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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.

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... Following Brunneel and Duysinx [3] study, Yang et al [4] and Ansola et al [5] proposed the modified ESO method to solve the topology [1] used the same method of the previous author to solve topology optimization problems of a structure subjected to body force loads (self-weight, centrifugal forces, and inertial forces). In their works, Han et al [1] and Huang and xie [6] stated that is almost impossible to obtain a {0,1} design using the power law material interpolation scheme. From this, the TOBS method arises as an alternative to overcome this statement. ...

... The TOBS method combines four numerical ingredients: sequential problem linearization, relaxation of constraints (motion limits), sensitivity filtering, and an integer programming solver. The method presents an improvement over the previously available discrete topology optimization methods, namely, the popular BESO method by Huang and Xie [6] and those by Svanberg and Werne [8] and Beckers [9]. Furthermore, the use of design variables {0,1} gives the TOBS method an advantageous potential in solving design-dependent physical problems. ...

... The design domain of a rectangular plate is illustrated in Fig. 2. This model is a benchmarking example for topology optimization with self-weight loads and it was studied by some other researchers (Bruneel and Duysinx [3], Ansola et al [5] and Huang and Xie [6]). The following material properties are assumed: Young's modulus of 200 GPa, Poisson's ratio of 0.3, and density of 7850 kg/m 3 . ...

The study of structures subject to self-weight loads is particularly important for the fields of civil, aeronautical, and aerospace engineering. Topology optimization emerges as a crucial tool in this analysis providing structures with non-intuitive conceptual designs and greater material savings. Binary methods are among the most established methods, where the design variables assume discrete values 0 and 1 for the void and for the solid material, respectively. In previous studies, it has been reported that topology optimization of structures subject to self-weight loads using binary methods is almost impossible to be employed without the RAMP material model. This article shows that binary topology optimization for self-weight loads depends on the formulation and not only on the material interpolation. To illustrate that, the classic SIMP material model is used together with the recently developed Topology Optimization of Binary Structures (TOBS) method for topology optimization of structures subject to self-weight loads. The algorithm was tested and verified to analyze two bidimensional benchmarking problems. The effect of penalty variation on the final topology was discussed for modified SIMP model. From the results, it was demonstrated that the modified SIMP model combined with TOBS allows to efficiently optimize structures subject to self-weight loads.

... Gao et al. [29] proposed and validated an improved RAMP model with a variable parameter in solving structural topology optimization under inertial loads. In 2011, Huang and Xie [30] developed a new BESO method utilizing the RAMP model to optimize structures with self-weight and demonstrated its capability to generate convergent optimal solutions with numbers of examples. Last but not least, in 2013, Xu et al. [24] introduced an Optimality Criteria (OC) method, namely, the Guide-Weight method (GW) into continuum structures optimization under body forces. ...

... However, in addition to the self-weight of the structure, the centrifugal forces, the magnitude and direction of the acceleration borne by structure also have a great impact on the topological structure. With the work by [24,30] as benchmarks, notably, this paper proposes to investigate topology optimization of continuum structures with density-dependent inertial loads (including self-weight, centrifugal forces, the magnitude and direction of the acceleration of inertial loads) by using an extended BESO method. This work presents an extended procedure to deal with maximum stiffness (minimum compliance) topology optimization of continuum structures, and the centrifugal forces, the magnitude and direction of the acceleration of inertial loads are considered. ...

... where ρ 0 and E 0 denote the density and Young's modulus of the solid material, and q is the penalty factor which is larger than zero for topology optimization problem. It is assumed that Poisson's ratio is independent of the design variables and the global stiffness matrix K can be expressed by the elemental stiffness matrix and design variables x i as (30) where denotes the elemental stiffness matrix of the solid element. From Eqs. (27), (29) and (8), the sensitivity of the compliance (structure with self-weight) can be obtained as (31) From Eqs. (27), (29) and (13), the sensitivity of the compliance (structure under centrifugal force) can be expressed as (32) Similarly, the sensitivity of the compliance (structure with self-weight and additional acceleration) can be obtained from Eqs. (27), (29) and (18), as (33) It is noted that the ratio between the first and second terms of Eqs. ...

This paper proposes to investigate topology optimization of continuum structures with density-dependent inertial loads which including self-weight, centrifugal forces, the magnitude and direction of the acceleration of inertial loads in order to make topology optimization more realistic. By using an extended bi-directional evolutionary structural optimization (BESO) method, this work presents an extension of this procedure to deal with maximum stiffness (minimum compliance) topology optimization of continuum structures when different combinations of inertial loads and fixed forces are applied. Notably, the magnitude and direction of the acceleration of inertial loads are considered. The computation of the element sensitivity numbers is deduced in detail in order to achieve the optimum design. MATLAB programming of proposed mathematical approach is done and compared with the conventional structural optimization problems with fixed external force. The procedure has been tested in several 2D and 3D benchmark examples to illustrate and validate the approach. The inertial loads have great influence on the topological structure, especially when the fixed external force is relatively small. The inertial loads change the magnitude and variation trend of objective function. The iterative curves converge to constant values stably, and the convergence rate is fast.

... It is noticed that certain numerical examples of structural optimization had shown different results between considering and not considering self-weight loads [4,5]. And since structural mass is used in eigenvalue analysis, structural self-weight loads must be considered in static analysis to strive for consistency. ...

... In Eqs. (3)(4)(5), λ * denote the multipliers of the optimal point A * . The dual programming problem of Eq. (1) can be expressed as ...

... The optimality conditions of Eq. (8) are still the same with Eqs. (3)(4)(5)(6)(7). The essential extremum condition of the dual function φ(λ) [6] in Eq. (8) is ...

For engineering structures, the limits of internal stresses, nodal displacements and fundamental frequencies must be simultaneously considered. This had been paid attention in the theory of structural optimization. Actually, most examples of only considering static constraints or only considering dynamic constraints were presented for simultaneously considering static and dynamic constraints. A few examples of considering both static and dynamic constraints were presented, but the advantage could not be presented. It is the reason that the singularity of structural optimization for considering dynamic constraints has not been discussed. To discover the singularity, an optimization model to simultaneously consider static and dynamic constraints is used for the truss size optimization. And according to the extremum conditions of the optimization problem, Ratio-Extremum method is proposed to solve the optimization problems of considering both static and dynamic constraints and only considering dynamic constraints, in which a new searching direction of design variables is to be discussed. Particularly, the step-size factors can be determined by formulas to iteratively solve Lagrangian multipliers and design variables. Numerical examples of 15-bar planar and 72-bar spatial trusses are used to show the singular solutions. On the convergent points, the optimization weights of only considering dynamic constraints are about 66.17% and 71.14% more than the weights of considering both static and dynamic constraints, respectively. The convergent solutions of only considering dynamic constraints are not the best results. However, additional static constraints can be helpful to obtain better results for considering dynamic constraints.

... Ansola et al. [15] presented a modified version of ESO in computing sensitivity numbers to obtain the optimum design for the self-weight problems. Huang and Xie [16] developed a BESO method to deal with self-weight problems using RAMP model. Level-set method is an alternative to solve this kind of problem. ...

... on is a piec he product en knot vec r derivative (16) where N refers to the matrix of B-spline basis functions presented in Section 2. When the exact imposition of the Dirichlet boundary conditions is made, Eq. (16) needs to be revised by means of weighted FCM [35,36]. By substituting Eq. (16) into Eq. ...

... 2L L Self-weight This example was extensively studied as a benchmark by several researchers [8,14,16]. Young's modulus and Poisson's ratio are assumed to be E=1 and μ=0.3, respectively. As shown in Fig. 8, a vertical force is applied at the middle point of the structure bottom. ...

In this paper, the inherent problem of the so-called parasitic effect of low density region caused by design-dependent loads is investigated for topology optimization. A CBS (closed B-splines)-based method is developed to solve efficiently the problem in the way of shape optimization. Compared to the standard density method, design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the CBS. As a result, design-dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure. To favour structural reanalyses and sensitivity analysis, the implicit form of the CBS, i.e., level-set function (LSF) is used in conjunction with the fixed computing grid of finite cell method (FCM). A variety of design-dependent body loads is considered for the mathematical formulation of the CBS-based method. Typical examples are given to illustrate that the CBS-based method is free of low density regions in the optimized topology owing to the proper definitions of design variables.

... Topology optimization of continuum structures with selfweight loads has been studied by Yang et al. [177] and Ansola et al. [4] using the modified ESO procedures (early versions of BESO). Huang and Xie [57] revisited later the same problem by using the soft/hard-kill BESO methods. The basic formulation of the topology optimization problem may also consist of minimizing the compliance of the structure with self-weight load subject to a constraint on the structural volume as given in (12) min ...

... Design domain of a simply supported rectangular plate subject to self-weight loading and the corresponding hard-kill BESO design result [57] where ∆f e is the change in the load vector due to the removal of the e-th element and v e is the element volume. ...

... As for BESO designs of the problem, Huang and Xie [57] have derived the sensitivity numbers from a strict sensitivity analysis with the adoption of the RAMP material interpolation model [132]. By the RAMP model, the density and the Young's modulus are defined in terms of the topology design variables as ...

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.

... They presented a correction factor to compute the sensitivities of the objective and to enhance the convergence of the optimization. Huang and Xie (2011) used the bi-directional evolutionary structural optimization approach with the RAMP material model (Stolpe and Svanberg 2001) for designing structures subjected to self-weight. Xu et al. (2013) proposed the guide-weight approach using the optimality criteria method. ...

... a CASE II and b CASE IVare close to 0-1 solutions. Topologies of these solutions resemble those reported in(Bruyneel and Duysinx 2005;Huang and Xie 2011;Novotny et al. 2021). Furthermore, these solutions (Fig. 5b and d)are symmetric with respect to the central y-axis of the domain and thus, symmetry nature of the problem is retained. ...

This paper presents a density-based topology optimization approach to design structures under self-weight load. Such loads change their magnitude and/or location as the topology optimization advances and pose several unique challenges, e.g., non-monotonous behavior of compliance objective, parasitic effects of the low-stiffness elements, and tendency to lose constrained nature of the problems. The modified SIMP material scheme is employed with the three-field density representation technique (original, filtered, and projected design fields) to achieve optimized solutions close to 0–1. A novel mass density interpolation strategy is proposed using a smooth Heaviside function, which provides a continuous transition between solid and void states of elements and facilitates tuning of the non-monotonous behavior of the objective. A constraint that implicitly imposes a lower bound on the permitted volume is conceptualized using the maximum permitted mass and the current mass of the evolving design. Sensitivities of the objective and self-weight are evaluated using the adjoint-variable method. Compliance of the domain is minimized to achieve the optimized designs using the Method of Moving Asymptotes. The efficacy and robustness of the presented approach are demonstrated by designing various 2D and 3D structures involving self-weight. The proposed approach maintains the constrained nature of the optimization problems and provides smooth and rapid objective convergence.

... They presented a correction factor to compute the sensitivities of the objective and to enhance the convergence of the optimization. Huang and Xie (2011) used the bi-directional evolutionary structural optimization approach with the RAMP material model (Stolpe and Svanberg, 2001) for designing structures subjected to self-weight. Xu et al. (2013) proposed the guide-weight approach using the optimality criteria method. ...

... The optimized designs of CASE II (Fig. 5b) and CASE IV (Fig. 5d) are close to 0-1 solutions. Topologies of these solutions resemble those reported in (Bruyneel and Duysinx, 2005;Huang and Xie, 2011;Novotny et al., 2021). Furthermore, these solutions ( Fig. 5b and Fig. 5d) are symmetric with respect to the central y−axis of the domain and thus, symmetry nature of the problem is retained. ...

This paper presents a density-based topology optimization approach to design structures under self-weight load. Such loads change their magnitude and/or location as the topology optimization advances and pose several unique challenges, e.g., non-monotonous behavior of compliance objective, parasitic effects of the low-stiffness elements, and unconstrained nature of the problems. The modified SIMP material scheme is employed with the three-field density representation technique (original, filtered, and projected design fields) to achieve optimized solutions~close~to~0-1. A novel mass density interpolation strategy is proposed using a smooth Heaviside function, which provides a continuous transition between solid and void states of elements and facilitates tuning of the non-monotonous behavior of the objective. A constraint that implicitly imposes a lower bound on the permitted volume is conceptualized using the maximum permitted mass and the current mass of the evolving design. Sensitivities of the objective and self-weight are evaluated using the adjoint-variable method. Compliance of the domain is minimized to achieve the optimized designs using the Method of Moving Asymptotes. The Efficacy and robustness of the presented approach are demonstrated by designing various 2D and 3D structures involving self-weight. The proposed approach maintains the constrained nature of the optimization problems and provides smooth and rapid objective convergence.

... The SIMP method has shown effectiveness in solving a broad range of optimum problems and its algorithm has been well accepted. However, the SIMP method using a given penalty exponent may result in a local optimum with "grey" regions meaning that the optimization bound is not clearly determined [18]. The HTO method has numerical instabilities such as checkerboard which is caused by the numerical calculation error, and the results is mesh-dependency. ...

... The structure optimization is focused on, reduc the robot, and then solving the problems of with large power and many other debugging Using the ESO method, the framewo structure of the IPR-I will be optimized a IPR-II will be designed. The ESO method approach [18] i distribution in the predetermined des the mean compliance of the struct function and the material volume c limit the maximum used materia Topology optimization methods are material so as to achieve the de element-based ESO method optimization problem is formalized where the mean compliance of are the global stiffness matrix and th V and V* are the optimized and the p volume. is the total number of variable denotes the density of th x min =0.001 is used to denote the vo used for solid elements. ...

Abstract—In order to interact with human flexibly, the robots
need lightweight structure to adjust their configuration
conveniently and further save operation energy. It is a challenge
in design when the robots are proceeding tasks with a huge and
heavy body. This paper presents an improved framework of the
humanoid robot optimized by the evolutionary structural
optimization (ESO) method for lightweight design. By
analyzing the force of the structure using the finite element
software, the location with maximum stresses much smaller
than allowable stresses of the materials was found and then
removed. By comparison, the weight of the optimized
framework achieved 50.15% less than the original one without
changing the stiffness and vibration performance, improving
the material utilization and extending the service time of the
battery.

... Ref. [5]) and, finally, structural self-weight (e.g.Ref. [6]). The interesting nature of the problem of design-dependent loads in topology optimization lies in the influence of optimization process on the location, direction and magnitude of the loads. ...

... [13]) was successfully adapted to topology optimization including self-weight loads, should also be mentioned. Moreover in Ref. [6] soft-kill and hard-kill BESO (Bi-directional Evolutionary Structural Optimization), and in Ref. [10] a new Guide-Weight methods were utilized. This paper presents the application of the Cellular Automata method into the discussed subject and thus expands a set of methods considered as an alternative ...

Topology optimization of structures under a design-dependent self-weight load is investigated in this paper. The problem deserves attention because of its significant importance in the engineering practice, especially nowadays as topology optimization is more often applied when designing large engineering structures, for example, bridges or carrying systems of tall buildings. It is worth noting that well-known approaches of topology optimization which have been successfully applied to structures under fixed loads cannot be directly adapted to the case of design-dependent loads, so that topology generation can be a challenge also for numerical algorithms. The paper presents the application of a simple but efficient non-gradient method to topology optimization of elastic structures under self-weight loading. The algorithm is based on the Cellular Automata concept, the application of which can produce effective solutions with low computational cost.

... Three numerical examples are taken to demonstrate the validity and efficiency of the proposed approach. The specimens are taken from the work of Garicia-Lopez et al. [8] and Huang and Xie [9]. All the models are under plane state of stress. ...

... The dimensions and support conditions of the design domain are shown in Figure 4. Due to the symmetry, only half of the design domain is discretized with 100x50 8node plane stress elements. The results are compared with the results of X.Huang et al. [9] who utilized BESO method for topological optimization. The material volume constraint is set to be 40% of the whole design domain. ...

This paper represents the optimal criteria method for topological optimization of isotropic material under different loads and boundary conditions with the objective to reduce mass of an existing material and study the different shape obtained. Topological optimization mainly comprises of a mathematical approach that optimizes the layout within a given design constraints, for a given set of loads and boundary condition such that the performance matches with the prescribed set of performance targets. Topological optimization solve the problem of distributing a given amount of material in a design domain subjected to load and supports conditions, such that the compliance of the structure is minimized while the stiffness of structure is maximized. For material distribution system solid isotropic with penalization approach is used. In all the structures objective function is compliance, design variable is pseudo density and state variables are the response of structures that is deflection. Objected function is subjected to volume constraint and by minimize the compliance stiffness of structures are maximize. Different numerical examples are taken to study the optimal criteria approach and validate the results obtained with SA-SIMP and BESO method. This paper work represents topological optimization for static and self-weight loading using finite element solver ANSYS. APDL (ANSYS Parametric Design Language) has been employed for utilizing the topological optimization capabilities of commonly used finite element solver ANSYS. 8 node 82 elements are used to model and mesh the isotropic structures in ANSYS.

... Especially the results with only self-weight coincide very well with results from literature, cf. [7] for example. ...

Topology optimization and additive manufacturing complement one another where the first one results in possibly complex structures, and the second one allows for manufacturing of those. For computing optimized components that also fit to the manufacturing limits, the building processes need to be accounted for already during the optimization process. A special characteristic of the additive manufacturing process is the step‐by‐step manufacturing. 1,3Herein, constructing large‐scale structures, as for example buildings or bridges, by assembling pre‐produced segments can also be considered as additive manufacturing. Especially, a design which also carries the manufacturing 1,3or assembling machine, as for example cranes or robots, on different positions during manufacturing is of interest. Therefore, we extend the established thermodynamic topology optimization for a sequential optimization process which considers changing manufacturing loads under structural self‐weight.

... Finally, taking homogeneous materials (a special case of FGM) as an example, the validity of the proposed method is verified by numerical simulations and experiments, finding that the proposed method can effectively improve the resonance frequency and dynamic stiffness. [5] 发展了一种功能梯 度结构拓扑优化设计新方法，实现了材料属性和 机械性能的同时优化。采用凸规划求解策略以及 周长控制方法，Li 等 [6] 对具有拉胀特性的新型功 能梯度蜂窝复合材料进行了拓扑优化设计。 邱等 [7] 利用 SIMP 法实现了功能梯度 MMB 梁和功能梯 度夹层结构夹芯的拓扑构型设计，并揭示了材料 性能和材料插值模型对结构优化中材料分布的影 响规律。近期，李等 [8] 对周期性功能梯度结构进 行了拓扑优化设计，获得了具有较好散热性能 FGM 结构。虽然上述方法可以很好地提升 FGM 结构的机械性能，但是面向重型机械装置、船舶 和航空航天装备领域中 FGM 结构的设计时，自 重载荷和动力学特性往往无法忽视。 已有研究对均一化材料组成的机械结构进行 了考虑自重载荷的拓扑优化设计。Chen 等较早地 [9] 提出了一种基于设计相关载荷的线弹性结构拓 扑优化方法。Bruyneel 等 [10] 研究了体积约束下柔 顺度最小化结构拓扑优化问题，他们发现当结构 自重占主导地位时， 会出现柔顺度非单调性行为、 最优结果无约束行为以及使用基于密度方法对低 密度寄生效应。 后来， 张等 [11] 利用 RAMP(Rational Approximation of Material Properties)材料插值模 型和平均敏度过滤技术很好地解决了该问题。还 有学者利用双线渐进结构优化方法和基于非均匀 有理 B 样条基函数插值的拓扑描述函数方法， 成功解决了考虑自重载荷的拓扑优化设计问题 [12,13] 。然而，上述研究大多针对均一材料的结构 拓扑优化设计。FGM 的材料密度不均匀分布，自 重载荷将对拓扑优化设计结果影响更大。 此外，考虑动力学特性的拓扑优化设计方法 在机械工程领域得到了广泛的关注。Oliver 等 [14] 提出了一种增广拉格朗日法，以解决一般动态载 荷作用下具有应力约束的结构拓扑优化设计问题。 基于一种新的凝聚函数策略和时域求解结构动态 响应的思路，Zhao 和 Wang [15] [15] ，e 为自然常数。η表示 单元密度即设计变量， p 为惩罚系数， 通常令 p=3， ...

... Several works using BESO have already been proposed to study the influence of different types of design-dependent loads. For instance, Huang and Xie (2011) proposed an improved BESO method using an alternative material interpolation scheme to SIMP in order to study self-weight loads. Regarding the same type of loading, an efficient sensitivity computation was proposed by Ansola et al. (2006) to enhance the convergence of the algorithm and, consequently, to achieve an optimized solution. ...

Topology Optimization (TO) is a powerful tool for designing lightweight and high-performance engineering structures. However, this kind of
problem is not straightforward to solve when thermal
gradients are considered. Thermoelastic stress loads,
caused by thermal expansion, depend on the solid
material layout and change along the optimization
process. This design-dependent nature brings several
extra challenges to topology optimization and is the
focus of this work. Numerical examples are analyzed
and presented by the classical approach to TO problems, pointing out several issues related to checkerboard patterns, intermediate densities, and convergence limitations. The present work proposes an
alternative procedure based on an evolutionary algorithm to overcome these issues, with a new approach
to the Bi-directional Evolutionary Structural Optimization (BESO) method being suggested. Two different schemes (hard- and soft-kill) are evaluated along
with a modified sensitivity number to account for
thermoelastic stress loads. Finally, the results coming
from the updated BESO method and from the classical approach are compared.

... However, the study of design-dependent loads was also performed by BESO method in the subsequent years. Focused on the study of design-dependent self-weight loads, Huang and Xie proposes a new BESO method using an alternative material interpolation to SIMP [Huang and Xie 2011]. Regarding the same type of loading, a efficient sensitivity computation was proposed by Rubén Ansola et al.. To enhance the convergence of the algorithm in order to achieve the optimum design, a correction factor was suggested to compute the sensitivities [Ansola et al. 2006]. ...

Structural optimization has gained popularity since its first studies in the late 19th century. Over the years, due to the improvement of technology, several works have been focused on its computational implementation. Among the most popular applications, topology optimization deals with the
non-homogeneous material distribution in a structure in order to optimize a
given structural objective. A compliance approach is usually carried out to
evaluate a topology optimization problem. Moreover, it is also considered
the Optimality Criterion and SIMP as the optimization method and the
material interpolation scheme, respectively.
In this work, several topology optimization problems are carried out
and evaluated, from a multi-objective approach, where thermal and mechanical analyses are simultaneously considered, to thermoelastic phenomena.
These problems are recognized for incorporating loads that depend on the
solution (in this case, thermal loads). Also known as design-dependent
loads, they depend on the material layout inside the structure and their
magnitude has a direct impact on the optimization process. Therefore, in
the resultant topologies, the instabilities associated with this type of loading
become evident.
The main focus of this work consists in introducing alternative ways
to prevent these issues and deal with the problems’ instability. Therefore,
an alternative procedure is proposed to control the problems that arise
from the mentioned analysis. An adaptation of the Evolutionary Structural
Optimization (ESO) method, also known as Bi-directional ESO, is implemented and the obtained results are compared with the conventional ones.
The development of a computational tool consists in an additional
outcome of this work and, therefore, the mentioned methodologies are
implemented considering a numerical simulation software, based on the
Finite Element Method (FEM), as background. Besides the FEM analysis,
the computational tool becomes capable of solving different types of
topology optimization problems.

... The objective function derived from the dynamic equation in the previous section is nonmonotonic and non-linear. Although some studies [51,52] have used the standard BESO method to solve problems with the non-monotonic objective function. However, the standard BESO method is numerically unstable when solving the objective function used in this paper, and the effects of modal control and vibration suppression are not obvious. ...

Although the structural vibration can be suppressed by adjusting eigenvalues outside the excitation frequency band, it becomes increasingly difficult to suppress vibration as the width of the excitation frequency band increases. Besides, for some high-precision equipment, e.g. space telescope mirror substrate and rocket motor casings, their complex and diverse modals directly affect the performance. Thus, a novel optimization algorithm was proposed, which applies to achieve eigenvector-based modal control and vibration suppression. The eigenvectors were defined as the objective function, and Nelson’s method without truncation error was used to calculate the sensitivity information. Due to the introduction of dynamic equations, the non-linearity of the objective function is prominent. Then, an improved solver that can handle this non-linear topology optimization problem was proposed. The optimization was performed under a multi-material framework, and the extended multi-material interpolation scheme was proposed to readily realize the optimization with three or more materials. In addition, the consistent mass matrix constructed by multi-material interpolation was used to describe the mass matrix without concentrated mass in the dynamic equation. Moreover, the modal assurance criterion was used to track jumping modals. Finally, the modal controllability was achieved through several numerical examples, which verify the effectiveness of the proposed method.

... Design-dependent loads have been considered for purely structural design, such as in self-weight problems [20,21], thermoelastic design [22,23] and pressure loads [24,25]. Surface loading, e.g. ...

Structural optimization is increasingly used across academia and industry because of the great design freedom it offers and due to the increasing availability of computational power. In this context, binary methods - which generate clear (0/1) designs - are an effective approach to solve optimization problems, especially multiphysics, wherein precise definition of the structural boundary is essential. This work adopts the Topology Optimization of Binary Structures (TOBS) method to solve structural optimization problems that consider buckling constraints and design-dependent loads, such as fluid pressure loading, a characteristic of submerged structures. Buckling constrained TO problems applied to design-dependent loads are not yet explored in the literature. Few optimization problems are investigated to demonstrate the effect of the buckling constraint on the optimized solutions as compared to that of the classical compliance minimization problem. The common issues associated with the eigenproblem characteristic of the buckling phenomenon are discussed. The method successfully considers design-dependent loads coupled with stability constraints, obtaining final solutions with significant improvement in buckling resistance and minimal stiffness loss when compared to the compliance design. It is concluded that the TOBS method presented promising results and potential application in stability problems of design-dependent loaded structures, such as those present in the offshore industry.

... -Body loads, which are distributed loads whose magnitudes are dependent on the location of material forming the optimized body or structure. These have for example been studied by Huang and Xie (2011), Kanno and Yamada (2017), and Fairclough et al. (2018). Most papers focus on body loads due to self-weight, but inertial, centrifugal and other types of loads may also be considered. ...

Transmissible loads are external loads defined by their line of action, with actual points of load application chosen as part of the topology optimization process. Although for problems where the optimal structure is a funicular, transmissible loads can be viewed as surface loads, in other cases such loads are free to be applied to internal parts of the structure. There are two main transmissible load formulations described in the literature: a rigid bar (constrained displacement) formulation or, less commonly, a migrating load (equilibrium) formulation. Here, we employ a simple Mohr’s circle analysis to show that the rigid bar formulation will only produce correct structural forms in certain specific circumstances. Numerical examples are used to demonstrate (and explain) the incorrect topologies produced when the rigid bar formulation is applied in other situations. A new analytical solution is also presented for a uniformly loaded cantilever structure. Finally, we invoke duality principles to elucidate the source of the discrepancy between the two formulations, considering both discrete truss and continuum topology optimization formulations.

... They proposed a new relation to calculate smoothing iterations during the optimization process that can overcome numerical instabilities in the traditional ESO. In another study on evolutionary methods, soft and hard kill in the BESO method was developed to account for self-weight (Huang and Xie, 2011). To find the optimum topology of trusses under the self-weight load, mixedinteger second-order cone programming was used by Kanno and Yamada (2017). ...

Purpose
Body forces are always applied to structures in the form of the weight of materials. In some cases, they can be neglected in comparison with other applied forces. Nevertheless, there is a wide range of structures in civil and mechanical engineering in which weight or other types of body forces are the main portions of the applied loads. The optimal topology of these structures is investigated in this study.
Design/methodology/approach
Topology optimization plays an increasingly important role in structural design. In this study, the topological derivative under body forces is used in a level-set-based topology optimization method. Instability during the optimization process is addressed, and a heuristic solution is proposed to overcome the challenge. Moreover, body forces in combination with thermal loading are investigated in this study.
Findings
Body forces are design-dependent loads that usually add complexity to the optimization process. Some problems have already been addressed in density-based topology optimization methods. In the present study, the body forces in a topological level-set approach are investigated. This paper finds that the used topological derivative is a flat field that causes some instabilities in the optimization process. The main novelty of this study is a technique used to overcome this challenge by using a weighted combination.
Originality/value
There is a lack of studies on level-set approaches that account for design-dependent body forces and the proposed method helps to understand the challenges posed in such methods. A powerful level-set-based approach is used for this purpose. Several examples are provided to illustrate the efficiency of this method. Moreover, the results show the effect of body forces and thermal loading on the optimal layout of the structures.

... See also earlier work by Turteltaub and Washabaugh (1999) and recent papers (Xu et al. 2013;Holmberg et al. 2015;Félix et al. 2020). Many approaches have been proposed, such as the use of mathematical programming, heuristics methods, and optimization criteria method (Ansola et al. 2006;Huang and Xie 2011;Xu et al. 2013). To introduce these ideas, we present a simple example into one spatial dimension. ...

Topology optimization of structures subject to self-weight loading has received considerable attention in the last decades. However, by using standard formulations based on compliance minimization under volume constraint, several difficulties arise once the self-weight of the structure becomes dominant, including non-monotonic behavior of the compliance, possible unconstrained character of the optimum, and parasitic effects for low densities when using density-based methods. In order to overcome such difficulties, a regularized formulation that allows for imposing any feasible volume constraint is proposed. The standard formulation based on compliance minimization under volume constraint is recovered when the regularizing parameter vanishes. The resulting topology optimization problem is solved with the help of the topological derivative method leading to a 0-1 topology design algorithm, which seems to be crucial when the self-weight loading is dominant. Finally, several numerical experiments are presented, showing the effectiveness of the proposed approach in solving a structural topology optimization problem under self-weight loading.

... Gao and Zhang (2010) studied the thermoelasticity problem and analyzed the element stiffness and thermal stress load of the finite element model through the penalty method. Huang and Xie (2011) were concerned with the non-monotonicity of the objective function with minimal compliance problems under fixed external loads. Wang et al. (2016) developed an improved topology description function (TDF) method using the non-uniform rational B-spline (NURBS) interpolation scheme, which can effectively solve these three problems. ...

This work proposes an improved topology optimization model for optimizing continuum structures with self-weight loading conditions. A modified Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, the penalty factor of the SIMP model is increased to maintain the activeness of the prescribed volume constraint and to drive the design domain to binary distribution. The optimization objectives include minimizing the total strain energy of the design domain and minimizing the total displacement of the fixed domain. The shape optimization procedure is used to furtherly enhance structural performance. The whole optimization procedure is implemented with a two-dimensional model under the loading conditions of self-weight and external force. The classic MBB beam and self-weight arch are utilized to verify the proposed method, and conceptual layout designs of the steel structure bridge are conducted. It is proved that the proposed model is effective for topology optimization of continuum structures including self-weight. And it is found that the optimal structural topology is affected by the ratio of the external force to self-weight.

... Nodal densities can be obtained using a heuristic filter similar to the one presented in BESO [71,72,73,74,75]: ...

Element-based topology optimization algorithms capable of generating smooth boundaries have drawn serious attention given the significance of accurate boundary information in engineering applications. The basic framework of a new element-based continuum algorithm is proposed in this paper. This algorithm is based on a smooth-edged material distribution strategy that uses solid/void grid points assigned to each element. Named Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT), the algorithm uses elemental volume fractions which depend on the densities of grid points in the Finite Element Analysis (FEA) model rather than elemental densities. Several numerical examples are studied to demonstrate the application and effectiveness of SEMDOT. In these examples, SEMDOT proved to be capable of obtaining optimized topologies with smooth and clear boundaries showing better or comparable performance compared to other topology optimization methods. Through these examples, first, the advantages of using the Heaviside smooth function are discussed in comparison to the Heaviside step function. Then, the benefits of introducing multiple filtering steps in this algorithm are shown. Finally, comparisons are conducted to exhibit the differences between SEMDOT and some well-established element-based algorithms. The validation of the sensitivity analysis method adopted in SEMDOT is conducted using a typical compliant mechanism design case. In addition, this paper provides the Matlab code of SEMDOT for educational and academic purposes.

... Greiner and Hajela [101] used multiobjective evolutionary algorithm using reunification criterion to increase search efficiency. Huang and Xie [102] used BESO utilizing an alternative material interpolation scheme. Huang et al. [103] used BESO to optimize the topology of PBC made of two-phase composites. ...

... Ever since the seminal paper of Bendsøe and Kikuchi [1], several topology optimization methods have been developed to address the design problem, such as, the Solid Isotropic Material with Penalization (SIMP) [2][3][4], Evolutionary Structural Optimization (ESO) [5][6][7][8], Level set method [9][10][11], recently presented moving morphable components (MMC) [12,13] and moving morphable voids (MMV) approaches [14,15] et al. The emergence of topology optimization has provided engineers with an unprecedented opportunity for designing novel [16][17][18], robust [19][20][21] and light products [22,23]. However, often statically determinate structures yielded from traditional topology optimization methods are sensitive to local failure due to there is almost no redundant material. ...

... Other structural performance measures have been suggested such as displacement or local stress based measures [40,41], stiffness optimization with multiple materials [38,42], periodic structures (i.e. honeycomb sandwich plates) [43,44], design dependent gravitational loading [45][46][47], compliant mechanisms [48], or to maximize the structure's natural frequency [49][50][51]. This thesis is solely concerned with stiffness and volumetric-based structural topology optimization because it's the simplest form and the developed techniques apply to all of the forms. ...

Industry 4.0 demands that the systems and processes in today’s product design and manufacturing not just be automated, but to be robust and containing many feedback mechanisms which enables it to be self-correcting. The hypothetical upcoming Industry 5.0 promises on demand and personalized products which this thesis aims to take a step in the direction of. It is proposed that an integrated and optimized process for structural topology optimization and subsequent additive manufacturing is possible for automated design and manufacturing starting from its problem definition. An improvement on the benchmarked topology optimization methods is shown which allows the user control over the optimization’s convergence characteristics which is then further studied to find a robust set of optimization parameters. The resulting topology of the structure is then analyzed for its optimal printing orientation based on a custom-made algorithm which minimizes manufacturing costs. Furthermore, the structure is then sliced for instruction generation of layer-based manufacturing techniques in a novel fashion which also serves to provide feedback of the manufacturing process planning to the topology optimization design stage.

... The multi-material structure can be regarded as a continuous multiple phases, and those phases that have larger Young's modulus always have higher sensitivity numbers than that of the rest constituent phases. Moreover, the design variables are either min or 1 in soft-kill BESO [63]. With these considerations, a heuristic iteration scheme can be applied to accomplish the topological evolution for the design of multimaterial structure. ...

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.

... Design-dependent physics problems can be classified by the type of loads considered, for instance, volumetric or surface loads. The first refers to problems where the loads depend on the volume the body occupies in the medium, e.g., thermal expansion, self-weight loads (Deaton and Grandhi 2016;Huang and Xie 2011). The second type of design-dependent loads acts on the boundaries or surfaces of the body. ...

A few level-set topology optimization (LSTO) methods have been proposed to address complex fluid-structure interaction. Most of them did not explore benchmark fluid pressure loading problems and some of their solutions are inconsistent with those obtained via density-based and binary topology optimization methods. This paper presents a LSTO strategy for design-dependent pressure. It employs a fluid field governed by Laplace’s equation to compute hydrostatic fluid pressure fields that are loading linear elastic structures. Compliance minimization of these structures is carried out considering the design-dependency of the pressure load with moving boundaries. The Ersatz material approach with fixed grid is applied together with work equivalent load integration. Shape sensitivities are used. Numerical results show smooth convergence and good agreement with the solutions obtained by other topology optimization methods.

... where This force distribution scheme is similar to the gravity loading scheme used by Huang and Xie (2011). A similar approach is used for defining a global torsional force vector F. For example, if a torsional moment 'T 0 ' is applied on a face, each element should be subjected to a constant angular displacement. ...

While physiological loading on lower long bones changes during bone development, the bone cross section either remains circular or slowly changes from nearly circular to other shapes such as oval and roughly triangular. Bone is said to be an optimal structure, where strength is maximized using the optimal distribution of bone mass (also called Wolff’s law). One of the most appropriate mathematical validations of this law would be a structural optimization-based formulation where total strain energy is minimized against a mass and a space constraint. Assuming that the change in cross section during bone development and homeostasis after adulthood is direct result of the change in physiological loading, this work investigates what optimization problem formulation (collectively, design variables, objective function, constraints, loading conditions, etc.) results in mathematically optimal solutions that resemble bones under actual physiological loading. For this purpose, an advanced structural optimization-based computational model for cortical bone development and defect repair is presented. In the optimization problem, overall bone stiffness is maximized first against a mass constraint, and then also against a polar first moment of area constraint that simultaneously constrains both mass and space. The investigation is completed in two stages. The first stage is developmental stage when physiological loading on lower long bones (tibia) is a random combination of axial, bending and torsion. The topology optimization applied to this case with the area moment constraint results into circular and elliptical cross sections similar to that found in growing mouse or human. The second investigation stage is bone homeostasis reached in adulthood when the physiological loading has a fixed pattern. A drill hole defect is applied to the adult mouse bone, which would disrupt the homeostasis. The optimization applied after the defect interestingly brings the damaged section back to the original intact geometry. The results, however, show that cortical bone geometry is optimal for the physiological loading only when there is also a constraint on polar moment of area. Further numerical experiments show that application of torsion along with the gait-analysis-based physiological loading improves the results, which seems to indicate that the cortical bone geometry is optimal for some amount of torsion in addition to the gait-based physiological loading. This work has a potential to be extended to bone growth/development models and fracture healing models, where topology optimization and polar moment of area constraint have not been introduced earlier.

... Self-weight loads do greatly affect on the process and result of structural optimization, and greatly change structural configuration. These are also verified by the results of continuous topology optimization considering self-weight loads [1]. ...

For structural self-weight loads being important component in the loads, ignoring self-weight loads cannot be considered as reasonable approximation. In addition, the constraints of stresses, displacements and frequencies must be simultaneously considered to satisfy the requirements of engineering structures. According to the saddle point theory, Ratio-Extremum method is firstly proposed to solve the size optimization problem of the truss simultaneously considering self-weight loads, static and dynamic constraints. It is a great advantage that the step-size factors can be directly determined. Two cases of 72-bar special truss are used to show the effectivity and rationality.

... Self-weight loads do greatly affect on the process and result of structural optimization, and greatly change structural configuration. These are also verified by the results of continuous topology optimization considering self-weight loads [1]. ...

... 29 Over the last three decades, many mathematical and heuristic optimization methods have been developed. [30][31][32][33] The works related to the optimization method have dealt with the optimization of structures with an isotropic material, even in some commercial programs of FEA, 34,35 topology optimization modules are included. However, so far there does not exist a topological algorithm applied to orthotropic materials, therefore, we had to propose and implement a new topology optimization criterion. ...

In this work, a topology optimization algorithm is developed and implemented to get an optimum composite patch shape. Typically, the design process consists of an iterative analysis, where the best solution is obtained from a comparative study. In this way, we propose a topology optimization algorithm applied to obtain the optimum composite patch shape. The algorithm is implemented in MatLab, and uses the commercial finite element code Abaqus/Standard. A numerical example is analysed to show the capability of the proposed method. The obtained results are compared with numerical results reported by other researchers, revealing the potential of the developed algorithm.

... In general, these methods can be considered gradient-based methods that rely on discrete design updates which result only in 0/1 solutions during the optimization process. Further works proved that the BESO method could efficiently solve different topology optimization problems such as for nonlinear structures [24], multiple materials design [25], natural frequencies maximization [28], self-weight loads [27] and the recents BESO applications on multiphysics [41,52,43] and multiscale [56,62,47,29,53] design problems. ...

This work presents an extended bi-directional evolutionary structural optimization (BESO) method applied to static structural design problems considering the interaction between viscous fluid flows and linearly elastic structures. The fluid flow is governed by incompressible and steady-state Navier-Stokes equations. Both domains are solved with the finite element method and simplifying conditions are assumed for the fluid-structure coupling, such as small structural displacements and deformations in a staggered method. The presented BESO method aims to minimize structural compliance in a so called “wet” optimization problem, in which the fluid loads location, direction and magnitude depend on the structural layout. In this type of design-dependent loading problem, density-based topology optimization methods require extra numerical techniques (usually mixed models with overlapping domains) in order to model the interaction of different governing equations during the optimization procedures. In this work, the discrete nature of the evolutionary topology optimization approach allows the fluid-structure boundaries to be modelled and modified straightforwardly by switching the discrete design variables between fluid and structural finite elements. Therefore, separate domains are used in this approach. Numerical results show that the BESO-based methods can be applied to this kind of multiphysics problem effectively and efficiently.

... Thus, the resulting topology of a structure evolves to an optimum. Although BESO was established based on this simple concept, it has been proven to be effective and reliable for various structural optimization problems [35][36][37]. In recent years, the BESO algorithm has been further extended to design the microstructures of materials with extreme mechanical properties [38,39], electromagnetic properties [30], and photonic crystals with large band gaps [40]. ...

This paper proposes a new topology optimization algorithm based on the bi-directional evolutionary structural optimization (BESO) method to design photonic crystals with broad all-angle negative refraction (AANR) frequency range. The photonic crystals are assumed to be two-dimensional periodical structures, which consist of dielectric materials and air. The conditions for the occurrence of AANR are identified and the design objective is to enlarge the AANR frequency range. The BESO algorithm is proposed based on finite element analysis for band diagrams of photonic crystals and the derived sensitivity numbers. Starting from a simple initial design without any AANR, BESO gradually re-distributes the dielectric materials within the periodical unit cell so that the AANR property emerges and its frequency range is enlarged accordingly. The numerical results show that the proposed BESO algorithm can effectively obtain AANR photonic crystals with novel patterns. The effects of dielectric permittivity contrast of two constituent materials, mesh-refinement and filter are discussed.

... A structural optimization is formulated as a mathematical programming problem with a design objective and a set of constraints, utilizing the level set models for the incremental shape changes. Huang and Xie [31] demonstrated the effectiveness and efficiency of the BESO method on the minimization compliance problem with fixed external loads. They considered the minimization of mean compliance for continuum structure subjected to design-dependent self-weight loads. ...

Nowadays, the casting structure of stamping dies is designed according to die design standards. These standards are usually not based on a structural optimization algorithm and often rely on high safety factors which cause the weight of die components to be more than required. This in turn calls for higher prices of dies and production energy required per part. Therefore, alternative methods to reduce the weight of these components are required. In this paper, a software package is presented which can design an improved structure of stamping dies with a substantial reduction in weight. This package implements Abaqus software and uses the bi-directional evolutionary structural optimization (BESO) method to create a new lighter structure which resembles the shape of the sheet metal part and applied forces in the operation. It obtains the desired optimum design by removing from and adding material to the die component structure. This method involves adding material to that part of the component where the structure is overstressed and simultaneously removing material where the structure is understressed. This procedure is carried out again and again until the objective function is minimized. Finally, the proposed structure can also be reconstructed by the designer to accommodate for a simpler casting method. The operation of the software is demonstrated by an example where the dies for a sheet metal part are studied. The die components are initially designed, analyzed, and compared with the standard die (the die which is in general use today). The final results show a reduction of 31 % of volume while the maximum displacement and stress of the die do not change approximately. This software package is developed in a Microsoft Visual C# programming environment with a link to Abaqus software to analyze finite element simulation processes.

... This chapter includes the recent research of the authors into various topology optimization problems (Huang and Xie 2009a;2009b;2009c;2009d;2009e;). The developed BESO algorithms can be directly applied to many practical design problems, some of which are presented in Chapter 9. ...

This paper investigates the topology optimization of structures subjected to self-weight loads with self-supporting constraints for additive manufacturing. The integration of topology optimization procedures and additive manufacturing techniques can make the most of their advantages, and there is significant interest today in integrating both approaches. Imposing overhang constraints in topology optimization has been addressed, but primarily for classical topology optimization problems with fixed external loads, not design-dependent loads. This work combines an effective numerical procedure for contour evaluation with a modified version of the power-law model for low densities to eliminate the problems that arise when self-weight loads are considered. The overhang edge detection is based on the Smallest Univalue Segment Assimilating Nucleus (SUSAN) method, and a variable mask size technique is used to avoid eventual dripping problems. The proposed constraint function evaluates the overhang globally and allows control of the formation of unsupported contours for maximum stiffness design problems when self-weight loads are present. Several numerical experiments demonstrate the proposed method's effectiveness and robustness.

This work proposes a method for optimizing the continuum structural topology under multiple load cases considering frequency constraints and the effect of self-weight. An improved Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, new matching smooth penalty functions on the element stiffness, volume and mass are constructed to greatly reduce the number of low-order pseudo-modes in the optimized structure. And low-order pseudo-mode identification and deletion measures are introduced to solve the pseudo-mode problem. The Heaviside three-field mapping scheme and two varied volume constraints are introduced to obtain a clear 0/1 distribution. Moreover, a volume change rate constraint measure of low-physical density elements is proposed to greatly improve optimization computation efficiency for the structural topology problem considering both frequency constraints and the effect of self-weight. It is concluded from examples that the proposed method is effective and robust for generating an optimal topology.

This article aims to propose an approach to the stress-based topology optimization of continuous elastic bi-dimensional structures subjected to design-dependent self-weight loads using the Bi-directional Evolutionary Structural Optimization (BESO) method. Topology optimization is developed through the minimization of P-norm von Mises stress while satisfying a volume constraint. To implement the algorithm, a consistent sensitivity analysis including design-dependent loads has been developed by the adjoint method. A series of tests has been performed to explore and validate the method through three numerical examples: an L-bracket; a doubly supported beam with one pre-existing crack notch; and a cantilever beam. Comparison between traditional compliance minimization and stress minimization analyses, including design-dependent self-weight loads, shows that the method is an effective way to reduce the maximum stress.

This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behaviour of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using NURBS and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with CAD. Two classes of optimization problems have been discussed, they are minimum compliance problems subject to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which is new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Topological optimization is an innovative method to realize the lightweight design. This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method. In the method, SIMP is used to update the chromosomes, which can accelerate convergence. The filtering scheme in the SIMP method can filter unconnected elements to ensure the connectivity of the structure. We studied the influence of varying the filtering radius on the optimized structure. Simultaneously, in the SIMP-GA method, each element is regarded as a gene, which controls the population number to a certain extent, reduces the amount of calculation, and improves the calculation efficiency. The calculation of some typical examples proves that the SIMP-GA method can obtain a better solution than the gradient-based method. Compared with the conventional genetic algorithm and GA-BESO (Bi-directional Evolutionary Structural Optimization) method, the calculation efficiency of the proposed method is higher and similar results are obtained. The innovative topology optimization method could be an effective way for structural lightweight design.

This chapter concerns the feature-driven optimization for structures under design-dependent loads. The implicit B-splines in closed and open forms are utilized as basic design primitives to describe the inner topology boundary and the moving pressure boundary. The design variables are associated with the shape and location parameters of the implicit B-spline curves and are thus unattached to the FE model. The sensitivities of design-dependent loads are analytically derived. The effectiveness of the presented methods is illustrated with several numerical examples.

Reverse shape compensation is widely used in additive manufacturing to offset the displacement distortion caused by various sources, such as volumetric shrinkage, thermal cooling, etc. Also, reverse shape compensation is also an effective tool for the four-dimensional (4D) printing techniques, shape memory polymers (SMPs), or 3D self-assemble structures to achieve a desired geometry shape under environmental stimuli such as electricity, temperature, gravity etc. In this paper, a gradient-based moving particle optimization method for reverse shape compensation is proposed to achieve a desired geometry shape under a given stimulus. The geometry is described by discrete particles, where the radius basis kernel function is used to realize a mapping from particle to density field, and finite element analysis is used to compute the deformation under the external stimulus. The optimization problem is formulated in detail, and MMA optimizer is implemented to update the location of discrete particles based on sensitivity information. In this work, self-weight due to gravity imposed on linear elastic structures is considered as the source of deformation. The objective of the problem is then to find the initial shape so that the deformed shape under gravity is close to desired geometry shape. A shape interpolation method based on Artificial Neural Network is proposed to reconstruct the accurate geometric prototype. Several numerical examples are demonstrated to verify the effectiveness of proposed method for reverse shape compensation. The computational framework for reverse shape compensation described in this paper has the potential to be extended to consider linear and non-linear deformation induced by other external stimuli.

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.

Inertial ampliﬁcation is a novel phononic band gap generation method in which wide vibration stop bands can be obtained at low frequency regions. The engineering importance for this novelty comes from the fact that the phononic band gap structures can be utilized as passive vibration isolators for the low frequency range. In this thesis, primarily the research is focused on the improvements achieved on stop band widths and depths via employment of structural optimization tools. To that end, size, shape and topology optimization studies are conducted on a compliant inertial ampliﬁcation mechanism, then with these compliant unit cell mechanisms, one and two dimensional periodic structures are formed. Consequently, by means of these periodic structures, it is demonstrated that the vibration transmission is inhibited for wide ranges at low frequencies. The work comprises analytical and numerical studies and more importantly experimental validation of the results. Moreover, topology optimization studies performed during the thesis lead to the development of a new fast topology optimization algorithm to obtain structures with maximized fundamental frequency, though this was not originally among the research objectives. Finally, explicit problem formulations and a comprehensive review on topology optimization are also presented.

Abstract.The Smoothing Evolutionary Structural Optimization (SESO) technique was extended to solve 2D elastic problems with strain energy criterion. Optimization procedure is based on the progressive reduction of the stiffness contribution of inefficient elements with the lower values of strain energy until it has no moreinfluence. A brief review of the linear elasticformulation is presented, where the goal is to maximize stiffness of a fixed-mass system by minimizing growth internal strain energy or equivalently external work. In the application examples, optimal topology of a classicalbeam is analyzed considering self-weight of the structure. A Short cantilever and Short corbel are also evaluatedwith the definition of a strut-and-tie model. The optimal settings obtained demonstrate the robustness of the optimization procedure.

Topology optimization of structure seeks to achieve the best material distribution in the Pre-determined design domain. In this paper, the effect of design parameters including length scale parameter and evolutionary volume ratio in improved bi-directional evolutionary structural optimization method with soft kill approach is discussed. The main aim of this method is searching for the stiffest structure with a given volume of material using finite element method. At each iteration of finite element analysis, sensitivity number is calculated for each individual element in design domain and then converted to the nodal sensitivity number. With Filter Scheme and using length scale, an improved sensitivity number is defined. This number is used as a criterion for rating each element in design domain and determining the addition and elimination (remove) of elements. To increase the convergence of the optimization process, the accuracy of the new elemental sensitivity numbers is improved by considering the sensitivity history. This method is convergent and mesh-independent and there are no checkerboard patterns and local solutions in optimal topologies. Using three design samples, a cantilever and classical beam and Michell type structure, affecting factors will be discussed on the final design of the structure. Change of length scale parameter produces various schemes in final structures in which, with increasing this parameter, more iteration is needed for convergent solution. Reducing evolutionary volume ratio forms different and even asymmetric topologies. Better optimized topologies are obtained with higher evolutionary volume ratios.

Purpose
To tackle the challenge topic of continuum structural layout in presence of random loads, and to develop an efficient robust method.
Design/methodology/approach
An innovative robust topology optimization approach for continuum structures with random applied loads is reported. Simultaneous minimization of the expectation and the variance of the structural compliance is performed. Uncertain load vectors are dealt with by using additional uncertain pseudo random load vectors. The sensitivity information of the robust objective function is obtained approximately via the Taylor expansion technique. The design problem is solved by Bi-directional Evolutionary Structural Optimization (BESO) method utilizing the derived sensitivity numbers.
Findings
The numerical examples show the significant topological changes of the robust solutions compared with the equivalent deterministic solutions.
Originality/value
A simple yet efficient robust topology optimization approach for continuum structures with random applied loads is developed. The computational time scales linearly with the number of applied loads with uncertainty, which is very efficient when compared with Monte Carlo-based optimization method.

The application of structural optimization to fluid-structure multiphysics systems has gotten huge attention of the researches in the last years. However, the evolutionary approach of the optimization methods has not been investigated in this class of problems. The present work aims to propose, implement, and validate an evolutionary topology optimization for elasto-acoustic systems. In this work, a finite element analysis of the proposed systems is carried out using the u/p mixed formulation. The structural domain is governed by the linear equation of elasticity and described in terms of the displacements, u, and the fluid domain is featured by the Helmholtz equation via the primary variable of pressure, p. The BEFSO (Bi-directional Evolutionary Fluid-structural Optimization) method, here proposed, follows the procedure of the evolutionary methods in which the material removal/addition in the system occurs in the discrete way. It means that the material density, the variable project, can be 1 or 0 for solid or void elements, respectively. As part of the proposed methodology, it is developed a procedure to remove/add solid materials in the system in order to keep the interface between the domains well defined during the optimization process. Examples of optimization for 2D and 3D elasto-acoustic systems are presented, through which can be verified the efficiency of the optimization procedure developed and implemented in this work, as well the feasibility for engineering problems solution.

When an inertia load constitutes a significant component of the loads in a truss, its effect on the bending stress in a bar element and how to control must be considered. For the problem, the analysis method for trusses was used to solve the axial stresses in the bars. And, the analysis method for a beam with uniformly distributed load is discussed to solve the maximum bending stress in a bar element. The two-component stresses are added to determine the total maximum stress in a bar. Then, it is proposed to use a method for controlling the slenderness ratio, to control the maximum bending stress caused by an inertia load. And, the corresponding buckling conditions are discussed to determine the allowable axial compressive stress. For example, the stress ratio method is used to minimize the truss mass under stress constraints for multiple loadings. And, a three-bar and a ten-bar truss with an inertia load are used to verify that the method for controlling the maximum bending stress is necessary and effective.

Structural topology optimization under inertial loads is a load-dependent problem. In the paper, Structural compliance is taken as objective function and volume of materials is taken as constraint variable. Based on homogenization method, an optimality criteria (OC) formula is derived that include the effect of inertial loads. Because the object function is non-monotonous, iterative process of optimization can produce oscillation and it is not easy to converge to optimal solution. So an updated scheme of OC is presented that improved the sensitivity compution about inertial load. The test examples show that the presented updated OC can steadily iterate to optimal topology solution. Finally, the topological structure of a rocket sled is optimized using the presented method.

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.

The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically
loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99
lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a
mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with
output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology
optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code
is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site
http://www.topopt.dtu.dk.

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

This paper proposes to investigate topology optimization with density-dependent body forces and especially self-weight loading. Surprisingly the solution of such problems cannot be based on a direct extension of the solution procedure used for minimum-compliance topology optimization with fixed external loads. At first the particular difficulties arising in the considered topology problems are pointed out: non-monotonous behaviour of the compliance, possible unconstrained character of the optimum and the parasitic effect for low densities when using the power model (SIMP). To get rid of the last problem requires the modification of the power law model for low densities. The other problems require that the solution procedure and the selection of appropriate structural approximations be revisited. Numerical applications compare the efficiency of different approximation schemes of the MMA family. It is shown that important improvements are achieved when the solution is carried out using the gradient-based method of moving asymptotes (GBMMA) approximations. Criteria for selecting the approximations are suggested. In addition, the applications also provide the opportunity to illustrate the strong influence of the ratio between the applied loads and the structural weight on the optimal structural topology.

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

This 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.

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 aim of this article is to evaluate and compare established numerical methods of structural topology optimization that
have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive
debate on this important topic.

We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly
elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint
on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the
material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material
used.
We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such
that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q.
This increases the probability to obtain a zero-one solution of the relaxed problem.

A common way to perform discrete optimization in shape or topology optimization is to use a method called the artificial power
law or SIMP. The focus of this paper is to show that this method gives a discrete solution under some conditions. Examples
from topology optimization are included for illustrative purposes.

The evolutionary structural optimisation (ESO) method has been under continuous development since 1992. Originally the method was conceived from the engineering perspective that the topology and shape of structures were naturally conservative for safety reasons and therefore contained an excess of material. To move from the conservative design to a more optimum design would therefore involve the removal of material. The ESO algorithm caters for topology optimisation by allowing the removal of material from all parts of the design space. With appropriate chequer-board controls and controls on the number of cavities formed, the method can reproduce traditional fully stressed topologies. If the algorithm was restricted to the removal of surface-only material, then a shape optimisation problem (along the lines of the Min–Max type problem) is solved. Recent research by the authors has presented and benchmarked an additive evolutionary structural optimisation (AESO) algorithm that, with appropriate decision making, starts the evolutionary optimisation procedure from a minimal kernel structure that connects the loading points to the mechanical constraints. Naturally this is unevenly and overly stressed, and material is subsequently added to the surface to reduce localised high stress regions. AESO only adds material to the surface, the present work describes the combining of basic ESO with the AESO to produce bi-directional ESO (BESO) whereby material can be added and removed. This paper shows that this method provides the same results as the traditional ESO. This has two benefits, it validates the ESO concept, and as the examples demonstrate, BESO can arrive at an optimum faster than ESO. This is especially true for 3D structures, since the structure grows from a small initial one rather than contracting from a, sometimes, huge initial one where around 90% of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Both 2D and 3D structures are examined and multiple load cases are applied.

After outlining analytical methods for layout optimization and illustrating them with examples, the COC algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members. The numerical results obtained are shown to be in close agreement (up to twelve significant digits) with analytical results. Finally, the problem of generalized shape optimization (finding the best boundary topology and shape) is discussed.

Optimal design from the hard-kill BESO method. Fig. 12. Evolution histories of mean compliances and volume fraction. Fig. 13. Computation time of each iteration for the hard-kill BESO method

- Fig

Fig. 11. Optimal design from the hard-kill BESO method. Fig. 12. Evolution histories of mean compliances and volume fraction. Fig. 13. Computation time of each iteration for the hard-kill BESO method. X. Huang, Y.M. Xie / Finite Elements in Analysis and Design 47 (2011) 942–948

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Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer, London,
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