Optimal Design of Tank Sheets Based on Parametric Modeling

LI Hao ,LI Chenggang ,LOU Yunfeng ,HAO Junjie ,ZU Qingming ,ZHANG Xing ,CHEN Mingliang

1 Aerospace System Engineering Shanghai,Shanghai 201109

2 Shanghai Academy of Spaceflight Technology,Shanghai 201109

Abstract:Optimal design of the tank has a significant effect on reducing the weight of a launch vehicle’s structure.In this paper,the key characteristics of a stiffened shell are identified from the design requirements,focusing on the influence of the internal pressure on the axial compression load-bearing capacity.The computing method of the ultimate load of the stiffened shell,the parametric modeling method and the surrogate modeling technique for optimal design are reviewed.An optimization process applicable to the stiffened shell was developed and applied in the optimization work for the tanks of solid-liquid bundled launch vehicle,so a better weight reduction effect could be achieved.

Key words:parametric modeling,surrogate model,optimal design,isogrid,orthogrid

1 INTRODUCTION

Currently,the first sub-stage propellant tanks have been designed to transmit the launch vehicle’s axial load while bearing the internal pressure.To achieve a higher ratio of load to mass,a stiffened structure form is widely used for the shell section of the tank.For the first generation launch vehicles in China,a 45° orthogrid was used as the configuration of the shell section,and a single section was generally made of four sheets welded together,each sheet was manufactured by a rolling bending and then milling process,and finally,multiple sections were welded together to form a whole tank with the bottoms and short shells.

With the development of new generation launch vehicles,China has been gradually using orthogrid and isogrid instead of traditional 45° orthogrid as the stiffening type of load-bearing tanks.On the one hand,most of the international launch vehicles in service use these two types,such as the H-2A,Delta,Titan,Atlas,and Falcon.On the other hand,in recent years,researchers have conducted considerable research work,which makes the application of the grid stiffened shell more mature.

HAN[1]studied the stability of the stiffened shell structure through theoretical analysis,the finite element method plus experiments,establishing that there is low-stress instability in the skin,which still has a certain load-bearing capacity after instability.ZHOU et al.[2]optimized the angle of stiffeners,the thickness of skin and stiffeners based on parametric modeling,combining with surrogate modeling techniques.MAO et al.[3]applied Ansys software to optimize the shell with orthogrid reinforcement and discussed the effect of structural parameters on the critical load and bending modes of the cylindrical shell.FAN et al.[4]used the equal stiffness smeared-out method,finite element method,and full-scale experiment to verify that the isogrid has higher structural efficiency than the 45° orthogrid.XU et al.[5]systematically summarized the nonlinear dynamic finite element method for tracing the post-buckling path of a thin-walled grid stiffened shell,and proposed the modeling and solution convergence control requirements.YUE et al.[6]proposed a high-precision modeling method for shell models based on Abaqus software by comparing simulation and test results.DONG et al.[7]investigated the variation of the bearing capacity of the stiffened cylinder under different diameters and internal pressures for orthogrid and isogrid.WANG and HAO et al.[8–10]proposed a hybrid optimization strategy based on a surrogate model and an equivalent stiffness model to investigate the suppression of imperfection sensitivity by the design of a cylindrical shell with varying amplitudes of hyperbolic generatrix shape,and developed a designable T-section for a bi-directional multi-stage stiffened configuration from the perspective of low imperfection sensitivity.

For this paper,key characteristics of the load-bearing stiffened shell structure were analyzed on the basis of the tank design requirements.A detailed description is provided on the ultimate load estimation methods,parametric modeling approaches,and surrogate modeling techniques involved in the optimization of the tank sheets for the LM-6A launch vehicle,as well as specific strategies for their application.

2 DESIGN REQUIREMENT ANALYSIS

2.1 Load Case

During the service of launch vehicle,the load conditions that the tanks are subjected to include air-tightness,hydraulic acceptance,transportation,lifting,propellant loading,ground-hold process and flight process.For the design of the shell section,the failure modes under the three loading conditions need to be considered simultaneously: stress failure under maximum internal pressure,instability under axial compression load with small or even zero internal pressure during the propellant loading process,and instability under axial compression load with considerable internal pressure during the flight process.

The so-called small internal pressure during the groundhold process is relative to the flight process,such that the superposition of internal pressure and axial compression load is not sufficient to cause the structure yielding,but the combination of the two types of load during the flight process will cause the cylinder sheets to yield.That is to say,different forms of loading correspond to different mechanisms exhibited in instability modes.

2.2 Manufacturing Process

The design of the launch vehicle tank section requires full consideration of manufacturing constraints.At present for manufacture,the sealed stiffened shells for aerospace applications generally use a thick plate milling process,with a roll bending process to form a single sheet.A single section of general diameter is generally made of four sheets welded together.Due to the decline in mechanical properties of the welded heat-affected zone,the sheets near the welded edge should be appropriately thickened.The design of the size of the stiffeners and skins should be considered in conjunction with the effects of clamping,processing stress,and orthopedic factors in the rolling,milling,and welding processes,to reasonably determine the permissible range of parameters.

3 KEY CHARACTERISTICS OF THE STIFFENED SHELL

3.1 Instability Mode

As mentioned earlier,different forms of loading correspond to different instability modes.During the propellant loading process,the internal pressure is zero or slightly pressurized,so the stress when the instability of the cylinder section occurs is within the elastic range of the material,it can be called the elastic instability,as shown in Figure 1.During the flight process,with the simultaneous effects of the internal pressure and axial compression,the region on both sides of the horizontal weld seam of the cylinder section generate large bending stress due to the discontinuity of stiffness,which makes the region vulnerable to yielding and thus plastic instability occurs when the axial pressure load increases beyond a critical value,as shown in Figure 2.The structures can be recovered after the load is removed considering elastic instability of structures,but the elastic instability is not allowed in design,and so as the plastic instability.

Figure 1 Elastic instability mode

Figure 2 Plastic instability mode

In addition,the elastic instability of the grid stiffened shell has two modes under different combinations of stiffener and skin parameters,local instability and global instability.When the thickness of skin is small or the spacing of grid is large,the skin is prone to local instability within the grid;and if the height-tothickness ratio of the stiffener is considerably large,local instability of the stiffener will occur.Therefore,the global instability of the grid stiffened shell is often following the first occurrence of local instability,so the local instability will inevitably have some influence on the ultimate load,which must be considered in the design prediction of the bearing capacity.

3.2 Imperfection Sensitivity

Cylindrical shell buckling theory and numerical methods for predicting ultimate loads have been developed over decades,and are now relatively well established,but there are still large discrepancies between the predicted values of ultimate loads and the test results.The reasons for the discrepancies include the inhomogeneity of dimensional and mechanical properties,the non-uniformity of force transmission due to incompatibility of loading or the stiffness mismatch of the constrained boundary,and the deviation of the physical object from the theoretical geometry of the shell.The most important of which is the last one,i.e.,the ultimate load depends largely on the amplitude of the initial geometry imperfection.

3.3 Determination of Knockdown Factor

The knockdown factor is used to quantitatively characterize the sensitivity of the grid stiffened shell to the initial geometric imperfection,and is also a correction factor used in engineering to scope the lower limit of load-bearing capacity.The NASA SP8007 manual gives an empirical formula for the knockdown factor based on the results of a large number of earlier tests on cylindrical shells with different ratios of diameter-to-thickness.However,it is feasible to use a lower envelope for scheme design,but it is too conservative if used for detailed design.A variety of simulation-based numerical methods for the quantitative determination of the knockdown factor have been developed and are available for engineering use,including the eigenmode-shape imperfection approach,single perturbation load approach,worst multiple perturbation loads approach,and FE model with measured realistic imperfection.

3.4 Ultimate Load under Internal Pressure

For the design of the tank section,in order to maximize the load-to-mass ratio of the structure,the influence of the internal pressure on the axial compression load-bearing capacity of the stiffened shell needs to be considered.The influence is considered from two aspects: on one side,the internal pressure acting on the bottom of the tank can cancel out part of the axial compression load,but on the other side,the effect of the lateral pressure is beneficial to reducing the imperfection sensitivity of the stiffened shell and improves the ultimate load of the structure.However,the beneficial effect on the axial compression load-bearing capacity is only applicable within a certain pressure range,when the internal pressure increases to a certain degree,the instability mode of the stiffened shell changes from elastic instability to plastic instability,at which time the greater internal pressure will reduce the ultimate load.

A typical set of isogrid stiffened shell parameters was selected,as shown in Table 1.The whole tank model was composedof 2 sections,2 tank bottoms and 2 short shells.The variation curve of the ultimate load of the stiffened shell against internal pressure was obtained by the nonlinear analysis algorithm.Figure 3 corresponds to the curve of the 1st set of parameters,where the axial compression bearing capacity has been subtracted from the part canceled out by the internal pressure.It can be seen that the beneficial effect on the bearing capacity is the greatest when the internal pressure is enough to change the instability mode of the stiffened shell from elastic instability to plastic instability.

Table 1 Parameters of isogrid stiffened shell

Figure 4 further compares the effect of internal pressure on the axial compression limit load bearing under different spacing of stiffeners,where the data points are the lower bound values of the load-displacement curve obtained from simulation.For comparison,the load-bearing limit under internal pressure at each point is divided by the load-bearing limit without internal pressure.Since the lower bound value of the simulation has a certain systematic error,curve fitting is performed on the data points to analyze the laws.It can be seen that the benefit of internal pressure on the ultimate load is more obvious when the spacing of stiffeners is relatively large,but at the same time,the pressure interval corresponding to the beneficial effect is relatively narrow.However,on the other hand,the beneficial effect corresponds to a relatively narrow pressure range.The beneficial effect of the isogrid stiffened shell with 320 mm spacing of inclined stiffeners decreases when the pressure exceeds 180 kPa,while the isogrid stiffened shell with 200 mm spacing of inclined stiffeners decreases only when the pressure exceeds 250 kPa.

Figure 3 Variation curve of ultimate load with internal pressure

Figure 4 Effect of internal pressure on axial compression limit load bearing under different parameters

4 OPTIMAL DESIGN FOR THE STIFFENED SHELL

4.1 Analysis Methods

The analysis methods for stiffened shells are divided into two categories: theoretical method and finite element method.The core concept of the theoretical method is that the stiffness equivalence,which equates to the effect of the stiffener to the in-plane stiffness A,bending stiffness D,and tensile-bending coupling stiffness B of the thin plate,after which the ultimate load is obtained according to the numerical solution of isotropic shell or orthotropic shell.Since the stiffener distribution of the cylindrical shell is periodic,microstructure treatment can be used for stiffness equivalence,typical methods include the representative body element method and the asymptotic homogenization method.CHENG and CAI[11]established the NIAH method,the analysis efficiency of the homogenization method can be improved from the finite element method with the guarantee of accuracy.WANG[12]combined the NIAH method with the Rayleigh-Ritz method to establish a fast buckling analysis method.The method takes into account the generality of the finite element method and the efficiency of the theoretical algorithm,which is convenient for engineering applications.The advantage of the theoretical algorithm is the high computational efficiency,which can be used with the global optimization algorithm to obtain the optimization results quickly and can be useful in the scheme design.However,there are certain shortcomings for this type of method,such as the prediction accuracy cannot be guaranteed in the case of local instability of the skin before the global instability of the shell,and it cannot be applied to the plastic instability problem under the combined effect of axial compression and internal pressure.

In contrast,the finite element method does not have the limitations of the theoretical method.It can not only predict the buckling critical load of the structure,but also track the post-buckling equilibrium path by nonlinear analysis.Along with the corresponding imperfection characterization method,it can make a more accurate prediction of the ultimate load for the actual product.The finite element algorithms used to predict the load-bearing capacity of stiffened shells can be divided into the following three categories: linear buckling procedure,nonlinear statics procedure,and nonlinear dynamics procedure.

The linear buckling approach,also known as eigenvalue analysis,is used to predict the critical buckling load of a rigid structure,especially when the instability type of the structure is of a branching type.Although a more accurate approximation of the critical load can be achieved by multi-step analysis,using the results of the previous solutions that take into account nonlinear factors like the base state,it is recommended to be used in the scheme design stage in combination with design experience in the preliminary prediction of structural load-bearing capacity as post-buckling information is not available.Another role of the eigenvalue analysis is that the obtained buckling modes can be introduced into the nonlinear analysis as initial imperfections.

In the nonlinear statics approach,the Newton-Raphson iterative method with displacement control can be used to obtain the ultimate load of the structure and can trace the post-buckling equilibrium path better than load control.However,for a stiffened shell,the NR iterative method will have convergence problems if the local instability of the skin is severe.The convergence problem can be solved by using stabilization control,which is essentially a method of balancing the control equations by applying an additional viscous damping force,so extra attention needs to be paid to the selection of damping coefficients when using it to avoid distortion of the results.The Riks method is also capable of tracing the post-buckling equilibrium path of the stiffened shell and is usually able to get closer to the true results.However,like the Newton method,the Riks method also suffers from convergence difficulties due to skin instability,and stabilization control cannot be applied.

The nonlinear dynamics approach,by introducing inertial forces into the equilibrium equations,can avoid the convergence problem in the static approach.Dynamics approaches are divided into the implicit method and explicit method,of which the explicit method is more commonly used because it does not require iterative solving and is more efficient when the scale of the model is large.However,the explicit method is conditionally stable and requires reasonable solution control to ensure the validity of the results.Engineering applications refer to the strategy given in the literature [5].When poor quality meshes inevitably exist in the finite element model,the solution efficiency can be improved by local mass scaling or by switching to a nonlinear statics approach.

4.2 Parametric Modeling

Parameter optimization is the process of exploring the design domain,which requires a large number of sampling analyses.For optimization based on FEA,whether the optimization program directly drives the analytical model or the corresponding approximate model is generated through experiment design first,and then the mathematical approximate model participates in optimization instead of FEA,parametric modeling techniques are needed to improve the efficiency of FEA model generation.

The current commercial finite element analysis software is provided with a secondary development interface,such as Abaqus software,which allows users to write scripts in Python language.The scripts can directly call kernel functions to automatically complete pre-processing work such as geometric modeling,meshing,properties assignment,boundary condition setting,solving parameters setting,and also automate the post-processing work to extract the ultimate loads in the analysis results.By using parametric modeling technology,the whole process of nonlinear finite element analysis of stiffened shell is programmed,which lays the foundation for further design of experiment and parameter optimization.

4.3 Surrogate Model Techniques

The nonlinear finite element method is more helpful to simulate the real response of the structure under load and is the preferred method for refinement analysis.However,when used for optimization,the time cost involved is unacceptable for engineering applications,so engineers must adopt the help of surrogate modeling techniques.The key to applying the surrogate modeling technique is to ensure the approximation accuracy of the model,which involves both sampling methods and model selection.

Sampling methods,also referred to as design of experiment(DOE),use probability theory and mathematical statistics as the theoretical basis to study how to conduct experiments rationally and analyze and process experimental results effectively.The number and quality of samples in DOE directly affect the degree of approximation of the surrogate model to the true solution.Commonly used DOE methods include Full Factorial Level,Orthogonal Array,and Latin Hypercube Sampling.Given the time-consuming characteristics of nonlinear algorithms,the Optimal Latin Hypercube method can obtain better sampling results in engineering,because it can achieve uniform filling of sample points in the design space.

The current widely used surrogate models include polynomial response surface,orthogonal polynomial,radial basis function neural network,Kriging model,and support vector regression.Among them,the surrogate models can be divided into two categories according to whether the approximation function passes through the sampling point or not.Polynomial response surface,orthogonal polynomial,and support vector regression are non-interpolated,while radial basis function neural network and Kriging model are interpolated.

In a general sense,simulation is different from testing,and the obtained analysis results are deterministic without random errors,so an interpolated model should be selected,such as the Kriging model,however,when the KDF is considered in the simulation results,for example,the lower bound value is selected more conservatively as the ultimate load of the stiffened shell,random errors are introduced to some extent,so a non-interpolated model can be selected at this time,such as polynomial response surface.It should be noted that the fitting accuracy of the response surface model will be greatly reduced when dealing with highly nonlinear,high-dimensional problems.

4.4 Process of Optimization

For an isogrid stiffened shell,based on the skin thickness determined according to the internal pressure,the parameters that can be optimized include the spacing of the rib,width of incline rib,width of longitudinal rib,and total height of sheet.Combined with the manufacturing process,the design interval for the width and height of the ribs in the above parameters is limited,which makes the size of the design domain acceptable.With a certain spacing of the ribs,a better approximation accuracy can be obtained by the Optimal Latin Hypercube sampling technique with the radial basis function surrogate model.However,when the spacing of ribs is included to form a higher dimensional design domain,the design domain becomes more complex due to the large variation interval of the spacing,and it is generally difficult to guarantee the approximation accuracy of the surrogate model even if the number of samples is increased,which can only be improved using sequential sampling.For the characteristics of the design domain,a dimensionality reduction approach can be adopted,i.e.,optimization based on the subspace of the design domain to reduce the reliance on the high-accuracy surrogate model.The optimization process is shown in Figure 5.As mentioned earlier,the subspace consists of the set of samples at the selected spacing of the ribs.After obtaining the optimal result for each given spacing,usually the minimum weight to meet the load bearing requirements,it can be combined with a step search algorithm to determine the new search subspace.Eventually,on one side the optimal solution A for the new subspace is obtained,on the other side,the Kriging model is generated and optimized based on the sample points of the subspace where this optimal solution A is located and the adjacent subspaces,and the optimal solution B is obtained.If the difference between the two optimal solutions is within the set tolerance,then the optimal solution is found,otherwise a new search subspace is further determined until convergence.

Figure 5 Optimization process of stiffened cylindrical shells by surrogate model

4.5 Engineering Applications

As the first solid-liquid bundled launch vehicle in China,the LM-6A has the feature of large flight overload compared with previous rockets.Due to this,the internal pressure of the tank covers a wider range,and structure optimization was needed in taking into account the influence of the pressure in the design interval of 200?500 kPa on the axial compression load-bearing capacity.

The optimization of the cylinder section sheet needed to meet both the axial compression bearing requirements without internal pressure and the bearing requirements under the axial compression and internal pressure.Combined with the load characteristics,after the optimization design based on parametric modeling,the first-stage liquid oxygen tank used an isogrid sheet,and the first-stage kerosene tank used an orthogrid sheet.

To verify the optimization effect,before the tanks were put into production,single section tests with two types of the grid were conducted respectively,as shown in Figure 6 and Figure 7,and the test results were found to be as expected.Finally,with the new type of grid sheets,the weights of the two tanks mentioned above were reduced by 9% and 29% respectively.

Figure 6 Test of isogrid section

Figure 7 Test of orthogrid section

5 CONCLUSION

In this paper,the key characteristics of the stiffened shell structure design are highlighted in conjunction with the requirement analysis of the propellant tank design.Based on the techniques of parametric modeling and surrogate model,an optimization method of dimension reduction approximation is proposed to solve the problem that surrogate model is difficult to ensure the approximation accuracy of high-dimensional design space.The method has been applied to the tank weight reduction design of the new generation launch vehicle in China,and good results have been achieved.

For the future development of propellant tanks,the author has the opinion that the following approaches can be made.

1) Further enrichment of the sealed thin-walled cylindrical shell configuration design,such as a corrugated sandwich shell and frame-and-truss-based shell.

2) Extension of theoretically mature configurations to engineering applications,such as hierarchical stiffened shells and double-sided laser-welded T-section sheets.

3) For imperfection sensitivity,it is necessary to accumulate a database of measured imperfections,combine statistical methods and machine learning algorithms for the reliable design of stiffened cylindrical shells.