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Structural shape and topology optimization has undergone tremendous developments in recent years due to its important applications in many fields. However, effectively controlling the structural complexity of the optimization result remains a challenging issue. The structural complexity is usually characterized by the distribution and geometries of interior holes. In this work, a new approach is developed based on the graph theory and the set theory to control the number and size of interior holes of the optimized structures. The minimum distance between the edges of any two neighboring holes can also be constrained. The structural performance and the effect of the structural complexity control are well balanced by using this approach. We use three typical numerical examples to verify the effectiveness of the developed approach. The optimized structures with and without constraints on the structural complexity are quantitatively compared and analyzed. The present methodology not only enables the designer to have a direct control over the topology of the optimized structures, but also provides diverse and competitive solutions.

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... Even if many complex structures can be printed by AM, there are still some manufacturing constraints that are needed to be addressed. In the AM-oriented TO, the most common manufacturing constraints considered in the literature are the self-supporting constraint (Gaynor and Guest 2016;Qian 2017;Guo et al. 2017;Langelaar 2017;Allaire et al. 2017;Wang et al. 2018;Johnson and Gaynor 2018;Zhang and Zhou 2018;Zhang et al. 2019;Fu et al. 2019;Zhang and Cheng 2020;Luo et al. 2020;Garaigordobil et al. 2021), minimum length constraint (Guest et al. 2004;Lazarov et al. 2016;Wang et al. 2011;Sigmund 2009;Zhou et al. 2015;Zhang et al. 2014Zhang et al. , 2017Guo et al. 2014;Xia and Shi 2015;Liu 2019) including solid-phase minimum length constraint, and that of void phase, connectivity constraint (Liu et al. 2015;Li et al. 2016;Zhou and Zhang 2019;Zhao et al. 2019;Wang et al. 2020;Xiong et al. 2020;Liang et al. 2022), and thermal residual stress/deformation constraint (Cheng et al. 2019;Misiun et al 2021;Xu et al. 2022). ...

... Following this approach, Wang et al. 2020 studied the parameter selection for suppressing the enclosed voids with an electrostatic model. Some non-SIMP approaches (Zhou and Zhang 2019;Zhao et al. 2019;Xiong et al. 2020;Liang et al. 2022) were also developed by special skills. ...

This paper studies the additive manufacturing (AM)-oriented minimum compliance structural topology optimization (TO) subject to four AM constraints: self-supporting constraint, connectivity constraint, solid-phase minimum length constraint, and void-phase minimum length constraint simultaneously. The essential novelty of this study is that we show that the connectivity constraint can be realized by imposing the void-phase self-supporting constraint. The corresponding proof is given in Appendix. The Elements Scheme (ES) method is used to construct the element-wise self-supporting constraint. By improving the constraint aggregating functions and aggregating a large number of element-wise self-supporting constraints on solid-phase and void-phase structures into three constraints, we propose a concise topology optimization formulation to effectively and simultaneously suppress the small overhang angle boundaries, hanging features (solid-phase upside-down triangles), voids with pointed tips (void-phase upside triangles), slim components, small voids, and enclosed voids in the optimized design. Numerical examples demonstrate the effectiveness of this formulation in comparison with other connectivity control methods.

... Xiong et al. [40] developed an approach to controlling the structural connectivity by generating tunnels. Based on the graph theory and set theory, Zhao et al. [41] developed a direct approach to explicitly controlling the structural complexity during the form-finding process. This approach has been successfully applied to the morphological optimization of biological organs [42]. ...

... The optimization process is implemented through Python code that links to Abaqus. A 2D structure can be treated as a degenerated 3D one (Fig. 5), and the SCC in 2D optimization can be realized by existing methods [36,38,41]. Therefore, this study only considers 3D optimization problems to demonstrate the effectiveness of Table 1. ...

Shape and topology optimization techniques aim to maximize structural performance through material redistribution. Effectively controlling structural complexity during the form-finding process remains a challenging issue. Structural complexity is usually characterized by the number of connected components (e.g., beams and bars), tunnels, and cavities in the structure. Existing structural complexity control approaches often prescribe the number of existing cavities. However, for three-dimensional problems, it is highly desirable to control the number of tunnels during the optimization process. Inspired by the topology-preserving feature of a thinning algorithm, this paper presents a direct approach to controlling the topology of continuum structures under the framework of the bi-directional evolutionary structural optimization (BESO) method. The new approach can explicitly control the number of tunnels and cavities for both two- and three-dimensional problems. In addition to the structural topology, the minimum length scale of structural components can be easily controlled. Numerical results demonstrate that, for a given set of loading and boundary conditions, the proposed methodology may produce multiple high-performance designs with distinct topologies. The techniques developed from this study will be useful for practical applications in architecture and engineering, where the structural complexity usually needs to be controlled to balance the aesthetic, functional, economical, and other considerations.

... The original intention of reducing low efficient materials are the same for both topology optimization and evolution of biological structures in nature. Topology optimization has not only been applied in engineering structural design but also for exploring the optimization mechanisms of biological materials (Zhao et al., 2018;Zhao et al., 2020aZhao et al., , 2020b. Apart from the advantages of achieving high structural performance and low material usage, beautiful structure appearance is also a by-product from topology optimization. ...

... It can produce 0-1 results where there is no transitional materials in the final design (Xia et al., 2018;. Recently, complex constraints on, e.g. the structural complexity/connectivity and the maximum principal stress have been successfully integrated into the BESO-based topology optimization (Zhao et al., 2020a(Zhao et al., , 2020bXiong et al., 2020;Chen et al., 2021). Novel approaches have developed based on the BESO method for generating diverse and competitive designs (He et al., 2020;Yang et al., 2019;Xie et al., 2019). ...

Purpose-Furniture plays a significant role in daily life. Advanced computational and manufacturing technologies provide new opportunities to create novel, high-performance and customized furniture. This paper aims to enhance furniture design and production by developing a new workflow in which computer graphics, topology optimization and advanced manufacturing are integrated to achieve innovative outcomes. Design/methodology/approach-Workflow development is conducted by exploring state-of-the-art computational and manufacturing technologies to improve furniture design and production. Structural design and fabrication using the workflow are implemented. Findings-An efficient transdisciplinary workflow is developed, in which computer graphics, topology optimization and advanced manufacturing are combined. The workflow consists of the initial design, the optimization of the initial design, the postprocessing of the optimized results and the manufacturing and surface treatment of the physical prototypes. Novel chairs and tables, including flat pack designs, are produced using this workflow. The design and fabrication processes are simple, efficient and low-cost. Both additive manufacturing and subtractive manufacturing are used. Practical implications-The research outcomes are directly applicable to the creation of novel furniture, as well as many other structures and devices. Originality/value-A new workflow is developed by taking advantage of the latest topology optimization methods and advanced manufacturing techniques for furniture design and fabrication. Several pieces of innovative furniture are designed and fabricated as examples of the presented workflow.

... Topology optimization is gaining exponentially growing applications in a wide range of industries such as civil, automotive, aerospace and others [1][2][3][4]. It has been recognized as an advanced design method for lightweight and high-performance structures [5,6]. The majority of topology optimization approaches are dedicated to linear elastic structures whose boundary value problem can be formulated as linear system of equations and solved efficiently. ...

... To give an example, several hyperelastic models under the assumption of uniaxial deformation [16] are given in Table 1. (5) in which the tangent modulus D tan e can be analytically calculated from prescribed constitutive model ME={(ε, σ)| σ=f(ε)}. Note that constant stiffness matrix K is usually used in replacement of K tan to avoid the time-consuming reassembly and inverse calculation of large-scale K tan in each iteration. ...

The application of structural topology optimization with complex engineering materials is largely hindered due to the complexity in phenomenological or physical constitutive modeling from experimental or computational material data sets. In this paper, we propose a new data-driven topology optimization (DDTO) framework to break through the limitation with the direct usage of discrete material data sets in lieu of constitutive models to describe the material behaviors. This new DDTO framework employs the recently developed data-driven computational mechanics for structural analysis which integrates prescribed material data sets into the computational formulations. Sensitivity analysis is formulated by applying the adjoint method where the tangent modulus of prescribed uniaxial stress-strain data is evaluated by means of moving least square approximation. The validity of the proposed framework is well demonstrated by the truss topology optimization examples. The proposed DDTO framework will provide a great flexibility in structural design for real applications.

... For instance, the level-set method uses higherdimensional implicit functions [22][23][24][25][26], and the moving morphable components method controls the shapes and layout of a set of structural components. Different kinds of new constraints have been imposed during the optimization process in recent years, such that further structural design problems can be addressed effectively and practically [27][28][29][30][31][32][33][34][35][36]. In addition, topology optimization has been applied in transdisciplinary research such as biomechanical morphogenesis [37][38][39] and metamaterial designs [40][41][42][43]. ...

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.

... In view of these drawbacks, many researchers have made many improvements to density-based method in recent years. Zhao et al. proposed a control method based on graph theory and set theory to control the number and size of internal holes in the optimized structure, which can well balance the structural performance and structural complexity control effect (Zhao et al. 2020). Jiu et al. developed a CAD oriented topology optimization method, the results show that this method is almost free from the limitations of conventional finite element methods in modeling and meshes (Jiu et al. 2020). ...

Spaceborne large aperture membrane microstrip reflectarray antenna has the characteristics of high gain, lightweight and small storage volume, which will be used in future space missions. However, there are two main reasons restricting its application. Firstly, the traditional dimensional optimization method cannot effectively affect the distribution of prestress in the membrane reflector, so it needs to increase too much mass to achieve the goal of stiffness improvement; secondly, low stiffness makes the membrane reflector more sensitive to various uncertainties. In view of this, this paper proposes a method to affect the distribution of prestress by sticking irregular shaped additional layer, and proposes a non-probabilistic uncertain topology optimization method to design the shape of additional layer. The effectiveness of the proposed methods is verified by numerical examples.

... However, experiments are time-consuming and significant resources are utilized. Topological optimization based on numerical simulation is another option [22][23][24]. However, the design space may be extraordinarily massive when the compositional and topological structures are complex [25]. ...

In this paper, optimization of vascular structure of self-healing concrete is performed with deep neural network (DNN). An input representation method is proposed to effectively represent the concrete beams with 6 round pores in the middle span as well as benefit the optimization process. To investigate the feasibility of using DNN for vascular structure optimization (i.e., optimization of the spatial arrangement of the vascular network), structure optimization improving peak load and toughness is first carried out. Afterwards, a hybrid target is defined and used to optimize vascular structure for self-healing concrete, which needs to be healable without significantly compromising its mechanical properties. Based on the results, we found it feasible to optimize vascular structure by fixing the weights of the DNN model and training inputs with the data representation method. The average peak load, toughness and hybrid target of the ML-recommended concrete structure increase by 17.31%, 34.16% and 9.51%. The largest peak load, toughness and hybrid target of the concrete beam after optimization increase by 0.17%, 14.13%, and 3.45% compared with the original dataset. This work shows that the DNN model has great potential to be used for optimizing the design of vascular system for self-healing concrete.

... Zhang et al. (2017) combined the level set method with the structural skeletons to explicitly control the holes for the 2D structures by endowing the independent level set function for each hole. Under the soft-kill bi-directional evolutionary structure optimization (BESO) method, Zhao et al. (2020) proposed a direct approach based on graph theory and set theory to control holes in structural optimization. Han et al. (2021) proposed a heuristic constraint for the hole-filling method based on BESO to achieve the maximum number of constrained holes in the structure. ...

Structural topology can be measured on the basis of its betti numbers. A fundamental feature of structural topology optimization is that it allows the structural topology to be changed during the optimization process. However, traditional structural topology optimization methods use indirect and nonquantitative approaches to change the structural topology during the optimization procedure. Therefore, these traditional methods leave the detailed implementation of optimization with nonintuitive parameters (e.g., filter radius) to adjust the final topology of optimized results. Choosing a suitable nonintuitive parameter for beginners is not straightforward, and makes the optimization procedure complex when applying structural topology optimization methods to engineering design tasks with a preferred level of complexity (number of structural holes). A 2D structure has two betti numbers, B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{0}$$\end{document} and B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{1}$$\end{document}, where B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{0}$$\end{document} and B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B}_{1}$$\end{document} correspond to the number of independent connected components and the number of holes in the structure, respectively. To solve the aforementioned problems, this paper explicitly quantitatively controls over the number of structural holes within the framework of the solid isotropic material with penalty (SIMP) interpolation of the design variable and the method of moving asymptotes (MMA) optimization algorithm in 2D, thus achieving direct unilateral constraint (constraining the maximum number of structural holes) over structural topology. The framework of SIMP and MMA is a powerful way because of its ability to handle more complex problems. Thus, the proposed topological control method based on SIMP and MMA is useful for structural topology optimization research field. For example, the proposed method is based on triangular meshing discretization of the initial design domain; therefore, irregular design domains can be easily processed, and adaptive meshes can be used to improve the geometric approximation of the design domains. Numerical examples show that the proposed method can effectively control the topology, the maximum number of holes (complexity) of the optimized structure.

... Recently, based on the description of the size, shape and number of holes of an optimized structure 1 , topology optimization has been implemented from a geometric complexity perspective in a given design domain. The number of holes in the structure is implicitly designed, and the set of elements in the holes is constrained [2][3][4] . In the 3D case, the number of holes is a topological invariant that is classified as the number of internal enclosed holes (or enclosed voids) and the genus (or the number of tunnels of a structure). ...

Structural topology constraints in topology optimization are an important research topic. The structural topology is characterized by the topological invariance of the number of holes. The holes of a structure in 3D space can be classified as internally enclosed holes and external through-holes (or tunnels). The genus is the number of tunnels. This article proposes the quotient set design variable method (QSDV) to implement the inequality constraint on the maximum genus allowed in an optimized structure for 3D structural topology optimization. The principle of the QSDV is to classify the changing design variables according to the connectivity of the elements in a structure to obtain the quotient set and update the corresponding elements in the quotient set to meet the topological constraint. Based on the standard relaxation algorithm discrete variable topology optimization method (DVTOCRA), the effectiveness of the QSDV is illustrated in numerical examples of a 3D cantilever beam.

... Zhao et al. [4] developed a new method to control topology in structural optimization, wherein the number and size of internal holes were explicitly controlled. The developed method improved the manufacturability of products resulting from structural optimization and provided designers with diverse and competitive solutions. ...

The rigidity and natural frequency of machine tools considerably influence cutting and generate great forces when in contact with the workpiece. The poor static rigidity of these machines can cause deformations and destroy the workpiece. If the natural frequency of the machines is low or close to the commonly used cutting frequency, they vibrate considerably, resulting in poor workpiece surfaces and thus shortening the lifespan of the tool. In this study, the finite element method was mainly used to analyze the structure, static force, modal, frequency spectrum, and transient state of machine tools. The results of the state analysis were verified and compared to the experimental results. The analysis model and conditions were modified to ensure that the analysis results were consistent with the experimental results. Multi-body dynamics analyses were conducted by examining the force of each component and casting of the machine tools and the load of the motor during the cutting stroke. Moreover, an external force was applied to simulate the load condition of the motor when the machine tool is cutting to confirm the feed. In this study, we used topology optimization for effective structural optimization designs. The optimal conditions for topology optimization included lightweight structures, which resulted in reduced structural deformation and increased natural frequency.

... Zhang et al. (2017) controlled the structural complexity through the constraints on multiple level-set functions in a level-set-based topology optimization. Zhao et al. (2020) applied void number and size constraint in evolutionary topology optimization. They used algorithms from graph theory and set theory to describe and adjust the status of voids formed by elements. ...

In powder-based additive manufacturing, the unused powder must be removed after printing. Topology optimization has been applied to designs for additive manufacturing, which may lead to designs with enclosed voids, where the powder will be trapped inside during printing. A topology optimization method incorporating a powder removal passageway is developed to avoid the powder being trapped inside the structure. The passageway is generated by connecting the entrance, all voids, and the exit sequentially. Each void is limited to have only one pair of inlet and outlet to guarantee a single-path flow to facilitate powder removal after the additive manufacturing. The path of the passageway is optimized to minimize its influence on structural stiffness. The proposed optimization method was applied to two practical case studies where the powder removal passageways were generated successfully.

... The structural connectivity was controlled by introducing a new auxiliary temperature field. With the foundation of graph theory and set theory (Zhao et al., 2020), the number and size of interior holes can be controlled by updated BESO (soft-kill bi-directional evolutionary structural optimization) method, and the conception of structural complexity control was proposed. Genus, the concept that describes topological invariant was used to directly control the maximum number of holes for the design domain. ...

Topological constraints have recently been introduced to structural topology optimization by the BESO method. However, for the classical and widely used SIMP-type optimization method, an implicit and continuously changing variable cannot express the topological characteristics directly during the optimization process. This is partly caused by missing well-defined boundaries to compute topological characteristics. To introduce topological constraints into the SIMP-type method, an auxiliary discrete expression of structural boundaries through the volume preservation projection method is used to compute topological characteristics, that is, the genus or number of holes. A topological control methodology based on persistence homology, a numerical calculation idea derived from topological data analysis, is introduced in this paper to implement topological constraints. With the help of the design space progressive restriction method, the proposed methodology shows that for the 2D static minimum compliance optimization problem, the inequality constraints on the number of holes can be satisfied. The effectiveness of the proposed topological control method for the SIMP-type framework is illustrated by several numerical examples.

... Recently, a transdisciplinary computational framework was established to reveal the developmental mechanisms of animal and plant tissues through biomechanical morphogenesis [18,19]. Besides, much effort has been directed toward increasing the resolutions of the design domain [20,21], enhancing the manufacturability of the optimized results [22], improving the multi-material compatibility of the optimization process [23], and controlling the structural complexity and connectivity [24,25]. ...

Topology optimization has rapidly developed as a powerful tool of structural design in multiple disciplines. Conventional topology optimization techniques usually optimize the material layout within a predefined, fixed design domain. Here, we propose a subdomain-based method that performs topology optimization in an adaptive design domain (ADD). A subdomain-based parallel processing strategy that can vastly improve the computational efficiency is implemented. In the ADD method, the loading and boundary conditions can be easily changed in concert with the evolution of the design space. Through the automatic, flexible, and intelligent adaptation of the design space, this method is capable of generating diverse high-performance designs with distinctly different topologies. Five representative examples are provided to demonstrate the effectiveness of this method. The results show that, compared with conventional approaches, the ADD method can improve the structural performance substantially by simultaneously optimizing the layout of material and the extent of the design space. This work might help broaden the applications of structural topology optimization.

... The original intention of reducing low efficient materials are the same for both topology optimization and evolution of biological structures in nature. Topology optimization has not only been applied in engineering structural design, but also for exploring the optimization mechanisms of biological materials (Zhao et al., 2018;Zhao et al., 2020aZhao et al., , 2020b. ...

Purpose
Furniture plays a significant role in daily life. Advanced computational and manufacturing technologies provide new opportunities to create novel, high-performance and customized furniture. This paper aims to enhance furniture design and production by developing a new workflow in which computer graphics, topology optimization and advanced manufacturing are integrated to achieve innovative outcomes.
Design/methodology/approach
Workflow development is conducted by exploring state-of-the-art computational and manufacturing technologies to improve furniture design and production. Structural design and fabrication using the workflow are implemented.
Findings
An efficient transdisciplinary workflow is developed, in which computer graphics, topology optimization and advanced manufacturing are combined. The workflow consists of the initial design, the optimization of the initial design, the postprocessing of the optimized results and the manufacturing and surface treatment of the physical prototypes. Novel chairs and tables, including flat pack designs, are produced using this workflow. The design and fabrication processes are simple, efficient and low-cost. Both additive manufacturing and subtractive manufacturing are used.
Practical implications
The research outcomes are directly applicable to the creation of novel furniture, as well as many other structures and devices.
Originality/value
A new workflow is developed by taking advantage of the latest topology optimization methods and advanced manufacturing techniques for furniture design and fabrication. Several pieces of innovative furniture are designed and fabricated as examples of the presented workflow.

... In the past three decades, several optimization techniques have been successfully established, including the solid isotropic material with penalization (SIMP) method (Bendsøe, 1989;Bendsøe, 1995;Sigmund and Maute, 2013), the level-set method (Allaire et al., 2002;Wang et al., 2003), the evolutionary structural optimization (ESO) method (Xie and Steven, 1993;Xie and Steven, 1997) and the bi-directional evolutionary structural optimization (BESO) method Xie, 2007, 2009). Most recently, imposing complicated constraints during the form-finding process has been realized (Chen et al., 2020;He et al., 2020;Xiong et al., 2020;Zhao et al., 2020a). By establishing transdisciplinary computational methods for biomechanical morphogenesis, Zhao et al. (2018Zhao et al. ( , 2020bZhao et al. ( , 2020c 4 have revealed the optimization mechanisms of, e.g., plant leaves and animal stingers. ...

The unique, hierarchical patterns of leaf veins have attracted extensive attention in recent years. However, it remains unclear how biological and mechanical factors influence the topology of leaf veins. In this paper, we investigate the optimization mechanisms of leaf veins through a combination of experimental measurements and numerical simulations. The topological details of three types of representative plant leaves are measured. The experimental results show that the vein patterns are insensitive to leaf shapes and curvature. The numbers of secondary veins are independent of the length of the main vein, and the total length of veins increases linearly with the leaf perimeter. By integrating biomechanical mechanisms into the topology optimization process, a transdisciplinary computational method is developed to optimize leaf structures. The numerical results show that improving the efficiency of nutrient transport plays a critical role in the morphogenesis of leaf veins. Contrary to the popular belief in the literature, this study shows that the structural performance is not a key factor in determining the venation patterns. The findings provide a deep understanding of the optimization mechanism of leaf veins, which is useful for the design of high-performance shell structures.

... Gholizadeh et al. (2020) introduced the Newton Metaheuristic Algorithm for discrete performance-based seismic design optimization of steel moment frames. As topology optimization for practical engineering problems, Zhao et al. (2020) proposed a direct approach to control the number and size of interior holes of the optimized structures. The method not only enables the designer to have a direct control over the topology of the optimized structures but also provides diverse and competitive solutions. ...

In this study, we present a novel non-parametric shape-topology optimization method for frame structures with multi-materials, aiming at designing more light and stiff frame structures. The sum of squared error norm for achieving the target displacements on the specified members is minimized under the volume constraints of multi-materials as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the material gradient functions for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The generalized solid isotropic material with penalization (GSIMP) method is employed to classify into the multi-materials in topology optimization. The gradient functions derived are applied to the unified H¹ gradient method for frame structures to determine the optimal shape and material variations. With this method, the optimal free-form and topology for arbitrary large-degree of design freedom frame structures with multi-materials can be obtained without any shape and topology parameterization. Numerical examples with various materials and different boundary conditions are demonstrated and the results are discussed.

... Topology optimization is a mathematical technique commonly used in free-form designs. Since its invention (Bendsøe and Kikuchi 1988), various TO-based design approaches have been developed (Jakiela et al. 2000, Wang, M. Y. et al. 2003, Juan et al. 2008, Schevenels et al. 2011, Guo et al. 2014, Zhang, W. et al. 2017, Zhang, X. et al. 2019, Zhao et al. 2020) and applied to design a wide range of structures and products such as automobile and aircraft parts/components (Cavazzuti et al. 2011, Zhu et al. 2016. The advent in additive manufacturing technologies has further broadened the application scope of TO. ...

Topology optimization (TO) is a common technique used in free-form designs. However, conventional TO-based design approaches suffer from high computational cost due to the need for repetitive forward calculations and/or sensitivity analysis, which are typically done using high-dimensional simulations such as finite element analysis (FEA). In this work, artificial neural networks are used as efficient surrogate models for forward and sensitivity calculations in order to greatly accelerate the design process of topology optimization. To improve the accuracy of sensitivity analyses, dual-model artificial neural networks that are trained with both forward and sensitivity data are constructed and are integrated into the Solid Isotropic Material with Penalization (SIMP) method to replace the FEA. The performance of the accelerated SIMP method is demonstrated on two benchmark design problems namely minimum compliance design and metamaterial design. The efficiency gained in the problem with size of 64 × 64 is 137 times in forward calculation and 74 times in sensitivity analysis. In addition, effective data generation methods suitable for TO designs are investigated and developed, which lead to a great saving in training time. In both benchmark design problems, a design accuracy of 95% can be achieved with only around 2000 training data.

... Furthermore, according to the working manner of the SLM technique, a large amount of metal powder will be left in the cavities of the rudder and should definitely be removed after manufacturing. In the academic community, design with closed voids is generally regarded as bad designs for additive manufacturing and several approaches have been put forward to eliminate closed voids in topology optimization [50][51][52][53]. In this work, the authors, from an engineering point of view, propose to open some powder-discharge holes on the ribs so that those cavities can be connected to outside space and the metal powder can be discharged. ...

In high-speed vehicles, rudders often endure both aerodynamic pressure and thermal loads. The innovative design of rudders is of great importance for the performance of the whole vehicle. In this work, thermo-elastic topology optimization is adopted to design a typical all-movable rudder structure. The compliance of the rudder skin is considered to be a new objective and the moment of inertia of the rudder is constrained during optimization to ensure its fast response to instructions of the control system. Then sensitivity analysis of the structural compliance and the moment of inertia is carried out. Optimization results show that thermal load has a great effect on the optimized configuration and minimizing the compliance of the rudder skin gives much better design than minimizing the global compliance. Subsequently, an engineering-oriented post-processing is conducted to make the optimized design suitable for additive manufacturing. An appropriate printing direction is selected based on the layout of the optimized ribs and certain ribs are reshaped with fillets to make the rudder free of internal support structures. Besides, according to a secondary topology optimization of the ribs and the stress distribution, a set of powder-discharge holes are properly opened on the ribs so that all cavities within the rudder are connected and the metal powder inside the rudder can be discharged with little effort after manufacturing. Finally, the optimized design is successfully printed using Selective Laser Melting, demonstrating the proposed post-processing is effective for additive manufacturing.

... As an important branch of topology optimization, the BESO technique is based on the simple concept of gradually removing inefficient material from a structure and adding material to the most needed locations at the same time [24]. It is widely 3 recognized owing to its high-quality topology solutions [25], ease of understanding and implementation [26], and excellent computational efficiency [27]. ...

Background: As an advanced design technique, topology optimization has received much attention over the past three decades. Topology optimization aims at finding an optimal material distribution in order to maximize the structural performance while satisfying certain constraints. It is a useful tool for the conceptional design. At the same time, additive manufacturing technologies have provided unprecedented opportunities to fabricate intricate shapes generated by topology optimization.
Objective: To design a highly efficient structure using topology optimization and to fabricate it using additive manufactur-ing.
Method: The bi-directional evolutionary structural optimization (BESO) technique provides the conceptional design, and the topology-optimized result is post-processed to obtain smooth structural boundaries.
Results: We have achieved a highly efficient and elegant structural design which won the first prize in a national competition in China on design optimization and additive manufacturing.
Conclusion: In this paper, we present an effective topology optimization approach to maximizing the structural load-bearing capacity and establish a procedure to achieve efficient and elegant structural designs. In the loading test of the final competition, our design carried the highest loading and won the first prize of the competition, which clearly demonstrates the capability of BESO in engineering applications.

... Topology optimization is a mathematical technique commonly used in free-form designs. Since its invention (Bendsøe and Kikuchi 1988), various TO-based design approaches have been developed (Jakiela et al. 2000, Wang, M. Y. et al. 2003, Juan et al. 2008, Schevenels et al. 2011, Guo et al. 2014, Zhang, W. et al. 2017, Zhang, X. et al. 2019, Zhao et al. 2020) and applied to design a wide range of structures and products such as automobile and aircraft parts/components (Cavazzuti et al. 2011, Zhu et al. 2016. The advent in additive manufacturing technologies has further broadened the application scope of TO. ...

Topology optimization (TO) is a common technique used in free-form designs. However, conventional TO-based design approaches suffer from high computational cost due to the need for repetitive forward calculations and/or sensitivity analysis, which are typically done using high-dimensional simulations such as Finite Element Analysis (FEA). In this work, neural networks are used as efficient surrogate models for forward and sensitivity calculations in order to greatly accelerate the design process of topology optimization. To improve the accuracy of sensitivity analyses, dual-model neural networks that are trained with both forward and sensitivity data are constructed and are integrated into the Solid Isotropic Material with Penalization (SIMP) method to replace FEA. The performance of the accelerated SIMP method is demonstrated on two benchmark design problems namely minimum compliance design and metamaterial design. The efficiency gained in the problem with size of 64x64 is 137 times in forward calculation and 74 times in sensitivity analysis. In addition, effective data generation methods suitable for TO designs are investigated and developed, which lead to a great saving in training time. In both benchmark design problems, a design accuracy of 95% can be achieved with only around 2000 training data.

... The ESO and BESO methods have been used for solving topology optimization problems in many areas of structural engineering. These problems include structural frequency optimization (Xie and Steven 1994), minimizing structural volume with a displacement or compliance constraint (Liang et al. 2000), structural complexity control in topology optimization (Zhao et al. 2020a;Xiong et al. 2020), topology optimization for energy absorption structures (Huang et al. 2007), design of periodic structures (Huang and Xie 2008), geometrical and material nonlinearity problems (Huang and Xie 2007a), stiffness optimization of structures with multiple materials (Huang and Xie 2009), maximizing the fracture resistance of quasi-brittle composites (Xia et al. 2018a), stress minimization designs (Xia et al. 2018b), biomechanical morphogenesis (Zhao et al. 2018(Zhao et al. , 2020b, stiffness maximization of structures with von Mises constraints (Fan et al. 2019), and diverse and competitive designs (Xie et al. 2019;Yang et al. 2019;He et al. 2020). ...

Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion. In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint. The present methodology will be useful for designing concrete structures.

... Based on different penalty methods, Yang et al. proposed five simple strategies to obtain diverse and competitive designs, which can be easily integrated into commonly used topology optimization techniques [34]. Zhao et al. developed an approach to control the number and size of the interior holes of structures [35]. ...

Topology optimization techniques have been widely used in structural design. Conventional optimization techniques usually are aimed at achieving the globally optimal solution which maximizes the structural performance. In practical applications, however, designers usually desire to have multiple design options, as the single optimal design often limits their artistic intuitions and sometimes violates the functional requirements of building structures. Here we propose three stochastic approaches to generating diverse and competitive designs. These approaches include (1) penalizing elemental sensitivities, (2) changing initial designs, and (3) integrating the genetic algorithm into the bi-directional evolutionary structural optimization (BESO) technique. Numerical results demonstrate that the proposed approaches are capable of producing a series of random designs, which possess not only high structural performance, but also distinctly different topologies. These approaches can be easily implemented in different topology optimization techniques. This work is of significant practical importance in architectural engineering where multiple design options of high structural performance are required.

... Most recently, Zhao et al. proposed an effective approach to controlling the structural connectivity in topology optimization [49]. Using this approach, the structural performance and the effect of the structural complexity control can be well balanced. ...

Topology optimization is increasingly used in lightweight designs for additive manufacturing (AM). However, conventional optimization techniques do not fully consider manufacturing constraints. One important requirement of powder-based AM processes is that enclosed voids in the designs must be avoided in order to remove and reuse the unmelted powder. In this work, we propose a new approach to realizing the structural connectivity control based on the bi-directional evolutionary structural optimization technique. This approach eliminates enclosed voids by selectively generating tunnels that connect the voids with the structural boundary during the optimization process. The developed methodology is capable of producing highly efficient structural designs which have no enclosed voids. Furthermore, by changing the radius and the number of tunnels, competitive and diverse designs can be achieved. The effectiveness of the approach is demonstrated by two examples of three-dimensional structures. Prototypes of the obtained designs without enclosed voids have been fabricated using AM.

This paper presents a study of the morphogenesis of brain corals based on an experimental investigation and a topological optimization method. The resistance to matter interchange was employed to allocate the optimal space for the growth of polyp colonies from the perspective of topological optimization, where the optimized structures are those of natural brain corals. Computational fluid dynamics simulations revealed that these complicated structures can provide shelter to protect polyps from ocean currents. A reverse mold was prepared from silica gel and used to cast models from mixtures of cement and calcium carbonate, where the mixture ratio was determined based on compressive strength and biocompatibility. Based on an acid corrosion experiment, the matter interchange capability was verifi�ed. This study also proved that the many folds in the structure of brain corals contribute to the circulation of seawater, thus maintaining the concentration of nutrients and hindering the deposition of harmful substances. This paper establishes an innovative methodology for the creation of artificial brain corals, which is important for environmental restoration.
Keywords: Brain corals, Topological optimization, Turbulent
hydrodynamics, Matter interchange

近年来，越来越多的设计师使用拓扑优化技术来寻找优美且新颖的建筑设计。然而由于无法直接满足建筑师与工程师提出的诸多设计需求，现有方法生成的拓扑优化设计往往很少在实际案例（特别是大型项目）中出现。本文指出了拓扑优化中惯用假设的局限性，并揭示了寻找设计多解的重要性。为了生成多样化、高性能且满足使用需求的设计，我们突破了这些限制并提出了面向建筑领域的拓扑优化新方法。与传统的拓扑优化不同，我们可以将荷载和边界条件作为额外的设计变量，以显著提高最终设计的结构性能。此外，改变设计域能带来更多的可能性，使设计者可以从诸多设计方案中的获得满意的结果。

In recent years, topology optimization has become a popular strategy for creating elegant and innovative forms for architectural design. However, the use of existing topology optimization techniques in practical applications, especially for large-scale projects, is rare because the generated forms often cannot satisfy all the design requirements of architects and engineers. This paper identifies the limitations of commonly used assumptions in topology optimization and highlights the importance of having multiple solutions. We show how these limitations could be removed and present various techniques for generating diverse and competitive structural designs that are more useful for architects. Unlike conventional topology optimization, we may include load and support conditions as additional design variables to enhance the structural performance substantially. Furthermore, we show that varying the design domain provides a plethora of opportunities to achieve more-desirable design outcomes.

The structural complexity (the number of holes) of the 2D or 3D continuum structures can be measured by their topology invariants (i.e., Euler and Betti numbers). Controlling the 2D and 3D structural complexity is significant in topology optimization design because of the various consideration, including manufacturability and necessary structural redundancy, but remains a challenging subject. In this paper, we propose a programmable Euler–Poincaré formula to efficiently calculate the Euler and Betti numbers for the 0–1 pixel-based structures. This programmable Euler–Poincaré formula only relates to the nodal density and nodal characteristic vector that represents the nodal neighbor relation so that it avoids manually counting the information of the vertices, edges, and planes on the surfaces of the structure. As a result, the explicit formulations between the structural complexity (the number of holes) and the discrete density design variables for 2D and 3D continuum structures can be efficiently constructed. Furthermore, the discrete variable sensitivity of the structural complexity is calculated through the programmable Euler–Poincaré formula so that the structural complexity control problem can be efficiently and mathematically solved by Sequential Approximate Integer Programming and Canonical relaxation algorithm Various 2D and complicated 3D numerical examples are presented to demonstrate the effectiveness of the method. We further believe that this study bridges the gap between structural topology optimization and mathematical topology analysis, which is much expected in the structural optimization community.

The paper presents a novel core design for sandwich panels and conducts an experiment to determine whether the mechanical strength of symmetric aluminum lattice core sandwich panels can be improved. Both Design of Experiments (DOE) and Response Surface Methodology (Box-Behnken) were used to establish a quantitative relationship between the strength-to-weight ratio and the input parameters. The thickness of the sheet, the height of sandwich panels, and the width of the seat were all considered design variables to achieve the optimal state. The maximum Initial Peak Crushing Forces (IPCF) were then determined using quasi-static axial flatwise compression tests. This study found that the model's predicted values were consistent with the experimental results. As a result, the parameters were optimized using the Design-Expert software to maximize the initial peak force while minimizing the weight. The results were validated using the Genetic Algorithm, NSGA2, and LINGO. The results indicated that the height of the sandwich panel and the thickness of the sheet had the most significant impact on the maximum force and panel weight. To this end, it is concluded that introducing a novel core design for the sandwich panel, utilizing a suitable Snap-Fitting method for attaching lattice parts rather than using a paste, and finally optimizing the core were the primary reasons for achieving this level of strength.

Topology optimization is a powerful tool for designing high-performance structures. However, the structures resulting from topology optimization usually have complex geometries, which makes them difficult or costly to fabricate. As a result, topology optimization is often used for the conceptual design of product structures. In this paper, a topology optimization method considering manufacturing constraints is proposed under the fixed finite element mesh. The minimum length scale and chamfer/round are controlled as required based on the floating projection topology optimization (FPTO) method, where the linear material interpolation scheme is adopted instead and the material 0/1 distribution is realized by applying sequential constraints on the elemental design variables through the floating projection. The minimum length scale is strictly controlled with the help of the structural skeleton, which is extracted from the structural topology by using a graphic thinning algorithm. Meanwhile, boundary filtering is proposed by using a variable filtering radius to control chamfers and rounds. Two-dimensional and three-dimensional numerical examples demonstrate that the proposed topology optimization algorithm is effective for designing the stiffest structures with smooth boundaries, desired minimum length scale and chamfers/rounds, so as to improve their manufacturability.

All-angle negative refraction (AANR) of phononic crystals and its frequency range are dependent on mechanical properties of constituent materials and their spatial distribution. So far, it is impossible to achieve the maximum operation frequency range of AANR theoretically. In this paper, we will present a numerical approach for designing a two-dimensional phononic crystal with broadband AANR without negative index. Through analyzing the mechanism of AANR, a topology optimization problem aiming at broadband AANR is established and solved by bi-directional evolutionary structural optimization method. The optimal steel/air phononic crystal exhibits a record AANR range over 20% and its refractive properties and focusing effects are further investigated. The results demonstrate the multifunctionality of a flat phononic slab including superlensing effect near upper AANR frequencies and self-collimation at lower AANR frequencies.

In the present paper, an explicit method for controlling the structural complexity of continuum structures in topology optimization is proposed. This method is devised under the Moving Morphable Component (MMC) based framework where the geometries of the basic building blocks for topology optimization are described explicitly. Compared to the existing structural complexity control approaches which are generally developed in an implicit geometry/topology description framework, the proposed method allows an explicit definition of structural complexity, which facilitates the construction of the controlling schemes significantly. The effectiveness of the proposed method is demonstrated by numerical examples shown at the end of the paper.

The evolutionary structural optimization (ESO) method developed by Xie and Steven (1993, [162]), an important branch of topology optimization, has undergone tremendous development over the past decades. Among all its variants , the convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) method developed by Huang and Xie (2007, [48]) allowing both material removal and addition, has become a widely adopted design methodology for both academic research and engineering applications because of its efficiency and robustness. This paper intends to present a comprehensive review on the development of ESO-type methods, in particular the latest con-vergent and mesh-independent BESO method is highlighted. Recent applications of the BESO method to the design of advanced structures and materials are summarized. Compact Malab codes using the BESO method for benchmark structural and material microstructural designs are also provided.

Truss optimization based on the ground structure approach often leads to designs that are too complex for practical purposes. In this paper we present an approach for design complexity control in truss optimization. The approach is based on design complexity measures related to the number of bars (similar to Asadpoure et al. Struct Multidisc Optim 51(2):385–396 2015) and a novel complexity measure related to the number of nodes of the structure. Both complexity measures are continuously differentiable and thus can be used together with gradient based optimization algorithms. The numerical examples show that the proposed approach is able to reduce design complexity, leading to solutions that are more fit for engineering practice. Besides, the examples also indicate that in some cases it is possible to significantly reduce design complexity with little impact on structural performance. Since the complexity measures are non convex, a global gradient based optimization algorithm is employed. Finally, a detailed comparison to a classical approach is presented.

Additive manufacturing (AM) technologies, such as selective laser sintering (SLS) and fused deposition modeling (FDM), have become the powerful tools for direct manufacturing of complex parts. This breakthrough in manufacturing technology makes the fabrication of new geometrical features and multiple materials possible. Past researches on designs and design methods often focused on how to obtain desired functional performance of the structures or parts, specific manufacturing capabilities as well as manufacturing constraints of AM were neglected. However, the inherent constraints in AM processes should be taken into account in design process. In this paper, the enclosed voids, one type of manufacturing constraints of AM, are investigated. In mathematics, enclosed voids restriction expressed as the solid structure is simplyconnected. We propose an equivalent description of simply-connected constraint for avoiding enclosed voids in structures, named as virtual temperature method (VTM). In this method, suppose that the voids in structure are filled with a virtual heating material with high heat conductivity and solid areas are filled with another virtual material with low heat conductivity. Once the enclosed voids exist in structure, the maximum temperature value of structure will be very high. Based upon this method, the simplyconnected constraint is equivalent to maximum temperature constraint. And this method can be easily used to formulate the simply-connected constraint in topology optimization. The effectiveness of this description method is illustrated by several examples. Based upon topology optimization, an example of 3D cantilever beam is used to illustrate the trade-off between manufacturability and functionality. Moreover, the three optimized structures are fabricated by FDM technology to indicate further the necessity of considering the simply-connected constraint in design phase for AM. © 2015, Higher Education Press and Springer-Verlag Berlin Heidelberg.

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

This paper presents a methodology for including fixed-area flexible void domains into the minimum compliance topology optimization problem. As opposed to the standard passive elements approach of rigidly specifying void areas within the design domain, the suggested approach allows these areas to be flexibly reshaped and repositioned subject to penalization on their moments of inertia, the positions of their centers of mass, and their shapes. The flexible void areas are introduced through a second, discrete design variable field, using the same discretization as the standard field of continuous density variables. The formulation is based on a combined approach: The primary sub-problem is to minimize compliance, subject to a volume constraint, with a secondary sub-problem of minimizing the disturbance from the flexible void areas. The design update is performed iteratively between the two sub-problems based on an optimality criterion and a discrete update scheme, respectively. The method is characterized by a high flexibility, while keeping the formulation very simple. The robustness and applicability of the method are demonstrated through a range of numerical examples. The flexibility of the method is demonstrated through several extensions, including a shape measure requiring the flexible void area to fit a given reference geometry.

This paper deals with recent developments of topology optimization techniques for application in some new types of design problems. The emphasis is on recent work of the Danish research groups at Aalborg University and at the Technical University of Denmark and focus is on the central role that the choice of objective functions and design parameterization plays for a successful extension of the material distribution approach to new design settings and to new types of physics models. The applications that will be outlined encompass design of laminated composite structures, design for pressure loads, design in fluids, design in acoustics, and design in photonics. A short outline of other design optimization activities is also given.

Purpose
This paper provides an overview of the different binding mechanisms in selective laser sintering (SLS) and selective laser melting (SLM), thus improving the understanding of these processes.
Design/methodology/approach
A classification of SLS/SLM processes was developed, based on the binding mechanism occurring in the process, in contrast with traditional classifications based on the processed material or the application. A broad range of commercial and experimental SLS/SLM processes – found from recent articles as well as from own experiments – was used to explain the different binding mechanism categories.
Findings
SLS/SLM processes can be classified into four main binding mechanism categories, namely “solid state sintering”, “chemically induced binding”, “liquid phase sintering – partial melting” and “full melting”. Most commercial processes can be classified into the latter two categories, which are therefore subdivided. The binding mechanism largely influences the process speed and the resulting part properties.
Research limitations/implications
The classification presented is not claimed to be definitive. Moreover some SLM/SLM processes could be classified into more than one category, based on personal interpretation.
Originality/value
This paper can be a useful aid in understanding existing SLS/SLM processes. It can also serve as an aid in developing new SLS/SLM processes.

The increasing complexity of engineering systems has sparked rising interest in multidisciplinary optimization (MDO). This paper surveys recent publications in the field of aerospace, in which the interest in MDO has been particularly intense. The primary c hallenges in MDO are computational expense and organizational complexity. Accordingly, this survey focuses on various methods used by different researchers to address these challenges. The survey is organized by a breakdown of MDO into its conceptual components, reflected in sections on mathematical modelling, approximation concepts, optimization procedures, system sensitivity, and human interface. Because the authors' primary area of expertise is in the structures discipline, the majority of the references focus on the interaction of this discipline with others. In particular, two sections at the end of this review focus on two interactions that have recently been pursued with vigour: the simultaneous optimization of structures and aerodynamics and the simultaneous optimization of structures with active control.

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

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

In recent years, the Evolutionary Structural Optimization (ESO) method has been developed into an effective engineering design
tool, allowing various structural constraints to be incorporated into the optimization process such as natural frequency,
buckling, stiffness, stress, displacement and heat. However, no attempts have been made to incorporate nonstructural constraints
such as the number of cavities in the final topology and manufacturing constraints. This paper introduces a modification of
the ESO method named Intelligent Cavity Creation (ICC) by which the number of cavities can be controlled. This method has
the additional benefit of eliminating the formation of checkerboard patterns. The proposed ICC algorithm is applied to several
optimization problems to show its effectiveness. It is also demonstrated that ICC produces more practical topologies.

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

Shape and topology optimization techniques are widely used to maximize the performance or minimize the weight of a structure through optimally distributing its material within a prescribed design domain. However, existing optimization techniques usually produce a single optimal solution for a given problem. In practice, it is highly desirable to obtain multiple design options which not only possess high structural performance but have distinctly different shapes and forms. Here we present five simple and effective strategies for achieving such diverse and competitive structural designs. These strategies have been successfully applied in the computational morphogenesis of various structures of practical relevance and importance. The results demonstrate that the developed methodology is capable of providing the designer with structurally efficient and topologically different solutions. The structural performance of alternative designs is only slightly lower than that of the optimal design. This work establishes a general approach to achieving diverse and competitive structural forms, which holds great potential for practical applications in architecture and engineering.

It remains a puzzling issue why and how the organs in plants living in the same natural environment evolve into a wide variety of geometric architecture. In this work, we explore, through a combination of experimental and numerical methods, the biomechanical morphogenesis of the leaves and stalks of representative emergent plants, which can stand upright and survive in harsh water environments. An interdisciplinary topology optimization method is developed here by integrating both mechanical performance and biological constraint into the bi-directional evolutionary structural optimization technique. The experimental and numerical results reveal that, through natural selection over many million years, these leaves and stalks have been optimized into distinctly different cross-sectional shapes and aerenchyma tissues with intriguing anatomic patterns and improved load-bearing performance. The internal aerenchyma is an optimal compromise between the mechanical performance and functional demands such as air exchange and nutrient transmission. We find that the optimal distribution of the internal material depends on multiple biomechanical factors such as the cross-sectional geometry, hierarchical structures, boundary condition, biological constraint, and material property. This work provides an in-depth understanding of the property–structure–performance–function interrelations of biological materials. The proposed topology optimization method and the presented biophysical insights hold promise for designing highly efficient and advanced structures (e.g., airplane wings and turbine blades) and analyzing other biological materials (e.g., bones, horns, and beaks).

A method is proposed for the control of minimum/maximum length scale in the level set based structural topology optimization. The minimum/maximum length scale of structure is characterized by using the concept of smallest/biggest maximal inscribable ball. In order to prevent trivial zero minimum length scale, the skeleton of structure is utilized and trimmed. The control of length scale is realized by constraining the distance from boundary to skeleton, and the distance is explicitly constructed by using the highly efficient fast marching method. Numerical examples in two dimensions are investigated.

Numerical layout optimization provides a computationally efficient and generally applicable means of identifying the optimal arrangement of bars in a truss. When the plastic layout optimization formulation is used, a wide variety of problem types can be solved using linear programming. However, the solutions obtained are frequently quite complex, particularly when fine numerical discretizations are employed. To address this, the efficacy of two rationalization techniques are explored in this paper: (i) introduction of ‘joint lengths’, and (ii) application of geometry optimization. In the former case this involves the use of a modified layout optimization formulation, which remains linear, whilst in the latter case a non-linear optimization post-processing step, involving adjusting the locations of nodes in the layout optimized solution, is undertaken. The two rationalization techniques are applied to example problems involving both point and distributed loads, self-weight and multiple load cases. It is demonstrated that the introduction of joint lengths reduces structural complexity at negligible computational cost, though generally leads to increased volumes. Conversely, the use of geometry optimization carries a computational cost but is effective in reducing both structural complexity and the computed volume.

Symmetry is not only one of the most fundamental concepts in science and engineering, but it is also an ideal bridging idea crossing various branches of sciences and different fields of engineering. In the past, symmetry has been considered important for its aesthetic appeal; however, this century has witnessed a great enhancement in its recognition as a basis of scientific and engineering principle. At the same time, the meaning and utility of symmetry have greatly expanded. It is not surprising that many valuable books are published in this field and regular annual conferences are devoted to symmetry in various fields of science and engineering. In the following, different definitions are provided for symmetry.

Through natural selection, many animal organs with similar functions have evolved different macroscopic morphologies and microscopic structures. Here, we comparatively investigate the structures, properties, and functions of honey bee stings and paper wasp stings. Their elegant structures were systematically observed. To examine their behaviors of penetrating into different materials, we performed penetration-extraction tests and slow motion analyses of their insertion process. In comparison, the barbed stings of honey bees are relatively difficult to be withdrawn from fibrous tissues (e.g., skin), while the removal of paper wasp stings is easier due to their different structures and insertion skills. The similarities and differences of the two kinds of stings are summarized on the basis of the experiments and observations.
© 2015. Published by The Company of Biologists Ltd.

In this paper, two novel methods are proposed for optimizing the layout of structural systems with embedding components considering the minimum/maximum distance constraints between the components. The key ideas are using level set functions to describe the shapes of arbitrary irregular embedding components and resorting to the concept of structural skeleton to formulate the distance control constraints explicitly. Numerical examples presented demonstrate that the proposed approaches can give a complete control of the layout of embedding components in an explicit and local way.

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

In this article, we propose a method to incorporate fabrication cost in the topology optimization of light and stiff truss structures and periodic lattices. The fabrication cost of a design is estimated by assigning a unit cost to each truss element, meant to approximate the cost of element placement and associated connections. A regularized Heaviside step function is utilized to estimate the number of elements existing in the design domain. This makes the cost function smooth and differentiable, thus enabling the application of gradient-based optimization schemes. We demonstrate the proposed method with classic examples in structural engineering and in the design of a material lattice, illustrating the effect of the fabrication unit cost on the optimal topologies. We also show that the proposed method can be efficiently used to impose an upper bound on the allowed number of elements in the optimal design of a truss system. Importantly, compared to traditional approaches in structural topology optimization, the proposed algorithm reduces the computational time and reduces the dependency on the threshold used for element removal.

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.

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.

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

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.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

Topology optimization is a computational tool that can be used for the systematic design of photonic crystals, waveguides, resonators, filters and plasmonics. The method was originally developed for mechanical design problems but has within the last six years been applied to a range of photonics applications. Topology optimization may be based on finite element and finite difference type modeling methods in both frequency and time domain. The basic idea is that the material density of each element or grid point is a design variable, hence the geometry is parameterized in a pixel-like fashion. The optimization problem is efficiently solved using mathematical programming-based optimization methods and analytical gradient calculations. The paper reviews the basic procedures behind topology optimization, a large number of applications ranging from photonic crystal design to surface plasmonic devices, and lists some of the future challenges in non-linear applications.

In this article, a modified (‘filtered’) version of the minimum compliance topology optimization problem is studied. The direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator. In this setting it is possible to establish the existence of solutions. Moreover, convergence of an approximation by means of finite elements can be obtained. This is illustrated through some numerical experiments. The ‘filtering’ technique is also shown to cope with two important numerical problems in topology optimization, checkerboards and mesh dependent designs. Copyright © 2001 John Wiley & Sons, Ltd.

Today, real-world crashworthiness optimization applications are limited to sizing and shape optimization. Topology optimization in crashworthiness design has been withstanding until today any attempt of finding efficient solution algorithms. This is basically due to the high computational effort and the inherent sensitivity of crash simulation responses to design scatterings. In this work, the topology optimization problem shall be addressed with a new approach, in such a way as mathematical graphs are used to describe the optimization sequence (including geometry, loads, design variables and responses, etc.). This design conception is a good advance in topological model flexibility and allows for the application of new (e.g. rule-based) topology optimization algorithms. In this contribution, the topology optimization of crash loaded flight passenger seats is presented. Therefore, we focus on the necessary workflow which includes the graph-based description of the structure´s topology, the CAD description of the structure and the formulation of the crash problem in LS-DYNA. This workflow is included in an optimization loop.

A new integrated layout optimization method is proposed here for the design of multi-component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non-overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal.
Finally, to validate the proposed design procedure, a variety of multi-component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright

A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non-linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user-defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.

Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or
sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and
void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple
projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based
on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.
KeywordsTopology optimization–Robust design–Compliant mechanisms–Manufacturing constraints

The aim of this article is to initiate an exchange of ideas on symmetry and non-uniqueness in topology optimization. These
concepts are discussed in the context of 2D trusses and grillages, but could be extended to other structures and design constraints,
including 3D problems and numerical solutions. The treatment of the subject is pitched at the background of engineering researchers,
and principles of mechanics are given preference to those of pure mathematics. The author hopes to provide some new insights
into fundamental properties of exact optimal topologies. Combining elements of the optimal layout theory (of Prager and the
author) with those of linear programming, it is concluded that for the considered problems the optimal topology is in general
unique and symmetric if the loads, domain boundaries and supports are symmetric. However, in some special cases the number
of optimal solutions may be infinite, and some of these may be non-symmetric. The deeper reasons for the above findings are
explained in the light of the above layout theory.
KeywordsTopology optimization–Non-uniqueness–Symmetry–Optimal layout theory–Trusses–Grillages

This paper presents a technique for imposing maximum length scale on features in continuum topology optimization. The design
domain is searched and local constraints prevent the formation of features that are larger than the prescribed maximum length
scale. The technique is demonstrated in the context of structural and fluid topology optimization. Specifically, maximum length
scale criterion is applied to (a) the solid phase in minimum compliance design to restrict the size of structural (load-carrying)
members, and (b) the fluid (void) phase in minimum dissipated power problems to limit the size of flow channels. Solutions
are shown to be near 0/1 (void/solid) topologies that satisfy the maximum length scale criterion. When combined with an existing
minimum length scale methodology, the designer gains complete control over member sizes that can influence cost and manufacturability.
Further, results suggest restricting maximum length scale may provide a means for influencing performance characteristics,
such as redundancy in structural design.

Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are presented.

A new method called the growing ground structure method is proposed for truss topology optimization, which effectively expands
or reduces the ground structure by iteratively adding or removing bars and nodes. The method uses five growth strategies,
which are based on mechanical properties, to determine the bars and nodes to be added or removed. Hence, the method can optimize
the initial ground structures such that the modified, or grown, ground structures can generate the optimal solution for the
given set of nodes. The structural data of trusses are manipulated using C++ standard template library and the Boost Graph
Library, which help alleviate the programming efforts for implementing the method. Three kinds of topology optimization problems
are considered. The first problem is a compliance minimization problem with cross-sectional areas as variables. The second
problem is a minimum compliance problem with the nodal coordinates also as variables. The third problem is a minimum volume
problem with stress constraints under multiple load cases. Six numerical examples corresponding to these three problems are
solved to demonstrate the performance of the proposed method.