Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.015694

ARTICLE

Bilateral Filter for the Optimization of Composite Structures

Yuhang Huo1, Ye Tian1, Shiming Pu1, Tielin Shi1 and Qi Xia1,2,*

1State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, China
2State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, China
*Corresponding Author: Qi Xia. Email: qxia@mail.hust.edu.cn
Received: 06 January 2021; Accepted: 08 March 2021

1  Introduction

Advanced manufacturing technologies of fiber-reinforced composite structures, for instance the automatic tape laying (ATL) and automatic fiber placement (AFP), allow composite structures to be manufactured with curvilinear fibers [1,2]. Therefore, stiffness can be different at different positions of the structure, and the freedom for improving the structural performance is larger than the constant stiffness composite structures [3–5]. However, the gap/overlap and excessive curvature of curvilinear fiber tows give rise to the appearance of manufacturing defects. The issue should be carefully dealt with at the design stage. When curvilinear fiber tows are not parallel [6–9], gaps and overlaps between adjacent fiber tows will appear. When the curvature of the fiber tow is too large, the tension and compression on the edges of the fiber tows will result in delamination and wrinkling [10,11].

How to avoid such defects has become an important topic in the design optimization of composite structures with curvilinear fibers, and many efforts have been made in recent years. Brampton et al. [8] employed the isolines of level set function to represent equally spaced fiber paths, hence preventing gaps/overlaps. Brooks et al. [11] treated fiber paths as the streamlines of a vector field, and gaps/overlaps and curvatures are respectively controlled by the constraints of the curl and divergence of this vector field. Hao et al. [12] proposed a multi-stage design strategy based on lamination parameters, in which the curvature and parallelism constraints were formulated as inequalities by using path functions. Hong et al. [13] developed an approach that controls the curvature of a fiber path through the gradient of lamination parameters. Tian et al. [14] proposed a parametric divergence-free vector field (pDVF) method for the optimization of fiber angle arrangement, and it ensures that fibers in one-ply do not cross each other.

In our previous study, within the Shepard-interpolation-based framework for design optimization, a gap/overlap constraint and a curvature constraint were proposed [15]. However, the two constraints should be defined at each design point, thus there are a large number of constraints, and the optimization is not efficient. In order to enhance the optimization efficiency, in [16] two filters were proposed to address the issue of gap/overlap and excessive curvature. At each design point, the sensitivity is first filtered in a rectangular region around the point, and by this means the fiber curvature is controlled. Then, in another rectangular region around the point, the filtered sensitivities are averaged to ensure fibers parallel to each other. Finally, the resulting sensitivity information is used to update the design variable.

In the present study, we propose to integrate the bilateral filter into the Shepard-interpolation-based fiber angle optimization (SFAO) [17,18]. According to the bilateral filter, a circular area is defined at each design point, and sensitivities at design points in the circular area are smoothed by both domain filtering and range filtering. The domain filtering is responsible for smoothing the magnitude of sensitivities, and the range filtering is responsible for adaptively adjusting the strength of smoothing according to the difference of fiber angles. The filtered sensitivities are used to update the design variables. As compared to the two-filter approach in our previous study [16], the bilateral filter approach is simpler and more convenient. In addition, the bilateral filter developed for image processing [19–21] has also been applied to the SIMP (Solid Isotropic Material with Penalization) [22,23] method for structural topology optimization, and it was proved to be effective to suppress the checkerboard pattern and simultaneously obtain a high-contrast black-white pattern of structure.

2  Optimization Problem

In this paper, the minimum compliance problem defined in Eq. (1) is considered

findθi(i=1,2,…,n)minc=FTUs.t.KU=Fθmin≤θi≤θmax(1)

where θi is the fiber angle at the i-th design point; n is the total number of design points; c is the objective function, i.e., the compliance of the structure; F is the global load vector; U is the global displacement vector; K is the global stiffness matrix; θmax and θmin are respectively the upper and lower bounds of the design variables, aiming to avoid the “π-ambiguity” issue of fiber angles [18].

Inspired by Kang et al. [24,25], the Shepard interpolation was proposed in our previous study to describe the fiber angles in the design domain. The fiber angles at finite element centers are computed by using a continuous function. This function is constructed by the Shepard method that interpolates the fiber angles at scattered design points, and it is given by [18]

Θ(x)=∑i∈Ixωi(x)θi(2)

where Θ(x) is the fiber angle at point x; Ix is the set of design points in the influence domain of point x; θi is the fiber angle at the design point at pi; ωi(x) is a weight function given by [26–28]

ωi(x)=∥x-pi∥-p∑i∈Ix∥x-pi∥-p(3)

where ∥x-pi∥ is the Euclidean distance between point x and design point pi; p is a positive parameter, and in the present study p = 2 because this makes ωi(x) infinitely differentiable [29]. This guarantees that the spatial variation of the fiber angle is continuous and smooth.

Another useful property of Shepard interpolation is expressed as

mini{θi}≤Θ(x)≤ maxi{θi}(4)

According to Eq. (4), when the fiber angle at any point in the design domain needs to be constrained as Θ(x)∈[θmin,θmax], this goal can be readily achieved by constraining the design variables θi as θi∈[θmin,θmax] .

The equilibrium equation KU=F is solved by the finite element method. The global stiffness matrix K is obtained by assembling the element stiffness matrix Ke given by

Ke=∫ΩeBTD(θe)BdΩ(5)

where B is the displacement strain matrix; D(θe) is the elastic matrix depending on the fiber angle (denoted as θe) in the e-th element, i.e.,

D(θe)=T(θe)D0T(θe)T(6)

where D0 is the elastic matrix when the fiber is not rotated, and T(θe) is the rotation matrix.

3  Sensitivity Analysis and Bilateral Filter

The sensitivity of the objective function with respect to design variables is given by [18]

∂c∂θi=-∑e=1NueT∂Ke∂θe∂θe∂θiue(7)

The derivative of the fiber angle θe with respect to design variable θi is obtained through Eq. (2) as

∂θe∂θi={ωi(xe),i∈Ixe0,i∉Ixe(8)

where xe is the coordinate of the centroid of the e-th element; Ixe is the set of design points in the influence domain of xe.

After the sensitivity of each design variable θi has been obtained according to Eq. (7), they are smoothed by the bilateral filter, as shown in Fig. 1. In order to improve the effects of filtering, the bilateral filter is usually applied to the sensitivities several times in each iteration of the optimization. The black dots represent the design points, and the yellow area is the bilateral filtering area. In this paper, the radius of the circular filtering area (denoted as rmin) is 10 times the grid size of design points to include more design variables.

The bilateral filtering of sensitivities is written as

∂c^∂θi=1H(pi)∑pj∈NiWc(∥pj-pi∥)Ws(|θj-θi|)∂c∂θj(9)

where Ni is the circular filtering area centered at the design point i; H(pi) is defined for normalization; Wc(∥pj-pi∥) is the domain filtering function; Ws(|θj-θi|) is the range filtering function. They are respectively given by [19,20]

H(pi)=∑pj∈NiWc(∥pj-pi∥)Ws(|θj-θi|)(10)

Wc(∥pj-pi∥)=e-12(∥pj-pi∥σd)2(11)

Ws(|θj-θi|)=e-12(|θj-θi|σr)2(12)

where σd and σr are the parameters of domain filtering and range filtering respectively. The two parameters directly affect the performance of bilateral filtering [19,21]. σd controls the strengthen of Gaussian filtering, and σr controls the discrimination ability of fiber angles.

Figure 1: Schematic diagram of the sensitivity bilateral filtering area (the black dots represent the design points)

4  Numerical Examples

In this section, the proposed optimization method is applied to several 2D structures subjected to in-plane loads. In these examples, the mechanical properties of the composite material are assumed as Ex = 1, Ey = 0.05, Gxy = 0.03, vxy = 0.3, vxy = 0.015. Plane-stress quadrilateral elements are used for the finite element analysis, and self-weight of the structure is not considered. The criterion of convergence is that the number of iterations is no more than 50. According to our experience, 50 iterations are enough for convergence. When the initial value of the fiber angle is set as 0∘, the upper and lower bounds of the design variables are set as θmin=-90∘ and θmax=90∘-ε; ε is set as a very small value to avoid “π-ambiguity” [18]. In this paper, ε is set as 1×10-8. When the initial value of the fiber angles is set as 90∘, the upper and lower limits of design variables are set as θmin=0∘ and θmax=180∘-ε.

The fiber angle distribution obtained by the optimization is post-processed by using the Tecplot software to generate fiber paths. In fluid dynamics, it is well known that the velocity of any point in a flow field is tangent to the streamline through the point. For the element e in the design domain, a vector at the element center is defined by

ve=(cosθe,sinθe)(13)

This vector is tangent to the fiber path through the element center. After importing the vector field constructed by Eq. (13), the Tecplot generates fiber paths.

4.1 Example 1

The first design problem is shown in Fig. 2. The size of the design domain is 1m×4m. The lower-left corner is fixed, and the lower right corner is fixed vertically. The center of the top edge is subjected to a downward concentrated load F of 1N. Because of the symmetry of the structure, only the right half is considered in the optimization.

Figure 2: Design problem of the first example

Firstly, 10×20 design points are evenly arranged in the design domain. The initial value of θi at each design point is set as 0∘, as shown in Fig. 3a. Then, the design domain is divided into 20×40 square elements. The initial fiber angles θe at the center points of all the elements are calculated by using the Shepard interpolation, as shown in Fig. 3b. The bilateral filter is applied to the sensitivities six times in each iteration of the optimization, and the parameters are set as σr=1 and σd=5. The results of optimization are shown in Fig. 4. The convergence history is shown in Fig. 5, and it can be seen that the optimization gradually converges after 20 steps.

Figure 3: The initial arrangement of fiber angles for the first example. (a) Initial fiber angles at design points, (b) initial fiber angles at center points of finite elements

Figure 4: The results of the first example with σr=1, σd=5 and six repetitions of bilateral filtering in each iteration, and the structural compliance is 178.14. (a) The optimized fiber angles, (b) the fiber paths obtained by Tecplot

Figure 5: The convergence history of the first example

Next, we investigate the influence of the number of bilateral filtering on the optimization results. We will also analyze the influence of domain filtering parameter σd and range filtering parameter σr on the optimization results.

When the number of bilateral filtering in each iteration is investigated, the parameters are set as σr=1 and σd=3. The optimization results with different numbers of repetitions of bilateral filtering in each iteration are summarized in Tab. 1. Without bilateral filtering, although the structural compliance can reach a smaller value of 72.39, severe overlaps of fibers can be found in the optimized structure, which does not meet the requirement of ATL or AFP manufacturing technology, and such results have no practical use. With the increase of the number of bilateral filtering in each iteration, the distribution of fiber paths becomes more uniform, since the fiber paths are almost parallel and the curvature of fiber paths are smaller. These results proved that bilateral filtering is effective.

Table 1: The optimization results with different numbers of repetitions of bilateral filtering in each iteration (with fixed parameters σr=1, σd=3)

Next, the influence of the domain filtering parameter σd and the range filtering parameter σr on the results are discussed. The number of repetitions of bilateral filtering in each iteration is set as six; the initial angle of the design point is set as 0∘; the results are shown in Tabs. 2 and 3. It can be seen from Tab. 2 that when σr is 0.1, the change of fiber angle is quite sharp; the fiber paths are not so smooth; the distance between fiber paths is not uniform. With the increase of σr, the spatial variation of the fiber angles becomes smoother. When σr increases from 3 to 5, the effect gradually weakens because the optimization results are almost the same. In addition, it can be seen from Tab. 3 that the larger value of σd leads to significant smoothing of fiber angles [19,21]. Therefore, when the bilateral filter is used to optimize fiber angle, the parameters σr and σd need to be properly set. According to our experience gained from numerical examples, we suggest that σr should be selected between 0.5~2 and σd should be selected between 3~5.

Table 2: The optimization results obtained with different σr

Table 3: The optimization results obtained with different σd

4.2 Example 2

The second design problem is shown in Fig. 6. The design domain is 1m×3m in size, fixed at the left edge. Also, it is subjected to in-plane load F of 1N, and evenly distributed along the boundary at the top right with a width of 0.5m. In the design domain, there are 10×30 design points uniformly arranged, and the initial values of all the θi are 0∘, as shown in Fig. 7a. The design domain is divided into 20×60 square elements. The initial fiber angle θe at the center points of all the elements are calculated by the Shepard interpolation, and they are shown in Fig. 7b. The bilateral filter is applied to the sensitivities six times in each iteration of optimization, and the parameters are set as σr=1 and σd=5. The results of the second example are shown in Fig. 8.

Figure 6: Design problem of the second example

Figure 7: The initial arrangement of fiber angles for the second example. (a) Initial fiber angles at design points, (b) initial fiber angles at center points of finite elements

Figure 8: The results of the second example with σr=1, σd=5 and six repetitions of bilateral filtering in each iteration, and the structural compliance is 195.63. (a) The optimized fiber angles, (b) the fiber paths obtained by Tecplot

It can be seen from Fig. 8b that the fiber paths are almost parallel to each other, which means that there exists no gap or overlap between adjacent fiber tows. In addition, the fiber paths are fairly smooth, which means that their curvatures are not large. At the same time, the optimization results also show that the suggested values of the parameters for the bilateral filter in the first example are reasonable.

Figure 9: Design problem of the third example

4.3 Example 3

The third design problem is shown in Fig. 9. The design domain is 1m×4m in size. The lower-left corner is fixed and the lower right corner is fixed vertically. The concentrated load F of 1N is applied at the middle point of the bottom edge. There are 10×40 design points uniformly arranged in the design domain, and the initial value of θi at all the design points are 0∘, as shown in Fig. 10a. The design domain is divided into 20×80 square elements. The initial fiber angle θe at the center points of all the elements are shown in Fig. 7b. The bilateral filter is applied to the sensitivities four times in each iteration, and the parameters are set as σr=1 and σd=5. The results are shown in Fig. 11. Through this example, the effectiveness of bilateral filtering in fiber angle optimization is proved again. As can be seen in Fig. 11b, the fiber paths are parallel, equidistant, and without large curvature.

Figure 10: The initial arrangement of fiber angles for the third example. (a) Initial fiber angles at design points, (b) initial fiber angles at center points of finite elements

Figure 11: The optimization results of the third example with σr=1, σd=5 and four repetitions of bilateral filtering in each iteration, and the structural compliance is 82.05. (a) The optimized fiber angles, (b) the fiber paths obtained by Tecplot

5  Conclusions

In this paper, the bilateral filter was integrated into the Shepard-interpolation-based method for the optimization of composite structures. According to the bilateral filter, sensitivities at design points in the filter area are smoothed by both domain filtering and range filtering. Then, the filtered sensitivities are used to update the design variables. Through several numerical examples, it was found out that the bilateral filter is useful to avoid gap/overlap between adjacent fiber tows or excessive curvature of fiber tows.

Funding Statement: This research work was supported by the National Natural Science Foundation of China (Grant No. 51975227) and the Natural Science Foundation for Distinguished Young Scholars of Hubei Province, China (Grant No. 2017CFA044).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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