-2D” solutions and an extension
to 3D systems is also suggested. Finally, the relevance of Voigt’s bound to extended optimality is discussed.

Recent results on optimal design with anisotropic materials and optimal design of the materials themselves are in most cases based on the assumption of linear elasticity. We shall extend these results to the nonlinear model classified as powerlaw elasticity. These models return proportionality between elastic strain energy density and elastic stress energy density. This is shown to imply localized sensitivity analysis for the total elastic energy, and for a number of optimal design problems this immediately gives practical, general results. For two- and three-dimensional problems the effective strain and the effective stress are defined from an energy consistent point of view, and it is shown that a definition generalizing the von Mises stress must be used. The optimization criterion of uniform energy density also holds for nonlinear materials, and several general conclusions can be based on this fact. Applications to size design illustrate this. For stiffness optimization the ultimate optimal material design problem is addressed. The validity of recent results are extended to nonlinear materials, and a simple proof based on constraint on the Frobenius norm is given. We note that the optimal material is orthotropic, that principal directions of material, strain and stress are aligned, and that there is no shear stiffness. In reality, the constitutive matrix only has one nonzero eigenvalue and the material therefore has stiffness only in relation to the specified strain condition. Results related to orientational design with orthotropic materials are also focused on. With respect to strength optimization, i.e. the more difficult problem with local constraints, we shall comment on the influence of the different strength criteria.

This review is principally concerned with the relatively slow speed (of the order of, say, 50 m/sec) dynamic impact of metallic structures and dwells on the large deformation plasto-mechanics of the simple structure elements frequently used as parts of complete devices. The design aim is to dissipate kinetic energy irreversibly rather than convert and store it elastically and in particular, restitution is to be avoided. The aim is to safeguard people, cargo, machinery or even the vehicle itself from suffering an excessively high rate of retardation or degree of damage. Devices used to this end are usually one-shot items, i. e. , once having been deformed, they are discarded and replaced. The implication of designing an energy-absorbing device on the basis of its quasi-static loading response is that inertia effects within the device itself are unimportant and hence the kinetic energy is considered converted into plastic work in a quasi-static deformation mode. Exceptions to this behavior will be mentioned where appropriate. Besides a brief statement concerning strain-rate effects and other general features, we conclude with some comments concerning nonmetallic systems for absorbing impact energy.

A computational scheme is presented for single or multiple eigenfrequency maximization of isotropic and composite plates. Eigenvalue maximization was achieved by means of an optimization process which sought to redistribute the material of the plate structure in an optimal way so that a bound on the total volume was satisfied. It was assumed that the plate structure possessed a repetitious microstructure and the homogenization theory was used to obtain equivalent elastic moduli. The structural eigenvalues and modes were computed via a finite-element analysis using a shear-deformable laminated finite element which was also applied to discretize a single-layered isotropic plate. Sequential linear programming was employed to perform the optimization task. Numerical examples are presented for clamped and simply supported plates for which the natural frequencies were extremized independently as well as simultaneously. For isotropic clamped plates, the optimality of the obtained results was verified by discretizing the resultant topology with a set of finite elements, computing its eigenvalues and then comparing them with eigenvalues from a uniform plate having the same volume.

This paper presents a method for optimal design of compliant mechanism topologies. The method is based on continuum-type topology optimization techniques and finds the optimal compliant mechanism topology within a given design domain and a given position and direction of input and output forces. By constraining the allowed displacement at the input port, it is possible to control the maximum stress level in the compliant mechanism. The ability of the design method to find a mechanism with complex output behavior is demonstrated by several examples. Some of the optimal mechanism topologies have been manufactured, both in macroscale (hand-size) made in Nylon, and in microscale (<.5mm)) made of micromachined glass.

In the past, the possibilities of structural optimization were restricted to an optimal choice of profiles and shape. Further improvement can be obtained by selecting appropriate advanced materials and by optimising the topology, i.e. finding the best position and arrangement of structural elements within a construction. The optimization of structural topology permits the use of optimization algorithms at a very early stage of the design process. The method presented in this book has been developed by Martin Bendsoe in co-operation with other researchers and can be considered as one of the most effective approaches to the optimization of layout and material design.

This paper presents three performance indices developed by using the scaling design approach for assisting the selection of optimal topologies for the minimum-weight design of continuum structures subject to stress and displacement constraints. These performance indices are incorporated in the Evolutionary Structural Optimization (ESO) method to monitor the optimization process from which optimal topologies can be identified. Examples provided demonstrate that the proposed performance indices are effective indicators of material efficiency and can be used to compare the efficiency of structural topologies generated by different optimization methods.

In this paper, topology optimization of both geometrically and materially nonlinear structure is studied using a general displacement functional as the objective function. In order to consider large deformation, effective stress and strain are expressed in terms of 2nd Piolar–Kirchhoff stress tensor and Green–Lagrange strain tensor, and constitutive equation is derived from the relation between the effective stress and strain. Sensitivity analysis of the general displacement functional is derived using the adjoint method. Numerical results of mean compliance design are compared under linear analysis, geometrical nonlinear analysis, material nonlinear analysis, and combined nonlinear analysis.

The effectiveness and efficiency of the Bi-directional Evolutionary Structural Optimization (BESO) method has been demonstrated on the minimization compliance problem with fixed external loads. This paper considers the minimization of mean compliance for continuum structure subjected to design-dependent self-weight loads. Due to the non-monotonous behaviour for this type of the optimization problems, the extended BESO method using discrete design variables has its difficulty to obtain convergent solutions for such problems. In this paper, a new BESO method is developed based on the sensitivity number computation utilizing an alternative material interpolation scheme. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures including self-weight loads.

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

The present work deals with topology design of 2D-frames with respect to crashworthiness. The objective of the optimization is to control the energy dissipation history of a structure during a crash and thereby minimizing the injuries of the passenger. For this purpose several objectives and constraints are suggested and tested numerically. The ground structure for the optimization consists of rectangular 2D-beam elements with plastic hinges where the height of each beam is a design variable. The elements can undergo large rotations, so the analysis accommodates geometric nonlinearities. Concentrated masses are applied to the structure. The time integration is done by the original Newmark method combined with an implicit backward Euler algorithm. The analytical sensitivities are computed by the direct differentiation method.

This paper presents an evolutionary method for structural topology optimization subject to frequency constraints. The evolutionary structural optimization (ESO) method is based on the idea that by gradually removing inefficient material, the residual shape of structure evolves toward an optimum. The method is further developed by allowing the material to be added as well as removed, and this new approach is called the bidirectional ESO method (BESO). BESO has been successfully used for problems of stress and stiffness/displacement constraints. Its application to frequency optimization is addressed in this paper. Three kinds of optimization objectives, namely, maximizing a single frequency, maximizing multiple frequencies, and designing structures with prescribed frequencies are considered. Four examples are tested by BESO and ESO. The objective functions yielded by the two methods are close, and BESO is computationally more efficient in most cases.

Evolutionary structural optimization (ESO) is based on a simple idea that an optimal structure (with maximum stiffness but minimum weight) can be achieved by gradually removing ineffectively used materials from design domain. In general, the results from ESO are likely to be local optimums other than the global optimum desired. In this paper, the genetic algorithm (GA) is integrated with ESO to form a new algorithm called Genetic Evolutionary Structural Optimization (GESO), which takes the advantage of the excellent behavior of the GA in searching for global optimums. For the developed GESO method, each element in finite element analysis is an individual and has its own fitness value according to the magnitude of its sensitivity number. Then, all elements in an initial domain constitute a whole population in GA. After a number of generations, undeleted elements will converge to the optimal result that will be more likely to be a global optimum than that of ESO. To avoid missing the optimum layout of a structure in the evolution, an interim thickness is introduced into GESO and its validity is demonstrated by an example. A stiffness optimization with weight constraints and a weight optimization with displacement constraints are studied as numerical examples to investigate the effectiveness of GESO by comparison with the performance of ESO. It is shown through the examples that the developed GESO method has powerful capacity in searching for global optimal results and requires less computational effort than ESO and other existing methods.

This paper presents topology optimization of geometrically and materially nonlinear structures under displacement loading. A revised bi-directional evolutionary optimization (BESO) method is used. The objective of the optimization problem is to maximize the structural stiffness within the limit of prescribed design displacement. The corresponding sensitivity number is derived using the adjoint method. The original BESO technique has been extended and modified to improve the robustness of the method. The revised BESO method includes a filter scheme, an improved sensitivity analysis using the sensitivity history and a new procedure for removing and adding material. The results show that the developed BESO method provides convergent and mesh-independent solutions for linear optimization problems. When the BESO method is applied to nonlinear structures, much improved designs can be efficiently obtained although the solution may oscillate between designs of two different deformation modes. Detailed comparison shows that the nonlinear designs are always better than the linear ones in terms of total energy. The optimization method proposed in this paper can be directly applied to the design of energy absorption devices and structures.

Recent results on the design of material properties in the context of global structural optimization provide, in analytical form, a prediction of the optimal material tensor distributions for two or three dimensional continuum structures. The model developed for that purpose is extended here to cover the design of a structure and associated material properties for a system composed of a generic form of nonlinear softening material. As was established in the earlier study on design with linear materials, the formulation for combined “material and structure” design with softening materials can be expressed as a convex problem. However, in contrast to the case with linear material, the optimal distribution of material properties predicted in the nonlinear problem depends on the magnitude of load. Computational solutions are presented for several example problems, showing how the optimal designs vary with different values assigned to data that fix the load and material parameters.

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.

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.

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.

An automated fully stressed design approach based on the Xie and Steven
algorithm is presented. With this algorithm a fully stressed
design is obtained by a gradual removal of low stressed material. By applying
this evolutionary procedure a layout or topology of a structure can be found
from an initial block of material. A fully integrated, interactive program is
presented which incorporates automatic mesh generation, finite element
analysis and the fully stressed design algorithm. The feasibility of the
approach is demonstrated using several examples.

Purpose
This paper surveys the current state and capabilities of three dimensional printing (3DP). A comprehensive review of 3D Printing applications is presented. The scope of the applications includes design, manufacturing, the medical field and architecture.
Design/methodology/approach
A large variety of manufacturing applications such as rapid pattern making and rapid tooling using the 3DP process directly or as core technology, as well as further implications in design and engineering analysis, medicine, and architecture are presented and evaluated.
Findings
Some research issues are also discussed. An attempt, based on the state of the art, to show weaknesses and opportunities, and to draw conclusions about the future of this important process rounds up this paper.
Research limitations/implications
The scope of this research survey is limited to evaluation and comparison of processes that may be characterised as 3D printing technologies.
Practical implications
The study is very useful as a basis for matching evaluated 3D printing machine and process capabilities to user requirements, and forms a framework on which future comparative studies can build.
Originality/value
A comprehensive overview of the capabilities of 3DP processes is presented and evaluated. It shows the application of 3D printing beyond concept modelling. The paper is valuable for researchers as well as individuals, who require adequate and relevant comparative information during decision making.

The bidirectional evolutionary structural optimization (BESO) method that allows the material removed and added to the structure is proposed for maximizing stiffness of nonlinear structures. The performance of the structures were gradually improved by removing and adding elements and the optimization process was stopped when both the objective volume and the prescribed. The BESO method provides a more robust and more efficient algorithm for searching for the optimal solution. The BESO method gains computational efficiency because the total elements become less. The improvements can be significant for problems involving local buckling. The numerical results show that the stiffness of structures optimized using combined geometrically and materially nonlinear modeling is higher than that using linear analysis.

\ This paper presents a method for and examples of topology optimization of energy absorption structures. The topology optimization problem is solved by using the elements as design variables. The sensitivity number of an element is derived from using an adjoint method to address two principal design parameters, namely absorbed energy per unit volume, e I, and absorbed energy ratio, e(2). Filter techniques are employed to smooth sensitivities in the design space and to eliminate unnecessary structural details below a certain length-scale. The bi-directional evolutionary optimization (BESO) technique is used to search for the optimal design in the whole design domain by gradually removing and adding material. Several examples are presented to demonstrate the capability and effectiveness of the proposed method.

A method is developed to systematically remove and reintroduce low density elements from and into the finite element mesh on which the structural topology optimization problem is defined. The material density field which defines the topology and the local ‘stiffness’ of the structure is optimally distributed via non-linear programming techniques. To prevent elements from having zero stiffness, an arbitrarily small lower bound on the material density is typically imposed to ensure that the global stiffness matrix does not become singular. While this approach works well for most minimum compliance problems, the presence of low density elements can cause computational problems, particularly in structures that exhibit geometric non-linearities, e.g. in compliant mechanisms. To resolve this problem, a systematic approach for removing and reintroducing low density elements is presented, and the substantial performance improvements both in design and computational efficiency of the method over current methods are discussed. Several structures and compliant mechanisms are designed to demonstrate the method. Copyright © 2003 John Wiley & Sons, Ltd.

Dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables, since in these problems the number of constraints is small in comparison to the number of design variables. The ‘raw’ dual algorithm, which was originally proposed for the minimum compliance design problem, worked well when a perimeter constraint was added in addition to the volume constraint. However, if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter, the algorithm tended to behave erratically. Recently, a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint; in particular the problem of checkerboarding which is frequently observed with the use of lower-order finite elements is eliminated. In this work, we show how the perimeter constraint can be incorporated in this improved algorithm, so that it not only provides a designer with a control over the topology, but also generates good topologies irrespective of the value of the upper bound on the perimeter. Copyright © 2002 John Wiley & Sons, Ltd.

This paper describes the use of topology optimization as a synthesis tool for the design of large-displacement compliant mechanisms. An objective function for the synthesis of large-displacement mechanisms is proposed together with a formulation for synthesis of path-generating compliant mechanisms. The responses of the compliant mechanisms are modelled using a total Lagrangian finite element formulation, the sensitivity analysis is performed using the adjoint method and the optimization problem is solved using the method of moving asymptotes. Procedures to circumvent some numerical problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Evolutionary structural optimisation (ESO) method is based on the idea that by gradually removing inefficient materials, the structure evolves towards an optimum. Bi-directional ESO (BESO) allows for adding efficient materials in the evolution. This paper investigates the ESO and BESO methods in solving the topology optimisation of continua structures with a constraint on the global stiffness. Based on the work on stiffness optimisation with fixed load conditions, this paper focuses on problems considering design dependent loads. The dependence can be due to transmissible loads, inclusions of structural self weight and surface loads. Sensitivity analysis and evolutionary procedure for problems of fixed load conditions are modified to accommodate the load variation condition. A number of examples are presented for verification. The results demonstrate that ESO and BESO are effective in solving the optimisation with design dependent loads. BESO has the flexibility of balancing solution quality and computing time.

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

A new method for non-linear programming in general and structural optimization in particular is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and solved. The generation of these subproblems is controlled by so called ‘moving asymptotes’, which may both stabilize and speed up the convergence of the general process.