Multi-objective optimization of different dome reinforcement methods for composite cases

Li ZU, Hui XU, Shijun CHEN,*, Jingxun HE, Qin ZHANG,Ping REN, Guiming ZHANG, Liqing WANG, Qioguo WU, Jinhui FU

aSchool of Mechanical Engineering, Hefei University of Technology, Hefei 230000, China

bAnhui Province Key Laboratory of Aerospace Structural Parts Forming Technology and Equipment, Hefei University of Technology, Hefei 230009, China

cXi’an Institute of Aerospace Propulsion Technology, Xi’an 710025, China

dThe 41st Institute of the Fourth Academy of CSAC National Key Lab of Combustion, Flow and Thermo-structure, Xi’an 710025, China

eCollege of Civil Engineering, Hefei University of Technology, Hefei 230000, China

KEYWORDSCarbon fibers;Damage mechanics;Failure;Filament winding;Mechanical properties

AbstractA multi-objective optimization method was proposed for different dome reinforcement methods of a filament-wound solid rocket motor composite case based on a Radial Basis Function(RBF) model.Progressive damage of the composite case was considered in a simulation based on Hashin failure criteria, and simulation results were validated by hydraulic burst tests to precisely predict the failure mode, failure position, and burst pressure.An RBF surrogate model was established and evaluated by Relative Average Absolute Error (RAAE), Relative Maximum Absolute Error(RMAE),Root Mean Squared Error(RMSE),and R2methods to improve the optimization efficient of dome reinforcement.In addition, the Non-dominated Sorting Genetic Algorithm(NSGA-II) was employed to establish multi-objective optimization models of variable-angle and variable-polar-radius dome reinforcements to investigate the coupling effect of the reinforcement angle, reinforcement layers, and reinforcement range on the case performance.Optimal reinforcement parameters were obtained and used to establish a progressive damage model of the composite case with dome reinforcement.In accordance with progressive damage analysis, the burst pressure and performance factor were obtained.Results illustrated that variable-angle dome reinforcement was the optimal reinforcement method compared with variable-polar-radius dome reinforcement as it could not only ensure the reinforcement angle’s continuous changing but also decrease the mass of composite materials.Compared with the unreinforced case, the reinforced case exhibited an increase in the burst pressure and performance factor of 36.1% and 23.5%, respectively.

1.Introduction

The composite case, as a key component of a solid rocket motor, is the propellant storage tank and the fuel combustion site where high temperature and high pressure exist during service.Such a complex service environment requires the case to have excellent strength and stiffness.Simultaneously,the composite case needs to keep its structure lightweight while undergoing internal pressure and axial pressure generated by the gas flow.Therefore, the mechanical properties of the composite case directly determine the performance of the solid rocket motor.1–2Since composite cases with unequal polar openings have the characteristics of non-geodesic winding and complex stress distribution, the carbon-fiber composite material has high brittleness and low fracture toughness, making it prone to low-stress burst led by dome damages.Reinforcing the domes of composite cases is the key to solving this problem.Hence,the reinforcement process and the corresponding burst prediction of reinforced cases have become a research hotspot in the field of composite cases of solid rocket motors.

At present, some investigations on composite cases have focused on the failure prediction of the composite material or the profile design of the dome part.Leh et al3employed the Hashin failure criterion to predict the burst pressure of a case by progressive damage analysis and verified the model through hydraulic burst tests.Burov4studied the influences of composite layer defects on burst pressure and predicted the damage failure modes of composite materials.Francescato et al5analyzed the prediction accuracy and calculation efficiency of burst pressure based on the Tsai-Wu failure theory.Liao et al6developed UMAT subroutines based on continuous damage mechanics and the Hashin failure criterion to study the damage evolution behaviors of fibers and resin matrices.Wang et al7established a progressive damage failure model for pressure vessels by considering the coupling effect of thermal and internal pressure and analyzed their failure behaviors.Kangal et al8established a progressive damage model to predict the burst pressures of pressure vessels.Eremin9established a micro-scale model of composite materials and calculated their mechanical property parameters.On this basis, a macroscopic model of a pressure vessel was established to analyze its damage failure process.Liu and Phoenix10established an improved mesoscopic model to analyze the damage propagation behaviors of composite layers.These investigations have shown that considering progressive damage in a simulation model could effectively improve the prediction accuracy of composite cases and thereby obtain more reliable data from numerical simulations on the burst experiments of composite cases.

As optimization algorithms being developed, an optimal design of the dome profile provides an approach to improving the strength of the dome part.Kru˙zelecki and Proszowski11used an annealing algorithm to optimize the dome meridian profile to improve the load-bearing capacity of a case.Hien et al12established a mathematical model of the dome profile based on the non-geodesic theory,and an optimal dome shape was obtained.Zhou et al13adopted a particle swarm optimization algorithm to optimize the shape of a non-geodesic dome.Paknahad et al14improved the structural strength and effective volume of a pressure vessel by optimizing the dome based on particle swarm optimization.Zu et al15adopted the multiisland genetic algorithm to optimize the dome reinforcement parameters and concluded that sectionalized reinforcement could improve the burst pressure and performance coefficient of a composite case.The existing literature on optimization of the dome of a composite case has still not clarified a way to realize quick convergence to the optimization process based on a finite element model, and important response indicators are not comprehensively considered, such as the performance coefficient and the burst pressure.

The above investigations have contributed knowledge from different aspects of dome reinforcement methods, namely laying reinforcement,16winding reinforcement, and dome cap reinforcement.Here,laying reinforcement means that a carbon cloth is dipped and directly placed between the winding layers of the dome, while winding reinforcement refers to cutting off the helical layer of the cylindrical section and retaining the winding layer of the dome part as the reinforcement layer.In regarding dome cap reinforcement, the dome is reinforced by a prefabricated dome cap made of fibers in accordance with the dome profile.Although these three reinforcement methods have been applied in the engineering field, actual dome reinforcement mainly relies on experimental experience, lacking the guidance of a basic theory.Determining the reinforcement position, angle, and layers to achieve the objective of dome reinforcement according to the stress distribution of a dome is also difficult.Hence, by developing different multiobjective optimization methods with Non-dominated Sorting Genetic Algorithm II (NSGA II) based on a Radial Basis Function (RBF) model incorporating progressive damage analysis for dome-reinforced composite cases in this article,the authors will address the following question: which reinforcement method is the best in terms of burst pressure and performance factor, and what are the optimal process parameters for the reinforcement methods?

In this study,different optimization methods for dome reinforcement of a composite case were investigated.On the basis of a refined finite element model, a method for constructing a surrogate model, as an approximation to the finite element model to realizing quick convergence in an optimization, was employed to establish a Radial Basis Function (RBF) model of the composite case with dome reinforcement.NSGA-II17–18was selected to establish optimization models of variableangle and variable-polar-radii dome reinforcements.The influence of the coupling effect of the reinforcement angle, range,and layers on the performance of the composite case was revealed, and the optimized design of reinforcement parameters was implemented to achieve the objectives of precise,lightweight, and efficient dome reinforcement.Progressive damage analysis was carried out to predict the burst pressure of the reinforced case to evaluate the reliability of the optimization model.By comparing the performance factors of the composite case and the fabrication process obtained by different reinforcement methods, a multi-objective optimization strategy of dome reinforcement that considers the process feasibility and the structural performance was established to provide basic theoretical support for engineering applications of composite cases.

2.Progressive damage modeling and experimental validation of composite cases

2.1.Progressive damage modeling of composite cases

2.1.1.Finite element modeling of composite cases

The netting theory was selected to design the lamination scheme of a composite case,and it was introduced as follows19:

where hφand hθare the thicknesses of hoop and helical layers,respectively; P is the burst pressure; R′is the radial distance from the center line to a point in the layer (note that its value here is generally taken as the same as the radius of the middle surface of the composite layers); σfis the effective fiber strength; α is the winding angle of the cylindrical section; ksis the stress equilibrium factor.

The winding angle on the dome of the composite case was calculated by the non-geodesic trajectories as follows20:

where R0is the radial distance from the center line to the turnaround point,n is the parameter,θtlis the winding angle at the tangent line,Rtlis the radius at the dome-cylinder tangent line,and δ is the difference in degrees between the frictionless winding angle and the winding angle calculated by the first term of Eq.(2).

The ply thickness of the dome is given by

where ttlis the thickness at the tangent line; θris the winding angle at the radius; rtlis the radius at the dome/cylinder tangent line; r0is the radius at the helical turnaround point; r is the radius at a given point on the dome; BW is the width of the helical fiber tow.

A finite element model of the composite case was established with the aid of ABAQUS software, as shown in Fig.1.Considering the computational cost, 1/18 of the full finite element model was established for numerical calculation.For this model, a cubic solid element (C3D8R) was selected for constructing the liner and metal boss, and an eight-node linear element (C3D8) was used for constructing composite layers.A zero-thickness cohesive element was inserted into the two adjacent layers to simulate the interlaminate damage.

2.1.2.Progressive damage failure criteria

The 3D Hashin criterion was used to predict the fiber and matrix damage initiation in this study and introduced as follows21for different failure modes:

For fiber tensile failure (σ11≥0),

where σij(i,j=1,2,3)is the effective stress tensor;τij(i,j=1,2,3)is the shear stress;XTand XCare longitudinal tensile and compressive strengths, respectively; YTand YCare transverse tensile and compressive strengths, respectively; Sij(i, j = 1,2, 3) is the shear strength.

The progressive degradation model proposed by Camonho and Matthews22was selected to present the damage status.The damage variables for fiber tension and compression as well as matrix tension and compression are presented in Table 1.

2.1.3.Interlaminar damage failure criterion

The delamination between composite layers was simulated by means of inserted cohesive elements.The delamination damage in cohesive elements includes two procedures: damage initiation and damage evolution.Damage initiation could be performed through stress and strain criteria, while damage evolution could be modeled based on energy or effective displacement.

The interface behavior was elastically linear before damage initiation occurred.The stress–strain relationship could be defined as23.

where σ and ε are the traction stresses and strains of the normal and two shear components,respectively,and Kjj(j=m,s,t)is the separation stiffness;σnis the traction force in the normal direction;σsand σtare the traction forces in the two shear directions, respectively.

In this study,a stress-based quadratic damage initiation criterion was employed as follows24:

Fig.1 Geometric dimensions and finite element model of the composite case.

2.2.Validation of the damage model

Table 1 Stiffness degradation criteria of composite materials.

where Nmax, Smax, and Tmaxare the corresponding strengths under crack modes I-III.

When damage initiation occurred, the stress–strain relations of the cohesive elements could be expressed as

where d is the stiffness degradation factor,which is determined by the B-K criterion given as25.

Fig.2 shows the progressive damage of the composite case without dome reinforcement.Matrix tensile failure occurred at the polar openings and the cylindrical section under internal pressure.Due to the deformation incongruity of composite layers at the shoulder of the metal boss,a stress concentration was prone to occur, which resulted in matrix failure.Compared with other positions of the dome, the curvature near the equator was the largest, and a bending moment was obvious.Therefore,the matrix withstood the bending moment and compressive stress simultaneously, which led to matrix compression failure, as shown in Fig.2(b).As illustrated in Fig.2(c),the cohesive elements at the equator and the shoulder of the metal boss were damaged.The reason could be that the helical and hoop layers were alternately wound at the equator,resulting in uncoordinated structural stiffness.In addition,the coupling effect of tension stress and bending stress at the shoulder led to matrix damage, which, in turn, caused interlaminar damage in the composite layers.Under the pressure of P = 23.9 MPa, fiber damage caused by tensile stress occurred at the shoulder of the metal boss, as depicted in Fig.2(e), and the burst position is in good agreement with experimental results.As shown in Fig.3(a), as the internal pressure increased, matrix failure occurred firstly.With the accumulation of matrix damage and the expansion of the damage area, interlaminate damage of the composite layers on the dome occurred,resulting in delamination between the composite layers and the instantaneous acceleration of fiber damage.According to Fig.3(b), the maximum displacement occurred in the cylindrical section,but the displacement at the shoulder changed suddenly under the pressure of 23.9 MPa with a radial displacement of 0.172 mm (Ds).Simultaneously, the displacement of the cylindrical section still maintained a linear increase,which indicated that damage occurred at the shoulder of the metal boss.

Hydraulic burst tests were carried out to validate the finite element model of the composite case without dome reinforcement, as shown in Fig.4(a).Strain gauges of the cylindrical section were equal-distantly pasted in the longitudinal and circumferential directions, and a dynamic strain gauge (model:DH5922D, collection frequency: 1 kHZ, error: within 0.5 %)was used to collect the axial and circumferential strains respectively.Strain gauges in the dome part were pasted respectively along the direction of the prepreg tape and perpendicular to the prepreg tape,and the strains in the fiber direction and perpendicular to the fiber direction were collected respectively.In order to obtain the axial and radial displacements of the composite case, displacement sensors (model: WTB-30, range: 0–30 mm) were set at both ends of the dome and the center of the cylindrical section respectively.

Fig.2 Progressive damage analysis of the composite case.

Fig.4(b)shows the load–displacement curve of the cylindrical section obtained by the hydrostatic burst tests and finite element simulations of the composite case.The deviation between the test results of the case numbered 1# and the numerical simulation results is relatively large, and the maximum deviation value at the same time is about 0.25 mm.The main reason for the larger error could be that the water pressure was unstable during the test process, resulting in the composite case shaking slightly and leading to the deviation of the contact position between the displacement sensors and the case surface.Thus,the accuracy of the test data decreased.Nevertheless, the test results of the case numbered 2# are in good agreement within a small error range.Fig.4(c) shows a comparison between the finite element results and the experimental results in terms of the stress–strain curves of the case shoulder.The results obtained by experiments and the finite element simulations are in good agreement.Therefore, the finite element model established in present article has good accuracy to predict the burst pressure of the composite case,and the maximum error is 8.1 %.

3.Multi-objective optimization methods for dome reinforcement of composite cases

As discussed later in Sections 3.1 and 3.2, design variables are discrete quantities so that a parametric optimization model should be the one to execute the optimization of the dome reinforcement methods of composite cases.An optimization model was developed specifically based on each reinforcement method, as detailed in the two subsections.A flow chart of the non-linear multiple-objective optimization was integrated to show the process as depicted in Fig.5.The optimization procedure was as follows.

Firstly, according to the dome reinforcement method,specific design and response variables were determined, and a parametric dome reinforcement finite model was established based on Python scripting language.Secondly, the optimal hypercube design (Opt LHD) experimental design method was used to select a sample point set in the design variable space.The Python script called ABAQUS to calculate the model and obtained the maximum fiber stress of the dome composite material layer and the mass of the composite material at the corresponding sample point.The RBF neural network model was used to fit and interpolate the sample data,and the model training and prediction accuracy were carried out.Finally, the NSGA-II algorithm was used to establish a multi-objective optimization model, the reinforcement parameters were optimized to obtain the Pareto front, then the optimal reinforcement parameter combination was obtained, and the reliability of the optimization model was verified.

Compared with other methods, the method proposed in this article would be capable of considering the multiobjective problem of reinforcement methods of composite cases for it is based on the NSGA II, a multi-objective optimization algorithm, and it also fits to solve a highly iterative optimization process with nonlinear properties and realizes quick convergence of an optimization process by incorporating a surrogate model base on an RBF algorithm.

3.1.Method of variable-polar-radius dome reinforcement

3.1.1.Multi-objective optimization model

Variable-polar-radius dome reinforcement is a method of filament winding reinforcement that is based on variable radii of polar openings, as illustrated in Fig.6.Compared with other filament winding reinforcement, variable-polar-radius dome reinforcement could effectively reduce the redundant mass of composite materials resulting from dome reinforcement.It also has good manufacturability.Therefore,an optimization model of the composite case based on this reinforcement was established to investigate the influences of the reinforcement parameters on the case performance and optimize the reinforcement layers and reinforcement range.

Fig.3 Variations of damage state variables and the load–displacement curves at the shoulder of the composite case with an internal pressure of the composite layer at the shoulder of the case.

In the process of variable-polar-radius dome reinforcement,the number of helical layers increased to at least two times of what the polar radius of winding reinforcement layers changed to.This increase resulted in the strength of the dome being sufficient enough not to need further optimizing the number of reinforcement layers.Supposing that the number of variablepolar-radius dome reinforcements was three, additional six helical layers were used to reinforce the dome.At this time,the ratio of helical layers to hoop layers of the composite case was 1.2:1, and the strength of the dome was sufficient enough not to need optimizing the reinforcement layers.However, if the number of variable-polar-radius dome reinforcements was only one, the ply thickness severely accumulated at the position of variable polar radii.Therefore, variable-polarradius dome reinforcement was carried out twice in this article.

For ensuring that the finite element model of the composite case with variable-polar-radius dome reinforcement was established successfully in the optimization process, two variable polar radii could not overlap, and the polar radius should not equate to the equator radius.Therefore, the dome reinforcement range was divided into two parts, of which one was from the shoulder of the metal boss to the polar opening and the other was from the shoulder to the equator of the composite case.In addition, as each variable-polar-radius dome reinforcement was equivalent to increasing two helical layers at least, the maximum reinforcement layer was six in this article.Hence,in the current optimization process,the design reinforcement parameters(key sensitive parameters)were specified as the reinforcement range and reinforcement layers, and the optimization objectives were the maximum fiber stress and mass of composite materials.Note that the influences of the two design parameters specified here will be analyzed later in Section 4.1.1.Finally, the objective function and the constraints of design variables are introduced below.

Fig.4 Hydraulic burst tests of the composite case without dome reinforcement and the corresponding results.

Front dome:

Back dome:

where rf1and rf2are the first and second polar radii of the front dome,respectively;n is the number of reinforcement layers in a winding cycle,and it stands for 2n helical layers;rb1and rb2are the first and second polar radii of the back dome,respectively.

Fig.5 Flow chart of the procedure of optimization based on the RFB surrogate model.

Fig.6 Schematic diagram of the case with variable-polar-radius dome reinforcement.

3.1.2.Error analysis of the optimization model

In accordance with the finite element model of the composite case with variable-polar-radius dome reinforcement, 100 samples were selected to establish the RBF surrogate model26–28on the basis of Opt LHD,29and the prediction accuracy of the surrogate model was evaluated, as shown in Fig.7.For the maximum fiber stress, the errors calculated by Relative Average Absolute Error (RAAE), Relative Maximum Absolute Error (RMAE), and Root Mean Squared Error (RMSE)methods30–31were 0.075, 0.29, and 0.11, respectively; for the mass of composite materials, the errors obtained were 0.028,0.091,and 0.011,respectively.The errors of the two optimization objectives were all within the acceptable level, and the R2values of the maximum fiber stress and the mass of composite materials were 0.88 and 0.985,respectively,which were close to 1, suggesting that the surrogate model had good prediction accuracy and could be used as an approximation to the refined element model for iteration in the optimization process.

3.2.Variable-angle dome reinforcement method

3.2.1.Multi-objective optimization model

When establishing a finite element model based on variableangle dome reinforcement, determining the reinforcement angle and ensuring a continuous variable are essential prerequisites for establishing an optimization model.In accordance with the distribution of the fiber stress of the dome, each element of the reinforcement layer was taken as the research object, and the optimal reinforcement angle of each element was searched in the range of [0°, 90°].This method could theoretically determine the reinforcement angle of each element in accordance with the direction of the fiber stress.However,establishing an optimization model is very difficult.If the reinforcement angle of each element is used as a design variable,the design variable also increases as the number of reinforcement layer elements increases.The element of the reinforcement layer should increase as much as possible to ensure a continuous variable of the reinforcement angle.Simultaneously,design variables may increase greatly in the optimization process, making it difficult to search for optimal solutions.Therefore, a finite element modeling method for variableangle dome reinforcement was proposed in this article to reduce design variables as much as possible and obtain the optimal solution, as shown in Fig.8.

By multiplying the element angles of reinforcement layers by the same coefficient, the problem of too many design variables could be resolved,and then the optimization of the reinforcement angle could be transformed into that of the reinforcement angle coefficient.Finite element modeling based on variable-angle dome reinforcement mainly includes the following steps.

Step 1.An initial finite model was established.An initial finite element model with reinforcement layers was established with the aid of ABAQUS software to generate element properties.As the angle of the reinforcement layer was a fixed value,its winding angle and material properties of the reinforcement elements must be realigned.

Step 2.The element properties of the helical layer of the dome were obtained.The element number and the corresponding material property of the initial finite model were obtained by writing a Python script program, and a text file that included the element number and its material properties was generated.

Step 3.Element sets and corresponding node sets of the helical and reinforcement layers of the dome were created.Each element and node number of the helical and reinforcement layers of the dome were extracted through the script program.By judging whether the node numbers of the helical and reinforcement layer are equal to each other,the elements of the helical layers in contact with the elements of the reinforcement layers were selected out and stored.

Fig.7 Error analyses of the surrogate model of the case with variable-polar-radius dome reinforcement.

Fig.8 Modeling process of the finite element model based on variable-angle dome reinforcement.

Step 4.Element properties of the reinforcement layers were assigned.In accordance with Steps 2 and 3,the winding angles and material properties of the helical layer sharing nodes with the elements of the reinforcement layers were assigned to the elements of the reinforcement layers, and a new finite element model with variable-angle dome reinforcement was established.

According to the operations of the above four steps, the optimization of the composite case based on variable-angle dome reinforcement could be realized by multiplying the angles of the reinforcement layers by the corresponding coefficients.

Considering that local dome reinforcement could improve the strength in the reinforced area and weaken the other areas of the dome,the entire dome was reinforced in this article,that is to say, the reinforcement range remained unchanged in the optimization process.Thus, it was not taken as a design variable.In addition, the coefficient ranged from 0 to 1 to adjust the reinforcement angles.Therefore, the reinforcement layer and the reinforcement coefficient of the dome were specified as the key sensitive parameters, and the response parameters were the same as those in the optimization model of variable-radius reinforcement.Likewise, the influences of the two design parameters specified here will be analyzed later in Section 4.2.1.Therefore, the objective functions of variableangle dome reinforcement are given by.

Front dome:

where cbis the coefficient of the back dome.

3.2.2.Error analysis of the optimization model

Fig.9 shows the errors of the surrogate model on the basis of variable-angle dome reinforcement.The errors obtained by RAAE, RMAE, and RMSE were all within the acceptable error level, and R2of the maximum fiber stress and mass of composite materials met the precision requirements proposed in this article.These findings suggested that the surrogate model had high accuracy in global approximation and nonlinear problem, thereby being fit to be used as an approximation to the refined element model for iteration in the optimization process.

Fig.9 Error analysis of the surrogate model based on variable-angle dome reinforcement.

4.Results and discussion

4.1.Results of variable-polar-radius dome reinforcement

4.1.1.Correlation between reinforcement parameter and response variables

Fig.10 shows the correlations between the design variables and the optimization objectives.A positive value indicates a positive correlation, that is, the values of optimization objectives increase as the values of design variables increase.Reversely, a negative correlation means that a design variable and an optimization objective are in an inversely proportional relationship.As shown in Fig.10,with an increase in the number of reaming layers,the maximum fiber stress decreased,and the mass of the composite materials increased.As the polar radius increased and the reinforcement range decreased, the maximum fiber stress increased.The reinforcement parameter with a significant correlation with the maximum fiber stress was the polar radius of the first reinforcement, followed by the reinforcement layers of the second reinforcement.Meanwhile,the reinforcement parameter that had an obvious correlation with the mass of composite materials was the reinforcement layers of the first reinforcement, followed by the polar radius.The correlation between the polar radius of the first reinforcement and the maximum fiber stress was greater than that of the second reinforcement,and the correlation between the reinforcement layers of the second reinforcement and the maximum fiber stress was greater than that of the first reinforcement.

Fig.10 Correlation coefficients between reinforcement parameters and optimization objectives based on variable-polar-radius dome reinforcement.

Therefore, for variable-polar-radius dome reinforcement,the polar radius of the first reinforcement should not be too large but be as close as possible to the original polar opening to improve the overall strength of the dome.If the variable polar radii are too large, it could only improve the strength of the local area of the dome.However, the strength of the non-reinforced area was weakened, which made it difficult to improve the pressure bearing performance of the dome.The second reinforcement should focus on reinforcing the local area of the dome, and appropriately increasing the number of reaming layers could effectively improve the reinforcing effect.Given that variable-polar-radius dome reinforcement is essentially a winding reinforcement in which the polar radius is variable, the number of reinforcement layers becomes the main parameter that affects the mass of composite materials.When the polar radius is determined in the dome reinforcement process, optimization of reinforcement layers is the key to decreasing the mass of composite materials.

4.1.2.Determination of optimal results

Fig.11 shows the Pareto frontier of the optimized model for variable-polar-radius dome reinforcement, where Smdenotes the maximum fiber stress and W denotes the mass of composite materials.The Pareto solution corresponding to the minimum fiber stress showed that the maximum fiber stress was 1408 MPa, and the mass of composite materials was 0.589 kg.Although the fiber stress of the dome significantly decreased, and the strength of the dome was effectively improved, the mass of composite materials was redundant when selecting this combination of solutions.A set of feasible solutions that was the closest to the design requirement of dome strength was selected as the optimal solution of the optimization model on the basis of variable-polar-radius dome reinforcement to meet the design requirement of dome strength and decrease the mass of composite materials led by the reinforcement layers as much as possible.The selected optimal solution showed that the maximum fiber stress was 1745 MPa,and the corresponding mass of composite materials was 0.549 kg.The optimal reinforcement parameters were obtained and are presented in Table 2.

4.1.3.Progressive damage analysis of the reinforced case

Fig.11 Pareto frontier of the optimization model based on variable-polar-radius dome reinforcement.

Table 2 Optimal reinforcement parameters based on variablepolar-radius dome reinforcement.

According to the optimal reinforcement parameters shown in Table 2, a finite element model of the composite case based on the variable-polar-radius dome reinforcement was established, as illustrated in Fig.12, which additionally shows the progressive damage analysis of the composite case on the basis of variable-polar-radius dome reinforcement.Compared with the unreinforced case, the pressure bearing ability of the reinforced case was effectively improved,and a fiber tensile failure mainly occurred in the cylindrical section.Interlaminar damage of the composite material in the cylindrical section occurred before the fiber failure.According to Fig.13(a),compared with the matrix tensile failure, the interlaminar damage had a great effect on the fiber damage,which was the main reason for the aggravation of the fiber damage.Fig.13(b) shows the load–displacement curve of the composite layer in the cylindrical section.The displacement increased sharply at the internal pressure of 33.3 MPa,suggesting that a burst occurred at the cylindrical section.According to the burst position of the reinforced case, the optimized process parameters of the variable-polar-radius dome reinforcement could meet the design requirements of the dome reinforcement.

4.2.Optimal results of variable-angle dome reinforcement

4.2.1.Correlation between reinforcement parameters and response variables

Fig.14 shows the correlations between the reinforcement process parameters and the optimization objectives for the variable-angle dome reinforcement.In accordance with the positive and negative correlation coefficients between the design variables and the optimization objectives, as the reinforcement angle coefficient increased,the maximum fiber stress of the front dome decreased, and the maximum fiber stress of the back dome increased.On the contrary, as the reinforcement layers increased, the maximum fiber stress decreased,and the composite mass increased.According to the absolute values of the correlation coefficients compared with those of the front dome, the correlations between the reinforcement angle coefficients of the back dome and the optimization objectives were more significant.Among all the design variables,reinforcement layers had the most obvious correlation with the optimization objectives.Given that the composite case had unequal polar openings, the dimension of the polar opening of the front dome was larger than that of the back dome.Therefore,as the internal pressure gradually increased,the circumferential expansion of composite layers at the polar opening of the front dome was more significant than that of the back dome.Increasing the reinforcement angle coefficient at the polar opening of the front dome facilitated increasing the hoop strength and improving the shear strength of the shoulder of the metal boss.Given that the polar radius of the back dome was small,the tensile stress at the polar opening was significant.Adopting small reinforcement-angle coefficients, that is, decreasing the reinforcement angle, could effectively improve the tensile strength at this position.Therefore, for the variable-angle dome reinforcement, a dome with a large polar opening should be reinforced with a large angle, and a small angle should be utilized to reinforce a dome with a small polar radius.When the reinforcement angle coefficient is determined, increasing the reinforcement layers is necessary to effectively improve the pressure bearing ability of the dome.

Fig.12 Finite element model of the composite case based on variable-polar-radius dome reinforcement and the corresponding progressive damage analysis of the case based on variable-polar-radius dome reinforcement.

4.2.2.Determination of optimal results

As shown in Fig.15, according to the Pareto frontier, several optimal solutions could be adopted for the maximum fiber stress and composite materials to meet the design requirements of the dome strength.Note that S denotes the maximum fiber stress and W denotes the mass of composite materials in this figure.A small change in the mass of composite materials has a slight effect on the case performance.Therefore,the Pareto frontier solution corresponding to the minimum fiber stress was selected as the optimal solution for variable-angle dome reinforcement optimization, which suggested that the optimal mass of composite materials and the maximum fiber stress were 0.519 kg and 1709 MPa, respectively.Optimal process parameters corresponding to the optimization objectives could be obtained, as shown in Table 3.

In accordance with the optimal reinforcement angle coefficient shown in Table 3, the variable-angle curve of the optimized reinforcement layer could be calculated by the initial winding angles of the dome, as shown in Fig.16.

4.2.3.Burst pressure prediction of the reinforced case

Fig.13 Damage analysis results.

Fig.14 Correlation coefficients between variable-angle reinforcement parameters and optimization objectives.

According to the progressive damage analysis of the variablepolar-radius dome reinforcement, the final burst position of the reinforced case occurred in the cylindrical section when satisfying the design requirement of the strength of the dome.For variable-angle dome reinforcement, the maximum fiber stress of the dome could be effectively decreased after optimization,and the strength of the dome is less than 1800 MPa,thus meeting the design requirement.Furthermore, the lamination scheme is the same for the reinforcement case on the basis of different dome reinforcement methods.Therefore, the burst pressure and burst position obtained by the variable-angle dome reinforcement is considered consistent with those of the variable-polar-radius dome reinforcement, and the burst pressure is 33.3 MPa.

Fig.15 Pareto frontier of the optimization model based on variable-angle dome reinforcement.

Table 3 Optimal process parameters of variable-angle dome reinforcement.

Fig.16 Angle curves of the dome with variable-angle reinforcement.

4.3.Comparison between different optimization methods

4.3.1.Comparison of case performance

In accordance with the optimization results and progressive damage analysis, the performance factors of the reinforced case were calculated to evaluate the performances of different dome reinforcement methods, as shown in Table 4.

In terms of numerical simulation, the variable-angle dome reinforcement method exhibited the best reinforcement effect.Compared with those of the unreinforced case, the burst pressure and performance factor increased by 31.4%and 23.5%,respectively.The variable-angle dome reinforcement is an equal-thickness and variable-angle reinforcement method without overlap between the reinforcement layers.Therefore,compared with variable-polar-radius dome reinforcement, themass of composite materials could further decrease, thereby improving the performance factor of the composite case.Although the reinforcement angle changed continuously in the process of variable-polar-radius dome reinforcement, the thickness of the composite layer gradually increased from the equator to the polar opening, resulting in a redundant mass of composite materials.In addition, for a large-dimension composite case, the waste of composite materials is severe when employing variable-polar-radius dome reinforcement.

Table 4 Comparison between results obtained by different dome reinforcement methods.

As the optimization results have shown, variable-angle dome reinforcement exhibits better performance produced by less materials consumption, better uniformity of thickness of the reinforcement layers,higher burst pressure,and higher performance factor.This reinforcement method has the potential of facilitating solid rocket composite cases to meet their high requirements of light weight and high strength.Performance enhancement could be the main focus under these requirements, and with the increasingly mature fiber placement process, the cost of variable-angle dome reinforcement could be reduced to a certain acceptable level.

4.3.2.Comparison between reinforcement processes

In terms of reinforcement process, the variable-angle dome reinforcement process has higher requirements on equipment because it includes two processes: winding and placement.The helical and hoop layers of the composite case are realized by the winding process,and a reinforcement layer is placed at a variable angle by placement equipment to achieve a constant thickness and a continuous variable angle.Therefore, the process cost of this dome reinforcement method is relatively high.The process of variable-polar-radius dome reinforcement could guarantee the continuity of fibers, and it is very mature but not suitable for large-dimension cases.The redundant mass of composite materials increases as the geometry dimensions of the case are enlarged.In addition, improving the performance factor of the composite case is difficult.Therefore,considering the performance of the composite case and its process cost, variable-angle dome reinforcement is taken as the most recommended dome reinforcement method when equipment conditions permit, although the three dome reinforcement methods generally fit in different fabrication processes and different requirements.

5.Conclusions

Based on the RBF model and NSGA-II, multi-objective optimization models of different dome reinforcement methods,such as variable-angle dome reinforcement and variablepolar-radius dome reinforcement, were established to reveal the coupling mechanism of reinforcement angle,reinforcement layers, and reinforcement range on the case performance.The optimal dome reinforcement optimization strategy was also determined.Conclusions were obtained as follows:

(1) For variable-polar-radius dome reinforcement, in the two dome reinforcement processes, the influence of the polar radius on the case performance is significantly greater than that of the number of reaming layers,which directly determines the mass of composite materials.Therefore, the key to improving the performance factor of the case is to reasonably select a variable polar radius.According to numerical simulation results, the area of the first dome reinforcement should cover the entire dome to improve the strength of the shoulder and the equator to avoid the effect of local area weakening due to an insufficient reinforcement range.The second dome reinforcement should focus on a local area ranging from the shoulder of the metal boss to the equator to improve the resistance to tensile stress and bending stress and reduce the mass of composite materials.

(2) Variable-angle reinforcement could ensure that the reinforcement layer has an equal thickness and a continuous variable angle.Optimization results have illustrated that the reinforcement angle increases continuously from the equator to the polar openings, indicating that a small angle should be used for reinforcement near the equator to improve the tensile and bending resistance,and a larger angle should be selected for reinforcement near the polar openings to improve the circumferential strength.

(3) Variable-angle dome reinforcement is the optimal reinforcement scheme in terms of reinforcement effect.It could not only ensure a continuous change of the reinforcement angle but also decrease the mass of composite materials.Compared with the unreinforced case, the reinforced case exhibited an increase in the burst pressure and the performance of 36.1%and 23.5%,respectively.This study solved the problems of uncontrollable quality of dome reinforcement and difficulty in improving reinforcement efficiency and provided new ideas and methods for dome reinforcement.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was co-supported by the National Natural Science Foundation of China (52175311, 52175133,12102115, and 52005446) and the Fundamental Research Funds for Central Universities in China (JZ2021HGTA0178 and JZ2022HGQA0150).

Data availability

The raw/processed data required to reproduce the findings cannot be shared at this time as the data is also a part of an ongoing study.