Improved proportional topology optimization algorithm for solving minimum compliance problem

Abstract

The paper proposes four improved proportional topology optimization (IPTO) algorithms which are called IPTO_A, IPTO_B, IPTO_C, and IPTO_D, respectively. The purposes of this work are to solve the minimum compliance optimization problem, avoid the problems of numerical derivation and sensitivity calculation involved in the process of obtaining sensitivity information, and overcome the deficiencies in the original proportional topology optimization (PTO) algorithm. Inspired by the PTO algorithm and ant colony algorithm, combining the advantages of the filtering techniques, the new algorithms are designed by using the compliance proportion filter and the new density variable increment update scheme and modifying the update way of the density variable in the inner and main loops. To verify the effectiveness of the new algorithms, the minimum compliance optimization problem for the MBB beam is introduced and used here. The results show that the new algorithms (IPTO_A, IPTO_B, IPTO_C, and IPTO_D) have some advantages in terms of certain performance aspects and that IPTO_A is the best among the new algorithms in terms of overall performance. Furthermore, compared with PTO and Top88, IPTO_A has some advantages such as improving the objective value and the convergence speed, obtaining the optimized structure without redundancy, and suppressing gray-scale elements. Besides, IPTO_A also possesses the advantage of strong robustness over PTO.

Appendices

Appendix A

Appendix B

In this section, we investigate the superiority of the new algorithms in avoiding the problems of numerical derivation and sensitivity calculation involved in the process of obtaining sensitivity information by solving the stress-constrained optimization problem.

As we all know, when employing the sensitivity-based optimization method to solve the stress-constrained optimization problem, the sensitivity information of the objective function and the stress-constrained function is required by the derivation method. However, it is difficult to obtain more rigorous sensitivity information without simplification of the derivation for stress problems in practical application duo to the characteristics of high non-linearity and the local nature of stress constraints (Le et al. 2010; Biyikli and To 2015; Verbart et al. 2016). Additionally, it is undeniable that obtaining sensitivity information of the stress-constrained optimization problem inevitably brings additional computational burden. To solve these problems, relevant scholars have proposed a variety of approaches such as the P-norm approach (Duysinx and Bendsøe 1998), the KS function approach (Yang and Chen 1996), and an enhanced aggregation approach (Luo et al. 2013). In fact, no matter what approaches are adopted in the sensitivity-based optimization method, the problems (where the sensitivity derivation is difficult and additional computation is required) cannot be gotten rid of in an ideal way. However, there is no need to consider the problems related to obtaining sensitivity information when using the new algorithms we proposed to solve the stress-constrained optimization problem. Taking the solution of the minimum volume optimization problem with stress constraints as an example, we will give a detailed explanation for the proposed new algorithms.

When the stress constraints are given, the topology optimization problem of minimizing the structural volume (Biyikli and To 2015) can be described as follows:

$$ \left\{\begin{array}{c}\min\ V\left(\mathbf{x}\right)=\sum \limits_{i=1}^N{x}_i{v}_i\kern3em \\ {}s.t\left\{\begin{array}{c}\mathbf{KU}=\mathbf{F}\kern5.75em \\ {}{\sigma}_i\le {\sigma}_{lim},\kern0.5em \mathrm{if}\ x>0\kern2em \\ {}0\le {x}_{\mathrm{min}}\le {x}_i\le {x}_{\mathrm{max}}\le 1\kern0.5em \end{array}\right.\end{array}\right. $$

(19)

where σ_i denotes the elemental stress measure, which is regarded as the von Mises at the center of element i, and σ_lim denotes the stress limit (e.g., material elastic limit) which are utilized to avoid structural failure by limiting the stress of any element in the system.

When solving the minimum volume optimization problem with stress constraints, we need to replace the functions (5–7) (in the density variable update of the new algorithms in the “Density variable update in the inner loop” section) with the new functions (20–22). Note that the difference between the two function sets is the interchange of intermediate variables related to the compliance C and the stress σ.

$$ \varDelta {x}_i^{out}={V}_{RM}\cdot {\tilde{x}}_i\cdot \varDelta {\tilde{\sigma}}_i $$

(20)

$$ \varDelta {\tilde{\sigma}}_i=\frac{1}{\sum \limits_{j\in {N}_i}{H}_{i,j}}\sum \limits_{j\in {N}_i}{H}_{i,j}\varDelta {\sigma}_j^q $$

(21)

$$ \varDelta {\sigma}_i=\frac{\sigma_i^{\lambda }}{\sum_{j=1}^N{\sigma}_j^{\lambda }{v}_j} $$

(22)

where Δσ_i and $\varDelta {\tilde{\sigma}}_i$ denote respectively the elemental stress proportion and the elemental filtered stress proportion, λ is the stress influence coefficient, and q is the stress proportion influence coefficient.

Combining the functions (2–4) and (8–15) in the “Improved proportional topology optimization algorithm” section on the basis of the new functions (19–22), the improved proportional topology optimization algorithms for solving the stress-constrained optimization problem can be constructed. Here, four improved proportional topology optimization algorithms are represented respectively by IPTOs_A, IPTOs_B, IPTOs_C, and IPTOs_D. Moreover, the algorithm (formed on the basis of the PTO algorithm) for solving the stress-constrained problem is represented by PTOs. To validate the effectiveness of the new algorithms, the MBB beam shown in Fig.1b in the “Numerical example and discussion” section is used here. Except that the maximum stress limitation σ_lim and the stress proportion influence coefficient q are respectively set to 1.08 and 1, the other parameters involved in solving the stress-constrained optimization problem for the MBB beam are consistent with the parameter settings in the “Numerical example and discussion” section. The optimized results of the MBB beam attained by each algorithm considering the parameter settings (λ = 0.75 and α takes value from 0.1 to 0.5 in increments of 0.1) are listed in Tables 5 and 6.

Examining Tables 5 and 6, it is found that the new algorithms (IPTOs_A, IPTOs_B, IPTOs_C, and IPTOs_D) can solve the minimum volume optimization problem with stress constraints and provide designers with an effective optimized structure of the MBB beam, and that the new algorithms have some advantages over PTOs in terms of certain performance aspects. For instance, IPTOs_A and IPTOs_C are obviously superior to PTOs in terms of improving the convergence speed of the algorithm and obtaining the optimized structure without redundancy. IPTOs_B has some advantages over PTOs in terms of improving the objective value and obtaining optimized structure without redundancy. In addition, comparing the optimized results obtained by PTOs, the advantages of IPTOs_D in some aspects have a certain dependence on the value of control parameters (λ and α).

According to the optimized results attained by solving the stress-constrained optimization problem for the MBB beam based on each algorithm listed in Tables 5 and 6, we can draw the conclusion that the proposed new algorithms have an advantage over the sensitivity-based optimization method in avoiding the problems of numerical derivation and sensitivity calculation involved in the process of obtaining sensitivity information.