Topology Optimization considering Nonsmooth Structural Boundaries in the Intersection Areas of the Components

In the structural topology optimization approaches, the Moving Morphable Component (MMC) is a new method to obtain the optimized structural topologies by optimizing shapes, sizes, and locations of components. However, the optimized structure boundary usually generates local nonsmooth areas due to incomplete connection between components. In the present paper, a topology optimization approach considering nonsmooth structural boundaries in the intersection areas of the components based on the MMC is proposed..e variability of components’ shape can be obtained by constructing the topology description function (TDF) with multiple thickness and length variables. .e shape of components can be modified according to the structural responses during the optimization process, and the relatively smooth structural boundaries are generated in the intersection areas of the components. To reduce the impact of the initial layout on the rate of convergence, this method is implemented in a hierarchical variable calling strategy. Compared with the original MMC method, the advantage of the proposed approach is that the smoothness of the structural boundaries can be effectively improved and the geometric modeling ability can be enhanced in a concise way. .e effectiveness of the proposed method is demonstrated for topology optimization of the minimum compliance problem and compliant mechanisms.


Introduction
Structural topology optimization aims to find the best distribution of materials within a prescribed design domain using an optimization algorithm in order to achieve some exceptional structural performance [1]. It breaks the limitations of the designer's thinking in traditional structure design and can get a more liberal, updated, and lighter conceptual design. At the same time, structural topology optimization is also considered a more challenging design approach compared with the size optimization and shape optimization. Since Bendsøe and Kikuchi [2] developed the topology optimization method based on the homogenization theory, after decades of development and improvement, topology optimization has gradually become an important method in the structural concept design stage and has been applied successfully to a wide range of physical disciplines [3][4][5][6]. So far, the solid isotropic material with penalization (SIMP) method [7][8][9] and level set method [10][11][12][13][14] are two popular strategies. For the other methods, the reader is referred to [15][16][17]. In the SIMP approach, the design domain is discretized by finite element mesh. en the pseudodensity of each element is set as a design variable to characterize the form of material distribution. It transforms the structural topology optimization problems into material distribution problems. e main advantages include simpler concepts, high computational efficiency, and easy to implement [18][19][20]. However, the densities of a certain number of elements are left to represent the final optimal results; the structure boundaries are easy to generate with many greyscale elements and jagged shapes [21][22][23]. In the level set method, the values of the level set function at the node points in the finite element mesh are often taken as the design variables.
e structural boundaries can be specified by drawing the contour of the level set function in one-higher dimensional space. e optimization process is represented by the evolution of specific faces. erefore, it often obtains the optimization result with smooth boundaries and easily handles topology changes compared to the traditional topology optimization methods [24][25][26]. But whether the SIMP approach or level set method, the design variables increase sharply with the mesh refinement, especially in the three-dimension problem, which will far exceed the solution capacity of the existing optimization algorithm [27]. In SIMP and level set methods, an implicit way is used to solve topology optimization problems; that is, without explicitly geometric information, it is implanted. It is difficult to accurately control the structural feature sizes, which is an important factor in considering manufacturability [28]. At the same time, the geometric/topology representation is not quite consistent with that in modern computer-aided design/engineering (CAD/CAE) modeling systems [29,30].
is treatment sets up a barrier in optimizing structural dimensions and direct link between the final topology results and CAD/CAE systems [31].
In order to overcome the above problems, recently, Guo et al. [27] proposed a more accurate and geometric topology optimization method based on the topology optimization framework, a so-called Moving Morphable Component (MMC). In this framework, a series of components with movable and deformable capabilities are used as building blocks of the topology optimization, and the optimal structural topology that meets specific performance can be obtained by varying the lengths, thickness, tilt angle, and center coordinates of these components. e same idea has also been applied in continuum-based topology optimization with discrete elements [32][33][34][35][36]. Although any curved structural parts can be approximated by precisely controlling a certain number of components with uniform thicknesses [27], so as to enhance the smoothness of the structural boundaries and the geometry modeling capability of the method, it is always highly desirable to include components with variable shapes in the MMC approach. Recently, Zhang et al. [37,38] improved the geometry modeling capability of the MMC approach by constructing the TDF with variable thickness components. Guo et al. [39] proposed the TDF of the component with a curved skeleton to constructed curved structural parts with a smooth boundary. In Meisam Takalloozadeh et al.'s method [40], the start and end of the component with two straight lines were modified as a concave/convex pointed shape by adding a new variable. Van-Nam Hoang et al. [33] and Deng et al. [32] constructed the rectangle moving component with semicircular ends and enhanced connectivity between components by adding constraints.
Although the geometric modeling capabilities of the above methods and the connection methods of components have been studied, in [37,40], there are still some obvious nonsmooth boundaries in the optimized structure using the straight lines or concave/convex pointed shapes at the start and end of a component. e method of [39] finds the optimal structural by constructing the components with curved skeleton and most of the boundaries of the optimized structure are smooth, but just like as described at the end of their paper, there still exist some local nonsmooth boundaries especially in the intersection areas of the components. In the optimization process of [32,33], the thickness of the component changed uniformly, which limits the geometry modeling capability of their methods. erefore, based on the MMC framework, this paper mainly studies how to use the variability of components' shape to solve the local nonsmooth problem in the intersection areas of the components. For this purpose, we control the length of components by adding new design variables in the TDF of the quadratically varying thicknesses. e variability of components' shape and the geometry modeling capability of the original MMC method are enhanced through the coordination of length and thickness variables. In the topology optimization process, the shape of components can be changed according to the structural responses, and relatively smooth structural boundaries are generated in the intersection areas between the components. In addition, considering the influence of the initial layout with more components on the rate of the convergence, the hierarchical variables calling strategy is proposed to optimize the layout of components in the initial stage. e effectiveness of the proposed method is verified by several numerical examples.

Problem Formulation
In the MMC approach, the topology description function (TDF, φ(x)) of components can be expressed in the following form: where x represents a set of points within the design domain; R represents the entire design area; Ω ⊂ R denotes a subset of R occupied by N components made of the physical material. If the number of components included in the design area is N, the TDF of the whole solid material in the structure area can be built as where i � 1, . . . , N is the number of components and the TDF of the i − th component can be written as where Ω i is the area occupied by the i − th and Ω � ∪ N i�1 Ω i . en the geometric representation of the corresponding structure is shown in Figure 1.

Shock and Vibration
Based on the MMC topology optimization framework, a typical topology optimization problem can be described as where J(R) and ξ k , k � 1, . . . , m, denote the objective function and constraint function, respectively, and U R is the admissible set that R belongs to.
Topology optimization for a mean compliance minimization under available volume constraint problem is considered, and the corresponding mathematical model can be written as where R i , i � 1, . . . , N, is the vector of the design variables of the i − th component. J is the objective function. f, u, t, and ε are the body force density, the displacement field, the prescribed surface traction on Neumann boundary Γ t , and the linear strain tensor, respectively. u is the displacement on Dirichlet boundary Γ u . υ is the corresponding test function defined on U I � υ | υ ∈ H(Ω S ), υ � 0 on Γ u , and the symbol H � H(x) means the Heaviside function. φ S (x; R) denotes the topology description function set for the overall component. q > 1 is an integer (in this work, q � 2 is used), and E � ( is the isotropic elastic tensor. E and v are Young's modulus and Poisson's ratio of the solid material, respectively, and Π and δ denote the symmetric part of the fourth-order identity tensor and the second-order identity tensor, respectively. V is the upper limit of the volume of the solid material.

TDF for the Variability of Components' Shape and Hierarchical Variable Calling Strategy
In this part, we will discuss how to construct the TDF for the variability of components' shape and implement the hierarchical variables calling strategy based on the MMC framework.

TDF for the Variability of Components'
Shape. e difference consists that, in the MMC method, it is possible to give an explicit description of the boundary and geometry features of a component. is cannot be achieved in the conventional level set method. In [27], the uniform thickness component can be described using the hyperelliptic TDF. In order to improve the geometry modeling capabilities of the MMC approach, the TDF of quadraticall varying thicknesses (QVT) can be expressed as where m is a relatively large positive integer, which takes (6) in this paper. Here, f(x ′ ) controls the shape of components x ′ where (x 0i , y 0i ), l i , and θ i denote the coordinate of the center, the half-length, and the inclined angle of i − th component, respectively. t i1 , t i2 , and t i3 are the three thicknesses for different positions of the i − th component, respectively, as shown in Figure 2.
It is worth noting that, in this method, the start and end of the component were two straight lines perpendicular to the direction of the component. When two or more components are connected at an angle, as shown in Figure 3 which plots the components with dashed boundary, the local Figure 1: e geometry representation based on the MMC framework.
nonsmooth region of the structural boundary is often caused. If we assume that the straight line of the start and end of the component can be bent and deformed, the variability of the shape of the component can be achieved by combining the thickness variations. en, the local smooth structural boundaries can be formed in the intersection parts of the components according to the structural response in the optimization process, as shown in Figure 3 which plots the components with solid boundary. e geometry description of the components' shape with variability is shown in Figure 4. e deformation of components' shape can be achieved by adjusting the length and thickness variables. e corresponding TDF can be constructed as where f(x ′ ) is the same as equation (7) and g(y ′ ) can be constructed as where l i1 , l i2 , and l i3 are the three lengths for different positions of a component as shown in Figure 4, respectively. (x s 0i , y s 0i ) and r are the center coordinates and radius of the arc of the start and end of a component, respectively, and we have By inputting different parameters in equation (9), the variability of components' shape can be easily achieved. It is worth noting that although the topology optimization process using the variability of components' shape based on the MMC framework is somewhat similar to the level set method (TDF has been used to represent the geometry of components), the proposed method can also possibly give an explicit description of the boundary and geometric features (such as length and thickness) of components, which cannot be achieved in the level set method that uses the free transformation of the level set functions to represent the boundary of a structure.

Hierarchical Variables Calling Strategy.
In this section, we will discuss how to optimize the initial layout of components to improve the rate of convergence. In MMC framework, although structural topology can be constructed by moving, deforming, disappearing, and overlapping of a set of components, the different initial layouts of components have different effects on the rate of convergence. In particular, when there are more components in the design domain, the effective force transfer path may be affected by improper positioning or deformation of some components, which will delay the rate of convergence.
In order to improve the rate of convergence of the optimization process, a hierarchical variables calling strategy is proposed. In this strategy, the optimization process is divided into two levels: the level 1 is the layout variables calling and the level 2 is all variables calling. e average variation of components' positioning (the central coordinates and the rotation angles) within the adjacent iterations is set as a constraint. In the initial stage of optimization, only the layout variables (the central coordinates and the rotation angles) are called, and the initial layout of the components can be modified according to the structural responses in this  process. When the constraint is satisfied, the full variables are called to construct the topology optimization structure. It is worth noting that the output of the level 1 is used as input for the level 2.
e constraint of the hierarchical variables calling strategy can be expressed as where j is the j − th iteration, τ and σ are the set maximum upper limits of change values, respectively, and generally, the smaller positive numbers are selected.
where d i top and ζ i tcp are the center coordinates (x 0i , y 0i ) and the rotation angles ζ i change value of the i − th component in twice adjacent iterations, respectively, as shown in Figure 6.

Numerical Solution Aspects
In this section, numerical solution aspects in the proposed approach will be discussed in detail.

Finite Element Analysis.
In this study, in order to analyze the structure constructed using the MMC method, the Eulerian mesh is utilized and TFD is used to calculate each node. In order to enhance the computational efficiency, the ersatz material model is used for FEM analysis. Young's modulus of every element can be calculated by equation (15), which was proposed in [37]: where q is an integer number, we take q � 2, E 0 is Young's modulus for the material of every element, and H is a regularized Heaviside function: where ε is the regularization parameter and a � 0.001 to prevent the occurrence of the singular of the global stiffness matrix, and the stiffness matrix of the i − th element can be obtained by equation (17): In the above formula, k S is the stiffness matrix of the element, regardless of the number of materials occupied by components.

Sensitivity Analysis.
Only design sensitivity analysis for the compliance minimization problem is discussed here. Other objectives/constraints can be achieved by resorting to adjoint sensitivity analysis. When we know the stiffness matrix of each element, the global stiffness matrix can be assembled. e structural compliance and the displacement vector will be obtained by using equation (18): When μ is assumed to be any one of the variables, then the sensitivity of the structural compliance of μ can be expressed as Since the force is independent of the design variables, equation (20) can be transformed into By equations (15) and (17), we can get equation (22): Shock and Vibration By substituting equations (21) and (22) into equation (19), which can be written as follows: where k i is the element stiffness matrix corresponding to φ e j � 1, j � 1, . . . , 4. Because of φ i (x) is a function of μ, the derivative of the structural compliance can be calculated using equation (23).
For the derivative of volume constraint, we also have where (zH(φ e j )/zμ) can be calculated easily since φ(x) is an explicit function of μ.
It is worth noting that, in the above derivations, as mentioned in [37], the nondifferentiable issue arising from the max operation when φ is constructed by multiple components overlapping is neglected. e numerical case also proves that this has no effect on the optimization process.

Numerical Examples
In this section, the effectiveness of the proposed method is verified by several representative numerical examples. Unless otherwise stated, all examples use a four-node bilinear square unit discrete design area. Young's modulus and Poisson's ratio of the solid material are taken as E � 1 and υ � 0.3, respectively. e two-dimensional topology optimization problem is considered.

e Compliant Mechanism Problem.
To demonstrate the effectiveness of the proposed method in dealing with local nonsmoothness problems in the intersection areas of the components, the compliant mechanism design problem is considered. e design objective is to minimize the geometric advantage of the compliant mechanism. e design domain, geometry data, and boundary conditions are shown in Figure 7. In this problem, the objective is to minimize the geometric advantage (GA) of the compliant mechanism GA � (−Δ out /Δ in ), where Δ in � F in u 1in + F out u 2in and Δ out � F in u 1out + F out u 2out , u jin , u jout , j � 1, 2 meaning the  Figures 9 and 10. It can be seen that the final structure obtained by the QVT method forms an obvious incomplete connection phenomenon in the intersection area between the components, which easily cause the local structural boundaries nonsmoothness (see Figure 9(d)). From the contour plot of the final structures shown in Figure 10, the components exhibit some flexibility in the form of the connection, and a relatively smooth structural boundary transition is obtained. In the case of the proposed method, the hinge-like connection can no longer be produced in the final optimized structure as illustrated in Figure 10(d). Figure 11 provides the convergence history of the objective function and the constraint function. It can be observed from Figure 11(b) that, for this example, the value of the objective function has dropped sharply in the earlier iteration step, which indicates that an effective loading transmission path can be faster constructed using the method. e comparison results are shown in Table 1. e values of the objective functions obtained by the two methods are Obj � −0.811 and Obj � −0.851, respectively. Compared with the QVT method, the proposed method has a faster convergence rate and less time consumption.

e MMB Problem.
In the literature on structural optimization studies, this example is often used to verify the validity of a topology optimization method. e design domain and boundary conditions are shown in Figure 12.
e unit vertical load is imposed on the middle point of the upper surface of the beam. e design domain with length and width of 6 × 1 is discretized by a 240 × 40 FE mesh. e upper bound of volume constraint is set to 0.4. Because of the symmetric property of the problem, only half of the structure is used to be optimized in this example.
Twenty-four components are placed in the initial design domain, and the overall number of design variables is only 24 × 9 � 216. e initial values of the design variables for each component of the QVT method and this paper method are set as 0.38 0.04 0.06 0.04 0.7 and 0.39 0.41 0.39 0.04 0.06 0.04 0.7 , respectively. In equation (12), the maximum upper limits of change values are τ � 0.38 and σ � 0.57. e initial layout of the components is shown in Figure 13.
Some steps of the optimization and the convergence histories of the MBB example are shown in Figures 14-16, respectively. From Figure 14(f ) and the 810th step in Figure 16, we can observe that the final topology optimization structures obtained by the two methods are very similar.

Shock and Vibration
However, it is worth noting that the QVT method shows a strong instability in the optimization process. In contrast, the proposed method has a more stable convergence process. It can be clearly seen that when equation (12) is not satisfied, the shape and size of the components do not change, as shown the 20th step contour plot in Figure 16. e entire process only optimizes the center coordinates and the rotation angles of components before the 40th iteration. Once equation (12) is satisfied, the components can be moved and deformed according to the structural response, as shown in the contour plot of the 41th and 42th step in Figure 16. e comparison results by using the two methods are presented in Table 2. e values of the objective function are Obj � 232.84 and Obj � 228.59, respectively. Compared with the QVT method, the value of the objective function obtained is slightly lower and the time consumption has been reduced with this paper method. e optimization process has been converged at the 810th iteration. It is worth mentioning that the proposed method enhances the stability of the solution process and accelerates the convergence rate.

e Bridge Problem.
In order to demonstrate that this paper method cannot only solve the structural boundaries nonsmoothness in the intersection areas of components but also realize the variability of components' shape to some extent, the bridge problem is considered in this section. e loading and boundary conditions are shown in Figure 17. e unit vertical load is evenly imposed on the upper surface of the design field. e design domain with length and width

Conclusions
In the present paper, a topology optimization method based on the MMC to solve the local nonsmooth problem of components connection areas is proposed. It enhances the variability of components' shape by coordinating multiple length and thickness variables, and the smooth structural boundaries can be formed in the connection areas between the components. is method preserves the advantages of the original MMC approach and enhances its geometry modeling capabilities. e effect of undesirable positioning or deformation of some components in a layout with more   Step 20 Step 40 Step 810 Step 151 Step 41 Step 42 Step 68 Step 250 Obj Vol     components on the rate of convergence can be optimized by the hierarchical variables calling strategy. Several numerical examples are used to verify the effectiveness of the proposed method. It is also observed that the initial layout of the components is very important for the rate of convergence. In principle, the hierarchical variables calling strategy can be applied to the structural topology optimization that needs to optimize the initial layout of components to improve the rate of convergence. But how to achieve a reasonable initial components layout way is a more challenging task, and the related research will be reported in separate works.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare there are no conflicts of interest regarding the publication of this paper.