A partition of unity level set method with moving knot CS-RBFs for optimizing variable stiffness composites

Gen LI, Ye TIAN, Kang YANG, Tielin SHI, Qi XIA

State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology,Wuhan 430074, China

KEYWORDSComposite structures;Moving knots;Partition of unity;Radial basis functions;Structural optimization

AbstractA partition of unity level set method with moving knot Compactly Supported Radial Basis Functions (CS-RBFs) is proposed for optimizing variable stiffness composite structures.The iso-contours of a level set function are utilized to represent the curved fiber paths,and the tangent vector of the iso-contour defines the orientation of fiber.The level set function of the full design domain is constructed according to the Partition of Unity (POU)method by a set of local level set functions defined on an array of overlapping subdomains, and they are constructed by using the CS-RBFs.The positions of knots are iteratively changed during the optimization to improve the performance of composite structures.Several examples of compliance minimization are presented.?2023 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

By using advanced manufacturing techniques, fiber-reinforced composite structures can be produced with curvilinear fiber tows,1,2which yields the so-called variable stiffness composite structures.Such structures have larger potential for improving the mechanical performance as compared to the ones with unidirectional fibers.3–5To fully explore the potential,many optimization methods were proposed, including the method based on fiber angles defined at element centers or design points,6–10the method based on fiber paths defined by analytical functions,11–13the method based on lamination parameters.14,15Although these methods are successful in some design optimization problems, there still exist several difficulties about the enforcement of manufacturing constraints,the dependence of priori knowledge about type of analytical function,the feasible domain of lamination parameters, or requirement of postprocessing to extract fiber paths.

Another successful method for optimizing composite structures with curvilinear fibers is to use the iso-contours of the level set function to describe the curved fibers.16–19The tangent vector of the iso-contour defines the orientation of fiber.Similar parameterization using the streamline function was also adopted in stiffener optimization.20–22The level set method naturally avoids the crossing of fibers and does not need post-processing to extract fiber paths.However, sensitivity analysis is often complicated.16,20–22

In our previous study,23a parametric level set method was proposed for optimizing composite structure with curved fibers.In this method, the level set function is constructed as the sum of a set of Compactly Supported Radial Basis Functions (CS-RBFs), and the coefficients of CS-RBFs are taken as the design variables.Besides the merits inherited from the previous level set method, the parametric level set method has additional benefit,i.e.,the sensitivity analysis for optimization is easy.

In the present study,we observed that when the positions of knots of CS-RBFs are changed, the iso-contours of level set function, i.e., the fiber paths, change accordingly.Therefore,the positions of knots of CS-RBFs are taken as the design variables in the design optimization,i.e.,the positions of knots are changed iteratively to improve the performance of composite structures.The idea of moving knot RBF was first proposed in Ref.24 where the evolution of structure boundary is linked to the time-dependent dynamic knots of RBFs,which is different from the present study.

In addition, in order to enhance the computational efficiency of the level set function, the Partition of Unity (POU)method is combined with the CS-RBF level set.The level set function of the full design domain is constructed according to POU by using a set of local level set functions defined on an array of overlapping subdomains,and they are constructed by using the CS-RBFs.24–29Generally,the techniques used for fitting and evaluation have important effects on the computational efficiency of the level set function, especially when the number of RBFs is large.Among many approaches for the enhancement of computational efficiency,30–36the POU method is attractive, since it is effective and easy to implement.24–29

2.Fiber angle and parametric level set function based on CSRBFs

where n is the quantity of CS-RBFs;αiis the expansion coefficient related to the i-th CS-RBF; φiis the i-th CS-RBF.The CS-RBF with C2continuity is used and given by37–40

where dsis a parameter that determines the support size of the CS-RBFs,ds>0;piis the knot related to the i-th CS-RBF;τ is a minimal normal number employed for avoiding division by zero, τ = 0.0001.

Now,the fiber angle can be computed according to Eq.(2),and we have

The positions of knots piof the CS-RBFs are also considered as the design variables and changed to optimized the composite structure.Therefore, we call it moving knot CS-RBF.The expansion coefficients αiremain unchanged throughout the optimization, and they are obtained before the optimization by solving an interpolation problem to enforce an initial fiber angle arrangement specified by the designer.

3.Level set function with POU

The parameterized level set method has many advantages.However, with the increase of the number of RBF knots, the size of the coefficient matrix of RBFs interpolation will increase and result in an increase in the computation time.The POU method has been proved to be an effective method to reduce the computation time, and is now widely used.41,42Therefore, in order to enhance the efficiency of computation,the POU method is applied to compute the level set function.Now, on the whole design domain D, the level set function Φ～consists of a group of local level set functions Φ～i(i=1,2,...,m)defined on an array of overlapping subdomains Di(i=1,2,...,m),24–29i.e.,

where niis the quantity of CS-RBFs in the i-th subdomain;φijis the j-th CS-RBF in Di.By combining Eqs.(10)and(9),Φ～is rewritten as

where Srand Trare the coordinates of the two corners S and T.In addition, the decay functions V with C2continuity is used:24–28

where d is the independent variable of the function.In the calculation, d will be replaced by P given in Eq.(14).

Now, we can compute the fiber angle by using Eq.(2).According to Eq.(11), we have24

According to Eq.(13), ?Wi/?x and ?Wi/?y are obtained by chain rule as

4.Finite element analysis of composite structures

The composite structure is discretized into rectangle finite elements.It is assumed that each element has a constant fiber angle, denoted as θe, and the angle is evaluated at the center of the element according to Eq.(2).The four corners of the rectangle element are also used as the knots of CS-RBFs.In the finite element analysis, Eq.(20) is solved.

where u is the global displacement vector to be solved; f is the external load;K is the global stiffness matrix,which can be calculated by assembling the element stiffness matrix Kewritten as

where Ωeis the area covered by the e-th element; B is the displacement–strain matrix;dΩ is the differential element of area;D(θe) is the elastic matrix related to the e-th element, which is rewritten as

where D0is the elastic matrix; T(θe) is the rotation matrix related to the e-th element, which is defined as

5.Design optimization problem

The compliance of composite structure is minimized by changing the positions of knots of the CS-RBFs.In other words,the compliance is taken as the objective function of a minimization problem,and the positions of knots of the CS-RBFs are taken as the design variables.The optimization problem can be represented by

where c is the compliance of composite structure;dpnis a constraint aboutζ is the upper bound of the constraint;N is the admissible region of moving knots piof CS-RBFs,which is a rectangle.In the design optimization problem, the positions of knots piof CS-RBFs are adopted as the design variables,i.e., the positions of knots are changed iteratively to minimize the compliance of composite structures.

The gradient reflects the change rate of the level set function value at a point on the surface.When the gradient norm at an arbitrary point is equal to 1, the iso-contours of the level set function are parallel and equally spaced.A simple example of such a level set function with=1 is shown in Fig.1.

In our previous study,23an aggregated constraint about the norm of gradient vector of the level set function was formulated to avoid overlaps and gaps between neighboring fiber tows.Such a constraint, i.e., dpn≤ζ, is also used in the optimization, and dpnis defined as23

where p is a positive parameter;E is the total quantity of finite elements in the global design domain; deis defined as18,23,39

where xeis the coordinates of the e-th element center; ?Φ～is defined as

Fig.1 An example of level set function whose norm of gradient vectors is equal to 1 almost everywhere.

6.Sensitivity analysis

The derivative of c with respect to the x-coordinate xijrelated to the CS-RBF knot pijis given by (the derivative of c with respect to the y-coordinate yijof pijcan be obtained similarly)

where texand teyare the components of the tangent vector at the center of the e-th element in the x and y directions.

When the POU is used, we have

7.Numerical examples

In all the numerical examples,mechanical properties related to composites are set as Ex= 1, Ey= 0.05, Gxy= 0.03,νxy= 0.3, νyx= 0.015, where Exis the tensile modulus along the fiber direction (i.e., x-direction) of the assumed composite material; Eyis the tensile modulus along the y-direction perpendicular to the x-direction; Gxyis the in-plane shear modulus; νxyand νyxare the Poisson’s ratios.The rest of the conditions and parameters are as follows:the plane stress state is assumed;the thickness of the structure is set to the constant of 1;self-weight of structure is not considered.The appropriate value of the parameter dsshould be chosen to ensure the nonsingularity of the CS-RBF interpolation and the computational efficiency.38Generally, a too small value of dswill lead to the numerical instability, and a too large value of dswill increase the computation efforts.43,44In our previous work,we found that it was appropriate for the value of dsto be set as 1.23The rectangular admissible regions N(pij) around the knots pijof CS-RBFs are all specified by a rectangle with lower-left corner and upper-right corner whose coordinates are respectively (–50, –50) and (50, 50).The knots of CSRBFs are evenly distributed in the overall design domain.In Method of Moving Asymptotes (MMA) algorithm,45,46the‘‘move”parameter is defined as 1 × 10–7.The optimization problem is considered as having been converged if the condition of Eq.(42) is meet.

where cerris the error of the compliance;c is the compliance;δcis the upper bound set as 0.1%.Furthermore,the optimization solution process will stop when the number of the iterations reaches 1000.

The optimization steps and processes are described as follows:

Step 1 Define the global design domain, and divide it into an array of overlapping patches.Set the load and boundary conditions.

Step 2 Specify the initial fiber angles; compute the expansion coefficients of RBFs by solving an interpolation problem that enforces the specified initial fiber angles.

Step 3 Perform the finite element calculations; compute the compliance and the partial derivative of compliance and constraint about the design variables.

Step 4 Update the positions of the RBF knots by using the MMA optimization algorithm.

Step 5 Check the converge of the optimization process.If it does not convergence,repeat Steps 3–5 until the convergence are met.If yes, stop the optimization and output the fiber paths.

7.1.Example 1

The first numerical example about optimization problem is illustrated in Fig.2.The size of global design domain is a 2 × 1 rectangle.An in-plane concentrated load F=1 is imposed vertically and downwards at the midpoint on the right side.A mesh with 24 × 28 square elements is applied in the Finite Element Analyses (FEA), and the number of knots of CS-RBFs is 25 × 49.The initial fiber paths are vertical in Fig.3(a).The upper bound ζ is set to 0.01.

Fig.2 Design problem of the first example.

Fig.3 Initial and optimized fiber paths.

In Example 1,several optimizations are conducted with different number of knots in the overlapping region of neighboring patches, and the quantity of patches in the vertical and horizontal directions are respectively set to 2 and 4.By this means, the influences of the quantity of overlapping knots on the optimization are investigated.The results of optimizations are displayed in Fig.3.As shown in Figs.3(b)–(d), the compliances of structures optimized with POU are smaller,and the fiber paths of structures optimized with POU are more curved.Fig.4 shows the convergence histories with different numbers of overlapping knots.

Table 1 shows the comparison of the results of optimizations.As one can see in Table 1, the compliance of the optimized structure decreases, i.e., the structure is stiffer, as the number of overlapping knots increases.Such a result shows that the POU is effective for obtaining a better result.In addition,when the number of overlapping knots is 5 ×5,the time cost by each iteration is smaller than a half of that without POU;the number of iterations and the compliance of the optimized structure are nearly the same as those optimized without POU.However, with the increase of overlapping knots, the computation time of every iteration also increases.The reason is explained as follows.Due to the increase of the number of nodes in each local domain, the size of coefficient matrix for RBF interpolation increases.Therefore, the matrix–vector multiplication required to compute the local level set functions needs more time, and also the sensitivity analysis needs more time.In addition, when the number of overlapping knots increases, the quantity of iterations becomes bigger than that without POU, and the total time cost by the optimization increases.We think that the increased computation time is the price that needs to be paid to obtain a better result of optimization (recall that the optimized structure is stiffer as the number of overlapping knots increases).

Fig.4 Convergence conditions with different number of overlapping knots.

7.2.Example 2

The second numerical example about optimization problem is illustrated in Fig.5.The size of global design domain is a 3×1 rectangle.A mesh with 24 × 72 square elements is applied in the FEA, and the number of knots of CS-RBFs is 25 × 73.The upper bound ζ is set to 0.05.The initial fiber paths are horizontal in Fig.6(a).

In Example 2,several optimizations are conducted with different number of patches,and the quantity patches in the horizontal and vertical directions are set to 2 and 4, respectively.By this means,the effects of the number of patches on the optimization are investigated.The optimized fiber paths are displayed in Figs.6(c)and(d).Convergence histories of the optimizations are given in Fig.7.

Fig.5 Design problem of the second example.

Fig.6 Initial and optimized fiber paths.

Fig.7 Convergence histories with different number of patches.

Table 1 Results of optimizations conducted with different numbers of knots in the overlapping regions of neighbouring sub-domains.

Table 2 Results of optimizations conducted with different numbers of patches.

Table 2 shows the comparison of the results of optimizations.As one can see in Table 2,the time required of every iteration decreases when the quantity of patches increases, which implies that the amount of the patches has an important effect on the computational time.However, with the increase of the number of patches, the compliance of the optimized structure increases gradually.

8.Conclusions

This paper proposed a partition of unity level set method with moving knot CS-RBFs for optimizing composite structures.The iso-contours of a level set function are utilized to describe the curvilinear fiber paths.The orientations of fibers are defined by using the orientations of the tangent vectors of the iso-contours.The level set function is constructed by using POU and CS-RBFs.The positions of knots of the CS-RBFs are taken as the design variables.Several examples verified the effectiveness of the proposed method.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research work was supported by the National Natural Science Foundation of China (No.51975227).The authors also gratefully thank Krister Svanberg for providing the MMA codes.