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Abstract

Manufacturing and machining property are the most important features of structures. The product with good processing property is not only to decrease its costs but also to increase its efficiency. It is very difficult to design and machine because of the complicated results of topology optimization. This study introduces a heuristic approach using an engineering constraint in topology optimization sensitivity process. With this special method, the result of topology optimization can meet the structural design requirement. Several numerical examples were provided to show that the positive influence of new engineering constraint method on the reasonable results of topology optimization was significant. It was a good way to control the material distribution of continuum structural topology optimization using this method.

Keywords

Engineering constraint topology optimization manufacturing and machining property sensitivity

Since early research by Bendsøe and Kikuchi,¹ topology optimization is recognized as an important technique to figure out the optimal structure layout within the given design domain. Recently, this technique has been recognized as a most efficient method in searching for optimum structure.² The application of topology optimization in structure design can be used in many fields, such as automobile and aerospace industries. However, practical structure optimization work does not take engineering constraint into account, which becomes an obstacle to process designing and manufacturing.^3,4

To solve industry applications, many numerical methods have been proposed to make the results of topology optimization more feasible. The singular value decomposition (SVD)⁵ was successfully applied in the model reduction. Topology optimization with sensitivity filter developed by Sigmund⁶ can provide mesh-independent, checkerboard-free results and minima size control; besides, the slope and perimeter constraint technique^7,8 can also solve this problem. A maximum length control approach was presented by Guest⁹ and was applied to restrict the size of material distribution in the topology optimization design domain. However, even if the minimum/maximum size control and instability problem are solved very successfully, there are still many difficulties about the transformation of the results of optimization to reality structure design, for example, to design the rib of wheel structure with rotational symmetry feature using topology optimization is still another great challenge.

Symmetry and repetitive feature is an important characteristic of the structure, whose function is to build the large body using a single structure and has the advantage of convenient manufacture. If the whole design domain of large structure is directly optimized by traditional way, the result of topology optimization may become very complex and is difficult to be product in the view of manufacture. As a result, good features may influence practicability of the design structure and the efficiency of manufacture. Therefore, many efforts have been made to improve the feasibility of the results of topology optimization. Wang et al.¹⁰ studied slender piezoelectric actuators with repetitive component patterns by unit-cell structure, which presents useful conceptual designs, and the optimal layout is easy to manufacture. A symmetry and pattern constraint method was developed as a tool for use in the continuous topology optimization by mapping the design variables.¹¹ Vatanabe et al.¹² presented manufacturing constraint techniques based on a domain of design variables projected approach, which illustrate the ability of the method can efficiently control the optimization solution. Nodal variables are implemented as the design variables to control over the thickness of structural members in topology optimization problems, which is achieved through the use of mesh independent.¹³

An optimal structure with good feature is one of the major content in current engineering design, which will not only improve its product efficiency and manufacturability but also increase the strength, frequency, and stiffness of component. In this article, the SIMP⁶ method with the engineering constraints using an efficient MATLAB code¹⁴ is introduced to make the results of topology optimization more designable, which will include pattern repetition, axis symmetry, and cyclical symmetry control.

A new engineering constraint method for topology optimization

Compliance optimization

Minima compliance of the Messerschmidt–Bölkow–Blohm (MBB) beam is a standard topic, and the mathematical equation of the optimization is defined in the following

$min_{x} : C (x) = U^{T} KU = \sum_{e = 1}^{N} E_{e} (x_{e}) u_{e}^{T} k_{0} u_{e}$ (1)

$s . t . : \frac{v (x)}{V} = f$

$KU = F$

$0 \leq x_{e} \leq 1$

where C is defined as objective denotes compliance; u_e is the element displacement vector; k₀ represents the element stiffness matrix; x_e is the design variable; N is the number of all design variables; U, F, and K are the displacement, load, and the global structure stiffness, respectively; V is the original volume; V(x) is optimal volume; and f is the global volume fraction.

E_e is the element Young’s modulus according to its density x_e that is modified by SIMP approach and is described in the following formulation

$E_{e} (x_{e}) = E_{min} + x_{e}^{p} (E_{0} - E_{min}), x_{e} \in [0, 1]$ (2)

where P is penalty factor, E₀ is Young’s modulus, and E_min denotes the low bound of Young’s modulus of density variable x_e. The new modified SIMP method has many outstanding capabilities to implement the extensional filters in topology optimization.¹⁵

Sensitivity filtering method

The sensitivities of the element densities x_j are needed by optimality criteria (OC) method and read as follows

$\frac{\partial c}{\partial x_{e}} = - {px}_{e}^{p - 1} (E_{0} - E_{min}) u_{e}^{T} k_{0} u$ (3)

Without control on sensitivities, checkerboard patterns and mesh dependency are two problems of the topology optimization with SIMP method. In order to solve this phenomenon, a filtering technique must be imposed in the process, which has been proved by numerous experiments in practice, although it has not yet been proved with mathematic theory.

The sensitivity filter modifies sensitivities (3) as follows

$\frac{\hat{\partial c}}{\partial x_{e}} = \frac{1}{max (γ, x_{e}) \sum_{i \in N_{e}} H_{ei}} \sum_{i \in N_{e}} H_{ei} x_{i} \frac{\partial c}{\partial x_{i}}$ (4)

$H_{ei} = max (0, r_{min} - Δ (e, i))$ (5)

$γ = 0.001$

where Δ(e, i) is the distance from element i to element e, N_e implies that filter radius minima influence element set, and H_ei is called as a weight factor. The small positive number γ is integrated in the sensitivity filter in order to avoid division by zero because x_e ranges from 0 to 1.

Engineering constraint method

To solve the results of topology optimization that is nondesignable, one approach^3,11 is to add design variable constraints according to the practicality requirement to the original optimization, which makes the single constraint problem become a multiple constraint problem, and cannot be solved by OC method. It should be solved by the advanced method of moving asymptotes (MMA)¹⁶ and others.

As discussed before, to avoid this defect in multiple constraints in initial problem, an engineering constraint method is improved to ensure the results of topology optimization with good design feature than before in MATLAB code program. The main idea is derived from sensitivity filter method, and sensitivity filter can be regarded as a heuristic principle. From this point, we can also guide sensitivities with special aims according to our requirements.

This engineering control method about plane symmetry reformulates element sensitivity with an average of each corresponding elements in a design domain according to their geometry correlation as shown in Figure 1.

Figure 1.

Symmtery geometry correlation of each element: (a) axis symmetry, (b) point symmetry, and (c) cyclical symmetry.

The engineering constraint method shown in Figure 1 is executed after the first-order sensitivity filter of the objective function using Sigmund method, so the engineering constraint method can be considered as an additional process of the first-order sensitivity filter.

Engineering constraint method has the following main steps:

Define design domain;

Parameter input;

Finite element analysis (FEA) solution;

Objective function and sensitivity analysis;

Sigmund sensitivity filter technique;

Sensitivity filter by engineering constraint method;

Variable update by OC;

Convergence if variable change <1% or go to step 3.

The important thing that should be pointed out is that checkerboard phenomena will appear without step 5, and even step 6 is performed later.

Numerical examples

In this section, typical cantilever beam will be used to illustrate the efficiency of the engineering constraint methods, because cantilever beam has been researched in many papers. There are two engineering constraint method examples including symmetry and pattern repetition control.

Symmetry problem

A structure shown in Figure 2 is now studied with engineering constraint method with different symmetry problems, respectively. A vertical force is loaded in the lower right corner; the boundary of left edge is fixed in two directions. The design domain is composed of square elements with four nodes; the final important thing is that the number of both nodes and elements is regularized identically in this article regardless of the coordinates of the design domain showed. Figure 2 illustrates the coordinates of the design domain with x- and y-axes.

Figure 2.

Cantilever beam design domain.

The main idea of symmetrical control is that the sensitivities of each element are symmetrical with respect to the special axis. For example, in single-axis symmetry about x-axis problem, element design sensitivities are symmetrical with respect to the x-axis and should follow this regulation as equation (6). By parity of reasoning, the y-axis symmetrical constraint formulation can be written as equation (7). Furthermore, for two-axis symmetry about x- and y-axis problem, element sensitivities should be modified by equations (6) and (7) successively. In another case of middle point symmetry, the correlation of the element sensitivity constraint is defined in equation (8)

$\frac{\tilde{\partial c}}{\partial x_{e}^{i, j}} = \frac{\tilde{\partial c}}{\partial x_{e}^{n_{x} + 1 - i, j}} = \frac{(\frac{\hat{\partial c}}{\partial x_{e}^{i, j}} + \frac{\hat{\partial c}}{\partial x_{e}^{n_{x} + 1 - i, j}})}{2}$ (6)

$\frac{\tilde{\partial c}}{\partial x_{e}^{i, j}} = \frac{\tilde{\partial c}}{\partial x_{e}^{i, n_{y} + 1 - j}} = \frac{(\frac{\hat{\partial c}}{\partial x_{e}^{i, j}} + \frac{\hat{\partial c}}{\partial x_{e}^{i, n_{y} + 1 - j}})}{2}$ (7)

$\frac{\tilde{\partial c}}{\partial x_{e}^{i, j}} = \frac{\tilde{\partial c}}{\partial x_{e}^{n_{x} + 1 - i, n_{y} + 1 - j}} = \frac{(\frac{\hat{\partial c}}{\partial x_{e}^{i, j}} + \frac{\hat{\partial c}}{\partial x_{e}^{n_{x} + 1 - i, n_{y} + 1 - j}})}{2}$ (8)

where $i = 1, \dots, n_{x}, j = 1, \dots, n_{y}$ .

In this numerical example, the design domain is discredited with 120 × 40 elements. The optimization volume fraction is executed to 50%; the penalization factor is identical to 3. The filter radius r_min equals to 1.5. The compliance minimal topology optimization is solved by traditional sensitivity filter method and engineering constraint method successively. Without any engineering constraint method, the results of topology is not distributed symmetrically as shown in Figure 3; however, a structure with x-axis symmetry control outputs a x-axis symmetrical feature in Figure 4. In this figure, the density of each element is symmetry about the x-axis. In the same way, we can use this engineering constraint to control other symmetry problems. Figure 5 has y-axis symmetrical result, while Figure 6 has the result symmetrical about both x- and y-axes and Figure 7 has middle point symmetrical distribution.

Figure 3.

Without any engineering constraint.

Figure 4.

Symmetry about x-axis.

Figure 5.

Symmetry about y-axis.

Figure 6.

Symmetry about both x- and y-axes.

Figure 7.

Symmetry about middle point.

There is critical problem associated with cyclical symmetry. In this particular issue, the number elements of domain in the horizontal and vertical directions should be identical, and the cyclical symmetry only includes four quarters or two halves because the design domain can only be formed as a square area. The element design sensitivities should be modified as the following function with four clockwise cyclical symmetrical parts. In automotive industry, wheel structure can use this method to optimize its body

$\begin{array}{l} \frac{\tilde{\partial c}}{\partial x_{e}^{i, j}} = \frac{\tilde{\partial c}}{\partial x_{e}^{n_{l} + 1 - i, n_{l} + 1 - j}} = \frac{\tilde{\partial c}}{\partial x_{e}^{j, n_{l} + 1 - i}} = \frac{\tilde{\partial c}}{\partial x_{e}^{n_{l} + 1 - j, i}} \\ = \frac{\frac{\hat{\partial c}}{\partial x_{e}^{i, j}} + \frac{\hat{\partial c}}{\partial x_{e}^{n_{l} + 1 - i, n_{l} + 1 - j}} + \frac{\hat{\partial c}}{\partial x_{e}^{j, n_{l} + 1 - i}} + \frac{\hat{\partial c}}{\partial x_{e}^{n_{l} + 1 - j, i}}}{4} \end{array}$ (9)

In this numerical example, the design domain is discredited with 60 × 60 elements. The boundary condition of this example is same as Figure 2. The traditional topology optimization results without engineering constraint are given in Figure 8; however, a new cyclical symmetry feature is formed in Figure 9 with a novel engineering constraint method.

Figure 8.

Without any symmetry constraint.

Figure 9.

Cyclical symmetry about middle point.

Pattern repetition problem

In pattern repetition design, the load, boundary condition, objective, and volume constraint of design domain are the same as Figure 2. Figure 10 indicates that the design domain has been split into several portions according to practical requirements with one master domain and some slave domains. This pattern repetition approach can be applied to design structures based on composite materials.

Figure 10.

Design domain divisions.

For this pattern repetition problem, each element sensitivity of slave domains should be equal to the identical element of master domain, whose relationship between them can be read as the following function

$\frac{\hat{\partial c}}{\partial x_{M}^{i, j}} = \frac{\hat{\partial c}}{\partial x_{S}^{i, j}}, i = 1, \dots, n_{x}, j = 1, \dots, n_{y}, s = 1, \dots, n$ (10)

where x_m is the master domain design variable and x_s is the slave domain variable.

Two different pattern repetition features are performed, consisting of three pattern repetitions and four pattern repetitions, as shown in Figures 11 and 12. It is clear that pattern repetition feature can be easily got from engineering constraint method.

Figure 11.

With three pattern features.

Figure 12.

With four pattern features.

The new optimal structure compliance will be deteriorated because the engineering constraint method for topology optimization provides extra restrictions to the original topology optimization. In this article, our purpose is to create a new feature for special structure design, while the cost of manufacture and manufacturability and not the structure performance may be more important in some engineering field. For each numerical example, iteration steps are listed in Table 1.

Table 1.

Iteration steps.

No.	Numerical example	Iteration steps
1	Classical (baseline)	24
2	x-axis symmetry	26
3	y-axis symmetry	25
4	x- and y-axis symmetry	49
5	Middle point	34
6	Cyclical	25
7	Three patterns	22
8	Four patterns	22

From Table 1, we can find that the iteration step is close to baseline except for example 4 and example 5. The root cause is that an extra filter is added in this example, which makes the topology program difficult to be converged.

Conclusion

A novel engineering constraint method in sensitivities is developed to solve an engineering application problem with SIMP method in MATLAB code program in this brief note. Several numerical examples have been provided in this article. This engineering method contributes to generate the results with good manufacturing and machining feature, which can be confirmed by commercial optimize program and is very important recently in industry. Therefore, it is very convenient and easy to be carried out in typical MBB beam or cantilever beam. In addition, it is shown that this engineering constraint method can extend the application of topology optimization in complicated structure, and it can play an important role in practical design.

Footnotes

Handling Editor: David R Salgado

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51375203 and 51675044).

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