1. Introduction

2. B-Splines and NURBS

3. Multi-Patch WSIGABEM

3.1. Weakly Singular Boundary Integral and Discretization

In a physical domain Ω with boundary Γ, the boundary integral equation of linear elastostatics in the absence of body forces is expressed as

$$C_{ij}\left(\xi \right)u_j\left(\xi \right)+{\int }_{\Gamma }{t}_{ij}^{*}(\xi ,x)u_j\left(x\right)d\Gamma \left(x\right)={\int }_{\Gamma }{u}_{ij}^{*}(\xi ,x)t_j\left(x\right)d\Gamma \left(x\right)$$

(9)

where the integral on the left side represents a Cauchy principal value integral, C_ij represents the jump term, and $t_{ij}^{*}(\xi ,\mathit{x})$ and $u_{ij}^{*}(\xi ,\mathit{x})$ are respectively the plane-strain fundamental solution kernels for 2D problems, i.e.,

$$t_{km}^{\ast }(\mathit{x},\xi )=-\frac{1}{4\pi G(1-\nu )r}\{\frac{\partial r}{\partial n}[(1-2\upsilon ){\delta }_{km}+2r_{,k}{r}_{,m}]-(1-2\upsilon )(r_{,k}{n}_{,m}-r_{,m}{n}_{,k})\}$$

(10)

$$u_{km}^{\ast }(\mathit{x},\xi )=\frac{1}{8\pi G(1-\nu )}[(3-4\upsilon )\ln \frac{1}{r}{\delta }_{km}+r_{,k}{r}_{,m}]$$

(11)

The integrals involving the kernel $t_{km}^{\ast }(\mathit{x},\xi )$ are usually strongly singular, and in conventional IGABEM it is usual to separate the jump term from the integral and produce a Cauchy principal-value integral [61], which is a rather complicated process.

In this paper, through the equation proposed by Liu [62,63], the jump term is transformed into the form of an integral, which has the following integral properties when the source point is at the boundary of the geometric model

$$C_{ij}(\xi )=-{\displaystyle {\int }_{\Gamma }{t}_{ij}^{*}(\xi ,\mathit{x})d\Gamma (\mathit{x})}$$

(12)

Applying the above equation in Equation (9), the following weakly-singular form of the BIE is obtained:

$${\displaystyle {\int }_{\Gamma }{t}_{ij}^{*}(\xi ,\mathit{x})[u_j(\mathit{x})-u_j(\xi )]d\Gamma (\mathit{x})}={\displaystyle {\int }_{\Gamma }{u}^{*}(\xi ,\mathit{x})t_j(\mathit{x})d\Gamma (\mathit{x})}$$

(13)

In Equation (13), both the integrals are weakly singular.

In the IGABEM, the shape functions, which are usually taken as general polynomials in the traditional BEM, are replaced with NURBS basis functions, thus maintaining all the geometric information of the original geometry model. And relationship $\mathit{x}=\mathit{x}(\xi )$ implies that in the parameter coordinates $\xi$, the displacements and tractions in Equation (13) are approximated by

$$u_j(\xi )={\displaystyle \sum _{l=1}^{p+1}{R}_{i,p}^{le}(\xi )d_j^{le}}$$

(14)

$$t_j(\xi )={\displaystyle \sum _{l=1}^{p+1}{R}_{i,p}^{le}(\xi )q_j^{le}}$$

(15)

where $d_j^{le}$ and $q_j^{le}$,denoting the displacement coefficients and traction coefficients of the lth control point of the eth element, are associated with a particular control point. Due to the localized support of the basis functions, only p+1 basis functions are non-zero for each element.

In this paper, the boundary of the geometric model is divided into multiple independent NURBS curves with the end points of the C⁰ continuity. The boundary is discretized into a set of non-overlapping N_h patches

$$\Gamma =\underset{e=1}{\overset{N_h}{\cup }}{\Gamma }_h,{\Gamma }_i\cap {\Gamma }_j=0,i\ne j.$$

(16)

Each patch N_h is then discretized separately into a set of non-overlapping N_eh elements

$${\Gamma }_h=\underset{e=1}{\overset{N_{eh}}{\cup }}{\Gamma }_{eh},{\Gamma }_i\cap {\Gamma }_j=0,i\ne j.$$

(17)

The discretized boundary integral equation is then rewritten as where ·^leh is the quantity on the lth control point of the eth element of the hth patch, and $\stackrel{-}{e}$ is the element in which the collocation point locates, and the Jacobian of transformation $J^{eh}(\stackrel{ʌ}{\xi })$ is given by [27] where ξ₂ and ξ₁ represent the upper and lower bounds of the element in the parameter space.

To to obtain the linear system, a series of source points (collocation points) are selected on the boundary

$$\mathit{Hu}=\mathit{Gt}$$

(20)

where H is the coefficient matrix of the displacement vector u and G is the coefficient matrix of the traction vector t. Rearranging this set of equations by placing all unknown components on the left-hand side and known components on the right, the following can be written [27]:

$$\mathit{Ax}=\mathit{b}$$

(21)

where b is the vector containing the unknowns d and q. Mallardo and Ruocco [31] give a general method for the imposition of boundary conditions. In the present paper, the approach in the work by Simpson et al. [27,28] is used for the imposition of boundary conditions for the simplicity.

3.2. Collocation Method

To obtain a system of equations in IGABEM, the collocation points must be given. In this paper, a new collocation method with no correction coefficient is proposed to move the points inside the patches. The collocation points are evaluated by the following formula

The number of total collocation points is usually not equal to the number of total control points when this method is used because some control points may be shared by multiple patches. Since each collocation point only belongs to one patch, and a simple method for merging equations handles the extra equations [32]

where where A_ip and A_jp represent the initial assembly matrix of two collocation points associated with corner points, which are directly added and combined.

4. Optimization Algorithm

4.1. PSO Algorithm

PSO is an optimization technique proposed by Kennedy et al. [64] based on the study of bird flock foraging behavior. Individuals of a flock of birds search for the optimal destination while at the same time making the group find the collective optimal destination by continuously sharing information.

As in Table 1, the flock foraging behavior and the algorithm principle are corresponded. In the PSO algorithm, the initial value is a set of particles randomly distributed within the D-dimensional solution space, each particle corresponds to an objective function value, and the position of the ith particle at the kth iteration is:

The position update vector of a particle is called velocity, and each particle has its own position and velocity. The individual historical optimum in an iterative step is called the individual extreme value (x_pbest), and the group optimum in this step is called the group extreme value (x_gbest). In PSO algorithm, each particle is regarded as a potential solution in the problem search space, and these particles move iteratively in the solution space and adjust their flight paths by tracking their own individual optimal positions as well as the global optimal position of the whole group. The particle position and velocity iteration formulas are [65] where k is the number of iterations; w is the inertia weight, which representing the dependence of the particle on its previous motion state; c₁ and c₂ are learning factors, which respectively adjust the maximum step towards to the individual best particle and the global best particle; r₁^k and r₂^k are two random numbers between 0 and 1.

A fitness function is a function, incorporating the objective function and constraints to assess the quality of solutions, defined as follow [59]

where f(x) is the objective function, g_j and h_i, respectively represent the inequality constraints and equality constraints, and ξ and ξ’ are the penalty factors for the constraints. In this study, the equality constraints are not considered, i.e., h_i = 0. The fitness functions are given in every optimization design example.

5. Numerical Examples of Multi-Patch WSIGABEM

[27]:

6. Optimization Design Based on the Multi-Patch WSIGABEM

In this section, three optimal designs of the shapes of the fillet corner, spanner and arch bridge based on the multi-patch WSIGABEM and PSO are present for the verification of the accuracy and effectiveness. For the simplicity, the materials in the examples are assumed to be the same with Young’s modulus E = 10⁷, and Poison’s ratio v = 0.3, while the yield strength different. The number of Gaussian’s points is 4. For the better accuracy, the order of the curves is p =3, and the mesh is refined to be mesh7 following the same procedure in Figure 2 and Figure 7. The von Mises stresses on the boundaries are calculated by the stress recover method explained in Appendix A. The learning factors c1 and c2 are 1.5, the maximum flight speed v is 1/5 of the search space, and inertia weights in PSO method is improved by: where the maximum weight w_max and the minimum weight W_min are respectively set to be 0.6 and 0.2 in three examples, i is the step of the current iteration, and i_max is the maximum number of iterations, the weights w are adjusted by using the adaptive linear method, which enables the algorithm to have a strong global search ability in the early stage, and a better local convergence speed in the later stage. These parameters are chosen to ensure the convergence and efficiency of the optimization algorithm. The accuracy of the algorithm is verified by the comparison of the optimal shape of the fillet corner by the present method and by the existed method [57,59]. Then the shape optimizations of the spanner and the arch bridge are investigated. The coordinates and weights of the control points of the shapes before the optimizations are given in Appendix B.

6.1. The Shape Optimization of the Fillet Corner

As in Figure 11a, a normal traction t = 100 is prescribed at the blue line, resulting a stress concentration at point N. The shape optimization of the fillet corner is conducted to find the minimum area while the von Mises stress is no greater than the yield strength 125. The symmetric condition is prescribed on the left and bottom sides, denoted as the red, since the shape in Figure 11 is only one-quarter of the whole fillet. The design parameters are chosen to be the coordinates and weights of the control point A and B, denoted as red points in the figure. The shape of the corner changes when the design parameters change. During the optimization the point A is assumed to be at the left top of point B.

The initial values of the design parameters are randomly distributed in the ranges listed in Table 4. The fitness function is defined as follows: where f(x) is the objective function, representing the area of the part, vonMises_max represents the maximum von Mises stress value on the boundary, and ξ is the penalty factor, which is chosen to be 1 after some trials. At the initial stage, there 20 sets of design parameters and the maximum number of iterations is 20. After the optimization procedure, the shape of the fillet is shown in Figure 11b. The area of the fillet with the minimum value of the fitness function on every iteration step is shown in Figure 12. The last column of Table 4 gives the values of the design parameters after the shape optimization. To speed up the iteration process, one set of the design parameters at the initial stage is chosen to be the same with the Figure 11a, i.e., A(11.17, 7.5), B(13.33, 6) and weights w_A = w_B = 1.

Figure 13a,b show von Mises stresses on the boundary before and after the optimization, respectively, with the first and last knot values of the closed curves corresponding to the coordinates starting from the point (0,0) counterclockwise to the end of the point (0,0). The similar shape optimization was conducted by Lian et al. [57] and Sun et al. [59], in which only the vertical coordinates of the control points served as the design parameters. The optimal area predicted by the former works was about 138, while in the present design all the coordinates and weights of the control points are served as the design parameters. The optimal area by the present method is 132.46, which is less than 138. Due to the PSO algorithm, the complicated analysis of sensitivity is not necessary, making the realization process simpler and easier.

6.2. The Shape Optimization of the SPANNER

Figure 14 shows the optimal design of the shape of the spanner. The boundaries denoted by the red lines are fixed and the shear traction t=10 is prescribed at the blue line. The vertical coordinates and the weights of the red points are the design parameters to be optimized. Since that the control points are symmetrically distributed, only the top 4 control points are set as the design parameters. Besides, for the simplicity, the weights of points A, D keep the same during the optimization. In summary, there are 6 design parameters, which are the vertical coordinates y₁, y₂, y₃, y₄ of points A, B, C, and D, and the weights w₁, w₂ of points B and C. To speed up the iteration process, one set of the design parameters at the initial stage is chosen to be A(-2,1.5), B(-1,3), C(1,1), B(5,1) and weights w₁ = w₂ = 2.

The shape optimization of the spanner is conducted to find the minimum area while the von Mises stress is no greater than the yield strength 825, and the fitness function is defined as follows

where f(x) is the area of the spanner and the penalty factor ξ = 10. At the initial stage, there 20 sets of design parameters. The maximum number of iterations is set to be 20. After the optimization procedure, the shape of the spanner is shown in Figure 14b, while the trajectory of the area in the iteration procedure is shown in Figure 15. The range of design parameters and the optimized design parameter values are shown in Table 5. The area of the spanner is reduced from 68.43 to 25.96, which is only 37.93% of the original shape.

6.3. The Shape Optimization of the Arch Bridge

The shape optimization of the arch bridge is shown in Figure 16. The fixed constraint is applied at the red line and the normal traction t=10 is prescribed at the blue line, which is at the center of the bridge. The design parameters are related to the red points. Due to the symmetry, only points A and B in the figure are optimized. Since the weight of point B is the main factor affecting the shape of the boundary, the weight of point A remains the same during the optimization. The design parameters are the horizontal coordinates of point A, and the coordinates and weight of point B. To speed up the iteration process, one set of the design parameters at the initial stage is chosen to be A(4,0), B(4,1) and weight w_B =1/$\sqrt{2}$.

The shape optimization of the Arch bridge is conducted to find the minimum area while the von Mises stress is no greater than the yield strength 50, and the fitness function is defined as follows

In this example, the penalty factor ξ = 1, the number of particles is taken as 20, and the maximum number of iterations is 20. After the optimization procedure, the shape of the arch bridge is shown in Figure 16b, while the trajectory of the area iteration process is shown in Figure 17. The range of design parameters and the values of the design parameters before and after the optimization are shown in Table 6. The area is reduced from 20.78 to 14.73, which is only 70.88% of the original.

7. Conclusions

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