Optimal dome design considering member-related design constraints

Abstract

This study proposes to optimize the design of geometrically nonlinear dome structures. A new Multi-objective Optimization Algorithm named Pareto Archived Genetic Algorithm (PAGA), which has an ability of integrating the nonlinear structural analysis with the provisions of American Petroleum Institute specification is employed to optimize the design of ellipse and sphere-shaped dome configurations. Thus, it is possible to investigate how the qualities of optimal designations vary considering the shape, size, and topology-related design variables. Furthermore, the computing efficiency of PAGA is evaluated considering six multi-objective optimization algorithms and eight quality measuring indicators. It is shown that PAGA has a capability of both exploring an increased number of pareto solutions and predicting a pareto front with a higher convergence degree. Moreover, the inclusion of shape-related design variables leads to a decrease in both the weights of dome structures and their load-carrying capacities. However, the designer easily determines the most requested optimal design through the archiving feature of PAGA. Thus, it is also demonstrated that the proposed optimal design procedure increases the correctness degree in the evaluation of optimal dome designs through the tradeoff analysis. Consequently, PAGA is recommended as an optimization tool for the design optimization of geometrically nonlinear dome structures.

Keywords

dome structure / geometric nonlinearity / multi-objective optimization / API RP2A-LRFD

Introduction

The dome structures have a capability of covering large areas without requiring any intermediate support. Therefore, the dome structures, which are constructed using different constituent material (wooden, stone, iron, concrete, etc.) have been seen in almost all of the historical buildings. When these traditional dome structures are examined, it is easily seen that their constructing costs are higher than the other structures. However, the configurations and forms of their skeletal structure along with their constituent material have been simultaneously evolved with the progression in structural engineering area. The construction costs of modern dome structures are considerably decreased compared to the traditional ones since their members, which are constructed using the ready steel profiles are automatically connected to each other through the robots in a manufacturing factory of steel construction. Therefore, the design of dome structure achieves to be one of the attractive application problems in engineering optimization field [¹–⁶]. Thus, the optimization techniques have been improved and developed comprising the different engineering applications areas [⁷–⁹]. This acceleration provides a further decrease in the constructing costs of dome structures. Nevertheless, the designer has to handle with some of uncertainties arisen from the design complexity of dome structures. Particularly, one of the bottlenecks in the design of dome structures is concerned with the possibility of including a slender ready steel profile into the current construction of dome structure due to the variation in the cross-sectional properties [³]. Thus, the dome structure can exhibit a structural instability leading to a significant configurational changes in its skeletal structure even when the member responses are well below a yield point of material [⁴,⁵]. In fact, the structural instability occurs when the tangent stiffness matrix of structural system is ill-conditioned. The ill-conditioned tangent stiffness matrix is not correctly solved using a classical numerical scheme, for example, “Newton-Raphson” method, since the entire load-displacement path is represented by a number of critical points (called as bifurcation, limit point, etc.). It was demonstrated that one of the proper nonlinear structural analysis techniques named ‘arc-length’ method has been successful in tracing the entire load-displacement path which comprises all critical points [³,⁶]. Particularly, due to associating the geometric nonlinearity with an occurrence of larger deflections in the nodes of structural systems, the nodal deflections have to be constrained by an upper limit for the serviceability purpose. Also, the load-carrying capacity of structural system is desired to be in a higher level. Although the larger deflection in the nodal points has a big advantage for the ductility issue which indicates about the degree of energy absorption capacity in the structural systems, the load-carrying capacity of structural system can be suddenly failed before reaching its ductility capacity to an enough level due to the stability problem. It is also noted that the cost of structural system has to be kept within the lower levels due to the economy-related reasons. These desired expectations are easily met with an integration of multi-objective optimization algorithm into the design stage of structural system. Thus, the designer has an opportunity to assessing his/her preferences thereby making a tradeoff analysis among the values of proposed objective functions. The tradeoff analysis increases the accuracy in the decision of designer [³].

A general multi-objective optimization problem consisting of K objective functions is formulated as:

(1)\min \mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}\max F(x)=\left\{f_{\mathrm{1}}\left(X\right),f_{\mathrm{2}}\left(X\right),\mathrm{...},f_{\mathit{K}}\left(X\right)\right\}.

A design (decision) variable set Xis defined in design variable space DS. At each run of MOA, a set of random solutions is obtained. Some of them are dominated by the other solutions (for example, f_i\left(X\right)\le f_{\mathrm{1}}\left(X\right)\mathrm{a}\mathrm{n}\mathrm{d}{f}_i\left(X\right)\ne\ne f_{\mathrm{1}}\left(X\right),\forall i\in (\mathrm{1},\mathrm{...},K)) [¹⁰]. Then, it is said that f_{\mathrm{1}}\left(X\right) is dominated the other f_i\left(X\right). This dominance is defined as pareto dominance in the context of multi-objective optimization and utilized to determine pareto solutions thereby comparing the current values of objective functions with each other. N inequality abbreviated as g_n and M equality constraints abbreviated as h_m are utilized along with these objective functions. Thus, a pareto solution P^{*} is described in DS as:

(2)P^{*}=\left\{f_i(X)\le f_i(X^{*}):\{\begin{array}{l}{f}_i\left(X\right)\ne f_i\left(X^{*}\right),\mathrm{}\forall i\in (\mathrm{1},\mathrm{...},K)\mathrm{a}\mathrm{n}\mathrm{d}\forall X\in DS\hfill \\ g_n\left(X^{*}\right)\le \mathrm{0},n=(\mathrm{1},\mathrm{...},N)\hfill \\ h_m\left(X^{*}\right)=\mathrm{0},m=(\mathrm{1},\mathrm{...},M)\hfill \end{array}\right\}.

Pareto front P{F}^{*} is formed by collecting pareto solutions together and defined as:

(3)P{F}^{*}=\left\{F\left(X\right),X\in P^{*}\right\}.

Although the current paretofront obtained in the end of execution of MOA may be improved thereby exploring new pareto solutions, a true pareto front is assumed that it cannot be improved further. However, it is difficult to obtain the true pareto front in a solution space represented with a discrete-type design variables. Particularly, it is practically impossible to obtain a certain true pareto front for the objective functions with the continuous-type design variables since the values of both objective functions and decision variables vary depending on their precision degrees [¹¹]. Therefore, the diversity, spread, spacing degrees of the current pareto solutions with respect to the true pareto front are assessed considering several quality measuring indicators, named hyper volume, spread, spacing, etc [¹²].

Through the development of new techniques, the MOAs algorithms have been increasingly developed and applied on a variety of applications [¹³,¹⁴]. Moreover, their exhaustive and recent reviews for the optimization of structural design problem in engineering field are found in Refs. [¹⁵,¹⁶]. There exist some works on the optimal design of dome structures in Refs. [¹⁷–²⁰]. Although these optimization algorithms are capable of generating the optimal dome designs, the nonlinear analysis against the inclusion of slender members is unfortunately not integrated with the available the design codes. This study proposes to fulfill this lack thereby both providing an integration of geometric nonlinearity into the optimization procedure along with the real world design provisions, namely American Petroleum Institute specification (API RP2A-LRFD) and increasing the reliability of optimal dome designs in a way of including an intelligent multi-objective optimization algorithm for a tradeoff analysis. Moreover, the optimality degree of dome designs is increased including the ellipse-shaped form along with shape-related design variable into the optimal design process using a new multi-objective optimization algorithm named PAGA. To properly evaluate the computing performance of PAGA coded in MATLAB, six different multi-objective optimization algorithms named NSGAII [²¹], SMSEMOA [²²], MOEA/D [²³], PESAII [²⁴], SPEA2 [²⁵], eMOEA [²⁶] are employed to optimize both three benchmark mathematical functions and five benchmark planar truss designs.

This study begins by introducing the proposed optimal design approach. The working principles of PAGA are described in Section 3. The details of proposed search methodology are presented in Section 4. The results are discussed in Section 5. The last section is reserved to summarize the primary results.

Proposed optimal design problem and size, topology, shape-related design variables

In this study, the nonlinear structural analysis involving the geometrical nonlinearity is integrated with the provisions of API RP2A-LRFD specification. Thus, it is possible to check the current strengths of dome members according to the allowable nominal strength at each incremental steps of nonlinear structural analysis. In fact, the nodal deflections are also checked for the higher serviceability of dome structure taking a predefined upper value into account. The responses of dome members to the external static loads are utilized to determine the load-carrying capacity of dome structure. In this regard, the structural responses of dome structure are computed by ANSYS software assuming its material behavior to be elastic. Then, the current strengths of dome members and the allowable nominal strengths are computed. However, the main difficulty is the inclusion of these strength-related computations into the proposed optimization algorithm. To overcome this barrier, a ratio of available strength of dome member to the allowable nominal strength and a ratio of maximum value of nodal deflection to a predefined value are utilized. These ratios are named Unity in this study. Thus, it is possible to include each of these Unity values into the proposed optimization algorithm as a design constraint. At this point, the main distinguished feature of the proposed optimal design procedure has a flexibility of integrating the geometric nonlinearity with one of the available specification provisions.

The proposed design optimization of the dome structures simultaneously minimizes the entire weight of dome structure for economically a higher profit along with nodal deflections for a higher serviceability and maximizes the member forces for a higher load-carrying capacity. The proposed optimal design approach in associated with the penalizing procedure, elements of which are based on provisions of API RP2A-LRFD specification is formulated as

(4)f_{\mathrm{1}}=\min \left(\mathrm{\sum }_{k=\mathrm{1}}^m{(w·l)}_k+P_{\mathrm{1}}\right),

(5)f_{\mathrm{2}}=\min (d_{ij}+P_{\mathrm{2}})\mathrm{}(i=\mathrm{1},\mathrm{...},\mathrm{12}\mathrm{a}\mathrm{n}\mathrm{d}j=\mathrm{1},\mathrm{...},n),

(6)f_{\mathrm{3}}=\max (f_{ij}+P_{\mathrm{3}})\mathrm{}(i=\mathrm{1},\mathrm{...},\mathrm{12}\mathrm{a}\mathrm{n}\mathrm{d}j=\mathrm{1},\mathrm{...},n),

where

(7)

(8)

The entire weight of dome structure, member forces and joint (node) deflections are represented by f_{\mathrm{1}} (m, number of dome member; w, unit weigh per member length l), f_{\mathrm{2}} (n, number of nodes; i, number of node freedom), and f_{\mathrm{3}}, respectively. While member-related design constraints are represented by Unity_Axial, Unity_Bending, etc., the nodal deflection-related design constraints are defined by Unity_Disp (see Eq. (7)). The terms, current generation number CGN and ϕ in Eq. (7) are dynamically adjusted depending on the generation number (see the further details about the use of these parameters in sub-section 3.2). In case of exceeding one of the constraint-related upper limit values, this unsatisfactory result is penalized by the penalty values P₁, P₂, and P₃ (see Eqs. (7)–(8)). It is noted that the extended details of member-related design constraints in Eq. (8) are presented in Refs. [²,³].

Sphere and ellipse shapes are used to form the geometrical configuration of dome structures (see Fig. 1). For this purpose, an automatic dome generating tool is employed to generate longitudinal, horizontal chord (arched) members and brace (diagonal) members in order to construct a dome structure. It utilizes three design variables, named size, shape, and topology for the generation of dome structures. Hence, depending on the shape, topology, and size-related design variables, three geometrical configurations are proposed for the generation of sphere and ellipse-shaped dome forms (see Tables 1–2). The main governing parameters of nonlinear structural analysis are also summarized in Tables 2–3. In this study, the circular hollow cross-sections are preferably utilized to represent the dome members with the ready steel profiles due to their higher resisting to the torsion-related effects compared to the open cross-sections. The properties of circular hollow cross-sections are employed to represent the size-related design variables Par_DV; shape and topology-related design variables are represented by the nodal coordinates and longitudinal-horizontal division numbers Par_LDN and Par_HDN, which are used to determine longitudinal and horizontal arched members of dome structures. It is noted that the parameter Par_LDN used to divide the longitudinal span of dome structures into small segments indicates about the numbers of longitudinal arched members located on semi sphere or ellipse- shaped dome structure. Thus, it is obvious that the number of size-related design variables Par_ND varies depending on the geometrical configuration of dome structure. The parameter Par_DV is assigned to the dome members from a steel profile database with 37 ready circular hollow cross-sections. Hence, the upper and lower limit of size- related design variables Par_DVU and Par_DVL is taken as 37 and 1. The topology-related design variables are represented by the upper and lower limits of longitudinal division numbers Par_LDNU and Par_LDNL and the upper and lower limits of horizontal division numbers Par_HDNU and Par_HDNL. The shape-related design variables Par_SDVx, Par_SDVy, and Par_SDVz are used to represent the spanning lengths of dome structure in x, y, and z directions. The shape-related design variables are represented by the upper and lower spanning lengths in x, y, and z directions Par_SDVxU and Par_SDVxL, Par_SDVyU and Par_SDVyL, Par_SDVzU and Par_SDVzL. In this regard, the nodal coordinates of sphere and ellipse-shaped dome structures are generated using the following basic equation:

(9)\begin{array}{l}x=Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}x}·\cos \left(\right(\mathrm{\pi }/\mathrm{2})·V)·\cos (\mathrm{\pi }·U)\hfill \\ y=Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}y}·\cos \left(\right(\mathrm{\pi }/\mathrm{2})·V)·\sin (\mathrm{\pi }·U)\hfill \\ z=Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}z}·\sin \left(\right(\mathrm{\pi }/\mathrm{2})·V)\hfill \end{array}\{\begin{array}{l}U=\mathrm{1}/Pa{r}_{\mathrm{H}\mathrm{D}\mathrm{N}},V=\mathrm{1}/Pa{r}_{\mathrm{L}\mathrm{D}\mathrm{N}}\hfill \\ Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}x}=Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}y}=Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}z}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{s}\hfill \\ Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}x}\ne Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}y}\ne Pa{r}_{\mathrm{S}\mathrm{D}\mathrm{V}z}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{s}\hfill \end{array}.

An introduction of PAGA

This section introduces the evolutionary search mechanism of PAGA. The first section is reserved for the introduction of basic definitions and assumptions in an evolutionary-based optimization procedure. The evolutionary search mechanism of PAGA is presented in the second sub-section.

Basic issues utilized by PAGA

It is mentioned that the use of MOA for the optimal design of geometrically nonlinear dome structures increases the correctness degree in the quality evaluation of optimal designs. However, the selection of efficient MOA is a difficult task due to the wide variety in MOAs.

Evolutionary algorithms with single objective were proven to be one of the most efficient MOAs for the structural engineering design problems since they don’t require any function derivatives [²⁷]. Evolutionary algorithms (EAs) mimic the Darwian’s evolutionary theory. The evolutionary based optimization search starts by randomly generating an initial population of children and terminates when a specified termination criteria (for example, a maximum generation number). In each generation, the new children are generated thereby applying a series of genetic operators named selection, crossover and mutation to the parents inherited from the previous generation. Hence, the performances of individuals (children or parents) affect the transmission of valuable genetic material to the next populations. The performance concept is explained by a qualitative measurement named fitness value obtained from a fitness function.

EAs with multiple objectives require an implementation of multi-criterion-based decision into their evolutionary search mechanism. Therefore, EAs with multiple objectives have to be handle a computing complexity arisen during the convergence of current pareto front to a true pareto front. In this regard, the multi-objective evolutionary optimization algorithms (MEAs) have been improved and developed using the basic evolutionary issues: the fitness assignment, individual migration, and population diversity [¹⁴].

MEAs are equipped with the rich features. Particularly, their flexible evolutionary search mechanisms lead to an increase in their reputation. One of the techniques utilized by MEAS is the implementation of quality measuring indicators into their evolutionary search mechanism [²²]. However, it is difficult to make a decision on the determination of quality measuring indicators. In this regard, the converging and diversity degrees of current pareto front with respect to a true pareto front have a big importance for the assessment of fitness values [²⁶]. Therefore, a simple but effective evolutionary implementation is required to immediately determine the elite pareto solutions and directly transmit the valuable genetic material to the next generations. In fact, genetic algorithms (GAs) are assumed to be a perfect optimization tool in meeting these lacks faced by a number of evolutionary-based approaches. But, the evolutionary search mechanism of GAs requires an additional implementation for the determination of elite solutions with the higher diversity and converging degree.

Search mechanism of PAGA

The fundamentals of PAGA are constituted on the determination and transmission of elite pareto solutions (see the flow chat of PAGA in Fig. 2). Thus, the new populations are generated thereby both exploring the current populations and exploiting the elite pareto solutions. The elite pareto solutions are consistently archived into a unique population. Then, this unique population is employed for an exploring and exploiting purposes through the sub-populations. In fact, these sub-populations is utilized to accordingly represent the unique population named “pareto-archived population”.

In this study, the multi-objective optimization problems have two and three objective functions. Therefore, the evolutionary mechanism of PAGA is governed by four and eight sub-populations which are employed to represent each location of pareto solution space with two and three objective functions (see Fig. 3). But, the evolutionary search strategy does not vary depending on the number of objective functions. In this regard, the evolutionary-based computing steps of PAGA are introduced for two objective functions. Thus, it is possible to present the important details without omitting the similar evolutionary-based computation with the different values of operator parameters due to an increase in the sub-populations of PAGA. PAGA with two objective functions utilizes the four sub-populations located on the four solution regions (see Fig. 3). While one of these sub-populations (Spop 2) is utilized for the exploring purpose, the other three sub-populations (Spop 1, Spop 3, and Spop 4) are exploited. The exploitation of valuable genetic material is accomplished in a way of adjusting both the sub-population sizes and the values of genetic operator parameters.

The evolutionary search of PAGA begins by initializing the values of genetic operator parameters. Then, the following the computation of fitness values of current population, the pareto solutions are determined and stored in a pareto-archived population named pareto_OF_ALL. A statistical data regarded to the pareto-archived population manages the adjustment of the population sizes (PS1, PS2, PS3, and PS4) (see Part A in Fig. 3). The elements of statistical data are max_pareto_OF1_ALL, min_pareto_OF1_ALL, max_pareto_OF2_ALL, min_pareto_OF2_ALL, each of which represents the bounds of each objective functions in the matrix of pareto_OF_ALL. Furthermore, middle_pareto_OF1_ALL and middle_pareto_OF2_ALL which is obtained by dividing the sum of max_pareto_OF_ALL and min_pareto_OF_ALL into a parameter named division number (div_num), is also included into this statistical data for the determination of promising solution regions to be concentrated the exploiting process of PAGA. Moreover, the sub-generation numbers (subGN1, subGN2, subGN3) and generation number GN, which are previously defined are employed to assign the values of genetic operator parameters including the parameter ϕ in the penalizing computation given in Eq. (7) (see Part B in Fig. 3). In this study, two genetic operators named simulated binary crossover and polynomial mutation are utilized for the combination and mutation-related operations of individuals. Thus, the genetic operators (crossover and mutation) are utilized to generate the new sub-populations (Spop 1, Spop 2, Spop 3, and Spop 4). While the crossover operator is managed by two parameters named crossover probability (crossover_prob) and crossover distribution index (crossover_dist), the mutation operator is governed by two parameters named mutation probability (mutation_prob) and mutation distribution index (mutation_dist). It is noted that the values of genetic operator parameter are dynamically adjusted (see Part B in Fig. 3). Following the application of genetic operators to the current sub-populations, the generated population is corrected considering the limits of design variables max_DV and min_DV if it is required. Then, the fitness values and corresponding pareto solutions are computed. The main loop is terminated once a maximum generation number is completed.

Search methodology

In this study, a new multi-objective optimization algorithm named PAGA is utilized for the design optimization of geometrically nonlinear dome structures. It is clear that the computing performances of PAGA and employed MEAs named NSGAII, SMSEMOA, MOEA/D, PESAII, SPEA2, eMOEA are highly sensitive to the parameter values of their evolutionary-related operators. To provide an equal competition among PAGA and the employed MEAs, the same genetic operators named simulated binary crossover and polynomial mutation are utilized for each of them. Whereas PAGA is ability of assigning different values for the genetic operator parameters (see PART B in Fig. 3), the certain values of genetic operator parameters are required for the employed MEAs. Following to execute a number of trials for each of the other employed MEAs, relatively better combinations of their governing parameter values are approximately determined as crossover_prob= 0.4, crossover_dist= 6, mutation_prob= 0.8, mutation_dist= 10. The population size and generation number is taken as 100 for both PAGA and the other employed MEAs. The other primary difficulty is the measurement of the computing performances of the employed MEAs since an optimization problem with multiple objective functions is represented by a true pareto front consisted of a number of pareto solutions. Therefore, the eight quality measuring indicators named “Distance Metric” DM, “Generational Distance” GD, “Hyper Volume” HV, “Inverted Generational Distance” IGD, “Normalized Hyper Volume” NHV, “Pure Diversity” PD, “Spread” and “Spacing” are utilized to evaluate the distance, volume, diversity, spread and spacing of the current pareto solutions with regard to the true pareto front. The maximum values of quality measuring indicators, DM, HV, NHV, and PD along with the minimum values of quality measuring indicators, GD, IGD, PD, Spread and Spacing indicate about the better quality of optimal solutions with regard to the true pareto front. To provide unbiased competition among the employed MEAs and PAGA, an experiment with 100 runs is conducted. Then, the average values of quality measuring indicators are utilized for the competition purpose. It is noted that there exists the code scripts of these algorithms and quality measuring indicators in the different computer languages (see JMetal [²⁸], MOEAframework [²⁹] in Java, MOA [³⁰] and PlatEMO [³¹], Gamultiobj [³²] in MATLAB).

Results and discussions

It is mentioned that PAGA is essentially proposed for the design optimization of geometrically nonlinear dome structures. The computing performance of PAGA is also demonstrated with respect to six trend MEAs named NSGAII, SMSEMOA, MOEA/D, PESAII, SPEA2, eMOEA. For this purpose, three benchmark mathematical functions named ZDT1, DTLZ2, and DTLZ7 along with five benchmark planar trusses with 756, 200, 4, and 2 bars are utilized. Then, three different geometrical configurations with sphere and ellipse forms are utilized to represent the topology of geometrically nonlinear dome structures (see Tables 2–3). Hence, the design of sphere and ellipse-shaped dome structures with six geometrical configuration named ellipse1, ellipse2, ellipse3, sphere1, sphere2, and sphere3 are optimized by PAGA. The optimal solutions obtained from the execution of the employed MEAs, are summarized in the following two subsections.

The assessment of computing efficiency of the employed MEAs for the optimization of benchmark mathematical functions and design examples

Two separate benchmark problem sets named “benchmark mathematical functions” and “benchmark planar truss design” are utilized for the evaluation of the employed MEAs and PAGA.

First, three benchmark mathematical functions with different computing complexities named ZDT1, DTLZ2, and DTLZ7 [¹⁷] are optimized. Whereas DTLZ2 and DTLZ7 with the concave and noncontiguous forms have 11 and 5 decision variables bounded within an interval [0,1], respectively, ZDT1 with several noncontiguous convex forms has 30 decision variables bounded within an interval [0,1]. Then, the designs of five planar trusses with 756 [³³], 200 [²], 4 [³⁴], and 2 bars [³⁵,³⁶] are optimized.

True pareto fronts and their current parato fronts with a better value of quality measuring indicator value chosen from 100 executions are represented in Figs. 4–6 for the benchmark mathematical functions and Figs. 7–11 for the benchmark design examples. The average values of quality measuring indicators are summarized in Tables 4 and 5. It is noted that the success degrees of employed MEAs tabulated in Tables 4 and 5 are ranked with a dark gray. Considering Tables 4 and 5, the computing capacities of PAGA for benchmark mathematical functions and benchmark planar truss designs are observed to be better than the other employed MEAs (see the decreased average values of indicators named GD, IGD, spacing and spread along with the increased average values of indicators named HV, NHV). However, the values GD for the benchmark planar truss designs with 2 bars and 200 bars, PD for the benchmark mathematical function DTLZ7, spacing for the benchmark planar truss designs with 2 bars and spread for both for the benchmark mathematical function DTLZ2 and DTLZ7 and benchmark planar truss designs with 2 bars indicates relatively a lower performance of PAGA compared to the employed MEAs. Considering the quality measuring indicator named DM, it is observed that eMOEA shows a better scattering among the optimal solutions than the other employed MEAs (see the values of DM with dark colors in Tables 4 and 5). Whereas the computing time of eMOEA is in the lowest values, PAGA is completed its computing steps in the highest values with regard to the other employed MEAs (see the run time with dark colors in Tables 4 and 5). It is also observed that the number of pareto solutions explored by PAGA is in the highest values than the other employed MEAs (see Figs. 4–11)

Application of PAGA for design optimization of geometrically nonlinear dome structure and preliminary results

In this section, PAGA is applied to optimize the design of a geometrically nonlinear dome structure, nothing that this dome example was first devised by Kaveh and Talatahari [³⁷] and also tackled in Talaslioglu [³]. The values of size, shape, and topology-related design parameters along with the governing parameter values of geometrically nonlinear analysis under static loading are tabulated in Tables 2–3. This dome structure with an Elastic modulus of 205 kN/mm² and a diameter of 20 m was first optimized for a minimization of its weight constraining the maximum values of nodal deflections to 30 mm. The cross-sectional properties used in sizing of dome members are assigned from a ready steel profile list consisted of 37 different cross-sections with circular hollows (see the details about the cross-sectional properties in Ref. [³⁸]).

PAGA is executed to optimize the design of proposed dome structure. Considering Figs. 12–14, the extreme optimal designs are listed for both sphere and ellipse-shaped dome structures in Tables 6–13. Noting that the member force values of dome structure are used to indicate its load-carrying capacity, the extreme optimal designs are also enlisted for three geometrical configurations of sphere and ellipse-shaped dome structures in Tables 6–13. Whereas it is shown that the minimum weight and maximum member force values utilized to indicate the load-carrying capacity are obtained by ellipse-shaped dome structure with geometrical configuration 1 as 3.35260 kN and 161111.77648 kN and sphere-shaped dome structure with geometrical configuration 1 as 3.73945 kN and 160002.81483 kN, the minimum nodal deflection values are displayed to be obtained by ellipse-shaped dome structure with geometrical configuration 1 as 1.30914 mm and sphere-shaped dome structure with geometrical configuration 3 as 3.00763 mm. It is clear that an increase in the weight of dome structure from 81.96421 kN to 95.92513 kN provides an increase in its load-carrying capacity from 160002.81483 kN to 161111.77648 kN. However, the stability issue has a big importance in the determination of load-carrying capacity of dome structure. For example, although the weight of dome structure is increased from 103.67229 kN to 311.83751 kN, its load-carrying capacity is decreased to 2514.27344 kN from 15310.77152 kN. The existence of stability problem is easily observed considering the variation in the nodal deflections, values of which are increased from 8.91583 mm to 26.61241 mm. An inclusion of a slender member into the construction of the dome structure causes to a failure in its stability. Thus, the maximum value of nodal deflections is increased without keeping its load-carrying capacity in a higher level.

The other interesting observation regarded to the stability issue is related with the case of geometrical configuration 2 which is devised to include the diagonal members into the skeletal system of dome structure. Although the inclusion of any diagonal members into its skeletal system is expected to lead to an increase in its load-carrying capacity and a decrease in its nodal deflections, this claim is unfortunately disproved considering the obtained optimal solutions. It is easily seen that the maximum values of nodal deflections as 25.29756 mm and 26.61241 mm (see Tables 6–13) is obtained by use of geometrical configuration 2 with Par_LDN = 2 and Par_HDN = 2 for ellipse-shaped dome structure, Par_LDN = 5 and Par_HDN = 4 for sphere-shaped dome structure, respectively. The most increased values of load-carrying capacity as 161111.77648 and 160002.81483 (see Tables 6–13) are obtained without the inclusion of diagonal members thereby using geometrical configuration 1 with Par_LDN = 2 and Par_HDN = 2 for both ellipse and sphere-shaped dome structure. The maximum deflections for geometrical configuration 1 for both ellipse and sphere-shaped dome structures as 9.35454 mm and 9.07981 mm are obtained using the topology-related design parameters with Par_LDN = 3, Par_HDN = 3, Par_LDN = 5, Par_HDN = 3. Also, the minimum weight for geometrical configuration 3 as 6.88694 kN is obtained utilizing Par_LDN = 2 and Par_HDN = 3 for sphere-shaped dome structure.

Considering the primary results it is easily seen that the use of a multi-objective optimization algorithm increases the correctness degree in the evaluations of optimal design for the design optimization of geometrically nonlinear dome structures according to the preference of designer. Although the inclusion of shape-related design variables into the proposed optimal design approach provides economically a profit for weight minimization, it is failed in providing an increase in the load-carrying capacity of dome structure. For example, it is observed that a decrease in the weight of dome structure from 350.1250 kN (obtained by use of size and topology-related design variables) to 6.88694 kN (obtained by use of size, shape and topology-related design variables) also causes to a decrease in its load-carrying capacity from 1562597.4696 kN to 2895.1755 kN (see Tables 10–13). The other important reason behind the decrease in the load-carrying capacity of dome structure is associated with the search mechanism of PAGA. The strong search mechanism of PAGA causes to obtain the pareto solutions with higher diversity and convergence degrees with respect to the true pareto front. Thus, the optimal dome designs with relatively lower divergence degree are correspondingly omitted. However, the main feature of PAGA is its ability of archiving the pareto solutions throughout the evolutionary search. Thus, it is possible to pick the most requested optimal design from this archive.

Conclusions

This study proposes to introduce a new multi-objective optimization algorithm named PAGA, asses its computing efficiency, and evaluate the optimal designs obtained by the proposed design optimization procedure. In this regard, some important results are summarized as:

1) The genetic-related parameter values of PAGA are consistently adjusted throughout the evolutionary search. The adaptive adjustment of genetic-related parameter values leads to an increase in the computing capacity of PAGA. This feature of PAGA has a big advantage for not only structural engineering problems but also different problem of engineering field.

2) It is shown that PAGA has an efficient evolutionary search mechanism considering the quality measuring indicator values obtained for the optimization of benchmark mathematical functions, benchmark design examples and designs of geometrically nonlinear dome structures. Although PAGA achieves to make a close approximation to the true pareto front thereby exploring an increased number of pareto solutions, the computing time of PAGA is in the highest value than the other employed MEAs.

3) An inclusion of the shape-related design variables into the proposed optimization approach leads to economically a profit in the weight minimization of dome structure, but cause to a decrease in its load-carrying capacity due to the possibility of including a slender member into the construction of dome structure. The other reason behind the decrease in the load-carrying capacity of dome structure is related the search mechanism of PAGA. Because, the exploration of optimal designs with relatively lower divergence degrees during the evolutionary search may be ignored since PAGA forces to immediately approximate the current pareto front to the true pareto front. However, the designer has an opportunity of determining the most requested optimal design through the archiving feature of PAGA. Thus, it is also demonstrated that using the multi-objective optimization algorithm for the geometrically nonlinear dome structure increases the correctness degree in the evaluation of optimal designs for the designer.

Consequently, PAGA is recommended to optimize the design of geometrically nonlinear dome structures. In the next study, new design constraints regarded to the joint strengths of dome structures will be also involved into the current design constraints in order to investigate the variation in the quality of optimal designs.

References

[1] Saka M P. Optimum geometry design of geodesic domes using harmony search algorithm. Advances in Structural Engineering, 2007, 10(6): 595–606

[2] Talaslioglu T. Weight minimization of tubular dome structures by a particle swarm methodology. Kuwait Journal of Science & Engineering (Issue 1B), 2013, 1(1): 145–180

[3] Talaslioglu T. Multi-objective size and topology optimization of dome structures. Structural Engineering and Mechanics, 2012, 43(6): 795–821

[4] Talaslioglu T. Multi-objective design optimization of geometrically nonlinear truss structures. Kuwait Journal of Science & Engineering (Issue 1B), 2012, 39(2): 47–77

[5] Saka M P, Kameshki E S. Optimum design of nonlinear elastic framed domes. Advances in Engineering Software, 1998, 29(7–9): 519–528

[6] Talaslioglu T. Global stability-based design optimization of truss structures using multiple objectives. Sadhana, 2013, 38(1): 37–68

[7] Ghasemi H, Kerfriden P, Bordas S P A, Muthu J, Zi G, Rabczuk T. Probabilistic multiconstraints optimization of cooling channels in ceramic matrix composites. Composites. Part B, Engineering, 2015, 81: 107–119

[8] Ghasemi H, Park H S, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62

[9] Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258

[10] Srinivas N, Deb K. Multi-objective optimization using non-dominated sorting in genetic algorithms. Evolutionary Computation, 1994, 2(3): 221–248

[11] Steuer R E. Multiple Criteria Optimization: Theory, Computation, and Application. New York: Wiley, 1986

[12] Beume N, Naujoks B, Emmerich M. SMSEMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 2007, 181(3): 1653–1669

[13] Talbi E G. Meta-heuristics from Design to Implementation. John Wiley and Sons, 2009

[14] Carlos A C C. Multi-objective evolutionary algorithms in real-world applications: Some recent results and current challenges. In: Greiner D, Galván B, Periaux J, et al., eds. Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences. Computational Methods in Applied Sciences, Vol 36. Berlin: Springer, 2015, 3–18

[15] Zavala G, Nebro A J, Luna F, Coello Coello C A. Structural design using multi-objective metaheuristics: Comparative study and application to a real-world problem. Structural and Multidisciplinary Optimization, 2016, 53(3): 545–566

[16] Zavala G R, Nebro A J, Luna F, Coello Coello C A. A survey of multi-objective metaheuristics applied to structural optimization. Structural and Multidisciplinary Optimization, 2014, 49(4): 537–558

[17] Kaveh A, Rezaei M. Topology and geometry optimization of different types of domes using ECBO. Advances in Computational Design, 2016, 1(1): 1–25

[18] Kaveh A, Rezaei M. Topology and geometry optimization of single-layer domes utilizing CBO and ECBO. Scientia Iranica, 2016, 23(2): 535–547

[19] Kaveh A, Zolghadr A. Optimal analysis and design of large-scale domes with frequency constraints. Smart Structures and Systems, 2016, 18(4): 733–754

[20] Kaveh A, Talatahari S. Optimal design of Schwedler and ribbed domes via hybrid Big Bang—Big Crunch algorithm. Journal of Constructional Steel Research, 2010, 66(3): 412–419

[21] Deb K, Pratap A, Agarwal S, Meyarivan T. A fast elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182–197

[22] Beume N, Naujoks B, Emmerich M. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 2007, 181(3): 1653–1669

[23] Qingfu Zhang, Hui Li . MOEA/D: A multi-objective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 712–731

[24] Corne D W, Jerram N R, Knowles J D, Oates M J. PESA-II: Region-based selection in evolutionary multi-objective optimization. In: Proceedings of the 2001 Genetic and Evolutionary Computation Conference. San Francisco: Morgan Kaufmann Publishers, 2001, 283–290

[25] Zitzler E, Laumanns M, Thiele L. SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Proceedings of the 5th Conference on Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems. Athens: International Center for Numerical Methods in Engineering, 2001, 95–100

[26] Deb K, Mohan M, Mishra S. Towards a quick computation of well-spread pareto-optimal solutions. In: Proceedings of the International Conference on Evolutionary Multi-Criterion Optimization. Berlin: Springer, 2003, 222–236

[27] Yang X S. Engineering Optimization: An Introduction with Metaheuristic Applications. John Wiley and Sons, 2010

[28] JMetal. A Java Framework for Multi-Objective Optimization. Version 4.5. Spain: GISUM, Grupo de Ingeniería del Software de la Universidad de Malaga, Malaga University, 2019

[29] MOEAframework. A Free and Open Source Java Framework for Multiobjective Optimization, Version 2.12. USA: Dave Hadka and Others, 2019

[30] Foster N. Multiobjective Optimization and Genetic Algorithms with MATLAB. USA: CreateSpace Independent Publishing Platform, 2016

[31] PlatEMO. Evolutionary multi-objective optimization platform. China: BIMK, Institute of Bioinspired Intelligence and Mining Knowledge, Computer Science Technology College, Anhui University, 2019

[32] MATLAB. Matrix Laboratory, Version 2015a, USA: The MathWorks, Inc., 2015

[33] Talaslioglu T. A new genetic algorithm methodology for design optimization of truss structures: Bipopulation-based genetic algorithm with enhanced interval search. Modelling and Simulation in Engineering, 2009, 2009: 1–28

[34] Coello Coello C A. A short tutorial on evolutionary multi-objective optimization. In: Proceedings of the 1st International Conference on Evolutionary Multi-Criterion Optimization, EMO 2001. Heidelberg: Springer, 2001, 21–40

[35] Coello Coello C A, Pulido G T. Multiobjective structural optimization using a microgenetic algorithm. Structural and Multidisciplinary Optimization, 2005, 30(5): 388–403

[36] Rao S S. Game Theory approach for multi-objective structural optimization. Computers & Structures, 1987, 25(1): 119–127

[37] Kaveh A, Talatahari S. Geometry and topology optimization of geodesic domes using charged system search. Structural and Multidisciplinary Optimization, 2011, 43(2): 215–229

[38] Carbas S, Saka M P. Optimum topology design of various geometrically nonlinear latticed domes using improved harmony search method. Structural and Multidisciplinary Optimization, 2011, 5: 1–23

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