Keywords

1 Instruction

In engineering design, the phenomenon that one structure consists of materials with different properties and characteristics is very common, and multi-material structure has potential to achieve specific function that single-material unable to achieve. For example, a multi-material structure with appropriate distribution of different materials tends to have better structural performance to withstand the stress than the same structure combined with single material. As an advanced technology, topology optimization can be employed to find the optimal layout of the structural material under the given design domain, given boundary conditions and loaded conditions, so that it can meet certain stress, volume constraints and the performance. In the past few decades, substantial number of studies are focusing on multi-material structural design using topology optimization.

Gao [1] proposed the multi-objective topology optimization model, which is attached to multi-material microstructure to solve relative optimization probelm. A new multi-material method including parameterizing process has been proposed by Bruyneel [2], where the weights are expressed acted by bilinear finite element shape functions in a weighted interpolation within the involving material. Lu [3] researches the method of topology optimization of continuum structure with using various materials, developing the innovative materials and neo-smart structures as well. An alternating active-phase algorithm was proposed by Tavakoli [4], who also used the techniques to divide a multiphase topology optimization cases into a number of relative binary phase topology optimization sub-cases. Jia [5] studied the hierarchical design of the microstructure of the heterogeneous material. Osanov [6] research the method of topology optimization for additive manufacturing. The level set method topology optimization for multi-material thermo-elastic problem with gradient has been proposed [7], but the numerical solution process is complex. Vogiatzis [8] researched the level set method for multi-material microstructure design. An efficient multi-material topology optimization is formulated with material nonlinearity considerations in [9], and the proposed formulation tackled with an arbitrary amount of candidate materials with flexible material properties. With the rely on the crack patterns, a compliance multi-material topology optimization design of continuum structures was employed by Banh [10], meanwhile, authors also creat a multi-material optimal topology and shape to guarantee it as an choice to prevent the propagation of crack patterns. To figure out the meso- and macro- scale multi-material lattice structures topology problems under any combination of material and load uncertainties, a kind of denstity-based robost topology optimization method was applied by Chan [11]. A polytree-based adaptive methodology is presented by Nguyen and Tran [12] for achieving multi-topology optimization. In order to cope with the concurrent design of multiphase composite structures under a certain range of excitation frequencies, a new multiscale topology optimization method has been presented and demonstrated by Li [13].

However, the above studies are based on the physical description of the topology optimization model, they can not be utilized to get clear and smooth structural boundaries straightforwardly. The level set method can get optimized structure with clear smooth boundaries compared to other topology optimization method, but numerical solution of level set method is complicated and unstable with high computation cost. Using SIMP method to optimize multi-material structure is simpler than level set method, but traditional SIMP method can’t get clear boundaries between different materials. To deal with the appearance of overlapping region between each materials and weak ability to obtain clear boundaries resulted by SIMP interpolation method, the proposed improved sensitivity filtering method for multi-material topology optimization will be a choice to solve.

In this paper, the numerical model for multi-material structure is presented in Sect. 2 and the original sensitivity filter model is also reviewed in Sect. 3 along with proposing the new sensitivity filter method, which consider the average of the four-node distance and weight function. In the last, two cases are listed to demonstrate the utilization and the meaning of this kind of the new filter method.

2 Numerical Model for Multi-material Structure

In order to solve problem related to SIMP method for multi-material structure with unclear boundaries, this paper firstly review the interpolation of multi-phase materials. And then, it considers the density gradient information and controls the number of filter elements in the multi-material topology optimization design. The importance of this method is how to constrain the boundary between multiphase materials. Initially, a three-phase material distribution issue is chosen to start the research about the multi-material topology. It is noted that the method of three materials can be extended to multi-material problems simultaneously. The materials model can be expressed as given as follow, and more detail about the model can be explored in [14].

$$ E_{i} \, = \,E\left( {x_{i} } \right)\, = \,x_{i1}^{p} E^{\left( 1 \right)} $$
(1)
$$ x_{i} \, = \,\left\{ {x_{ij} } \right\}\left( {i\, = \,1,2,3 \ldots n;j\, = \,1} \right) $$
(2)

The symbol i has a design variable \( x_{i1} \) which can display whether the element is void or solid. The topology optimization problem of maximum the compliance of the structure can be described as follow.

$$ \left\{ {\begin{array}{*{20}c} {find:\,\{ x_{i1} \} \left( {i\, = \,1,2,3 \ldots n} \right)} \\ {minimize:C\, = \,{F}^{T} u} \\ {subject\, to:F\, = \,Ku } \\ {0\, < \,x_{min} \, < \,x_{i1} \, < \,1} \\ {volume \,constraint:\,\sum\nolimits_{i = 1}^{n} {V_{i} x_{i1} \, \le \,vf_{1} \cdot\sum\nolimits_{i = 1}^{n} {V_{i} } } } \\ \end{array} } \right. $$
(3)

The material consumption can be ensured by controlling the volume of the material 1. The parameter \( vf_{1} \) stands for the volume of material 1, and \( C \) is the compliance of the whole structure. \( u \) and \( K \) are element displacement and stiffness matrix. As usual, stiffness matrix is a symmetric matrix, where \( \left\{ {k_{ij} } \right\}\, = \,\{ k_{ji} \} \). To avoid the singularity in matrix calculation, \( x_{min} \) is set as a value near to 0 like 0.001.

Similarly, for the structural design problem with two-phase materials and voids, Young’s modulus penalty model can be described as follow.

$$ E_{i} \, = \,E\left( {x_{i} } \right)\, = \,x_{i2}^{p} \left[ {x_{i1}^{p} E^{\left( 2 \right)} \, + \,\left( {1 - x_{i1}^{p} E^{\left( 1 \right)} } \right)} \right] $$
(4)
$$ x_{i} \, = \,\left\{ {x_{ij} } \right\}\left( {i\, = \,1,2,3 \ldots n;j\, = \,1,\,2} \right) $$
(5)

where parameter \( x_{i1} \) is used to determine the material existence in the element, while parameter \( x_{i2} \) is used to clarify that the existence of material 2 in the element. As shown in Fig. 1, the relation between the value set and material distribution has been demonstrated clearly.

Fig. 1.
figure 1

Three-phase material distribution

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$$ \left( {x_{i1} ,\,x_{i2} } \right)\, = \,\left\{ {\begin{array}{*{20}c} {\left( {1,0} \right) \,Material \,1} \\ {\left( {1,1} \right) \,Material \,2} \\ {\left( {0,0} \right) \,Void } \\ \end{array} } \right. $$
(6)

Simultaneously, the corresponding structural optimization problem can be edited logically as follow.

$$ \left\{ {\begin{array}{*{20}c} {find:\{ x_{i1} ,x_{i2} \} \left( {i\, = \,1,2,3 \ldots n} \right)} \\ {minimize:C\, = \,F^{T} u} \\ {subject \,to:F\, = \,\varvec{K}u } \\ {0\, < \,x_{min} \, < \,x_{ij} \, < \,1 \left( {j\, = \,1,2} \right)} \\ {volume \,constraint:\sum\nolimits_{i = 1}^{n} {V_{i} x_{i2} \, \le \,\left( {vf_{1} + vf_{2} } \right)\cdot\sum\nolimits_{i = 1}^{n} {V_{i} } } } \\ {\sum\nolimits_{i = 1}^{n} {V_{i} x_{i1} \, \le \,vf_{2} \cdot\sum\nolimits_{i = 1}^{n} {V_{i} } } } \\ \end{array} } \right. $$
(7)

The consumption of material is controlled by the two volume constraints. The first constraint is the sum of material 1 and material 2. Otherwise the second constraint is the consumption of material 2.

Then, the sensitivity of objective function about \( x_{i1} \) and \( x_{i2} \) can be formulated, and \( \varvec{U} \) is the whole displacement matrix in the structure. With the relationship of complicance between the whole structure and element, the code is performed.

$$ \frac{\partial C}{{\partial x_{i1} }}\, = \,F^{T} \varvec{K}^{ - 1} \left( {\frac{\partial F}{{\partial x_{i1} }} - \frac{{\partial \varvec{K}}}{{\partial x_{i1} }}\varvec{U}} \right)\, = \, - \varvec{U}^{T} \frac{{\partial \varvec{K}}}{{\partial x_{i1} }}\varvec{U}\, = \, - px_{i1}^{p - 1} u^{T} \varvec{K}u $$
(8)
$$ \frac{\partial C}{{\partial x_{i1} }}\, = \, - \varvec{U}^{T} \frac{{\partial \varvec{K}}}{{\partial x_{i2} }}\varvec{U}\, = \, - \frac{{px_{i2}^{p - 1} \left( {E^{\left( 1 \right)} - E^{\left( 2 \right)} } \right)}}{{E^{\left( 2 \right)} + x_{i2}^{p} \left( {E^{\left( 1 \right)} - E^{\left( 2 \right)} } \right)}}u^{T} \varvec{K}u $$
(9)

The sensitivity of the constraint function can be expressed as

$$ \frac{\partial V}{{\partial x_{i1} }}\, = \,\frac{\partial V}{{\partial x_{i2} }}\, = \,1 $$
(10)

3 Sensitivity Filter Comparison

3.1 Original Sensitivity Filtering Method

Sensitivity filter is always carried out as these two steps: Calculate the sensitivity of each design variable \( \partial f/\partial x_{i} \) firstly, then update the sensitivity by weight function. The weight function is as follow.

$$ H_{i} \, = \,R - dist\left( {i,k} \right),\{ x\, \in \,N|dist\, \le \,R,k\, = \,1,2 \cdots N\} $$
(11)

Where \( R \) and \( N \) are filter radius and the numbers of elements within the filter radius respectively. \( dist\left( {i,k} \right) \) represents the distance between element \( i \) and element \( k \). This filtering scheme has been defined in Fig. 2.

Fig. 2.
figure 2

Filter scheme

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Sensitivity filter formula is defined by Sigmund [15] as \( \frac{{\partial \hat{f}}}{{\partial x_{i} }} = (\sum\nolimits_{i\, \in \,N} {H_{i} x_{i} \frac{\partial f}{{\partial x_{i} }})} /x_{k} \cdot\sum\nolimits_{i\, \in \,N} {H_{i} } \). To obtain clearer topology optimization boundary, Borrvall [16] proposed a modified sensitivity filter formula as \( \frac{{\partial \hat{f}}}{{\partial x_{i} }}\, = \,(\sum\nolimits_{i \in N} {H_{i} x_{i} \frac{\partial f}{{\partial x_{i} }}} )/\varSigma_{i \in N} H_{i} \cdot x_{k} \). In the topology optimization of multi-physics fields, the sensitivity in the formula leaves out as \( \frac{{\partial \hat{f}}}{{\partial x_{i} }} = {{\left( {\mathop \sum \limits_{i \in N} H_{i} \cdot\frac{\partial f}{{\partial x_{i} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\mathop \sum \limits_{i \in N} H_{i} \cdot\frac{\partial f}{{\partial x_{i} }}} \right)} {\sum\nolimits_{i \in N} {H_{i} } }}} \right. \kern-0pt} {\sum\nolimits_{i \in N} {H_{i} } }} \) [17].

Through the above three methods of numerical experiments, there exists a short consideration of intermediate density elements in the sensitivity formulation and the topologically outcome is not fully smooth with checkerboard problems [15,16,17]. For further improved, a method considering density gradient information and elements numbers in the filter radius in topology optimization is proposed.

3.2 Improved Sensitivity Filtering Method

The proposed method considers density gradient message and elements numbers in the filter radius, the former focus on modifying the weighting function and the later focus on modifying the elements’ density within the filter radius. So, the new sensitivity can be presented as follow.

$$ \frac{{\widehat{\partial f}}}{{\partial x_{ij} }} = \frac{{\mathop \sum \nolimits_{i \in N} \hat{H}_{kj} \cdot H_{k} \cdot x_{kj} }}{{\left[ {\left( {x_{ij} } \right)^{\eta } + \left( {\mathop \sum \nolimits_{g = 1}^{4} x_{gj} } \right)/4} \right]}}\frac{\partial f}{{\partial x_{kj} }} $$
(12)

Where,

$$ \hat{H}_{kj} \, = \,\left\{ {\begin{array}{*{20}c} 1 & { \left| {x_{kj} - x_{ij} } \right|\, \le \,\alpha } \\ {0.001} & { \left| {x_{kj} - x_{ij} } \right|\, > \,\alpha } \\ \end{array} } \right. $$
(13)

The subscripts \( i,j,k \) are used to demonstrate the central element number, distinguish the design variable and imply the element number within the filter radius. As a new design variable with the whole part \( \left[ {\left( {x_{ij} } \right)^{\eta }\, + \,\left( {\sum\nolimits_{g = 1}^{4} {x_{gj} } } \right)/4} \right] \) in the denominator in Eq. (12), the consideration about the elements near the filtered element is more complete than original one. Apparently, this proposed method has two advantages. Initially, it identifies the elements near the boundary of the structure and modify the weight function with the threshold set to decrease the intermediate density of the structure boundary, at the same time, the weak effect appears. Next, it modifies the elements’ density within the filter radius by controlling the number of elements, thus, to enlarge the value of the non-boundary sensitivity. The parameters are set as \( \eta \, = \,0.5,\,\alpha \, = \,0.65 \), thus, to choose the average of the biggest four elements’ density as the element’s density. We choose the parameter \( \bar{x} \) to present the number of intermediate elements in the code programming, the smaller \( \bar{x} \), the smaller number of intermediate elements, the better filter effect.

$$ \bar{x}\, = \,\sum\nolimits_{i = 1}^{ne} {x_{i} \left( {1 - x_{i} } \right)} $$
(14)

4 Numerical Validity

With the same material properties in Table 1, two widely studied examples are presented by different load and support to demonstrate the effectiveness of the proposed method through working on the MATLAB 2017b. The design area is discrete into four-node rectangular plane element unit.

Table 1. Material properties of three-phase case
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4.1 Middle Force Case

In this case, the lower left corner of the structure is fixed, and the lower right corner is simply supported. The external force is loaded in the middle of the upper surface of the structure with the magnitude of 600 N (Fig. 3). In addition, the whole design area is meshed by 120 × 60.

Fig. 3.
figure 3

Design area (Middle Force)

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The objective function value (Compliance) decreases rapidly at the start and end all the iteration after 100 times, it tends to keep static after 12 steps with the magnitude of 9702.0651 (Fig. 4, Table 2), and end all the iteration within 100 times. From groups of optimization process (Fig. 5), with the void (blue part), the strong material is distributed in the outer of the structure to bear the force while the weak material is included by strong material. We can consider that there is a clear material boundary between two materials and the topologically outcome is feasible to manufacture.

Fig. 4.
figure 4

Variation of total compliance (Middle Force)

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Table 2. Compliance value (Middle Force)
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Fig. 5.
figure 5

The results of each iteration (Middle Force)

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4.2 Right Middle Force Case

In this example, the fixed area is in the left of the whole structure, and the external force is 600 N applied in the middle of the right boundary in vertical direction (Fig. 6). This design area is meshed by 120 × 60.

Fig. 6.
figure 6

Design area (Right Middle Force)

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From the variation of total compliance diagram in Fig. 7 and Table 3, the compliance tends to be stable after 14 iterations with the compliance value 52396.0337 and the whole iteration ends at 78. As these optimization results imply (Fig. 8), the strong material has been distributed in the key force path, otherwise the weak material distributed in the secondary force path. It can be observed that the material utilization has been greatly improved.

Fig. 7.
figure 7

Variation of the total compliance (Right Middle Force)

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Table 3. Compliance value (Right Middle Force)
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Fig. 8.
figure 8

The results of each iterations (Right Middle Force)

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4.3 Comparison with Original Sensitivity Filter

To make comparison with the original filtering method that is proposed by Sigmund [15], the outcome with different filtering method will be shown by two groups of pictures. Based on the consideration of the precision and computational cost, 60 iterations is set as the limitation to compare the effect of the different filtering technique. As shown in Fig. 9 and Fig. 10, the design outcome with original filtering method show weak connectivity between distinguished materials, and the element with intermediate density cannot be eliminated in the optimization process. The main reason is the improved filtering equation consider the number in the filtering domain and use the parameter \( \bar{x} \) to constrain the number of the intermediate element while the original one only considers the general element number in the filtering process. Furthermore, in Eq. (12), the density information of 4-node has been equalized calculated by introducing this term in denominator, which will affect the result of the filter outcome deeply.

Fig. 9.
figure 9

Compared outcome under middle force

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Fig. 10.
figure 10

Compared outcome under right middle force

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5 Conclusion

In this paper, an improved method is proposed to update the original sensitivity filtering function by considering the density gradient information and modifying the elements’ density within the filter radius. The proposed method was applied in the multi-material topology optimization problem in order to obtain a fully clear material boundary between two materials. Result shows that the clear and smooth structure boundary could be achieved, and the topological outcome converges within 100 times in both two case, which demonstrates that low-cost performance in topology optimization to make the material distribution in a proper way. Furthermore, it is concluded from the comparison that the original sensitivity is inadequate to solve multi-material topology optimization. These conclusions also prove that the use of materials can be more efficient by the multi-material topology optimization method with an improved sensitivity filtering.