Border-search and jump reduction method for size optimization of spatial truss structures

Abstract

This paper proposes a sensitivity-based border-search and jump reduction method for optimum design of spatial trusses. It is considered as a two-phase optimization approach, where at the first phase, the first local optimum is found by few analyses, after the whole searching space is limited employing an efficient random strategy, and the second phase involves finding a sequence of local optimum points using the variables sensitivity with respect to corresponding values of constraints violation. To reach the global solution at phase two, a sequence of two sensitivity-based operators of border-search operator and jump operator are introduced until convergence is occurred. Sensitivity analysis is performed using numerical finite difference method. To do structural analysis, a link between open source software of OpenSees and MATLAB was developed. Spatial truss problems were attempted for optimization in order to show the fastness and efficiency of proposed technique. Results were compared with those reported in the literature. It shows that the proposed method is competitive with the other optimization methods with a significant reduction in number of analyses carried.

Keywords

optimum design / sensitivity analysis / reduction method / spatial trusses / OpenSees

Introduction

In the past years, new studies and investigation tend to focus on the development of novel techniques that primarily based on nature-inspired techniques such as simulated annealing [¹], genetic algorithm [²], Tabu search [³], ant colony optimization [⁴], particle swarm optimization [⁵], cuckoo search [⁶], leap frog [⁷], flower pollination algorithm [⁸], fruit fly optimization algorithm [⁹], and ideal gas molecular movement [¹⁰]. These algorithms are progressively analyzed or improved by researchers in many different areas of structural optimization (see, e.g., Refs. [¹¹–²⁶]). However, there is not yet a specific algorithm to quickly perform the best solution for all structural optimization problems. This threat is very serious for the real-world application and structural problems where the scale of the problem is mostly large and each analysis is too time consuming. The main defect of these methods for solving structural problems is that a large number of parameters need to be tuned for every type of problem. Besides, mostly trial and error techniques are utilized and often experience is needed for parameters setting.

One could use a local or global approach to do sensitivity analysis task. The present article attempts to present a local sensitivity-based method for finding global optimum solution of spatial trusses, which is often around feasible region borders. Using the concept of sensitivity analysis of constraint violation as much as the change in the size of design variables in structures, the complexity order of problems is reduced into one-dimensional series. The technique allows a fast approach towards the near-global optimum which could be on the vicinity of feasible and non-feasible (FNF) border inside the feasible region. More details of sensitivity analysis approaches could be found in Refs. [²⁷,²⁸].

Finally, to globalize the proposed approach to be applicable for optimizing of every type of structures such as continuum structures, framed structures, the optimization package was prepared, in which the optimization algorithm was coded in MATLAB and OpenSees software was used as structural analyzer.

In the following, we concentrate on the developments of border-search operator (BSO) and jump operator (JO) and proposed technique of combined BSO and JO and show its application for optimum design of spatial trusses.

Structural sizing optimization problems

Size optimization of structural systems involves arriving at optimum values for member cross-sectional areas vector A that minimize an objective function f, usually the structural weight W. This is expressed mathematically as

(1)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}f=W\left(\mathit{A}\right)={\displaystyle \sum _{j=\mathrm{1}}^n{A}_j{L}_j{\rho }_j},

where A_j, L_j and r_j are the cross-sectional area, length and unit weight of jth truss member, respectively, and n is the total number of members. The vector A is selected between lower A^L and upper A^U bounds. Equation (1) is subjected to the following normalized constraints [²⁹]:

(2)\{\begin{array}{c}\hfill s_m\left(\mathit{A}\right)=\max \left(\mathrm{0},{\displaystyle \frac{{\sigma }_m}{{\sigma }_{m,\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{d}}}}-\mathrm{1}\right),m=\mathrm{1},\mathrm{2},\mathrm{...},NE\hfill \\ \hfill b_m\left(\mathit{A}\right)=\max \left(\mathrm{0},{\displaystyle \frac{{\lambda }_m}{{\lambda }_{m,\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{d}}}}-\mathrm{1}\right),m=\mathrm{1},\mathrm{2},\mathrm{...},NE\hfill \\ \hfill d_k\left(\mathit{A}\right)=\max \left(\mathrm{0},{\displaystyle \frac{u_k}{u_{k,\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{d}}}}-\mathrm{1}\right),k=\mathrm{1},\mathrm{2},\mathrm{...},ND\hfill \end{array},

where NE and ND denotes the number of members and the number of degrees of freedom of truss, respectively, s_m, b_m, and d_k are respectively, the member stress, member buckling and nodal displacement normalized constraint functions, s_m and l_m are the stress and the slenderness ratio of mth member, s_m,allowed and l_m,allowed are the allowable axial stress and allowable slenderness ration for mth member, respectively, u_k,allowed and u_k are the allowable displacement and nodal displacement of kth degrees of freedom, respectively. In Eq. (2), all the normalized constraint functions are activated when the violated constraints have values larger than zero. In this paper, the function of normalized degree of constraints violation Z and penalty function γ could be determined as Eqs. (3) and (4), respectively [³⁰]

(3)Z={\left({\displaystyle \sum _{m=\mathrm{1}}^{NE}(s_m+b_m)+}{\displaystyle \sum _{k=\mathrm{1}}^{ND}{d}_k}\right)}^{\mathrm{2}},

(4)\mathit{\gamma }=\mathrm{1}+Z\mathrm{.}

Then, the penalized objective function of the structure is considered as Eq. (5):

(5)F^{\prime }=f\mathit{\gamma }\mathrm{.}

Numerical sensitivity analysis

In the proposed optimization framework, a local type of sensitivity analysis is employed as the key to find the structural members that do not have an important role in the optimization procedure and can be neglected in some iterations of optimization task. Doing this important causes, the order of problem to be reduced effectively and helps to reach the most probable zone of global solution in early stages of optimization iterations.

Consider a sizing optimization problem that adding a value to every design variable will cause a value of the normalized degree of constraints violation Z, to be decreased or remains fixed. If \mathit{X} is the current design vector of member cross-sectional areas, then if the design is perturbed to X–Dx_i, where Dx_i is a perturbation value, using backward difference method for Z, we can write the sensitivity value Sens for ith variable as [²⁹]

(6)Sen{s}_i={\displaystyle \frac{Z(\mathit{X}-\Delta x_i)-Z\left(\mathit{X}\right)}{\Delta x_i}}\mathrm{.}

In Eq. (6), X is considered to be a feasible design that causes the above equation to simplify as Eq. (7):

(7)Sen{s}_i={\displaystyle \frac{Z(\mathit{X}-\Delta x_i)}{\Delta x_i}},

where Dx_i is called the dynamic perturbation value, given by Eq. (8):

(8)\Delta x_i={\displaystyle \frac{x_i}{\alpha }},

where a is the sensitivity analysis step parameter. To scale the value of sensitivity of each design variable to the value of other design variables, normalized sensitivity values (NSV), are determined as

(9)NS{V}_i={\displaystyle \frac{Sen{s}_i}{{\displaystyle \sum _{i=\mathrm{1}}^{n}Sen{s}_i}}},

where n is the total number of design variables. The feature of Eq. (9) is that it shows the sensitivity rate of a variable to other variables.

An efficient random strategy

One of the most important sections of the presented method is the strategy related to finding a first local point at early stage of Phase 1 that includes starting from the feasible region. To do so, uniform random number concept is used. First, each design variable of the primary points is generated from an integer power of 10 between two space orders s as given in Eq. (10):

(10)\mathit{X}={\mathrm{10}}^{s-\mathrm{1}}+{\mathrm{10}}^s\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d},

where X is design vector, s is space order of ten, and \mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d} denotes the random vector as the Eq. (11). Equation (10) has the advantage of producing one design in each design domain as well as it makes possible of searching the whole searching domain with a few number of analyzes. This matter may be so effective for structural problems where each analysis is too time consuming.

(11)\mathbf{r}\mathbf{a}\mathbf{n}\mathbf{d}={\left\{\begin{array}{c}\hfill rand.uni\hfill \\ \hfill rand.uni\hfill \\ \hfill ⋮\hfill \\ \hfill rand.uni\hfill \end{array}\right\}}_{n\times \mathrm{1}},

where n is the number of design variables, and rand.uni expresses the uniform random numbers generated between 0 and 1. Generating primary point in such manner has the advantage that the number of generated primary points is independent from the number of design variables and there is no need to increase them together with the increase in design variables. This method may be highly effective for the problems where the upper and lower limits are not provided for the variables and/or it aims to find the best limit of solutions. Generating these primary random points is continued until the first point without any constraints violation Z (using Eq. (3)) is spotted, though other primary points are deleted.

For example, in Fig. 1, the second point which is the first random design that is feasible as well as it possesses a relatively local optimum value of penalized objective function Z at this stage, is selected as the starting point from now on.

A description of border-search operator

At this stage, by having the NSV for each design variable, BSO is used until a local optimum is found. The variables with less sensitivity play the main role of allowing the searching procedure. The pseudo-code of Fig. 2 clarifies the local point computing procedure using BSO.

In Fig. 2, {{\displaystyle \lambda }}_{\mathit{var}} is called border-search step-length which is determined by reducing value of the varth design variable x_var as given in Eq. (12):

(12){\lambda }_{\mathit{var}}={\displaystyle \frac{x_{\mathit{var}}}{e}}

where e parameter is related to adjusting accuracy to find a sequence of points on the FNF, that is, the greater e is, the smaller the step length {{\displaystyle \lambda }}_{\mathit{var}} will be, as a result of which the border of a FNF region will be found causing a fast step towards the local optimum.

The proposed border-search and jump reduction method

In this section we present the basic concepts and implementation of the proposed border-search and jump reduction method for optimum design of spatial truss problems. Figure 3 shows the procedure of the proposed algorithm, which consists of Phases 1 and 2.

Carry out Phase 1: Find the first local optimum solution {\mathit{X}}_{\mathrm{L}}^{\mathrm{p}\mathrm{1}}

Phase 1, Step 1: Use an efficient random strategy (Section 4) as illustrated in Fig. 4 to find the first feasible space s and initialize the values of each design variable equal to 10^s to construct the start design vector {\mathit{X}}_{\mathrm{start}}^{\mathrm{p}\mathrm{1}}, where superscript p stands for “phase”.

Phase 1, Step 2: Determine the point without constraints violation near the FNF border by using the stair-wise step-by-step formulation of Eq. (13):

(13){\mathit{X}}_{step}^{\mathrm{p}\mathrm{1}}={\displaystyle \frac{{\mathit{X}}_{\mathrm{start}}^{\mathrm{p}\mathrm{1}}}{step}},

where step is an integer number starts from 1. The concept of this step is shown in Fig. 5.

Phase 1, Step 3: Determine the values of NSVs at {\mathit{X}}_{step}^{\mathrm{p}\mathrm{1}} using Eqs. (7) and (9) and sort them in ascending order.

Phase 1, Step 4: Compute the first local optimum design {\mathit{X}}_{\mathrm{L}}^{\mathrm{p}\mathrm{1}} at the end of Phase 1 using BSO considering pseudo-code of Fig. 2.

Carry out Phase 2: Find the global optimum solution X^G

Phase 2, Step 1: Set iteration number t=1, {\mathit{X}}_{\mathrm{L,}t}^{\mathrm{p2}}={\mathit{X}}_{\mathrm{L}}^{\mathrm{p}\mathrm{1}}, set best design {\mathit{X}}^{\mathrm{G}}={\mathit{X}}_{\mathrm{L}}^{\mathrm{p}\mathrm{1}}, set best objective function f^{*}=f\left({\mathit{X}}_{\mathrm{L}}^{\mathrm{p}\mathrm{1}}\right), where p and L stand for “phase” and “local”, respectively.

Phase 2, Step 2: Determine the values of NSV at {\mathit{X}}_{\mathrm{L},t}^{\mathrm{p}\mathrm{2}} using Eqs. (7) and (9) and sort them in ascending order.

Phase 2, Step 3: Update the most sensitive variables (MSV) considering the archived values of NSV from previous step. Updating of variables is carried out for the MSV with the corresponding NSVs greater than the mean value of NSV for all design variables. Here, JO is introduced as it causes a movement back to the feasible domain using Eq. (14):

(14)\underset{x_i\in MSV}{x_i^{t+\mathrm{1}}}=x_i^t+\Delta {\tau }_i^{t+\mathrm{1}},

where \Delta {\tau }_i is called jump length value for ith variable and could be determined using Eq. (15):

(15)\Delta {\tau }_i^{t+\mathrm{1}}=x_i^t(\mathrm{1}+NS{V}_i)\mathrm{.}

Phase 2, Step 4: Determine the local optimum design {\mathit{X}}_{\mathrm{L},t+\mathrm{1}}^{\mathrm{p}\mathrm{2}} using BSO as shown in Fig. 2.

It should be noted that the BSO is performed sequentially for the variables with the low values of sensitivity to high values, which lets the algorithm explores for better cases of optimal solutions. In other words, this type of border searching, first determine the values of non-effective design variables in optimization iterations, and then lets the values of MSV change, to reach the border. This type of searching in vicinity of FNF border helps in structural problems with large number of design variables in order to reduce the number of structural analyzes and run time, mutually.

Phase 2, Step 5: Compute f\left({\mathit{X}}_{\mathrm{L},t+\mathrm{1}}^{\mathrm{p}\mathrm{2}}\right), if f\left({\mathit{X}}_{\mathrm{L},t+\mathrm{1}}^{\mathrm{p}\mathrm{2}}\right) is better that f^{*} then set {\mathit{X}}_{}^G={\mathit{X}}_{\mathrm{L},t+\mathrm{1}}^{\mathrm{p}\mathrm{2}},\hspace{0.17em}{f}^{*}=f\left({\mathit{X}}_{\mathrm{L},t+\mathrm{1}}^{\mathrm{p}\mathrm{2}}\right).

Phase 2, Step 6: Check for convergence. If convergence is occurred, stop and save results, else set t=t+1 and go to Step 2 of Phase 2.

Figure 6 shows a kind of jump and search that are defined for Phase 2 of proposed approach.

Numerical examples

In this section, in order to show the robustness and efficiency of proposed method, three numerical benchmark examples of spatial trusses previously treated by other investigators are presented: A 22-bar space truss, a 72-bar space truss and 120-bar dome truss, all subjected to multi loading conditions. For these examples, the parameters a and e were set as a fixed value of 100 and the maximum number of iterations of Phase 2 is limited to 100. As OpenSees is used for structural analysis, results precision may be greater than some other results reported in the literature; i.e., some optimum solutions reported by other researchers were observed to have some constraints violations.

Sizing optimization of 22-bar space truss

Figure 7 shows a 22-bar cantilever space truss. This problem has been studied by researchers including Khan et al. [³¹], Li et al. [³²], Lee and Geem [³³], and Talatahari et al. [³⁴]. The material density was 0.1 lb/in³ (2767.99 kg/m³) and the modulus of elasticity was 10000 ksi (68.950 GPa). This space truss was subjected to three loading conditions as shown in Table 1 and members were linked into seven groups as given in Table 2. Also, displacement constraints of ±2.0 in (±5.08 cm). were imposed on all nodes in all directions and minimum member cross-sectional area of 0.1 in² (0.645 cm²) was enforced. The truss members were subjected to the stress limitations shown in Table 2.

The bar graph of Fig. 8 shows the values of NSV at {\mathit{X}}_{step}^{\mathrm{p1}}. As the diagram shows, Variables 2–4 are least sensitive, that causes these variables to be as the first candidates for BSO and not for JO in most of iterations. Comparing the results of this sensitivity analysis and optimum results of Tables 3, it is clear that mentioned variables have lower values than other ones. In addition, Table 3 contains the results for the same optimization task from different research efforts. The minimum weight obtained from the proposed method is 1025.802 lb (1 lb=453.592 g) with no constraint violation. Table 4 shows the weights and the number of analyses.

The proposed algorithm provides good results when compared with other results, however, those obtained from the heuristic particle swarm optimization [³²], harmony search algorithm [³³] and the multi-stage particle swarm algorithm [³⁴] are lighter than the results from the proposed method, but they violated some of constraints. Figure 9 provides the convergence history of the optimum truss weight of Phases 1 and 2 of algorithm. It illustrates that the proposed algorithm converges much faster than those reported in the literature. It takes about 1800 analyses after 45 iterations for proposed method to converge.

Sizing optimization of 72-bar space truss

Figure 10 shows a 72-bar space truss with its node and element numbering schemes. This space truss, has been recently optimized by Perez and Behdinan [³⁵], Degertekin [³⁶] and Kaveh et al. [³⁷]. The problem had complex and non-linear search space as it had 320 nonlinear constraints (72 tensions, 72 compressions, 8 positive displacements, 8 negative displacements for each loading case). The material density and modulus of elasticity were considered 0.1 lb/in³ (2767.990 kg/m³) and 10000 ksi (68.950 GPa), respectively. The constraints contained the stress of ±25 ksi (±172.375 MPa) and displacement limitations of ±0.25 in for uppermost nodes in all directions. This space truss was subjected to two loading conditions: Condition 1, in which P_X=5.0 kips, P_Y =5.0 kips (22.25 kN), and P_Z= –5.0 kips (–22.25 kN) on Node 17; and Condition 2, that P_X= 0.0 kips, P_Y= 0.0 kips, and P_Z =–5.0 kips (–22.25 kN) were acted on Nodes 17–20. For design and manufacturing considerations truss members are classified into sixteen groups as shown in Table 5. Table 6 shows the weights and the number of analyses. For this problem, the value of lower bound limitation of design variables was 0.1 in².

The bar graph of Fig. 11 shows the values of NSV at {\mathit{X}}_{step}^{\mathrm{p1}}. As the diagram shows, Variables 3, 4, 7–9, and 11–13 are least sensitive, and they have been assigned the lower allowable value of the bar cross-sectional area at optimum design. Table 5 contains the results for the same optimization task from different research efforts. In addition, the truss weight convergence history is given in Fig. 12. As could be observed, first local optimum point {\mathit{X}}_{\mathrm{L}}^{\mathrm{p1}} with the corresponding truss weight 524.98 lb was achieved after 100 analyzer calls. The proposed method found the best solution after a few 1250 analyzer calls. The best weight obtained by the proposed method for this space truss is slightly lighter than the optimum truss weight results by other researchers.

From convergence history of Fig. 12 and optimum results of Table 5, it can be seen that the convergence rate of the proposed algorithm is apparently higher than the other methods reported in the literature. The proposed method achieves optimum solution after only 1250 analyses that is a fraction of total 10500 analyses reported by Kaveh et al. [³⁷].

Besides as evident in Table 5, one could see that some of the optimal solutions presented by others have not completely satisfied all the constraints.

Sizing optimization of 120-bar dome truss

The topology and nodal numbering of the 120-bar dome truss are shown in Fig. 13. This dome truss was first addressed by Soh and Yang [³⁸] as sizing and shape optimization problem to obtain minimal weight of truss, however, in this study, only sizing variables were considered. The example was also studied by Kazemzadeh Azad [³⁹] and Hadidi et al. [⁴⁰]. The structure is subjected to vertical loading at all unsupported nodes. The loads are taken as –13.49 kips (–60 kN) at Node 1, –6.744 kips (–30 kN) at Nodes 2 to 14, and –2.248 kips (–10 kN) in the rest of the nodes. The minimum allowable cross sectional area of each member is limited to 0.775 in² (5 cm²). The allowable tensile stress was 0.6F_y and the compressive stress constraint {\sigma }_i^bof member i was as follows:

where E is the modulus of elasticity of steel, F_y is the yield stress of steel, {\lambda }_i is the slenderness ratio ({\lambda }_i=k{L}_i/r_i), C_c is the critical slenderness ratio C_{\mathrm{c}}=\sqrt{\mathrm{2}{\mathrm{\pi }}^{\mathrm{2}}E/F_y}, k is the effective length factor, L_i is the ith member length and r_i is the radius of gyration. The material has a modulus of elasticity of 30450 ksi (210 GPa), mass density of 0.288 lb/in³ (7971.81 kg/m³) and yield stress of 58 ksi (400 MPa). Here displacement constraint contains maximum displacement limitations of ±0.1969 in (±0.5 cm) imposed on all nodes in x and y directions.

Table 7 compares the optimal results of the present study with those of the other optimization techniques. Table 8 shows the weights and the number of analyses. Figure 14 shows the primary sensitivity results at {\mathit{X}}_{step}^{\mathrm{p1}}. As illustrated in Table 7 and convergence history of Fig. 15, the proposed method required a total of 1100 analyses to find a minimum global weight of 19905.45 lb (9028.96 kg) after 39 iterations. It suitably agrees with the results obtained by other researches, while at the same time it required lower computational effort.

Conclusions

In this article, a new and relatively fast optimization method for sizing optimization of spatial trusses has been introduced. First, it finds the most possible zone of the location of the global optimum point, using an efficient random strategy, second, the first local optimum point is found doing a few number of analyzes utilizing sensitivity-based concept and BSO (referred to as Phase 1), and third, it repeats using BSO and JO based on sensitivity analysis until the global optimum design is determined (referred to as Phase 2).

Having applied the method into some spatial trusses, a major advancement of the technique was observed on the optimum solution in terms of number of analyses required for optimization and a minor improvement on the accuracy. Open source software of OpenSees was prepared for truss analyzer, which helps to get more accurate structural responses. Investigations on the convergence history and results emphasis that this method always performs feasible global optimum design.

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