Computational techniques are widely used to solve complex optimization problems in different fields such as engineering, finance, biology, and so on. In this paper, the Human Conception Optimizer (HCO) is proposed as a novel metaheuristic algorithm to solve any optimization problems. The idea of this algorithm is based on some biological principles of the human conception process, such as the selective nature of cervical gel in the female reproductive system to allow only healthy sperm cells into the cervix, the guidance nature of mucus gel to help sperm track a genital tracking path towards the egg in the Fallopian tube, the asymmetric nature of flagellar movement which allows sperm cells to move in the reproductive system, the sperm hyperactivation process to make them able to fertilize an egg. Thus, the strategies pursued by the sperm in searching for the egg in the Fallopian tube are modeled mathematically. The best sperm which will meet the position of the egg will be the solution of the algorithm. The performance of the proposed HCO algorithm is examined with a set of basic benchmark test functions called IEEE CEC-2005 and IEEE CEC-2020. A comparative study is also performed between the HCO algorithm and other available algorithms. The significance of the results is verified with statistical test methods. To validate the proposed HCO algorithm, two real-world engineering optimization problems are examined. For this purpose, a complex 14 over-current relay based IEEE 8 bus distribution system is considered. With the proposed algorithm, an improvement of 50% to 60% in total relay operating times is observed comparing with some existing results for the same system. Another engineering problem of designing an optimal proportional integral derivative (PID) controller for a blower driven patient hose mechanical ventilator (MV) is examined. A significant improvement in terms of response time, settling time is observed in the MV system by comparing with existing results. Analytical optimization methods are complex and time-consuming processes to get an optimal solution of a complex optimization problem. Again, heuristic optimization methods are problem-dependent techniques7. They need particularities of an optimization problem. They are too greedy to get trapped in a local solution. Meta-heuristic methods are problem independent. They can provide an acceptable solution without guaranteeing optimality8. A simple concept can be implemented easily to make a metaheuristic algorithm to solve a complex problem quickly. Such algorithms can be applied in any optimization problem without altering the structure of the algorithm. In comparison to analytical based optimization algorithms, a metaheuristic algorithm is free from derivation action to find optimal solution. Thus, a real time problem can be solved by any metaheuristic algorithm where it needs only the information of input and output of the system9. Therefore, researchers are giving priority to develop metaheuristic algorithms using natural concepts such as the concept of evolution, the behaviour of natural creatures and hunting procedure followed by animals, and so on11.The optimization method is a numerical computational method to find the optimal solution of a real-time problem in a diversified field such as engineering, management, finance and so on11. In the exploitation phase of the metaheuristic algorithm, a new solution is produced based on the best solution available in the population. Thus, metaheuristic algorithms use an exploration and exploitation process to avoid local trapping problems and converge towards the optimal solution. Moreover, by striking a proper balance in the exploration and exploitation phases of such an algorithm, the local optimality problem of traditional methods can be avoided13.In metaheuristic algorithms, they start with exploring new solutions and transmitting them to exploit the best solution for a given problem14. In Ref.15, the authors have proposed a sperm motility algorithm (SMA) using the principle of chemoattractant secreted by the ovum to guide sperm movement in the female reproductive system. In Ref.16, the authors have proposed a Sperm Swarm Optimization (SSO) algorithm based on the temperature guidance for sperm to search for the egg in the female reproductive system. In Ref.17, the authors have proposed a multi-objective sperm fertilization procedure (MOSFP) as a modified form of SSO for multiobjective optimization problems. The general principles of natural conception have been used to develop novel algorithms17. The conception process from the point of view of sperm movement towards the egg in the female reproductive system is modeled in this paper. The concept of the natural selection of healthy sperm allowed by cervical gel to enter into the cervix, their unsymmetrical trajectories during movement, sperm guidance mechanism, flaggers’ moving characteristics with very spatial hyperactivation principle of sperm during the fertilisation of a mature egg are utilised as the ideas of the proposed Human Conception Optimizer (HCO) algorithm. Such concepts are utilized for the first time to develop a metaheuristic algorithm. The efficiency of the proposed algorithm is validated with standard IEEE CEC-2005 and CEC-2020 benchmark functions. A comparative study is also performed between the HCO algorithm and some existing algorithms for the benchmark functions. The statistical significance of test results is also studied with two types of non-parametric tests, such as the Friedman test and the Wilcoxon signed rank test. The applicability of the HCO algorithm in engineering problems has been validated for two different cases. In the first case, the optimal coordination of over-current relays in a power distribution network is studied for an IEEE 8-bus system. In the second case, an optimal PID controller is designed for the human respiratory ventilation system.In this paper, the Human Conception Optimizer (HCO) is proposed as a novel metaheuristic algorithm to solve any optimization problems. The HCO algorithm is inspired by the biological principle of natural human conception. For successful natural conception by a female, the fittest sperm of a male must fertilise a mature egg. The movement of sperm in the genital tracking path towards the egg is a unique characteristic. Sperm also use a unique technique to avoid environmental obstacles in the female reproductive system3. There are mainly three types of metaheuristic algorithm found in literature such as: physics based, swarm intelligence based and evolutionary metaheuristic algorithm. In evolutionary metaheuristic algorithms, the laws of natural evolution are used12. The search technique begins with a random generation population where the best solutions are combined and mutated to form new solutions. Genetic algorithm (GA)18 is one of evolutionary algorithms based on the Darwinian evolution concept. Other evolutionary algorithms are such as Evolution Strategy (ES)19, Genetic Programming (GP)20 etc. Some metaheuristic algorithms are inspired by well-known physical laws of the nature. Some of them are Simulated Annealing (SA)21, Gravitational Search Algorithm (GSA)22, Big-Bang Big-Crunch (BBBC)23, Atomic orbital search (AOS)24, Charged System Search (CSS)25 etc. Some swarm-based metaheuristic algorithms are inspired by the social behaviour of animals such as Particle Swarm Optimization (PSO)6, Ant Colony Optimization (ACT)26 etc. A swarm intelligence based algorithm employs a large number of particles to cover a search space, and the optimum answer is discovered by following the best location along their pathways27. Particles with their best solutions and the best one obtained so far in the swarm are used to update the particle position. Many other swarm based algorithms are found in literature such as Whale Optimization Algorithm (WOA)28, Grey Wolf Optimization (GWO) algorithm5, Sailfish Optimizer (SFO)29, Bottlenose Dolphin Optimizer30. Some human behavior-based metaheuristic methods are also found in literature such as Teaching Learning Based Optimization (TLBO)31, Group Search Optimizer (GSO)32, Imperialist Competitive Algorithm (ICA)33, Class Topper Optimization (CTO)4, Criminal search optimization34 and so on.In the literature, different analytical solutions are found, such as Quadratic Programming (QP), Dynamic Programming (DP), Lagrangian method for optimization problems. All such methods are based on deferential operators. They start searching for an optimal solution nearest to the initial point. Insufficient gradient data may lead them to a local solution. Thus, a limited application of such methods is found for real-world, complex optimization problems. In this regard, metaheurestic methods are found to be better than analytical methods in the literature13. Another issue in metaheuristic algorithms is the trapping problem in local solutions during the exploration and exploitation stages of such algorithms. Thus, the improper selection of initial value or position of search agents and local stack problems of such metaheuristic methods need to be solved to get an efficient optimizer. In Ref.35, authors presented a theory named the No Free Lunch (NFL) theory and proved that there is a universal best optimization method as all such methods perform similarly for all possible optimization-based problems. Therefore, many authors are involved in developing specific problem-based optimizers with the aim of getting global and local search strategies. In this regard, an attempt has been made to solve such issues by developing the Human conception optimizer (HCO). The unique features of the human conception process justify the development of such an algorithm. The HCO algorithm can solve the issues as stated above by resembling some spatial techniques of the conception process, which are discussed in the next section. A list of some existing optimization methods has been presented in Table Generally, a metaheuristic algorithm starts with a random initialization of the search variables within specified range. The convergence performance of such an algorithm depends on the correct selection of the initial value or position of the searching agent. An improper selection of initial value, which may be in a different direction where the actual solution may exist, can lead towards the wrong solution. Thus, the selection of the initial position of the search variable has an impact on the convergence performance of a metaheuristic algorithmThe core of this paper is to establish a nature-inspired optimizer named the Human Conception Optimizer (HCO). Some unique features of the human conception process are utilized to develop the algorithm to solve any optimization problems. A method of generating healthy populations at the start of the HCO algorithm is modelled by replicating the concept of sperm selection by cervical gel according to the fitness of each one. A probability function is defined for this purpose. The probability function is formulated by considering a fraction of sperm (position of solutions or searching agents) lies between the best and worst positions in the population. During the generation of a healthy population, the concept of possible egg position in either ovary is also utilized. Thus, the initial generation of the population will also be based on the best combination of a randomly generated search variable and its oppositional directional (sperm positions) based search variable. Thus, the initial sperm positions (positions of solutions or searching agents) in the healthy population are already formed with the possible best direction where global solution may exist. Moreover, sperm oriented far away from the global solution or those that are towards the opposite direction of the global solution are ignored at the initial stage. Thus, the optimal solution can be searched within a healthy population with the possibility of getting the best solution quickly. Therefore, the issue of random initialization of the position of the search variables, which may be in different directions or far away from the global solution, is avoided in the HCO algorithm. Such a velocity profile will balance exploration and exploitation based on the fitness of the best sperm cell (position of search variable) and the fitness of the average sperm cells in an iteration. This will happen during the updating of positions of search variables (sperm cells). A hyperactivation function is also formulated by replicating the concept of flagellar oscillation during the hyperactivation stage of a sperm fertilizing an egg in the Fallopian tube. This function will help the algorithm escape from the local optimal solution.In this section, the inspiration and the mathematical modeling of the Human Conception Optimizer are explained in detail.54. The process begins with millions of sperm released into the female reproductive tract. All sperm cells compete to fertilize a single egg as presented in the Fig. 56. The method of sperm meeting egg for successful fertilization is explored in detail below.Human conception happens when a healthy sperm cell meets the egg in the Fallopian tube57. Initially, sperm takes a random position in the vagina and stay inside the fluid called semen. According to the fitness of sperm, a swarm of the fittest sperm cells is able to enter the cervix. During their journey to the egg, sperm perform several outstanding navigational tasks. The sperm tail (flagellum) aids sperm swimming towards the egg by creating an irregular and oscillating beat pattern, as shown in Fig. Human conception occurs when a sperm cell is able to meet a mature egg, interact, and fuse in the female reproductive system58. Sperm can collect physical and chemical information to identify the egg in the female genital system with the help of some mechanism, such as60:Rheotaxis-sperm orientation against the fluid to move upstream.Thermotaxis-sperm sense temperature variation in the reproductive system. It swims against a temperature gradient in a higher temperature zone near oviduct.Chemotaxis-the movement of cells up to a concentration gradient of chemoattractant. Sperm move toward increasing chemical concentration.Chemotaxis was suggested in the literature as an active sperm guidance mechanism61. Sperm can sense the change in liquid concentration in the uterus. In thermotaxis, sperm move toward a higher temperature in the female reproductive system. The contractions of mucus in the female reproductive zone may also guide the sperm towards the egg.Sperm cells move with different hydrodynamic modes  on the basis of environmental conditions such as temperature and viscosity inside the female reproductive system62.Active sperm use a stroke called hyperactivation to cross the barrier of cumulus cells surrounding the egg. A fraction of sperm is able to become hyperactive. The flagellar beats of hyperactivated sperm have high curvature and a wider amplitude, leading to a highly active motility. Such a pattern of hyperactivity may create forces to facilitate sperm detachment and migration. The sperm have to pass another barrier called zonal pellucida (a layer of egg). The sperm cells undergo a process called the acrosome reaction, an enzyme deposited at the head of the sperm. It helps to break the zonal pellucida barrier to fertilise the eggAmong millions of sperm cells, only a single sperm cell is able to fertilize the matured egg in the challenging environment of the female reproductive system. The complete process is so challenging and unique that it motivates us to utilise the selection principles of winner sperm to develop a nature-inspired metaheuristic algorithm. In the next section, the detailed modelling of the proposed algorithm is discussed.58  will be examined by considering the right ovary as the place in a positive movement in the search area where the egg  may be found. The left ovary is considered as the place in a negative movement in the search area where the egg  may be found. The mucus fluid dynamics will be used to model the velocity of the sperm cells (sperm) to update their position during the exploration and exploitation stage of the proposed algorithm.After being released on the vagina, sperm cells enter the cervix, where their journey begins in a hostile environment. Only healthy sperm cells can enter the uterus and fallopian tubes58 Fig. a. In a f62. This concept will be realised to model the sperm movement through a curvature tracking path during the searching procedure of the algorithm. At each iteration, the best position achieved by each sperm cell along the curved path will be evaluated and called the present best position or solution gained by each cell.The tail of the sperm creates a jerking like movement which helps the sperm move into the uterus. Sperm cells starts following the curvature path caused by flagellar movement to reach the egg62. This sensing technique of liquid concentration in the reproductive system will be utilised to mimic the position update of sperm with respect to the best position of sperm achieved by any sperm cell in the population till the present iteration.The tail of sperm can sense the concentration of liquid in the reproductive system. According to that, it changes the position60. This concept will be used to overcome the local stuck problem of the algorithm.Sperm cells overcome the barrier across the egg by a hyperactivation process. They have to pass another barrier called zonal pellucida. To pass such a barrier, sperm must undergo a process called acrosome reaction. This is an enzyme deposited at the top of sperm cells. It will break the zonal pellucida barrier, allowing sperm to penetrate the eggThe detailed modeling of the HCO algorithm is given below.In this section, the biological principles of human conception are mathematically presented to develop the HCO algorithm. Generally, a set of natural facts and assumptions are considered to formulate the HCO algorithm. The concept of HCO is summarized as follows:64. In HCO, each search agent resembles the position of the sperm cells. In any metaheuristic algorithm, the performance of a swarm-based optimization method depends on the initialization of the population. In HCO, the initial position of sperm cells will be generated randomly within a search space with a higher population size. From the initial population, a fitter population will be produced, which will follow the other steps of the proposed algorithm.During intercourse, millions of sperm cells are discharged into the female menstrual system. All cells try to enter the cervix. The liquid inside the cervix will allow only healthy cells to enter into the cervical tracking path. Therefore, there is a natural selection of initial healthy sperm cells where only fit cells can start the journey from the cervix towards the eggStep 1: initial population generation Let, there are X) is defined as follows:Let initial position of sperm cells  is the fitness matrix with fitness value of all sperm (sperm).The algorithm first examines the fitness of all randomly generated initial search agents. The fitness values of all initial sperm (sperm cells) are defined as follows:Position of opposite directional solution The population of opposite directional solution will be calculated as follows:a and b are lower and upper boundary of search agent respectively.Fitness Thus, the initial population based on egg position will be as follows:61. Typically, a single egg is released at a time. This concept can be modelled as for a single-objective optimization HCO algorithm. In some cases, more than one egg may be released, sometimes resulting in the conception of multiples (twins). This concept leads to the multiobjective HCO algorithm. To simplify, the present paper is discussed as a single objective HCO algorithm. The twins may be produced by fertilising a mature egg with two sperm cells. In the HCO algorithm, among two close solutions, the best one will be selected, ignoring the twin solution.In a fertile female, a mature egg is produced by either the right or left ovary for fertilisation every month during ovulationStep 3: selection of healthy population In the natural fertilization process, only healthy sperm cells can enter into the cervix to fertilize a mature egg. In HCO algorithm, the initial population size is taken as high as possible from where an initial fittest population will be selected according to a probability function. The fittest population will be allowed to follow the further steps of the proposed algorithm.w is a weight factor.The best answer is assigned as the initial best solution (fittest sperm cell). The worst solution is also identified. The fitness of others will be compared with the fitness of the initial best with a probability of Therefore, the healthy population will be chosen as:n is the size of fit population.Thus,The fitness of initial fit population for an objective function depending on optimization problem will be as follows:The healthy or fit population will be used as the fittest initial population to search for the best solution for an optimization problem. In HCO, this step to find the fittest population from the initial randomly generated population will be done only once.Algorithm 1: Pseudo-code of HCO for generation of initial healthy populationInput: Set population size of sperm position, other constants.Generate initial random particle             Generate initial population for each variable randomly within a range of search space by using .Evalute fitness             Evaluate fitness     if     Select     else     Select endif    Select initial best and worst particle             Find the best fitness s matrix .Derive the probability function using .     if      Update fit population using .     else discard and check for next healthy sperm.endif     Output: Initial healthy populationSperm orientation can be a replica of particle orientation. Some sperm may be towards the global solution and some may be alongside the boundary of the search space. Some of them may be in the opposite direction of the global solution. In HCO, the initial fittest population is chosen with sperm (position of search agents) oriented towards the egg .The male reproductive cell, sperm, has a single flagellum or a tail. To achieve fertilization, sperm needs to move up the oviduct. The sperm’s tail produces a distinctive, jerky motion that pushes the head of the sperm backward and sideways while simultaneously propelling the sperm forward. The cells migrate through the fluid in the cervix by moving backwards and sideways. The sperm cell is aided in its journey toward the egg by this combination of actions. They can’t swim backwards due to the nature of flagellar movement. The moving trajectory of the sperm cell is shown in the Fig. 62. There are three types of sperm swimming guidance mechanisms: thermotaxis (based on temperature gradient), rheotaxis (swimming against a fluid flow), and chemotaxis (based on chemoattractant concentration gradient)55. Sperm cells will move against the mucus flow, which is a rheotaxis mechanism. They assume the egg position based on the concentration of liquid change near the egg. In HCO, the rheotaxis mechanism of sperm guidance towards the egg is used to find the velocity of sperm in fluid against the flow. The flagellar asymmetric movement is taken as a sinusoidal curvature path in the HCO algorithm.Human sperm use various sensing mechanisms to gather physical or chemical signals to spot the egg. During the fertilisation process, sperm cells move along the narrow cervical tracking path towards the oviduct. Mucus in the cervix helps sperm move through the uterus and oviducts66.The human spermatozoa can sense a flow of liquid and change the direction of their path against the flow. It performs positive rheotaxis and orients itself against an oncoming flow. Mucus flow (like as a sperm cell flow in fluid) can be described by the Poiseuille profile, where the speed increases quadratically with the distance to the compartment boundary. The Poiseuille profile is used to find the speed of the sperm cells. It tells how fast the sperm cells are moving at each point within the uterusIn HCO, the Poiseuille velocity profile is used to model the velocity of sperm to update their position. The Poiseuille velocity profile for sperm movement in the female reproduction tracking path is shown in Fig. r) in the fluid can be calculated by measuring the distance of the point from the centre of the tube, as graphically shown in Fig. r), the velocity is formulated as66:P is the pressure difference, L is length of a pipe with radius a, The velocity profile shows the amplitude of velocity according to the position of a particle in a fluid. According to the Poiseuille velocity profile, the velocity at a point, called specific radius  Evalute velocity of sperm             Evaluate velocity of each sperm in the healthy population using .Update velocity of sperm             Update velocity of sperm using .Update position of sperm             Update the position of each sperm using .    Repeat step 5 to 13 till the termination criterion reached or maximum number of iteration.Output: Fittest sperm or global solution.The flowchart of the proposed algorithm is presented in Fig. 67. To cross this barrier of cumulus, the sperm cells must use a very specific stroke called hyperactivation. It is characterised by an asymmetrical flagellar beat pattern which rises to a whip-like movement of the flagellum that can produce circular figure-eight swimming trajectories. The change in motion and force of the tail movement in the trajectory enable the sperm to escape from the epithelium67.In human conception, sperm cells conform a obstacle of cumulus cells around the egg. Before reaching the egg, the sperm cells are often trapped in epithelial cells in the fallopian tube. They are rendered inert unless they undergo hyperactivationIn HCO, the concept of sperm hyperactivation process will be adapted when the best solution is found stuck in a position for a long time before reaching termination criteria. The position of the hyperactivated particle will be compared with the best solution achieved before the hyperactivation process. Among the hyperactivated solutions and the non-hyperactivated solutions, the best one will be assigned as the current global solution for the population. To model the hyperactivation process, eight (8) shaped beat patterns are used. The new position of the best hyperactivation particle is modeled as follows:Concept of healthy initial population The HCO algorithm replicates the concept of sperm selection by the female reproductive system to allow them in the cervix and the position of the egg in either of the ovaries. The initial population in the HCO algorithm is not assigned directly to a randomly generated initial sperm within a search space. In this algorithm, a healthy population is generated at the initial stage by neglecting the sperm in the population oriented so far from the optimal position. Using the concept of egg position in the right or left ovary in the Fallopian tube, the fittest of all randomly generated sperm is evaluated along with the fitness of their opposite directional sperm in the search space. Thus, the healthy population will be based on the best possible solution in the positive or negative direction in the search space. The healthy population will ensure the best initial fit population within which the optimal solution will be found by the algorithm. Healthy populations will include initial positions of sperm based on their initial fitness and the best side of the position of the mature egg  by checking a sperm position and its opposite distortional position.Velocity update based on Poiseuille Velocity profile During the updation of the velocity of sperm cells, the position-based velocity profile is used, called the Poiseuille Velocity profile. The advantage of using such a velocity profile in the HCO algorithm is that the velocity of each sperm or search variable at an iteration will be calculated with the fitness value of the best position of a sperm or search variable in that iteration along with the average fitness in the population. Therefore, a good balance can be maintained between the exploration and exploitation stages of the algorithm.Hyperactivation for local optima avoidance Like sperm’s hyperactivation process to fertilize egg, a hyperactivation function is used in the HCO algorithm to avoid local solution trapping problems.HCO provides some advantages which makes it unique from some others algorithm. Some spatial features are as follows: 24.A metaheuristic algorithm must have some capabilities to solve complex optimization problems. An optimizer must exhibit a good balance in exploration and exploitation stages, local optima avoidance, and smooth convergence capability. To check the achievement of the HCO algorithm, two suites of test functions are taken in the study, such as 23 numbers of classical test functions from the CEC 2005 special session and ten number of 30 and 500 dimensional benchmark or test functions from the CEC-202024. Such functions are minimization functions. They can be grouped as: unimodal, multimodal, and fixed-dimensional multimodal. The details of such BMFs can be found in the CEC 2005 technical report27. The termination of the algorithm is set at a fixed iteration. The other parameters for the HCO algorithm are presented in Table SD) of each BMF are examined. The convergence performance for each BMF with the HCO method is carried out and compared with PSO6, CTO4, GWO5, WHO28, and SFO (Sailfish Optimizer)29.In this section, the response of the HCO algorithm is verified with CEC 2005 BMFsSD of objective values are examined. The output of the BMFs is presented in the Table 27. By looking at the Table For each BMF, the population size is considered as 500, and the maximum iteration is 500. Other constants of the HCO algorithm are tabulated in Table 27 exhibit multiple local optimal. The exploration feature of an optimization method may be verified with multimodal functions. By analyzing the Table 24 is found by the HCO algorithm where others are not able to find the same. For function 27, HCO is superior to SFO, CTO, and PSO. The convergence curve of the HCO algorithm for some BMFs is analyzed. Some BMFs such as The multi-modal functions such as 6, CTO4. For simulation purposes, each function is tested in a population of 50 sperm and run for 500 iterations for 30 dimensional problems and 500 dimensional problems. The convergence performance of the HCO for some selected 30 dimensional CEC-2020 benchmark functions is shown in the Fig. The HCO algorithm is verified with CEC-2020 BMFs to explore the achievement of the algorithm in terms of exploration, exploitation, convergence, and local optima avoidance. It includes unimodal, multi-modal, hybrid, and composite functions to validate the proposed algorithm. Each function has been tested with two conditions: with a 30 variable based optimization problem and another one with a 500 variable problem, and simulated 20 times in LabVIEW©2015 platform. The results in terms of average value and standard deviation are computed after 20 tests run in each of the MBF. The achievements of the HCO algorithm for each MBF are compared with some existing methods as reported in the literature, such as PSOThe use of non-parametric tests is also tested HCO algorithm for the CEC2020 benchmark functions. Such tests have a wide variety of applications in literature27. In this paper, the Friedman and Wilcoxon signed test are carried out CEC 2020 benchmark functions as presented in Tables 27. In the Wilcoxon signed test, In this section, the performance of the proposed HCO algorithm is validated for two constrained engineering optimization problems. In the first case, a over-current relay coordination based optimization problem is chosen. In this problem, the HCO is used to get an optimal setting of the over-current relays used in the protection scheme of a power distribution network. In the second case, an optimal PID controller is designed for the human respiratory ventilation system. In this problem, the proposed HCO algorithm is used to tune the PID controller for a blower type ventilator model.68. To get the optimal operating time of the protective relay system, the PS (plug setting) and TDS  are optimized. The test system is executed with HCO in LabVIEW©2015 platform. The convergence performance for the relay coordination problem is shown in Fig. 68 with some existing results is carried out. For this purpose, some well-known methods, such as BBO-LP69, BIP70, HWOA71, WOA72, MWCA73 and SA-LP74 are considered for the same system and presented in Table In this optimization problem, optimal coordination is established among over-current relays used in a power distribution network to supply uninterrupted power. To validate the proposed HCO algorithm for such a real-world engineering problem, an IEEE 8-bus distribution network with 14 over-current relays is considered. It consists of 20 numbers of selectivity limitations at three phase fault. The range of relay time dial setting (TDS) is between 0.1 and 1.1 s. The co-ordination time interval is 0.3 second. Plug setting is between 0.5 and 2.5. The other specifications such as current transformer ratio, short circuit current at fault locations are taken from Ref.75. For simulation purpose, the transfer function  used in the mechanical ventilator (MV) system used in the intensive care unit (ICU). The parameters of the conventional PID controller must be efficiently set such that the ventilator system can provide sufficient air to maintain a stable air-pressure in the lung system. The details of the mathematical modeling, and associated constraints MV are studied from Ref.75:The transfer function of blower system  and TDS  for 14 relays in the IEEE 8 bus system is efficiently selected during the generation stage of a healthy population as proposed in the algorithm  is developed to solve real-world optimization problems. The proposed algorithm is simple to understand and effective. It is based on a natural process that exists because of the evolution of human beings. The ability of the proposed algorithm has been tested with classical CEC-2005 and CEC-2020 benchmark functions (BMFs). A comparative analysis of the HCO algorithm with some existing results has also been performed for both sets of benchmark functions. The simulation results showed the superiority of the proposed algorithm. As observed in Table Moreover, the practical applications of the proposed algorithm have some limitations with the size of the real-world complex optimization problems, which is clearly observed from the simulation results. For the classical benchmark functions of CEC 2005, the HCO algorithm smoothly converges for most BMFs, as shown in Fig.


The λq at the outer target from the simulation case with all drifts could well match with the multi-machine scaling proposed by Eich and the DOIA of λq is in reasonable agreement with the scaling proposed by Goldston.The BOUT++ transport code is run to study the effects of plasma drifts on the divertor out-in asymmetries (DOIAs) of particle and heat fluxes and their decay widths for EAST lower single null H-mode discharges. The diamagnetic drift seems to have no effects on the DOIAs of total particle and heat fluxes due to its divergence-free nature. However, it could significantly increase the DOIAs of peak particle and heat fluxes and the flux decay widths. The E × B drift is found to induce a large plasma flow to the divertor region, enhancing the DOIAs of both total and peak particle and heat fluxes and the flux decay widths. Both the radial and poloidal components of the E × B drift are necessary in increasing the DOIAs, however, the radial E × B drift seems to play a more important role. The effects on the DOIAs caused by both diamagnetic and E × B drifts are reversed with the reverse of toroidal magnetic field. The heat flux decay width  The particles and energy from the core plasmas along the magnetic field lines in the scrape-off layer (SOL) to the divertor, are usually found to be asymmetrically distributed between the outer and inner divertor targets2, which make the divertor particle and heat loads more of an issue to the tokamak fusion community. The DOIA is a general concept representing the unbalances of particles and energy at the outer and inner divertor targets. The larger DOIAs mean the differences of particles and energy between the outer and inner targets are larger. Understanding the DOIAs of particles and energy is of great importance to the design and operation of future high-power and long-pulse tokamaks like ITER and China Fusion Engineering Test Reactor (CFETR).The excessive particle and heat fluxes onto the plasma facing components (PFCs), especially the divertor targets, poses a severe threat to the lifespan of the divertor materials and the sustainability of the advanced steady state operation for tokamak fusion devices5, JET8, ASDEX-Upgrade10, LHD12, JT-60U14 and TCV15. The basic finding is the reversed asymmetric behavior with the reverse of toroidal magnetic field. The particle flux to the inner divertor target is usually larger than that of the outer divertor target for lower single null (LSN) discharges with normal toroidal magnetic field (ion B × ▽B direction towards the lower X-point), while the asymmetry is reversed with reversed toroidal magnetic field (ion B × ▽B direction directed away from the lower X-point). The DOIA of heat flux is usually opposite to that of particle flux14, resulting in a much larger heat flux to the inner or outer divertor target than the other one. A common explanation to the strong DOIAs of particle and heat fluxes is due to the existence of various types of plasma drifts, including the diamagnetic and E × B drifts20. Presently, the E × B drift has been widely recognized as the most important factor in enhancing the DOIAs. However, there is still dispute over which component  of the E × B drift playing the leading role. Rozhansky et al. presented that the poloidal E × B drift was the main factor in enhancing the DOIAs of plasma density and particle flux based on the study of sheath boundary condition at targets19. However, Chankin found that the radial E × B drift was playing the dominant role by analyzing the convective flows caused by the poloidal and radial components of the E × B drift based on a series of EDGE2D-EIRENE modeling20. It should be noted that most of the previous studies are focused on the DOIA of density or total heat flux without shedding light on the detailed distributions of particle and heat fluxes, especially the peak flux and the flux decay widths which are also of great importance to the safety and maintainability of target materials. Besides, although the diamagnetic drift is considered not to be important in inducing the DOIAs due to its divergence-free nature23, its effects on the DOIAs of target parameters have not been widely discussed yet.The studies of DOIAs have been carried out worldwide on present tokamaks such as DIII-D26 and DOIA of particle flux for the H-mode discharges28 have been carried out recently. The results also show that the DOIAs could be reversed by reversing the toroidal magnetic field. The simulations of EAST discharges with SOLPS also demonstrate similar asymmetric behaviors with the experiment31. However, these modeling studies are mainly focused on the qualitative agreement between the simulations and the experiment without much analysis on the quantitative level. There is still a lack of effort on the effects of different types of drifts on the detailed distributions of particle and heat fluxes at the outer and inner divertor targets, especially the peak flux and the flux decay widths . BOUT++ is a framework for implementing 2D and 3D plasma/fluid simulation in curvilinear geometry37 and is shown in Fig. 19 m−3 and 450 eV respectively. The “U” shaped profiles of transport coefficients are set in the simulation as shown in Fig. 39. The ion thermal diffusivity is set to be the same as the electron thermal diffusivity for simplicity. Note that the transport coefficients are by default poloidally constant in the simulation.An EAST H-mode discharge #48,337 is chosen for the simulation. It is a LSN discharge with ion B × ▽B direction towards the X-point. The divertor is in a high-recycling regime for the discharge. The plasma current and toroidal magnetic field for this discharge are 0.4 MA and 2.2 T respectively. The simulation grid is generated based on the magnetic equilibrium of this discharge from the kinetic EFIT26 and the DOIA of particle flux for H-mode discharges28 have been carried out in EAST in recent years. There is still a lack of study on the DOIA of heat flux for EAST H-mode discharges due to the unreliability of heat flux measurement from the divertor Langmuir probes. For the H-mode discharges, since the heating power is relatively high, resulting in the arcing faults in some of the probe channels, which make the heat flux profiles from the divertor Langmuir probes unreliable. For the DOIA of particle flux for EAST H-mode discharges, the main finding is the reversed asymmetry behavior with the reverse of toroidal magnetic field27. In this section, we will study the effects of diamagnetic and E × B drifts on the DOIAs of particle and heat fluxes through modeling. The effect of toroidal magnetic field direction on the DOIAs will also be investigated for the same discharge  by artificially reversing the direction of toroidal magnetic field.As aforementioned, the experimental studies of the DOIA of heat flux for L-mode dischargesThe first set of simulations is carried out with only diamagnetic drift, and the simulation without drifts is also included to make comparison. The heat flux at one of the targets acts as an indicator to check whether a steady-state solution is achieved. Figure 23. In the low field side of the SOL, the diamagnetic drift velocity directs towards the outer divertor target, adding to the parallel particle transport from the upstream at the low field of the SOL to the outer target. Since the particle flux decay width is determined by the competition between the parallel and perpendicular transport, the larger parallel transport induced by diamagnetic drift could lead the larger peak particle flux and smaller particle flux width at the outer target as shown in Fig. 21 and has confirmed the divergence-free nature of diamagnetic drift. The numbers of particles across separatrix into the SOL induced by diamagnetic drift are also calculated for the two cases with diamagnetic drift. The results show that there are 6.37 × 1017 particles across separatrix into the SOL for the case with normal toroidal magnetic field. The number is −9.66 × 1017 for the case with reversed toroidal magnetic field. The numbers of particles across separatrix by diamagnetic drift for the two cases are more than 2 magnitudes smaller than the numbers of total particles to the divertor targets. The results above indicate that the diamagnetic drift could cause particles to cross flux surfaces and this process should be mainly happened in the divertor region. The diamagnetic current should be perpendicular to both the magnetic field lines and the pressure gradient. The pressure gradient in the parallel direction in the SOL couldn’t cause a diamagnetic current. However, the pressure gradient in the poloidal direction in the SOL should be able to cause a cross-field diamagnetic flow. In the upstream SOL, the pressure gradient along the poloidal direction in the SOL is very small, meaning that the cross-field particle transport induced by diamagnetic drift is weak, which may explain why the total particles and energy across separatrix due to the diamagnetic drift are small. In the downstream SOL of the divertor region, there is significant pressure loss in the poloidal direction, which causes a large pressure gradient and a large cross-field diamagnetic flow in the divertor region. This large cross-field flow leads the parallel flow induced by diamagnetic drift to change direction near the target, which results in almost no net diamagnetic flow to the target. Detailed analyses of this process are presented experimentally21 and numerically22 in previous literatures. Note that the total particles and energy to the outer boundary  are much smaller than those of the targets for all of the three cases as shown in Table Figure Another set of simulations is done with only E × B drift. Figure 20 particles across separatrix into the SOL for the case with normal toroidal magnetic field. The number is 2.09 × 1020 for the case with reversed toroidal magnetic field. The numbers of particles across separatrix by E × B drift for the two cases are much larger than the counterparts that caused by the diamagnetic drift in previous simulation cases with diamagnetic drift.To study in detail on which component of the E × B drift plays the main role in inducing the strong DOIAs of particle and heat fluxes, the total particles across the interface ➀ to ➄ by E × B drift are calculated for the two simulation cases with different toroidal magnetic field directions and are shown in Table The above simulations are done either with only diamagnetic drift or with only E × B drift. To study the effects of all drifts on the DOIAs of particle and heat fluxes, two simulation cases with normal and reversed toroidal magnetic field respectively, are done with all drifts switched on. Figure mentclass2pt{minimThe above simulations show that the DOIAs of total particles and energy are mainly induced by the E × B drift while the diamagnetic drift almost has no effects on them. However, both the diamagnetic and the E × B drifts could affect the DOIAs of peak particle and heat flux and the flux decay widths. Figure 40. Equation (S is the spreading width which characterizes the particle or heat dissipation into the PFR. fx, which is defined to be the ratio of a specified extent at OMP to the extent at divertor target. s0. qBG is the background flux intensity. Figure As we mentioned previously, both the diamagnetic and E × B drifts could significantly affect the particle and heat flux decay widths from the qualitative point of view. However, it is necessary to study the effects of both drifts on the flux decay widths in the quantitative point of view. The particle and heat flux decay widths could be obtained by fitting the particle and heat flux profiles at divertor targets using the fitting function proposed by Eich et al.Equation  is the fFigure 41. Eich et al. also proposed a multi-machine scaling law for the 42. These scaling laws could act as alternatives to compare with the above simulation results. Figure 43. The fact that we reproduce reasonable agreement with Goldston’s result when turning down both heat and particle diffusivities indicates that drifts may dominate the heat transport compared to diffusion. Figure Since the particle and heat flux profiles measured by divertor Langmuir probes for this discharge on EAST are not smooth enough to carry out the fitting for the flux decay widths and spreading widths, it is hard to make comparisons of DOIAs of flux decay widths and spreading widths between the experiment and simulation. However, Goldston has proposed a drift-based model showing that the ratio of A comprehensive study of the effects of plasma drifts on the DOIAs of particle and heat fluxes and their decay widths is carried out for EAST H-mode discharges with the BTC. The diamagnetic drift is found to have almost no effects on the DOIAs of total particle and heat fluxes due to its divergence-free nature. While the E × B drift could significantly affect the total particles and energy to the inner and outer targets, thus changing the DOIAs of total particles and energy. For the simulation with normal toroidal magnetic field, the E × B drift could lead to much larger particle fluxes to the inner target than the outer one, while the DOIA of energy is opposite to that of the particles. Detailed analysis shows that both the poloidal and radial components of the E × B drift are necessary in inducing the DOIAs of total particles and energy, however, the radial E × B drift seems to play a more important role. Both the diamagnetic and E × B drifts could affect the DOIAs of peak particle and heat flux and the flux decay widths. For the simulation with normal toroidal magnetic field, the diamagnetic drift could lead to the increase of peak particle flux at the outer target and the decrease of peak particle flux at the inner target. In general, the paper focuses on the quantitative studies of the effects of drifts on the DOIAs of particle and heat fluxes and their decay widths. The results are in reasonable agreement with the related scaling laws. However, more efforts are still needed to figure out the effects of other factors such as ballooning-like transport, plasma geometric effects from the targets themselves and divertor plasma conditions e.g., on the DOIAs of particle and heat fluxes, which will be our next-step work.


Usually for non-destructive testing at high temperatures, ultrasonic transducers made of PZT and silver electrodes are used, but this could lead to damage to or malfunction of the ultrasonic transducer due to poor adhesion between PZT and silver. Soldering is one of the most common types of bonding used for individual parts of ultrasonic transducers , but silver should be protected using additional metal layers (copper) due to its solubility in solder. A mathematical modelling could help to predict if an ultrasonic transducer was manufactured well and if it could operate up to 225 °C. The observed von Mises stresses were very high and concentrated in metal layers (silver and copper), which could lead to disbonding under long-term cyclic temperature loads. This paper presents a multilayer ultrasonic transducer , which was heated evenly from room temperature to 225 °C and then cooled down. In the B-scan, it was observed that the amplitude of the reflected signal from the bottom of the sample decreased with an increase in temperature. However, after six heating–cooling cycles, the results repeated themselves and no signs of fatigue were noticed. This ultrasonic transducer was well manufactured and could be used for non-destructive testing when the environment temperature changes in cycles up to 225 °C. The manufacturer states that a piezoelectric charge coefficient (31-d) of Pz27 slightly increases when the temperature rises (25–225 °C). Therefore, it also demonstrates good stability of performance in temperature ranges corresponding to our tasks. Based on its known properties and our experience with it, Pz27 was chosen as the piezoelectric material for our developed ultrasonic transducer [Ultrasonic transducers used for measurement or non-destructive testing purposes in general are like sandwiches of different components, namely a piezo element, electrodes, matching layer, and backing. The adhesion between those layers is the most important parameter defining the survivability of a transducer during exploitation in high-temperature conditions. Usually, commercially available piezoelectric ceramic elements are coated with silver or gold electrodes ,2,3. Altansducer ,12.Another important question is how to join reliably individual parts of an ultrasonic transducer . For this purpose, different techniques are used, such as soldering, welding, gluing, etc. . To protTo predict the performance of an ultrasonic transducer at elevated temperatures, and to predict possible failures, mathematical modelling can be applied. For this purpose, a finite element method could be used to calculate and predict possible stresses and deformations in a structure of the transducer sandwich at different temperatures ,19. MathThe objective of this paper was the development and investigation of the ultrasonic transducer performance in a temperature range of up to 225 °C during multiple heating and cooling cycles, as well as the identification of key issues that determine the survivability of the probe.In this paper, the structure of the high-temperature transducer is presented first, paying primary attention to the improvement of the adhesion of the electrodes to the piezoceramic element. For that purpose, the silver electrodes are coated by copper and the techniques used to do so are demonstrated. In the following section, the expected stresses in the structure of the ultrasonic transducer are assessed using numerical modelling. The final section is devoted to the experimental investigation of the performance of the developed transducer during temperature increase/decrease cycles.For this developed ultrasonic transducer, a commercially available Soft PZT (Navy II) Type Pz27 piezo element , with two already applied silver electrodes (~10 µm thickness), was used . On bothAdditional metal layers on the silver electrodes were placed to prevent silver electrodes from possible damage during silver soldering because of the chemical and physical properties of silver . Copper 2CO3) solution and ultrasonic bath were used. An electroplating solution was made by mixing copper sulfate (150–250 g/L) and sulfuric acid (15–60 g/L). A 1–10 A/dm2 current  were applied to the external edges of the object under investigation. Stationary structural analyses were performed for different ambient temperatures from 40 °C to 200 °C in increments of 40 °C.The proposed structure consists of different layers whose materials have a wide range of thermal expansion coefficients. Under the influence of high temperature, the materials of the structure’s layers expand differently, and this causes thermal stress in the structure. The easiest way to evaluate stresses in such a multilayer structure is through numerical modelling using the finite element technique. Numerical modelling of thermal stress in the coating of the piezoelectric ceramic disc was performed using COMSOL finite elements software. Due to cylindrical structure of the object under investigation, the model was simplified to a 2D axisymmetric model. A graphical representation of the finite element model and its dimensions is presented in Thermal stresses in the investigated structure were evaluated in a term of von Mises stress. The von Mises stress is scalar quantity and criterion of yield very often used for metals. If the von Mises stress is greater than the yield strength of the material, then the material will yield. The distribution of von Mises stress in the piezoelectric ceramic disc coated with silver and a copper layer at 200 °C is presented in Due to significant differences in the thermal expansion coefficients of the piezoelectric ceramic and coated layers, the obtained stresses are very high and concentrated in the coated layers. The resulting stresses greatly exceed the yield strength of silver and copper, which are 45 MPa for silver and 33 MPa for copper. Stress distributions in the silver layer along the radius of the piezoelectric ceramic at different temperatures are shown in In order to test the performance of the developed ultrasonic transducer during multiple heating–cooling cycles, the experimental set-up needs to be prepared. The signals were measured on a carbon steel block  to which the transducer was attached using a specially developed holder ensuring a constant pressure force throughout the duration of the experiments . The couThe next stage of this research was to investigate the performance of the developed high-temperature ultrasonic transducer under six temperature variation cycles. Using the experimental set-up presented in It was observed that in less than 1 h, the temperature inside the oven reached 200 °C, while the temperature of the sample was only ~80 °C. After 6 h of heating, the sample reached 200 °C and the temperature of the environment was ~225 °C. The temperature measurement of the sample and inside the oven stopped after 8 h of cooling. It can be noticed that, even after 8 h, the cooling temperatures of the test block and oven were still high at 55 °C and 40 °C, respectively. These results were crucial for this experiment because the carbon steel sample did not heat up as quickly as the environment and it was important to track the temperature changes inside the sample to be able to understand the changes in the signals.The signals reflected from the bottom of the sample at different temperatures  and their spectra are presented in A frequency-based analysis of the received echo signals in the range of experimentally tested temperatures showed no influence of temperature on the shape of the frequency spectra of the received signals in general b. The teA comparison of the transducer’s reverberation signals in the temperature range a shows teT is the time at which the measurements were performed, and t is the propagation time of the reflected ultrasonic signal. Before heating the sample, the amplitude of the reflected signal from the bottom of the sample was at its highest (7 × 10−3 V). The more the temperature increased, the more the amplitude of the reflected signal decreased. The amplitude of the reflection was at its lowest (2 × 10−3 V) after 6–8 h of heating when the temperature of the sample was 200 °C. When the carbon steel cooled to 160 °C, the amplitude of the reflected signal began to increase. At the end of this experiment (after 8 h of cooling down), it was almost the same as it was at the beginning because the temperature of the sample was higher than it was before heating .In the B-scan, a dependence between the amplitude of the signal and the temperature is presented, where According to the temperature curve of the sample  after 6–The observed slight hysteresis in After six repeated cycles of heating and cooling the sample, the results repeated themselves, and no signs of fatigue were noticed, so it could be concluded that there were no residual deformations. This developed ultrasonic transducer was well manufactured, and could be used for non-destructive testing when the environmental temperature changes in cycles up to 225 °C.The developed high-temperature ultrasonic transducer may operate up to 225 °C and is resistant to multiple heating–cooling cycles, which is characteristic for many non-destructive testing applications. Such a result was achieved by improving the adhesion quality between internal components of the transducer by means of additional copper layers. The prediction modelling of the thermal expansion of the developed ultrasonic transducer demonstrated that very strong deformations and stresses could be expected at the edges of the sandwich structure, which can lead to the delamination of the electrodes during heating–cooling cycles. However, the developed technology of electroplating electrodes yielded the softening of stresses arising at the edges of the piezo element and electrodes. In such a case, the survivability of the ultrasonic transducer during multiple heating–cooling cycles increased.On the other hand, the experiment revealed that the amplitude of the signals propagated through the steel block possessed significant amplitude variations. During heating up to 225 °C, it dropped by 12 dB, but it completely recovered when the transducer cooled down to the room temperature. There is no clear explanation for this, but possible reasons include a variation in the properties of the coupling liquid, different diffraction conditions of the waves propagating in the steel block with temperature gradients, and variations in the electromechanical coefficients of the piezoceramic. However, we think that variations in the electromechanical coefficient cannot significantly affect the amplitude of the reflected signal because our measurements of the electromechanical coefficient in a temperature range of up to 225 °C showed that they were insignificant. Nevertheless, the experiments demonstrated relatively good repeatability of the amplitude variations during different heating cycles. As such, it can be stated that the developed ultrasonic transducer demonstrated good survivability and performance at elevated temperatures.


To study the noise spectrum characteristics of marine pump units induced by different excitation sources, a computational aeroacoustic (CAA) model of the internal and external field noise of a marine pump was established. The coupled acoustic-vibration method was used to obtain the spectrum characteristics of internal and external field noise. The accuracy and feasibility of the simulation method for noise prediction were confirmed through a noise test. Due to the different mediums in the internal and external fields of the marine pump, an external field acoustic model was established based on the automatically matched layer (AML) technology. The spectral characteristics of different excitation sources and the spatial distribution of the radiated sound field were analyzed, and the contribution of different sound source excitations to the internal and external sound field was revealed. The results show that the main frequency of the internal field noise generated by different excitations is at the blade passing frequency, and the internal field noise induced by the dipole acoustic excitations dominates at 180.6 dB. For the external field noise, the main frequency is still located at the blade passing frequency. The radiation noise induced by the fluid excitation (139.2 dB) is higher than that induced by the dipole excitations . Marine pumps produce loud noise during operation, and the noise generation mechanism is complex. The noise level of marine pumps is critical, especially for military vessels. Vibration and noise are inseparable during pump operation. Vibration generates noise, and noise also affects vibration. There are many noise sources for pump units, and the most common source is the noise caused by the vibration of the pump units, which is structural vibration noise. The noise generated by fluid flow is called hydrodynamic noise5. Subsequently, Williams and Hawkings6 applied the governing equation to the boundary problem of solid motion and proposed the famous FW-H equation to divide hydrodynamic noise sources into monopole, dipole and quadrupole sound sources. For ease of comprehension, scholars classified noise into broadband noise and discrete noise, namely, single-tone noise7. Among the flow-induced noise of the pump, monopole source noise is induced by the volume compression effect of the pump cavitation and is a discrete noise. The dipole source is mainly caused by the fluid's unsteady fluctuating force acting on the structure's surface, which includes broadband noise and discrete noise. The quadrupole source is caused by the turbulence generated by the high-speed fluid flow, which is classified as broadband noise8.The concept of hydrodynamic noise was initially developed due to the Lighthill acoustic analogy theory9 studied the pressure pulsation and radial force characteristics in the unsteady flow process at different cavitation stages through tests. Howe11 pointed out that the main sound source of rotating machinery is the dipole sound source caused by the unsteady force and proposed that the flow field can be solved first and then the sound field can be further solved according to the obtained flow field results. Zhou et al.12 proposed that the sources of hydrodynamic noise are similar when the fluid Reynolds numbers are similar. The main acoustic calculations for the internal and external fields of pumps are the boundary element method (BEM) and the finite element method (FEM). Si et al.13 used the direct BEM to calculate the sound field in a centrifugal pump and found that the blade passing frequency and multiplier are the characteristic frequencies of the fluid-induced noise. Cai et al.14 and Yu et al.15 also used this method to calculate the internal noise of submersible sewage pumps and vortex self-priming pumps. Allen16 proposed the coupled FEM/BEM to calculate the structure radiated noise by the boundary layer, using fluctuating pressure on the wall to define the excitation on the structure-acoustic system. Warszawski et al.17 used the FEM/BEM to study the propagation characteristics of acoustic waves generated by fluid–structure interactions. Han et al.18 realized the acoustic-vibration coupling calculation for the con-shell structure by the FEM/BEM, obtained the sound pressure level at the measuring point, and verified the accuracy of the calculation result through experiments. Liu et al.19 studied the influence of the blade outlet angle and the width on the fluid-induced noise of the centrifugal pump by the direct BEM method. Dai et al.20 also calculated the external field noise of a centrifugal pump by coupled acoustic vibration.Dong et al.22 predicted the underwater low-frequency noise of a small-scale stiffened shell based on the acoustic finite element method and proved the effectiveness in predicting the infinite sound field. Majda23 proposed the absorption boundary condition, which replaces the infinite external sound field with a convex fluid domain. The sound absorption boundary condition is imposed on the convex fluid domain. Berenger24 used the perfectly matched layer (PML) method to calculate the fluctuation of the electromagnetic field. Because this method requires experienced engineers to complete the division of the absorption layer, the automatically matched layer (AML) technology is proposed. The AML method automatically generates matching layer cell domains based on the structural finite element area, and the size of the cell domain varies according to the analysis frequency, which can improve the calculation efficiency.Although the mediums are different inside and outside the marine pump, the boundary element can define only one fluid property, so this method cannot effectively solve the noise problem in the external field of the pump. In contrast, the acoustic finite element method is effective to calculate radiation acoustics. Xie et al.There have been few studies on the effects of fluid excitation and dipole excitations on pump noise, and the primary mechanism of pump noise generation is unclear. Therefore, according to the generation and transmission mode of the noise for marine pump units, the present paper established a computational aero-acoustic model for the internal and external radiated noise generated by fluid flow. Considering the coupled structure-sound field, the acoustic finite element method was used to calculate the internal field noise under fluid excitation and dipole acoustic excitations. The accuracy and feasibility of the coupled acoustic-vibration method for predicting internal field noise was confirmed by test comparison. Due to the different mediums in marine pumps' internal and external fields, an external acoustic model based on AML technology was established. The spectral characteristics of different noise sources and the spatial distribution of the radiated sound field were analyzed, and the contributions of different sound sources to the external sound field were revealed.Qd = 25 m3/h, head H = 35 m, rotational speed n = 2950 r/min. Table A marine centrifugal pump with a specific speed of 66.7 was used as the research object. The flow rate k-ε model, and pressure-inlet and mass-flow-outlet conditions were adopted. The interface between the domains used the general grid interface (GGI). The convection term adopts the second-order upwind. The time step was set to ΔT = 0.565 × 10–4 s, i.e., every 1° rotation of the impeller. The pressure pulsation data of 10 stable cycles of impeller rotation were extracted as the excitation source, and the total time was set to 0.25 s. The residual was set as 10–4.The entire flow field calculation model includes the inlet pipe, elbow, impeller, clearance (between the impeller and the volute), volute and outlet pipe, as shown in Fig. LMS Virtual.Lab 13.6 software was used to calculate the internal field noise of the marine pump. The noise of the marine pump can be divided into (1) flow-induced noise, which is the fluid excitation source radiating sound to the structure, causing the structure to vibrate and thus generating radiated noise, and (2) flow noise, which is directly radiated by the dipole excitation source. Moreover, there are two primary dipole excitation sources for the pump: a rotating dipole excitation on the impeller surface and a surface dipole excitation on the volute wall.For the calculation of flow-induced noise, the pressure pulsation action on the pump wall was used as the excitation. The data on the cell nodes were transferred to the grid of the pump structural model, and the coupled acoustic-vibration algorithm was used to calculate the structural vibration of the pump and obtain the flow-induced noise. For the noise induced by the surface dipole, the pulsation of the pump wall was first used as the boundary condition of the acoustic dipole. Then, the geometric interpolation method without energy loss was used to transfer the fluid field information to the acoustic grid, the sound pressure on the inner wall was calculated, and the noise induced by the surface dipole was calculated. For the noise induced by the rotating dipole on blades, the time-domain pulsation data on the blade were extracted, and the discrete blade source was used to calculate the sound field. The impeller was divided into four parts, and the load and its acting position are defined on each part. The sound pressure on the wall of the internal field pump was calculated, and then this result was used as the acoustic excitation for the coupled acoustic-vibration calculation to obtain the noise excited by the rotating dipole.Z = ρc = 1.5 × 106 kg/(m2·s), and the speed of sound was taken as 1500 m/s. The internal field acoustic-vibration model and the locations of the inlet and outlet monitoring points are shown in Fig. p is the sound pressure and pref is the reference value of the underwater sound pressure, 1 μPa.The inlet and outlet of the infield acoustic model were defined as full sound absorption properties, and the remaining surfaces were defined as full reflection walls. The characteristic acoustic impedance The coupled acoustic-vibration calculation of the marine pump is carried out based on the modal space. It is necessary to comprehensively consider the structural and acoustic modes to calculate the internal field noise accurately. The internal acoustic mode of the marine pump is shown in Fig. Figure By comparing the inlet and outlet sound pressure spectra, it can be found that the law and peak values are consistent from 0 to 590 Hz (2BPF). The inlet and outlet sound pressure spectra are almost identical at the twice blade passing frequency and subsequent frequency bands, but the broadband peak is different. Overall, below 1000 Hz, the sound pressure level at the outlet is higher at rated operating conditions. However, the inlet sound pressure level is higher at some special frequencies between 1000 and 2000 Hz.Figure The finite element method is used to calculate the internal sound field coupled with the acoustic excitation of the rotating dipole, as shown in Fig. The spectral curves of the field noise at the inlet and outlet induced by fluid excitation and acoustic dipoles are shown in Fig. Figure This paper uses the RHSA-10 hydrophone to measure the noise signal of marine pumps under rated operating conditions. The hydrophone has an operating frequency range of 20 Hz to 200 kHz. The sound pressure sensitivity is − 210 dB, the horizontal and vertical directivities are ± 2 dB and ± 2.5 dB, respectively, and the housing material is stainless steel. We install the hydrophone vertically at six times the pipe diameter of the pump inlet flange. This installation method can effectively measure the dipole sound source caused by the fluid pulsation in the pump, avoid the interference of other sound sources, and at the same time have little effect on the fluid flow in the internal field. The hydrophone and its installation location are shown in Fig. There are interferences from other factors in the test process, such as valve noise, pipeline resonance and mechanical structural vibration, affecting the infield noise test results.The sound waves inside the pump interact with the complex internal structure, resulting in reflection and scattering of the sound wave. There is also loss when the sound waves propagate. Moreover, the reflection and scattering of sound waves are ignored, which causes errors in the sound pressure level.Figure Lp is introduced. The formula is as follows:fi is the minimum resolution. f0 and fmax are the initial and limit values of the calculated frequency, respectively, pa and pi are the root mean square (RMS) of the sound pressure, Pa, and P0 is the reference sound pressure, generally P0 = 1 × 10–6 Pa in water, and P0 = 2 × 10–5 Pa in air.To clearly compare the contribution of different excitation sources to the noise of the marine pump units, the total sound pressure level Table According to the method of fluid-induced noise, the external field noise of marine pumps mainly includes the following: (1) The vibration and noise generated by the internal fluid directly acting on the pump. (2) The sound pressure of the dipole acoustic source acting on the pump structure, causing the structure to vibrate and radiate noise to the external space. The excitation is the same as the internal field noise, but the sound waves propagate in a different direction and form. In addition, the internal and external field mediums of marine pumps are different. The internal medium of the pump is water, and the external medium is air; therefore, the FEM/AML sound-vibration coupling method is used to calculate the external field noise. Figure 1 is set up to analyze the frequency response curve of the external field sound pressure.Inside the AML is the finite element model of the structure. The first layer of the AML mesh fits tightly with the outer surface of the internal structure shell, and a symmetrical surface is used as the ground. Sound propagation has pronounced directivity, and the sound pressure levels measured at different positions are also different. To obtain the sound pressure level distribution in the external field of the marine pump units, radial and axial monitoring surfaces are established with the model coordinate origin. Thirty-six monitoring points are set at 1 m from the coordinate origin, and the angle between each monitoring point is 10° to analyze the directivity distribution of external field noise. As shown in Fig. The external field noise induced by fluid excitation directly acts on the internal surface of the pump, causing vibration and noise. Figure The spatial distribution of the external field radiated noise by the fluid excitation at the characteristic frequency is shown in Fig. Figure The spatial and directivity distributions of the external field noise generated by the surface dipole at the characteristic frequency are shown in Figs. Figure The spatial and directional distributions of the external field noise generated by the rotating dipole at the characteristic frequency are shown in Figs. Figure The sound pressure level at characteristic frequencies and the total sound pressure level are used to intuitively evaluate the contribution rate of different excitations to the external field noise, as shown in Table The main frequency of the internal field noise induced by different excitation sources is distributed at the BPF (295 Hz). The second main frequency is at 2BPF (590 Hz), and each characteristic frequency is distributed at each multiple of the BPF. At the main frequency, the internal field noise induced by the dipole occupies a dominant position. Above 3BPF (885 Hz), fluid excitation gradually dominates.The broadband spectrum of the internal field noise induced by the surface dipole agrees with the test result. The total sound pressure level errors of the surface dipole, rotating dipole and fluid excitation are 1.1%, 1.25%, and 1.4%, respectively. The contribution to the total sound pressure level of the infield noise is ordered as follows: surface dipole excitation (180.6 dB) > rotating dipole excitation (180.4 dB) > fluid excitation (180.2 dB).The main frequency of the external field noise is the blade passing frequency, where the rotating dipole excitation contributes the most at 51.6 dB, and the lowest is the surface dipole at 41 dB. The external field noise induced by fluid excitation is dominant, especially between 1200 and 2000 Hz. Moreover, the external sound field distribution under different excitations is generally the same. The external field's total sound pressure level contribution is ordered as follows: fluid excitation (129.2 dB) > rotating dipole excitation (137.3 dB) > surface dipole excitation (136.3 dB).The acoustic finite element method is used to calculate the internal field noise by fluid excitation and acoustic excitations, and the external field noise by different excitation sources is calculated based on the AML method. The spectral characteristics of different excitation sources and the distribution of the radiated sound field are analyzed. The contributions of different excitation sources to the internal and external sound fields are revealed. The conclusions are as follows:


The uncertainty of information is an important issue that must be faced when dealing with decision-making problems. Randomness and fuzziness are the two most common types of uncertainty. In this paper, we propose a multicriteria group decision-making method based on intuitionistic normal cloud and cloud distance entropy. First, the backward cloud generation algorithm for intuitionistic normal clouds is designed to transform the intuitionistic fuzzy decision information given by all experts into an intuitionistic normal cloud matrix to avoid the loss and distortion of information. Second, the distance measurement of the cloud model is introduced into the information entropy theory, and the concept of cloud distance entropy is proposed. Then, the distance measurement for intuitionistic normal clouds based on numerical features is defined and its properties are discussed, based on which the criterion weight determination method under intuitionistic normal cloud information is proposed. In addition, the VIKOR method, which integrates group utility and individual regret, is extended to the intuitionistic normal cloud environment, and thus the ranking results of the alternatives are obtained. Finally, the effectiveness and practicality of the proposed method are demonstrated by two numerical examples. Multicriteria group decision making (MCGDM) is an important research topic in modern decision science, and its theory and methods have been widely used in economic and social fields . Many scThe information aggregation of multiple experts is another key problem to be solved in MCGDM . The cloIn the research of MCGDM methods, the determination of criteria weight is a key factor affecting the objectivity of decision making. Information entropy, as a tool to measure the amount of information, can reduce the influence of subjective factors in the decision-making process, and has been widely used in weight determination ,29,30. EIn MCGDM problems, the ranking of alternatives is also an important part of the decision-making process. Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) is a compromise ranking method that integrates group utility and individual regret, proposed by Opricovic in 1998 . VIKOR cIn summary, the motivation and contribution of the work in this paper are as follows. To fully consider the fuzziness and randomness of multicriteria group decision making in the intuitionistic fuzzy environment and overcome the deficiency of existing backward cloud generation algorithms that have difficulty handling the intuitionistic fuzzy numbers, we designed the backward cloud generation algorithm of intuitionistic normal cloud for aggregating expert evaluation information, which can avoid the loss and distortion of information. To overcome the shortcoming that information entropy ignores group consistency when measuring information quality in the cloud environment, we propose the concept of cloud distance entropy, which extends the flexibility of the traditional entropy method in decision making. In addition, we propose the VIKOR method in the intuitionistic normal cloud environment, which provides a new idea to solve the multicriteria group decision-making problem where the decision information is in the form of cloud representation.The rest of the paper is structured as follows. In Intuitionistic fuzzy set, as a generalization and extension of fuzzy sets, describes the essential properties of things in more detail through nonmembership functions and hesitation functions. It is defined as follows:Definition 1.Letbe a universe of discourse, The IFS in can be defined as follows:whereandare membership function and nonmembership function of elementto set, respectively, under the condition, for all .LetXThe cloud model is a mathematical model that can relate qualitative linguistic values to quantitative values through uncertain relationships. The model can better describe the relationship between the ambiguity and randomness of things. It is defined as follows:Definition 2.Let be a quantitative universe represented by exact numerical values and be a qualitative concept on . If the quantitative value is a random realization of the qualitative concept , the degree of certainty of on is a random number that tends to be stable, where , then the distribution of on the universe is called a cloud, and each is called a cloud droplet..Let U The numerical characters of the cloud are represented by three numbers : expectaThe intuitionistic normal cloud overcomes the deficiency that the normal cloud model cannot reflect the degree of nonmembership and hesitation of fuzzy concepts by combining the intuitionistic fuzzy theory and cloud theory. Its basic definition is as follows:Definition 3.is a given domain of discourse, and is a qualitative concept related to the domain of discourse. is called the intuitionistic normal cloud defined on corresponding to the concept , where expectation , entropy , and hyperentropy have the same meanings as the normal cloud model, and are the membership degree and nonmembership degree of , respectively. Furthermore, when and , the intuitionistic normal cloud model degenerates into the normal cloud model..X is aDefinition 4.Letbe a set of INCs on the same universe of discourse, and define the intuitionistic fuzzy cloud weighted arithmetic averaging (INCWAA) operator as follows:whereis the weighted vector of .LetYiThe backward cloud generation algorithm is able to aggregate extended forms of data describing qualitative concepts into a dense form describing the same concept. In the context of group decision making, the backward cloud generation algorithm can aggregate evaluation information provided by different experts into a single cloud, which takes into account both tendency and spread of the assessments. We designed a backward cloud generation algorithm for INCs, the basic idea of which is to transform a set of intuitionistic fuzzy numbers into a corresponding intuitionistic normal cloud. The backward cloud generator algorithm can be implemented by the following steps.Step1: Calculate the sample mean Step2: Estimate the value of Step3: Estimate the value of Output: The estimated value In information theory, information entropy is an effective way to measure the degree of information uncertainty and system disorder , which cAccording to competitive equilibrium theory in economics , the marDefinition 5.Let there be sequencesin the fusion set, and each sequence containsclouds.is the optimal cloud of sequence.is the distance between the cloudand the optimal cloud. Take the ratio oftoas the probability ofoccurs. Then the cloud distance entropy of sequenceis defined as:In particular, when The entropy weight method based on cloud distance entropy can comprehensively reflect the preference and quality of objective decision information, and provide the guarantee of accuracy and objectivity for the determination of criterion weights in the cloud environment.Distance is an effective way to measure the difference between cloud models and an important technique to apply cloud models to solve practical problems ,43. In tDefinition 6.Let and be two INCs, then the distance measurement is defined as follows:In particular, when Furthermore, when Theorem 1.Let and be two INCs; is the distance between two clouds, then satisfies the following properties:(1) ;(2) ;(3) If and only if (4) Ifis an arbitrary INC, then.In order to verify the validity of the proposed distance measure, the proof process of Theorem 1 is given as follows.Proof of Theorem 1.(1) According to Definition 6 we can obtain(2) According to Definition 6 we can obtain(3) When When (4) According to Definition 6 and Cauchy–Schwarz inequality, we can obtainIn this paper, the proposed cloud distance measurement is used to calculate the distance between each alternative and the positive and negative ideal solutions, and then the VIKOR method in the intuitionistic normal cloud environment is proposed.For an intuitionistic fuzzy MCGDM problem, let Step1: According to the backward cloud generation algorithm for intuitionistic normal clouds, all expert evaluation information is transformed into the intuitionistic normal cloud matrix.Step2: Determine the positive ideal solution Step3: The distance between the alternative cloud Step4: Calculate the cloud distance entropy Step5: Calculate the weight Step6: Calculate the group utility value Step7: If one of the above two conditions is not satisfied, multiple compromise alternatives are obtained:(1) If condition 2 is not satisfied, then both (2) If condition 1 is not satisfied, The flow of the proposed method is shown in Suppose that in a supplier selection decision, four alternative suppliers The calculation steps based on the proposed method are as follows:Step1: The decision information of all experts is aggregated into the intuitionistic normal cloud through the backward cloud generation algorithm for intuitionistic normal clouds . The criterion values of the four alternative suppliers under the three criteria are expressed as the intuitionistic normal cloud matrix in Step2: Determine the positive ideal solution and negative ideal solution of each criterion. According to Equation (14), we can obtain the positive ideal solution of the decision cloud matrix as follows:According to Equation (15), we can obtain the negative ideal solution of the decision cloud matrix as follows:Step3: Determine the criterion weight. According to Equation (16), we can obtain the distance matrix between each scheme cloud and the ideal optimal cloud as follows:According to Equations (17) and (18), we obtain the criterion weight asStep4: Take the compromise coefficient According to the value of The effectiveness of the backward cloud generation algorithm directly affects the accuracy of the aggregated expert evaluation information, so the error analysis of the algorithm is required. Considering that the backward cloud generation algorithm based on intuitionistic fuzzy numbers has not been found in the existing research, it cannot be analyzed by the method of similar comparison. We designed experiments based on the intuitionistic normal cloud correlation theory to analyze the proposed backward cloud generation algorithm. The experimental steps are as follows.Step1: Generate a normal random number Step2: Generate a uniformly distributed random number Step3: Calculate membership degree Step4: Output the intuitionistic fuzzy number Step5: Repeat Step1–Step4 until a sufficient number of intuitionistic fuzzy numbers are generated.In this experiment, let (1)Algorithm validity analysisThe above method is used to generate 1000 intuitionistic fuzzy numbers as cloud droplet samples, and then the estimated numerical features of intuitionistic normal clouds are calculated by the proposed backward cloud generation algorithm. Let the number of experiments be 100, and the absolute error of each numerical feature in each experiment is shown in From (2)Algorithm adaptability analysisHyperentropy is an uncertainty measurement of entropy that can be used to describe the thickness of a cloud, and its magnitude affects the state distribution of the cloud. As hyperentropy increases, the state of discrete cloud droplets transforms from normal to pan-normal distribution and then transitions to the atomized state. To verify the adaptability of the backward cloud generation algorithm, let hyperentropy take a range of values The mean absolute error of the estimates of the five numerical features increases as (3)Effect of cloud droplet number on errorIn the backward cloud generation algorithm, the number of cloud droplets directly affects the comprehensiveness of cloud model information. In order to analyze the effect of cloud droplet number on the error, the range of values of cloud droplet number is set to As the number of cloud droplets increases, the errors of the estimates of the five numerical features keep decreasing and converge to 0. Meanwhile, the estimated numerical features with high accuracy can be obtained at the initial stage when the number of cloud droplets is small. This feature determines that the proposed backward cloud generation algorithm can have good application scenarios in intuitionistic fuzzy multicriteria group decision problems, thus improving the accuracy of decision making without increasing the workload.In the multicriteria group decision problem, we need to test the robustness of the proposed decision model by sensitivity analysis. Specifically, we need to observe the impact of potential changes in criterion weights on the ranking results, which is the key to effectively utilizing the model and enabling quantitative decision making. We used the perturbation method  for sensAccording to Equation (22), we can obtain Obviously, the Through a review of existing research, the Monte Carlo simulation is the main method to deal with multicriteria group decision problems in the intuitionistic normal cloud environment ,45,46. TFrom To further verify the superiority and limitations of the proposed method and to avoid the coincidence brought by a single numerical example, we added a bigger general example to increase the convincing power. This numerical example contains eight alternatives According to the method of determining the criterion weight based on cloud distance entropy, we can obtain the criterion weight asThe ranking results obtained according to INC-GRA, INC-TOPSIS, INC-Monte Carlo, and the proposed method are shown in From In addition, the coefficient of ranking similarity is a useful way to compare the ranking of different methods in decision-making problems . To compectively . The obtThe all measures show the same relationship between rankings, i.e., Spearman’s coefficient Moreover, VIKOR can take into account both group utility and individual regret. Its significant advantage over other methods is its ability to reflect the decision maker’s preferences through different compromise coefficients in decision making, which improves the flexibility and elasticity of decision making. When the compromise coefficient is small, it indicates that the decision maker prefers individual regret value in decision making and vice versa for group utility value. To analyze the influence of decision preferences on the assessment results, the results obtained by selecting different compromise coefficients are shown in As shown in In this study, a multicriteria group decision-making method based on intuitionistic normal cloud and cloud distance entropy is proposed to solve the group decision-making problem with unknown criterion weights in an intuitionistic fuzzy environment, and the effectiveness of the proposed model is verified by two numerical arithmetic examples and comparative analysis. The main contributions of the proposed model are as follows: (1) The proposed backward cloud generation algorithm deals with intuitionistic fuzzy decision-making information based on a comprehensive consideration of the fuzziness and randomness of the information. This feature facilitates the solution of group decision-making problems of different scales without loss and distortion of information. (2) The proposed concept of cloud distance entropy combines the advantages of cloud distance measure in dealing with group consistency measurement and entropy in describing information quality, which extends the application of the traditional entropy method in the field of decision making in the cloud environment. This advantage effectively solves the problems of incomplete information and insufficient objectivity in group decision making in the cloud environment. (3) The proposed intuitionistic normal cloud distance measurement is general and can accurately describe the distances and differences between cloud models, which has good practicality in the field of uncertainty decision making. (4) The model integrates group utility and individual regret, and the resulting decision results have higher stability than existing methods while maintaining validity. Future research will focus on the following aspects: (1) It would be very interesting to develop appropriate group decision-making models for solving evaluation problems in other domains. (2) Differences in the capabilities of different experts can be considered in the process of constructing the cloud. (3) Other intuitionistic normal cloud weighted aggregation operators such as the weighted geometric mean operator can be developed to accommodate decision-making problems in different environments. (4) Integrating the psychological behavior of decision makers into the decision-making process is also worth exploring.