# Ultimate Bearing Capacity Analysis of Manned Submersible Based on the Genetic Algorithm Discontinuous and Galerkin Finite Element Method

Authors: Lu Jiang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The pressure hull of deep manned submersible is the most basic component to ensure its intended function. It is necessary to study the influence of initial geometric defects on the bearing capacity of pressure hull of manned submersible with different depths. According to the idea of discontinuous Galerkin finite element method, the theoretical model is constructed and the corresponding algorithm is designed, and the genetic algorithm is combined with discontinuous Galerin finite element method to establish the inverse method to obtain the ultimate bearing capacity of manned submersible. First, the discontinuous Galerkin finite element model is constructed, the inversion model is also established through combing the discontinuous Galerkin finite element method and genetic algorithm, and then the corresponding solution algorithm is designed. Moreover, then, the ultimate bearing analysis of manned submersible for different deep is carried out based on the inversion model combing discontinuous Galerkin finite element method and genetic algorithm. The effect of defect parameters on ultimate bearing capacity of manned submersible is obtained.


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## Body

## 1. Introduction

With the continuous development of national economy and the continuous increase of world population, the exploration and development of marine resources has become an important support for the development of human society in the 21st century. Manned submersible is a kind of deep-sea carrier equipment and is the cutting-edge technology in the deep-sea strategy, which has been highly valued by the developed countries in the world. Manned submersible is an important vehicle for deep-sea resource exploration and scientific research and plays an important role in the future development of marine resources in the world. However, due to the complex working environment of manned submersible, its structure will be damaged by the changing load and external impact, At the end of each voyage, various degrees of damage were found in the frame of the submersible. A total of 161 defects were detected. Among them, the cracks occurred at the joint of the components, that is, the weld. Therefore, the maintenance and operation management of the manned submersible has become particularly important [1].As a part of the most advanced marine equipment in the world, manned submersible can carry scientists deep into the ocean to conduct scientific investigation and research work intuitively. It plays an irreplaceable role in the study of marine geology, resource investigation, deep-sea biology, and marine chemistry. The pressure spherical shell of manned cabin is one of the key components of manned submersible. It is necessary to ensure the safety of personnel and the normal operation of equipment under the condition of alternating hydrostatic pressure and unknown deep sea environment. Therefore, the pressure spherical shell needs to have enough strength, but at the same time, the weight should be reduced as much as possible, so as not to affect the overall performance and weight index of the submersible. The design and manufacturing methods of the pressure hull of the submersible are described in the specifications of the submersibles and diving systems of the major ship class societies [2].For manned submersible, its pressure hull is a medium thickness shell. When the structure reaches the limit of bearing capacity, its material has already entered the nonlinear stage, so the conventional elastic theory cannot solve the problem of ultimate bearing capacity of the structure. In the existing literature, approximate formulas are used to check the ultimate bearing capacity of the structure. These literature only roughly estimate the manufacturing defects of the pressure hull of the structure. Some scientists have carried out the finite element analysis of the geometric defects caused by the manufacturing, but only limited to the pressure resistant shell with a depth of 7000 in. In this paper, with the help of the numerical tool ANSYS, the initial geometric defects caused by the manufacturing of several different depths of manned submersible pressure hull are studied, and the Atlas of titanium alloy pressure hull affected by the defects is obtained, which can be used by designers [3, 4].In order to improve the ultimate bearing capacity analysis effectiveness of manned submersible, the discontinuous Galerkin finite element method can be applied. The discontinuous finite element method was proposed by Reed et al. The discontinuous Galerkin finite element method has many advantages, such as the accuracy of arbitrary high order can be constructed within the grid element, the boundary can be easily handled and parallel computation can be carried out. In recent years, it has been widely used in aviation, ocean, meteorology, oil exploration, and many other fields [5].The discontinuous Galerkin finite element method overcomes the disadvantages of finite volume method and finite element method and combines the advantages of the two methods; the numerical flux and limiter of finite volume method is integrated in finite element method, which is suitable for processing situation with discontinuous solution. It can process complex domains and has uniform high order precision; therefore, the discontinuous Galerkin finite element has developed rapidly. The discontinuous Galerkin finite element method uses the discontinuous piecewise polynomial to construct approximate function and weight function space; therefore, it is easy to implement adaptive mesh refinement, interpolation function ascending, and parallel computing. Due to the relaxation of strong continuity requirements between units, the discontinuous Galerkin finite element method can effectively avoid all kinds of self-locking phenomenon; therefore, it has the advantages of good convergence and high accuracy. In addition, the numerical flux is introduced into the inner boundary of the unit; therefore, it has the ability to eliminate pseudoshock at discontinuity, which can deal with discontinuous and high gradient problems with high efficient and good numerical stability. It is feasible to analyze the ultimate bearing capacity of manned submersible based on discontinuous Galerkin finite element method [5].In recent years, artificial intelligence methods suitable for solving optimization problems, such as genetic algorithm (GA). Compared with the traditional optimization methods, they generally have high functions of nonlinear mapping, adaptation and self-organization and have unique advantages in solving local optimization and global optimization and improving problem-solving efficiency. In this paper, the joint inversion method of genetic algorithm and finite element is proposed. The finite element program is embedded into the genetic algorithm program as a separate module to solve the initial stress field of nonlinear rock mass, and an example is given to illustrate the stress field, the rationality, and practicability of this method. The finite element model is transformed to a unconstrained unimodal quadratic function optimization problem, and the genetic algorithm is used to cope with optimal model, and the optimal solution can be obtained. Then, the analysis precision and efficiency can be improved.


## 2. Theoretical Model of the Discontinuous Galerkin Finite Element Method

Discontinuous Galerkin finite element method (dgfem) is a recently developed finite element method for solving time-dependent problems. Its main feature is that the finite element discretization is adopted in the space domain and time domain at the same time. The node basic knowledge vector and its time derivative vector in the semidiscrete (space discrete) control equation of the problem are interpolated independently in the time domain, and they are allowed to break between discrete time periods, during which the break value is determined by the variational principle.The algorithm process of discontinuous Galerkin finite element method is listed as follows:The computing domain is divided into several subdomains that do not overlap each other. The subdomain can be selected as the arbitrary unstructured mesh shape, and the finite element space of discontinuous function is defined by [6].(1)Vhp=vh∈L2Ωm:vh|Ωe∈Vpm;∀Ωe∈Ω,where Ωe denotes the sub domain space; Vhp denotes the local function space, which takes p order polynomial set, p=1,2,….Let us suppose that the approximate solution of discontinuous function in finite space is defined byUh, the trial function Φ is multiplied by two sides of equations, and fractional integral formula is used to obtain weak solution equation, which is expressed by the following:(2)ddt∫ΩeΦhQhdΩ−∫ΩeF⟶iQ⟶h⋅∂Φh∂xidΩ+∫ΓeFi⟶Q⟶h⋅∂Φh∂xidΓ=∫ΓeFviQh,∇Qh⋅n⟶iΦhdΓ−∫ΩeFiv⟶Q⟶h,∇Q⟶h⋅∂Φh∂xidΩ,∀Φh∈Vp,where Qh denotes the finite element approximate solution of precision solution Q and Φh denotes the finite element approximate solution of trial function Φ.n⟶i=nx,ny,nz is the unit outer normal vector of element boundary Γe, and Φh is expressed by basis function [7]:(3)Uh=∑j=1NUjtϕjpx,(4)Φh=∑j=1NΦjϕjpx,where ϕjpx denotes the p polynomial basis function.The basis function equally satisfies the expression (2), and the following expression can be obtained [8]:(5)ddt∫ΩeBjQhdΩ−∫ΩeF⟶iQ⟶h⋅∂Bj∂xidΩ+∫ΓeF⟶iQ⟶h⋅n⟶iBjdΓ=∫ΓeFviQh,∇Qh⋅n⟶BjdΓ−∫ΩeF⟶ivQ⟶h,∇Q⟶h⋅∂Bj∂xidΩ,∀Φh∈Vp,F⟶iQ⟶h⋅n⟶i is replaced by the Riemann flux function H⟶Q⟶hL,Q⟶hR,n⟶, Q⟶hL is the left unit solution, Q⟶hR is the right unit solution, finally the finite element equation is deduced, which is expressed by the following:(6)MdQ⟶dt=RQ⟶.The mass matrixM is diagonal matrix using Mij as diagonal block, RQ⟶ denotes the residual. The diagonal matrix Mij is expressed by [9]:(7)Mij=∫emϕ1pϕ1pdV∫emϕ1pϕ2pdV⋯∫emϕ1pϕNpdV∫emϕ2pϕ1pdV∫emϕ2pϕ2pdV⋯∫emϕ2pϕNpdV⋮⋮⋱⋮∫emϕNpϕ1pdV∫emϕNpϕ2pdV⋯∫emϕNpϕNpdV.The standard orthogonal basis function can makeMij have a simple form, which can reduce the computational storage. It also can simplify the matrix structure to reduce the calculation amount of whole flow field.Time discretization is calculated based on GMRES (m), and the linear equations are obtained, which is expressed by [10].(8)AΔUn=Rn.The Krylov sub space is constructed, which is listed as follows:(9)Km=Spanr⟶0,Ar⟶0,A⟶m−1r⟶0,Lm=AKm,Lm=SpanAr⟶0,A⟶2r⟶0,A⟶mr⟶0.The Arnoldi algorithm constructs a set of standard orthonormal bases, and the corresponding calculation procedure is listed as follows [11]:Step 1:U0∈Rn is given, the residual is calculated by the following:(10)r⟶0=R⟶0−A⟶Q⟶0.Letv⟶1=r⟶0/r⟶0.Step 2: The appropriate size ofm is selected to complete Arnoldi process, the vectors vii=1m and H¯m are obtained.Step 3: Minimization ofβe⟶1−H⟶¯my⟶ can get y⟶m.Step 4:Q⟶m is calculated by [12](11)Q⟶m=Q⟶0+V⟶my⟶m.Step 5:rm is calculated by the following:(12)rm=R⟶n−A⟶⋅Q⟶m.If the calculation results satisfy the condition, the algorithm is over; otherwiseQ⟶0=Q⟶m, v⟶1=r⟶m/r⟶m, return to step 2.


## 3. Basic Theory of the Genetic Algorithm

Genetic algorithm uses Darwin’s theory of biological evolution and Mendel’s law of heredity to simulate the phenomena of selection, crossover and mutation in nature. Starting from the randomly generated initial population, through the action of selection, crossover and mutation operators, cattle produce individuals with higher fitness. This cycle continues, making the solution obtained by genetic algorithm gradually move towards the direction of the optimal solution in the solution space, Until we finally get a group of individuals with the highest degree of fitness [13].The main operation process of genetic algorithm is as follows:Coding: The process of mapping the solution in the solution space into a series of data strings is coding, that is, the process of mapping phenotypes into genotypes. Different data strings represent different genotypes, which means different individuals.Generation of initial population: The program first randomly generates a certain number of initial individuals (data strings) according to the requirements of coding. These individuals form the initial population. The number of individuals represents the size of the population. It can also be described as setting the number of iterationst=0 and the maximum number of iterations t to randomly generate an initial population with n individuals, which we record as P0 [14].Fitness value evaluation and detection: The function that measures the value of individual objective function in the population is called fitness function. The value of fitness function indicates the value of individual objective function and individual “environmental adaptability.” Individuals with high fitness are more likely to be selected in the next generation, and individuals with low fitness are less likely to be selected in the next generation. According to the needs of specific problems, we can choose the appropriate form of fitness function to calculate the fitness value of individuals in populationPT.Basic operators of genetic algorithm are listed as follows:


### 3.1. Choose

Selection is to select individuals with high fitness from the current population according to the value of fitness function and certain principles. The principle of selection is that individuals with higher fitness have a greater probability of contributing to one or more offspring, and individuals with lower fitness have a smaller probability of contributing to offspring. The selection operator solves the problem of how many individuals are selected from the parent and how many children are generated [15].


### 3.2. Overlapping

Crossover operation is the main means for genetic algorithm to generate new individuals, which determines the global search performance of genetic algorithm. The crossover operator first pairs the individuals in pairs. The pairing principle is that if the population size is even, all individuals can complete the pairing. If the population size is odd, they are paired in an odd and even way, and the last individual does not participate in the pairing. Then, for the individuals after pairing, we determine the appropriate intersection location according to a certain probability (i.e., crossover probability), and finally exchange the information on both sides of the individual intersection, so as to complete the crossover operation. Cross reflects the idea of information exchange.


### 3.3. Variation

Mutation operation is an auxiliary means for the genetic algorithm to generate new individuals, which determines the local search performance of genetic algorithm. The mutation process is to select the location of the mutation point according to a certain probability and then replace the gene at the mutation point with its allele [16].The inversion model is constructed through combing the discontinuous Galerkin finite element method and genetic algorithm. It is to be noted that the key pressure value of the on-site measuring point isσi′i=1,2,…,n, and the stress value of the corresponding measuring point calculated by finite element simulation is σii=1,2,…,n. The inversion problem of ultimate bearing capacity of manned subsidiary can be transformed into the optimization problem of the mathematical model shown in (13), [17, 18].(13)ψ=−∑i=1nσi−σi′.The optimization problem described in formula (13) is solved by the genetic algorithm, and the steps are as follows:Step 1: The finite element calculation model and calculation domain are determined according to the test data.Step 2: The evolutionary algebraT of genetic algorithm is determined, and the possible load type, quantity, and value range of the finite element calculation model are determined. Within this range, a certain number of individuals are randomly generated, and each load case corresponds to one individual as the initial group [19].Step 3: The simulated stress value at the measured point corresponding to each individual in the initial population is calculated by the finite element method, and then the correspondingψ value is calculated according to formula (14) according to the measured stress value as the individual fitness.Step 4: Select according to the fitness of each individual in a certain way, and send the selected individual to the pairing library for breeding the next generation.Step 5: Two individuals are selected from the pairing library to cross, and a new individual is generated.Step 6: A certain mutation operation is carried out for each individual, and a certain number of new individuals are generated [20].Step 7: Using the finite element method, the simulated stress value at the measured point corresponding to the new generation of individuals is calculated, and its fitness is calculated. If the algebra is less thanT, go to (4), otherwise, take the individual with the maximum fitness obtained in the evolution process as the output of the optimal solution and stop the calculation. The optimal solution generated by evolution shall be judged manually. If the accuracy requirements are not met, the genetic information of generation T shall be taken as the first generation, and the next evolution process shall be started from Table 3. Otherwise, the calculation shall be terminated.In order to improve the global search ability, local search ability and search efficiency of genetic algorithm, a priori method and adaptive mutation technology are adopted on the basis of retaining the optimal value in the process of population renewal. In the process of producing the initial population, the mechanism of randomly generating individuals by genetic algorithm is not fully used, but a priori method is used to determine the individuals who meet all constraints, which not only speeds up the optimization process and global convergence of genetic algorithm but also plays a reference role in the selection of penalty factor and weighting function in the objective function. Adaptive technology makes the crossover rate and mutation rate change adaptively with the individual fitness in the process of genetic operation. At the initial stage of the algorithm, the mutation probability is high, and new individuals can be generated by using the mutation probability; in the later stage of the algorithm, it is small, so it can be used for local search.


## 3.1. Choose

Selection is to select individuals with high fitness from the current population according to the value of fitness function and certain principles. The principle of selection is that individuals with higher fitness have a greater probability of contributing to one or more offspring, and individuals with lower fitness have a smaller probability of contributing to offspring. The selection operator solves the problem of how many individuals are selected from the parent and how many children are generated [15].


## 3.2. Overlapping

Crossover operation is the main means for genetic algorithm to generate new individuals, which determines the global search performance of genetic algorithm. The crossover operator first pairs the individuals in pairs. The pairing principle is that if the population size is even, all individuals can complete the pairing. If the population size is odd, they are paired in an odd and even way, and the last individual does not participate in the pairing. Then, for the individuals after pairing, we determine the appropriate intersection location according to a certain probability (i.e., crossover probability), and finally exchange the information on both sides of the individual intersection, so as to complete the crossover operation. Cross reflects the idea of information exchange.


## 3.3. Variation

Mutation operation is an auxiliary means for the genetic algorithm to generate new individuals, which determines the local search performance of genetic algorithm. The mutation process is to select the location of the mutation point according to a certain probability and then replace the gene at the mutation point with its allele [16].The inversion model is constructed through combing the discontinuous Galerkin finite element method and genetic algorithm. It is to be noted that the key pressure value of the on-site measuring point isσi′i=1,2,…,n, and the stress value of the corresponding measuring point calculated by finite element simulation is σii=1,2,…,n. The inversion problem of ultimate bearing capacity of manned subsidiary can be transformed into the optimization problem of the mathematical model shown in (13), [17, 18].(13)ψ=−∑i=1nσi−σi′.The optimization problem described in formula (13) is solved by the genetic algorithm, and the steps are as follows:Step 1: The finite element calculation model and calculation domain are determined according to the test data.Step 2: The evolutionary algebraT of genetic algorithm is determined, and the possible load type, quantity, and value range of the finite element calculation model are determined. Within this range, a certain number of individuals are randomly generated, and each load case corresponds to one individual as the initial group [19].Step 3: The simulated stress value at the measured point corresponding to each individual in the initial population is calculated by the finite element method, and then the correspondingψ value is calculated according to formula (14) according to the measured stress value as the individual fitness.Step 4: Select according to the fitness of each individual in a certain way, and send the selected individual to the pairing library for breeding the next generation.Step 5: Two individuals are selected from the pairing library to cross, and a new individual is generated.Step 6: A certain mutation operation is carried out for each individual, and a certain number of new individuals are generated [20].Step 7: Using the finite element method, the simulated stress value at the measured point corresponding to the new generation of individuals is calculated, and its fitness is calculated. If the algebra is less thanT, go to (4), otherwise, take the individual with the maximum fitness obtained in the evolution process as the output of the optimal solution and stop the calculation. The optimal solution generated by evolution shall be judged manually. If the accuracy requirements are not met, the genetic information of generation T shall be taken as the first generation, and the next evolution process shall be started from Table 3. Otherwise, the calculation shall be terminated.In order to improve the global search ability, local search ability and search efficiency of genetic algorithm, a priori method and adaptive mutation technology are adopted on the basis of retaining the optimal value in the process of population renewal. In the process of producing the initial population, the mechanism of randomly generating individuals by genetic algorithm is not fully used, but a priori method is used to determine the individuals who meet all constraints, which not only speeds up the optimization process and global convergence of genetic algorithm but also plays a reference role in the selection of penalty factor and weighting function in the objective function. Adaptive technology makes the crossover rate and mutation rate change adaptively with the individual fitness in the process of genetic operation. At the initial stage of the algorithm, the mutation probability is high, and new individuals can be generated by using the mutation probability; in the later stage of the algorithm, it is small, so it can be used for local search.


## 4. Analysis of Ultimate Bearing Capacity of Manned Submersible

In this study, the deep manned submersible is considered, and the safety reserve factor can be appropriately reduced. The safety factorK of the design scheme of manned submersible is 1.32. The calculated pressures of 5000  m and 11000  m deep submersibles are 55.76 MPa and 141.22 MPa, respectively.The stress of shell plate of spherical shell should satisfy the following formula (Victor et al., 2020; Lou et al., 2019)(14)σ≤σs,where σs is the yield strength of the material and the yield strength of titanium alloy is 950 MPa.The inner diameter of 5000 m deep submersible is 2.4 m and the thickness of shell plate is 55 mm; the inner diameter of 8000 m deep submersible is 2.4 m and the thickness of shell plate is 82 mm; the inner diameter of 10000  m deep submersible is 2.4 m and the thickness of shell plate is 120 mm.With the help of inversion model combing discontinuous finite element method and genetic algorithm, the ultimate bearing capacity of pressure hull of manned submersible with different depths is calculated under the condition of local defects. The curves of pressure spherical shells with 5000  m and 11000  m depths affected by local defects are obtained, which are shown in Tables1–4, respectively.Table 1
Amplitude of defect on critical load (5000  m deep).
Amplitude of defect (mm)Critical pressure (MPa)Defect size = 6 mmDefect size = 10 mmDefect size = 14 mm171.268.365.4269.467.264.2368.466.563.6466.364.461.4Table 2
Range of defect on critical load (5000  m deep).
Range of defect (°)Critical pressure (MPa)Defect size = 1 mmDefect size = 2 mmDefect size = 3 mm570.669.267.41067.965.763.11564.863.661.22067.866.364.2Table 3
Amplitude of defects on critical load (1100  m deep).
Amplitude of defect (mm)Critical pressure (MPa)Defect size = 6 mmDefect size = 10 mmDefect size = 14 mm1124.7123.6122.52123.5121.8119.53122.8120.7117.94121.3118.8117.4Table 4
Range of defects on critical load (11000  m deep).
Range of defect (°)Critical pressure (MPa)Defect size = 1 mmDefect size = 2 mmDefect size = 3 mm5123.7122.6121.510122.4119.7117.515121.6117.5115.420122.6120.5118.6It can be seen that the pressure hull structure of deep submersibles is not very sensitive to local defects; with the increase of depth and shell thickness, the difference between the ultimate bearing capacity between the perfect structure and the weakest structure with local defects and the defect range of critical arc length becomes smaller and smaller. The ultimate bearing performance analysis results of manned submersible can provide the benefit guidance for designing and managing it.


## 5. Conclusions

In this paper, the inversion model through combining the discontinuous Galerkin finite element method and genetic algorithm is used to simulate and analyze the influence of defects on the pressure hull of manned submersible at different depths. With the increase of diving depth, the thickness of pressure hull increases, and the influence of geometric defects caused by manufacturing on the ultimate bearing capacity of pressure hull is weakened.The influence of geometric imperfections on the ultimate bearing capacity of the titanium alloy pressure spherical shell is not obvious. For the deep manned submersible titanium alloy pressure spherical shell structure, under the condition of meeting the construction specifications, the plate thickness and overall roundness have a certain impact on the ultimate bearing capacity of the structure, and the thickness deviation of shell plate has a more obvious influence than the overall roundness deviation on the bearing capacity of the structure. The discontinuous Galerkin finite element calculation is embedded in genetic algorithm as an independent module. The complexity of the inversion problem that this method can solve completely depends on the finite element calculation module. The joint inversion method of genetic algorithm and finite element method has a strong practicability and can be applied to the inversion analysis of ultimate bearing capacity of manned subsidiary in practical engineering. The ultimate bearing performance of manned submersible can be obtained based on the proposed model, which offers basis of optimal designing of manned submersible. In future, the other intelligent algorithm can be used to improve the analysis accuracy and efficiency of the discontinuous Galerkin finite element.



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*Source: 8524479-2022-05-27.xml*