Particle swarm optimization: an alternative in marine propeller optimization?

ABSTRACT This article deals with improving and evaluating the performance of two evolutionary algorithm approaches for automated engineering design optimization. Here a marine propeller design with constraints on cavitation nuisance is the intended application. For this purpose, the particle swarm optimization (PSO) algorithm is adapted for multi-objective optimization and constraint handling for use in propeller design. Three PSO algorithms are developed and tested for the optimization of four commercial propeller designs for different ship types. The results are evaluated by interrogating the generation medians and the Pareto front development. The same propellers are also optimized utilizing the well established NSGA-II genetic algorithm to provide benchmark results. The authors' PSO algorithms deliver comparable results to NSGA-II, but converge earlier and enhance the solution in terms of constraints violation.


Variables and notation of optimisation algorithms β
Reduction parameter N G Number of time steps N I Number of particles in the swarm p mut Mutation probability v ij Velocity vector of the ith particle w Inertia weight x ij Input parameter vector of the ith particle x pb ij jth input parameter, personal best of the ith particle x sb j jth input parameter, best solution so far xPB Swarm archive of personal best solutions xSB Archive (size = 1) of best solution obtained so far

Introduction
The combination of more available computer resources in personal desktop computers and emerging readily applicable algorithms inspired by biological phenomena have enabled optimization in many real-world problems. Especially the genetic algorithms (GAs) introduced in the mid-1970s, e.g. Holland (1975), are suitable in complex and constrained engineering design processes. Unlike classical optimization, GAs are not based on the gradient and continuity of an objective function, and most importantly for multi-objective design tasks, they provide a set of Pareto optimal solutions, from which the final design may be chosen according to the designer's strategy.
One of the most popular GAs is the non-dominated sorting genetic algorithm-II, NSGA-II (Deb et al., 2002), which has successfully been used in marine propeller applications, e.g. Jung et al. (2007), Puisa and Streckwall (2011) and Ghassemi and Zakerdoost (2017). However, GAs have their shortcomings: they are population based and require a large number of evaluations before convergence can be achieved and are often hindered by applied constraints which isolate feasible solutions. Marine propeller design demands generally multi-disciplinary evaluation and consideration of numerous requirements. The most limiting factor is, however, the lead time of a propeller design, which restricts the number of design alternatives; an optimization algorithm is thus under an obligation to manage to converge with fewer evaluations, which aggravates the use of population based optimization algorithms. The problem is on the other hand too complex to solve the optimization with deterministic approaches such as linear programming or gradient based methods. Hence, in this article a variation of the particle swarm optimization (PSO) algorithm is investigated to enhance the convergence and search efficiency, considering constrained optimization in propeller design.
PSO emerged in the mid-1990s, introduced by Kennedy and Eberhart (1995) and Eberhart and Kennedy (1995). The concept relies, similarly to GAs, on a population, yet the set of prospective solutions (particles) are 'flying' through the hyperspace, and the concept incorporates the idea of a social component that controls the development of the swarm. Kennedy and Eberhart (1995) developed the algorithm while attempting to simulate the behaviour of swarms. Thus, it is biologically related to the motion of, for example, birds and their ability to search the environment to find food sources and avoid predators by information sharing. This motivates the high search efficiency of PSO. Hassan, Cohanim and de Weck (2005) found PSO to result in the same high solution quality, but with less effort for function evaluation than the applied genetic algorithm. Li (2003) demonstrated a high consistency of his PSO algorithm, which converged in 10 out of 10 runs to the Pareto front, and which is an improvement on NSGA-II counterpart. However, PSO appears to be problem dependent, regardless of whether the margin in computational efficiency is large or small (Hassan, Cohanim and de Weck, 2005).
In the current article, three variations of the PSO algorithm are proposed and tested on generally excepted optimization benchmark functions (Deb et al., 2002). Their use for application in the shape optimization of marine propellers is evaluated, which is typically a multi-objective optimization problem to both improve performance and reduce nuisance. Moreover, depending on the type of vessel, the propeller design problem changes character, which is a challenge for a general optimization procedure. The PSO algorithms presented here are extended for multi-objective optimization through a non-dominated sorting procedure, and two different perturbation methods are compared to introduce diversity, all in an attempt to increase the robustness of the methodology.
For the performance measure of their deterministic PSO, Pellegrini et al. (2014) utilized a measure of the proximity of a solution to a reference Pareto front. This measure provides a straightforward comparison between the algorithm's performance on test functions. However, the true Pareto front is not explicitly known for most engineering problems and has to be determined by extensively enumerating the design space (van Veldhuizen and Lamont, 1999). This type of result assessment is thus not applicable in engineering optimization. Therefore, the focus of this article is the evaluation of the final Pareto optimal solutions, the convergence and the geometry. The assessment is based on comparing the trends of the PSO variants by comparing the median of each generation and the geometrical features of the denoted optimal solutions. NSGA-II is used as the reference for the performance measure.
The analysis tools used to evaluate the propeller performance are introduced first, followed by an outline of the developed PSO algorithm modifications. The optimization set-up is described in Section 2.3. The first part of the results section is then dedicated to presenting the results on the benchmark Zitzler-Deb-Thiele function (ZDT2) (Zitzler, Deb and Thiele, 2000) and the Osyczka and Kundu function (OSY) (Osyczka and Kundu, 1995), while the second part presents the results of the propeller optimizations, categorized into performance, convergence and geometry of the denoted optimal propellers.

Propeller design objective and analysis tools
Each propeller design is unique for the particular ship and requires a comprehensive consideration of several constraints such as strength, noise and vibration, as well as cavitation. Cavitation directly influences the propeller efficiency and the propeller induced pressure pulses, which are normally the two performance measures, thus forming the optimization objectives. The common philosophy for many years has been to avoid cavitation within typical operating conditions as far as possible. Nevertheless, the demand for highly efficient propellers makes this no longer feasible. Today, the design is exploited for efficiency, which requires a smaller margin for erosive cavitation, and thus cavitation becomes a design factor that needs to be considered carefully.
The development and behaviour of cavitation are physically profoundly complex, and a detailed assessment demands highly unsteady and computationally costly calculations. Yet during the design process, within a limited time frame, it is necessary to apply less accurate but faster methods, e.g. potential flow methods. These provide fairly accurate performance predictions (as in e.g. Lu et al. [2012]), and, with elaborated constraints, have been shown to be sophisticated enough to perform commercial design tasks. Thus, here the vortex lattice method 'MPUF-3A' (He, Chang and Kinnas, 2010) is utilized for the propeller performance, including the prediction of sheet cavitation. The pressure pulses are calculated using a boundary element method, 'HullFPP' (Sun and Kinnas, 2007), which solves the diffraction potential on a flat hull dummy surface to compute the pressure amplitudes of the first three blade harmonic frequencies. It should be remarked, however, that the performance of the optimization methods does not per se depend on the chosen tools.

The multiobjective PSO algorithm
Since Kennedy and Eberhart (1995) presented the PSO algorithm, several modifications have been proposed to improve the original algorithm, e.g. the inertia weight by Shi and Eberhart (1998) to balance the global and local search. Li (2003) presented a modification for multi-objective handling-a non-dominated sorting PSO related to NSGA-II, since it uses the same non-dominated sorting and rank allocation. Similarly to Li (2003), a non-dominated sorting PSO is suggested in this article, which utilizes elitism and dominance comparison with crowding distance to sort the entire swarm. However, here the global best solution is retained while iterating until a better solution appears to replace it, unlike Li (2003), and only then when the best solutions of the current particle are compared and the personal best is determined. Li (2003), on the other hand, also compares with all the other particles' bests. Three approaches, PSOp, PSOv and PSOt, are proposed here which differ in determination of best solutions and from which PSOt in itself contains dominance ranking to enhance the selection towards non-dominated solutions.
The standard PSO algorithm begins with an initialization of particles in the design space in which each of the N I particles is assigned a random position (Equation 1) and velocity (Equation 2) in the n-dimensional design space: Once all particles are assigned with a position, their initial objectives and constraints can be calculated. Subsequently, the particle's personal best location (x pb ij ), and the best solution (x sb j ) obtained so far, can be determined. The standard algorithm, which is in general a single-objective approach, readily compares the objective value and replaces the best if a better solution is obtained. These best solutions can also be referred to as a cognitive component (personal best) and a social component (best solution so far in the complete swarm). When updating the particle's velocity for the next time step (Equation 3), the cognitive and social components determine the degree of a particle's self-confidence and its trust in the ability of other members in the swarm to find good solutions, The velocity update contains q and r as random numbers [0, 1], and the constants c 1 and c 2 which are commonly set to two to achieve a statistical mean of one for the terms c 1 q and c 2 r (Wahde, 2008). Although this parameter selection is not necessarily optimal for the problem, a variation is beyond the scope of this article, but is discussed, for example, by Peri and Tinti (2012). The inertia weight w (Shi and Eberhart, 1998) is introduced to assign significance to either the cognitive and social component or to exploration. When w > 1, the particle velocity is assigned a higher significance, and the particle is attracted to explore the design space rather than exploit formerly found best solutions. Since exploration is more important at an early stage of the optimization, it is natural to adapt the inertia weight during optimization advancement towards exploitation. This is achieved by gradually reducing w by a constant factor β ∈ ]0, 1[. The new velocity is restricted such that |v ij | < v max , and eventually the particle position is updated. Figure 1 depicts the two-dimensional interpretation of the velocity update on particle i from time step k−1 to time step k. The resulting position is hence a combination of the cognitive and social components and the inertia at time step k.
In this article, a PSO algorithm is proposed for multi-objective optimization that introduces ranking according to non-domination and crowding distance. The pseudocode for the proposed nondominated PSO algorithm is shown as Algorithm 1. After the random initialization of particles and their evaluation (Steps 1 and 2 in Algorithm 1), each particle is ranked and assigned with a crowding Number of particles per time step Require: w > 1 Inertia weight Require: w b < 1 weight bound 1: P 0 ← Initialize Random initialization of particles' positions and velocities 2: Evaluate(P 0 ) 3: AssignRank&Distance(P 0 ) 4: x PB , x SB ← UpdateBests(P 0 ) 5: for k ← 1 to N G do 6: UpdatePositions(C i ) 10: Mutation of particles' positions or velocities 11: AssignRank&Distance(C) 13: 14: x PB , x SB ← UpdateBests(P k ) distance factor as in NSGA-II, before the positions of the best solutions (x pb ij and x PB j ) are determined. The distance in each objective dimension is calculated as the average distance between one particle and its adjacent particles in ascending order. The crowding distance is then the accumulation of distance values in each objective, for one particle (Deb et al., 2002). The following steps (6-14) are then repeated until the maximum number of iteration steps (N G ) is accomplished: introduce a candidate swarm (C), update the velocities and positions in C (8 and 9), apply modifications by mutation (10), evaluate and rank C (11 and 12), combine the candidate swarm and the current swarm and sort the particles into the new swarm P k according to their level of non-domination and distance (13 and 14).

Tournament selection based on non-domination rank
The new swarm C is introduced as a copy of the previous swarm P k−1 and aims to preserve elitism. Updating of the velocities and positions as well as modifications are only applied on each particle in swarm C, which is equivalent to the child generation in GAs. Subsequent to the updating of positions and velocities, modifications are introduced by mutation (Step 10). Two of the proposed algorithms (PSOp and PSOt) modify the particle position and one (PSOv) the particle velocity. As in NSGA-II, the perturbation is caused by a polynomial mutation operator (Deb and Agrawal, 1995).
The update of bests is crucial to determining the best performance of variants. Compare Table 1 for an overview of the applied update procedures. The first approach, denoted as the PSOp algorithm, consists, beyond the mutation of particle position, also in the updating of the bests by an arbitrary single objective approach, i.e. updating the best particles if any of the objectives is improved. In PSOv, the second update approach of x pb ij and x sb j evaluates an aggregate objective function, as a linear combination of the system performance f ij and x sb j , respectively. However, this leads to a biased relation between the objectives when their magnitudes differ. Even if normalized values are applied, bias may still not be dispelled when the cluster of variants used for the normalization contains outliers. Hence the third approach, PSOt, assesses each particle according to its non-domination rank and crowding distance in a tournament selection. xSB ← Tournament(xSB, C i ) return xPB, xSB Algorithm 2 describes the update procedure using tournament selection. For the determination of best solutions, two new archives are introduced; one of the size of N I particles for all personal bests (xPB) and a second archive containing only the global best solution found so far (xSB). The nondomination rank and crowding distance of each particle in the xPB archive is assigned with respect to the new swarm C before tournament is conducted with the particles in C. This ensures the correctness of data before allowing the solution C i to compete with its former best xPB i and the former best xSB.

Optimization application examples
In the following, the optimization behaviour of the proposed algorithm variations is investigated on five benchmark propellers of different designs and for different types of vessels (see Figure 2). The designs are provided by the Rolls-Royce Hydrodynamic Research Centre (RRHRC), Kristinehamn, Sweden, and correspond to a preliminary design. Propeller parameters like the diameter and the operating condition at the design point are, however, already determined. The design task at hand is thus  about finding the wake adapted optimal blade geometry that maximizes the propeller efficiency, minimizes the hull pressure pulses and fulfils the design requirements. Starting from an early project phase, this task rarely leads to a satisfactory design in a single optimization step. The intention of the exemplified optimizations, with selected geometry variation and limits, is rather to guide the designer faster to the next design iteration to finalize the design synthesis. At this point in most design processes the geometry is described by standard distribution curves based on experience and the company's design philosophy. Parameters like the blade area ratio or the skew angle at the blade tip are used to modify the corresponding curves along the radius, and thereby change the propeller geometry. The process is here implemented in the automated optimization procedure, where in total eight parameters are applied in the optimization to change the distribution curves of Chord, Skew, Blade thickness, Rake and Pitch (see Table 2). These parameters are only partly known in the design problem, while Chord, Skew, Blade thickness and Rake are prescribed by the parametrized distribution curves, and the radial distribution of pitch and the radial and chord-wise distribution of camber are to be determined with respect to the inflow. Here, a vortex lattice-based propeller lifting surface design method (Greeley and Kerwin, 1982) is applied, which also uses a lifting line method (Lerbs, 1952) to ascertain that pitch and camber are optimally aligned with the given circumferential mean inflow. Thus the parameters to control the pitch are factors to unload the root and the tip region, respectively, rather than the actual pitch distribution. Absolute values of decision variables and objectives are confidential to protect the commercial interests of Rolls-Royce AB.
In the propeller analysis problem, constraints are selected according to the Rolls-Royce common design procedure. This includes a constraint for a minimum K T to meet the required thrust and constraints for the dynamic blade stress which is calculated at three different positions on the blade, and from which the maximum von Mises stresses are taken at MCR conditions. The cavitation predicted by MPUF-3A is constrained by the maximum cavity volume and cavity length on the blade suction side. For all propeller cases, the constraint thresholds are set to improve the design by 5% of the initial values. Furthermore, constraints are applied for the corresponding class rules and geometric circumstances such as blade collision control (for CPP). The optimization problem is thus to find the best compromise of eight input parameter to min(f 1 (x), f 2 (x)) as subject to g i (x) ≥ ∨ ≤ limit i , i = 1, . . . , J = 6 inequality constraints and box limited decision variables. An overview of the applied parameters, objectives and constraints and their limits is given in Table 2.
The designs represent contemporary designs of merchant and pleasure vessels. The different ship types imply different margins of the objective function and thereby deviating Pareto fronts. Such variety demands an optimization algorithm of high flexibility, that can deal with different levels of possible improvements and constraints violation. The designers perspective on the objectives are possibly different for a container vessel than for a Ro-Pax ferry, but the optimization should still lead to a Pareto front to select from.

Optimization benchmark functions
Two benchmark functions are used to test the modifications of the PSO algorithm, considering practical settings for the optimization of propeller geometry. The test functions ZDT2 (Zitzler, Deb and Thiele, 2000) and OSY (Osyczka and Kundu, 1995) are selected because of a large number of decision variables and the characteristics of the Pareto optimal front which may prevent the algorithm from finding Pareto optimal solutions (Zitzler, Deb and Thiele, 2000). A practical set-up for an optimization that utilizes computationally expensive evaluations requires that the algorithm is able to find the Pareto optimal solutions with fewer individuals and converges as early as possible. Hence, an algorithm (e.g. NSGA-II, PSOp, PSOv, PSOt) is prudent if it has the ability to find variants close to the Pareto front, at an early stage, and with a small number of individuals. The number of individuals per generation and the number of iterations are therefore limited to 20 and 500, respectively. The euclidean distance between a variant and an approximated Pareto front (P*) is evaluated to measure the performance. Figure 3 provides the minimum distance to P* for each generation of both benchmark functions. This illustrates an advantage of all PSO algorithm variations for ZDT2 which reduce the minimum distance quicker and with lasting consequence compared to NSGA-II. However, the assessment requires a metric to calculate the non-uniformity of the obtained Pareto optimal solutions, e.g. Deb et al. (2002). Consulting the scatter plot of the generated variants in objective space (provided in Figure A1 of the online Supplemental data, which can be accessed at http://dx.doi.org/10.1080/ 0305215X.2017.1302438) clarifies that particularly PSOt manages to get close to P* with an evenly distributed set of variants, even closer than NSGA-II. And from the colour scale it is obvious that these variants are obtained earlier than the closest solutions from NSGA-II. PSOv converges prematurely outside the range shown towards one end of the Pareto front. PSOp outperforms PSOv in terms of finding the Pareto optimal solutions, even with an arbitrary single-objective update method for the best particles.
The more complex Pareto front of the constrained OSY benchmark function is not entirely located by either NSGA-II or the proposed PSO algorithms. Yet the PSOt algorithm provides the smallest distance to P* up to generation 100 (Figure 3(b)) and the PSOv algorithm accumulates the most variants close to P* and considers the constraints. Particularly PSOp violates the constraints and generates several variants beyond the constraint limitations in the lower left corner of the figure ( Figure A2 in the online Supplemental data).
The optimization configuration with minimal variants and constraints is a difficult task for all algorithms. However, the results of the test functions indicate a potential benefit of the PSO algorithms and motivate their application in marine propeller optimization. PSOv is advantageous on the OSY test function while PSOt performs best on the ZDT2 function where PSOv is defeated. All PSO algorithms together outperform NSGA-II in terms of finding solutions close to the Pareto front at an early phase of the optimization. This can provide the engineer with valuable information on the parameter-objective relation in an early project phase before the optimum needs to be known.

Optimization of propeller design
Optimizations are performed for the five test propellers (Figure 2), utilizing the three PSO algorithms (PSOp, PSOv and PSOt). Results are compared with those of the NSGA-II algorithm. Thus, in total 20 optimization runs were conducted, each with the same settings for population size (N I = 24), number of generations (N G = 40), mutation probability (p mut = 0.75) and for the PSOs' inertia weight (w = 0.4). A large number of generations and smaller population size satisfy mainly NSGA-II requirements for exploration and exploitation of 'good' variants. It should be noted that PSO algorithms possibly perform better with a larger swarm size. The results are summarized by medians of the population generations and swarm size, respectively. The median is considered to be a more robust measure of the location parameter for a skewed or non-normal distribution, compared with the mean and standard deviation. Deploying the median as the location measure reduces the importance attached to outliers. The dispersion is accordingly given by the median absolute deviation (MAD), MAD = median i |X i − median j X j | . To evaluate the performance of the optimization algorithm, the assessment is classified into the categories performance, convergence and geometry. This classification facilitates the rating of the optimization, and provides a confidence level for the ability to actually find the global optimum. A converged geometry will confirm an optimum, which ideally should be the same regardless of the optimization algorithm used.

Performance
Performance refers to the capability of the method to find a valid and optimal solution. Thus the analysis is initialized by discussing the validity of the results, i.e. to what extent the algorithms find feasible solutions. Then the Pareto fronts and the feasibility of the Pareto efficient solutions are compared.
For validity, the validity ratio between valid variants and the population size ( Figure 4) is complemented by an analysis of the behaviour in more detail using an aggregated constraints violation measure (CVM, as proposed by Vesting, Gustafsson and Bensow [2016]). Each result of the six constraint functions g i (x) for an individual is assigned a constraints violation which is relative to the maximum or minimum values of g i (x) in the current population (Equations 4 and 5). Accordingly, a CVM becomes negative in the case of a violation of the constraints, zero if the constraint value matches the limit, and positive if the constraint value is feasible. A solution can therefore balance a violation in one constraint by one or more positive (feasible) constraint violations, Eventually, a variant is considered to be infeasible if the CVM is negative. In Figure 5, the CVM value is presented by the generation median. A first general observation is that the ability of an algorithm to find valid variants is case dependent and varies considerably between the five propellers presented here. A high validity is normally achieved within the first generations but, as the optimization proceeds, validity is sacrificed for exploration. This is reasonable since it relates to the non-dominated sorting algorithm, which prefers a feasible solution over an infeasible but possibly non-dominated solution. Once enough valid solutions are reproduced, the algorithm starts exploring towards the boundaries, and thereby again generates more invalid variants. Opt-5 in Figure 4(b) provides an example of this behaviour. Overall, the NSGA-II algorithm performs best, with the highest ratio of valid results in all cases except Opt-1. For Opt-1, NSGA-II provides a dramatic example of the decrease in the number of valid designs as optimization proceeds (Figure 4(a)). The validity ratio suddenly declines from about 0.6 to 0.3 between generations 12 and 13. Analysing this in detail demonstrates that this is the consequence of a continuous decrease in K T (the constraint of minimum thrust, Figure 5(c)). The algorithm counteracts this to maintain the K T , which yields designs with increased cavitation volume-by 5% at generation 13-with the lowest stress reduction of all the algorithms. The combined performance of K T , CavVol and blade stress yield a negative CVM and thereby a majority of infeasible variants ( Figure 5(a)).
The PSO algorithms on the other hand yield high validity for optimization cases Opt-1 and Opt-2, but do not perform as well for cases Opt-5 and Opt-7. In the first case, the number of valid solutions doubled within the first 10 generations for all PSOs, which is significant. PSOv provides, e.g. for Opt-1 and Opt-2, an improvement of validity within the first five generations (Figure 4(a) and Figure B1 in the online Supplemental data). However, the ability to maintain or even slightly improve the validity declines with the progression of iterations.
PSOp, with mutation of the particle position, improves stagnation. Once a decreasing trend is followed, PSOp features possible restoration towards higher validity, as in optimization case Opt-1 after generation 10 and upwards or in case Opt-2 at generation 15 (see Figure B1 in the online Supplemental data).
PSOp in optimization case ) and PSOt for case Opt-7 (see Figure B1 in the online Supplemental data) indicate results where no valid solutions were obtained. Yet PSOt improved the constraints violation measure (Figure 5(b)) primarily by reducing blade stresses by 30%, which is significant compared with the NSGA-II and PSOp results (see the online Supplemental data). For this optimization case the maximum blade stress is the most severe violation of constraint limits. Though PSOt converges towards a lower level of objective improvement, it thereby finds the true Pareto front for the given constraints and parameters. The PSOt result is therefore considered of better quality since it uncovers important information for the designer to determine the next design iteration.
Both validity and optimality are shown in Figure 6. This plot unifies the exploration of the algorithms in objective space and the feasibility of the solutions. This makes it possible to determine which algorithm explored which region of the objective space. Ideally all algorithms will accumulate their valid individuals along the Pareto front during the progress. However, this is only explicitly the case for Opt-2, and to some extent for Opt-4. Figure 6(a) clearly shows a trend for invalid solutions with increasing η. All algorithms explored that region, ascertained the violations, and finally converged to valid CVMs. However, they arrived at different locations with their final generation, which is indicated in Figure 6 by the median of the last generation (N G ).
From Figure 6(a) for Opt-2, an advantage of the NSGA-II algorithm over the PSO algorithms can be noted. Almost all individuals that are close to the Pareto front are generated by NSGA-II. Correspondingly, the median of the final generation (N G NSGA-II) is closer to the Pareto front than any of the PSO algorithms. This also becomes clear from Figure D1 in the online Supplemental data, where the actual Pareto optimal solutions are plotted. The Pareto fronts of optimization case Opt-2 are similarly densely distributed and have 8 to 12 particles along the front, the densest among the test cases. NSGA-II provides the best trade-off front.
In summary, it should be pointed out that the overall performance of the algorithms can be considered successful. In Table 3, the performance of selected solutions for each case and algorithm is summarized. This shows that each optimization case could be improved by all algorithms for both objectives. Each result is given relative to its corresponding initial performance. All variants are taken from the Pareto front except for cases Opt-2 and Opt-5. For these two cases, the solutions marked '*' are selected to fulfil the constraint of thrust inequality (K T ≥ K Baseline T ). Pareto optimal solutions do not always manage to avoid violation for some constraints although the CVM is positive. It should also be noted that the individual optimal designs given in Table 3 cannot reflect the overall algorithm performance. The quality of the final solutions is not entirely reflected by this comparison; the convergence rate is only loosely given by the number of variants -particularly in optimization case Opt-7, where the Pareto optimal solutions of PSOt do not correspond the converged solutions which yield a more favourable design due to stress reduction, than the Pareto optimal designs. NSGA-II, on the other hand, balances its constraint violation measure by an increase in K T of 3.5%, and converges further towards solutions with higher-efficiency low-pressure pulses yet unacceptable blade stresses. In general, the PSO algorithms converge earlier to a certain geometry and therefore require less evaluation.
Interrogating the progress of the Pareto fronts between the generations and swarm iterations respectively, indicate the development of the algorithm. In Figure 7 such a Pareto front development is exemplified by the optimization case Opt-7 for NSGA-II and PSOt. The side-bar scale in this figure is related to the generation, i.e. from first to last generation (0, . . . , 1). NSGA-II presents a progress towards higher efficiency and lower pressure pulses (Figure 7(a)) while PSOt develops away from the initial Pareto front with high improvements in both objectives when the algorithm progresses. The PSOt algorithm develops clearly to a different Pareto front which, as pointed out above, satisfies the constraints.

Convergence
For each algorithm, the progress is shown for objective medians of the algorithms through the generation median (see the online Supplemental data). A final value of the objectives is approximated for the majority of test runs. NSGA-II provides the most consistent trade-off between the two objectives, yet often NSGA-II is attracted to improve p more than η. The PSO algorithms generally achieve a more pronounced improvement for one objective than the other, e.g. for Opt-2 (Figures 8(a) and 8(b)), where PSOp achieved the highest η, but the least reduction in pressure pulses, and PSOt and PSOv developed towards the lowest pressure pulses, but actually achieved a decrease in efficiency. This behaviour is very clear for PSOv despite the best solutions being determined based on the aggregated objective function. The plots in Figure 8 and in the online Supplemental data also reveal the preserved behaviour of the PSO algorithms to develop a clear and steep trend towards a certain objective value at the beginning of the progress. This abates rapidly within the first 10 generations, and the algorithms may often converge prematurely. This is particularly the case for the PSOv algorithm. In Figure 8 and the online Supplemental data, the tendency of PSOv is to settle within the first five generations. PSOv has also, together with PSOt, the smallest dispersion in the last generations, which is especially clear for Opt-2 and Opt-7 (see the online Supplemental data), which is the result of an early and distinct convergence of these algorithms.
Comparisons between the median absolute deviation of the two objectives in Figure 9 and the online Supplemental data frequently show a higher dispersion obtained for the pressure pulse improvement than for the efficiency. The highest dispersion among the optimization cases can be found for pressure pulses in case Opt-5 (Figure 9), which also provides the smallest relative improvements on pressure pulses (see Table 3) and the smallest validity ratio (see Figure 4). In this case, as in all other optimization runs, a high dispersion is achieved particularly by NSGA-II and PSOp.
The high dispersion is certainly related to a higher margin for the pressure pulse improvement, but the pressure pulse improvement ratio (IR) between the final swarm P N G and P N G −1 of each optimization indicates that pressure pulses have not settled completely ( Figure 10). From Figure 10, a difference in pressure pulses up to 5% is repeatedly noticeable between the generations, while the efficiency varies only within less than 1%.

Geometry
The development of geometry parameters is presented over the progress of the optimization as the generation median in Figure 11 for the blade area ratio (EAR). The algorithms yield similar geometrical modifications in only 5 out of 40 possible parameter-case combinations. A similar geometry is achieved in optimization cases Opt-2 and Op-4 ( Figure 11). For these cases there is a very rapid convergence within the first 10 to 15 generations towards the parameter limits.
In general, it can be noted that the algorithms frequently agree on the final geometries, whereas one of the PSOs involved settled very early in the optimization process towards the parameter limits.   Such behaviour of a PSO is then enforced through a high importance of that parameter, e.g. the EAR parameter in Opt-5. Comparing the EAR parameter development and the objective behaviour in Figure 12 shows a sudden increase of EAR for NSGA-II for the last four generations, which yields a similarly sudden decrease in efficiency. Comparing the geometry development between the different algorithms, the most frequent agreement of parameters is achieved between NSGA-II and the PSOp algorithm. Both converge in total 12 times to the same final geometry changes.
Even if the algorithms did not converge towards the same geometry in all cases, similar trends between the test cases could be observed. For instance the loading of the blade, controlled by parameters for unloading the tip and the root, develops a new balance during the optimizations. For Opt-1 and Opt-2, a higher unloading of the root leads to a reduced unloading at the tip, and vice versa for cases Opt-5 and Opt-7. Only in case Opt-4 are both root and tip further unloaded.
Further trends are observed regarding skew and EAR which hold for several cases. Propellers with initially high values for skew (Opt-1 and Opt-5) and EAR (Opt-5) tend to reduce their values during the optimizations. The influence of EAR on the objectives is particularly obvious and almost linear in case Opt-5. In this case, NSGA-II yields designs with the smallest pressure pulses (the development of objectives for this case is provided in Figure E1 of the online Supplemental data), only by rake increase, thickness reduction and unloading the blade tip, although the skew develops towards a smaller angle at the tip by a reduction of 6%. In case Opt-7, PSOt is the only algorithm that reduces the blade stresses at the mid-span trailing edge by a 10% increase in EAR and an unloaded blade tip; consequently η declines.

Discussion
In this article, three versions of a multi-objective PSO algorithm adapted for constrained propeller design have been presented. These three versions differ in the application of disturbance and the nomination of best solutions. Further, a classified assessment of the optimization outcome in terms of performance, convergence and geometry by interrogating the median values of parameter, objective and decision variables has been focused upon. It should, however, be pointed out that the optimizations performed are intended to be applied in an early design cycle, with parameters that are too coarse to finalize the design.
For the update of the best solutions, two of the proposed PSO algorithms utilize either a pure single objective or an aggregate objective function, as applied e.g. by Pellegrini et al. (2014). The corresponding results showed partly premature convergence and convergence towards an improvement of only one of the objectives. Therefore, tournament selection based on the non-dominated rank and distance was introduced to the algorithm to maintain a selection pressure towards non-dominated solutions as nominated best solutions for the PSO cognitive and social contributions.
Finally, the performance of all algorithms was evaluated in comparison with the optimization results of the benchmark algorithm NSGA-II. A true Pareto front comparison, as used by Li (2003) and Pellegrini et al. (2014) cannot be determined for the applied problem definition of propeller optimization. However, the NSGA-II algorithm can be considered as a mature evolutionary optimization algorithm, and is frequently used as a benchmark, e.g. by Cabrera and Coello Coello (2010).
Dispersion given by the median absolute deviation (MAD) indicates whether the algorithm is still in an exploration phase, as in Opt-5. The main development of the objectives or parameters could be described clearly. The PSOv algorithm suffers from a significantly smaller diversion, resulting in an insufficient exploration. Instead, the PSOp algorithm showed a higher MAD, and hence an improved capability to enhance the validity during the algorithm's progress. Hence, the dispersion criteria together with an improvement ratio (IR) can also be interpreted as a measure for convergence. A high MAD together with an IR = 1.0 imply results that are not yet converged. This can also be seen from the generation-wise Pareto front determination, which gave a quantitative overview of the development of the algorithm's progress if its advance towards an expected Pareto front stagnates or if it develops a deviant Pareto front. A more qualitative measure of convergence rate could possibly be appointed from the distance between the generation-wise Pareto fronts.
Most frequently, the PSO algorithms provided extreme solutions for one objective, whereas NSGA-II always provided the best trade-off between the objectives. Geometrical agreement between the algorithm outcomes could be found particularly when a high importance of input parameter was assigned and in the case when the algorithm settled at an early stage of the optimization. Hence the PSOv and PSOp algorithms offer a semi, non-dominated optimization that focuses on one objective. This can, however, be a useful quality for propeller design in the case of the two objectives propeller efficiency and pressure pulse, where the margin of possible improvements is often significantly higher for pressure pulses.
Additionally, PSOt was tested with a 30% larger swarm size on case Opt-2, and provided the exact same development regarding objectives. This indicates a sufficiently large swarm size, which is significantly smaller than that applied in e.g. Pellegrini et al. (2014) and comparable with that in Cabrera and Coello Coello (2010). The PSOt algorithm provided the best solutions among the PSO algorithms, whereas PSOp and PSOv could possibly have improved their performance by utilization of a larger swarm size. The algorithm settings could have been optimized further for each algorithm, but with a constant setting for all algorithms, comparison is straightforward.
It should be noted that in all cases there was a higher possible improvement in pressure pulses than in efficiency. This could partly be related to the fact that the pressure pulse objective was further supported by constraints on cavitation, which has a significant influence on pressure pulses. However, in two cases there was either virtually no cavitation predicted (Opt-4) or only wetted simulation applied (Opt-7).

Conclusion
The results of the proposed multi-objective PSO algorithms lag behind those of NSGA-II in terms of finding the best trade-off quality. However, the proposed PSOt yields comparable results. PSOt in two cases outperforms the benchmark algorithm (NSGA-II) based on the Pareto solutions. It also created comparable feasible sets of solutions and additionally managed severe violations of an active preferable constraint. Thus PSOt is highly suitable in constrained propeller optimization.
The proposed assessment parameters, the generation median, improvement ratio or validity ratio can be readily monitored by the designer during an optimization to ensure optimization quality. There is, however, no evidence that an improved optimization algorithm guarantees a better result. It is still the designer's endeavour to set up a suitable combination of objectives, constraints and decision variables to achieve a successful optimization. With an improved and faster algorithm, the designer would be able to proceed towards the next design cycle in a shorter time.