Integrated hydrobulging of prolate ellipsoids from preforms with multiple thicknesses

The integrated hydrobulging of stainless-steel prolate ellipsoids from preforms with two thicknesses was investigated. The produced ellipsoids were closed with two 16 mm thick closures and had nominal semiminor and semimajor axes of 89 and 125 mm, respectively. The ellipsoidal preforms comprised eight conical segments inscribed inside the target perfect ellipsoid. The four end and middle segments of the preforms had nominal thicknesses of 0.67 and 0.83 mm, respectively. The hydrobulging of these preforms was explored analytically and numerically and was compared with that of prolate ellipsoids with constant thickness. Two nominally identical ellipsoidal preforms were fabricated, measured, and hydrobulged to confirm the theoretical predictions. The results indicated that varying the preform thicknesses is an efficient method of overcoming insufficient hydrobulging of the ends of prolate ellipsoids in other methods. Moreover, hydrobulging instability can be effectively monitored by measuring geometric dimensions, such as axial height.


Introduction
Prolate shells of revolution have long attracted substantial interest from researchers owing to their high performance and loading capacity. These shells are often applied as internal pressure vessels, external pressure hulls, or decorative objects. They can have ellipsoid [1,2], barrel [3,4], egg [5,6], or teardrop [7,8] forms in accordance with the desired application. Among these forms, the prolate ellipsoid is the most commonly used due to its well-known mathematical formula and the relative ease of relevant calculations.
Typical-shaped shells are widely used as internal pressure vessels or as heads and bottoms of cylindrical tanks and vessels [9]. Reinforced concrete and structural steel domes of buildings, air-supported rubber-fabric shells, and underwater pressure vessels are also made in the form of ellipsoidal shells. The design and manufacture of other typical-shaped shells can be provided an important reference through exploring the geometry and stress-strain state of ellipsoidal shells. In addition, compared with spherical shells, ellipsoidal shells are more suitable to be used as water tanks and pressure vessels for storing volatile liquids due to the advantages of the small wind area and lower centroid. For example, Caldwell Company built a 500,000-gallon water storage tower (WST) in Hollywood and a 400,000-gallon tank in Charlotte Hall. The ellipsoidal water storage towers can add to the capacity in their respective areas and improve firefighting capabilities [10]. The glass dome of Palace of President of Georgia in Tbilisi was very similar to an ellipsoidal shell [11]. Muttoni et al. described the design and construction of a shell in the form of an ellipsoid (93 × 52 × 22 m). Its thickness varied between 100 and 120 mm [12]. A communication tower was built from two ellipsoidal spheres formed by integrated hydrobulging forming. The major axis of the larger ellipsoidal shell was 5 m [13]. The main dome of a nuclear center near Munich, Germany (1957) was in the form of an ellipsoidal shell of revolution. It was the most effective form of a dome, which satisfied the design functional requirements [14]. In addition, a religious building in Jerusalem, Israel (1957) was erected in the form of a domelike monolithic reinforced concrete shell on a square plan with rounded angles. An upper ellipsoidal part of the shell passed smoothly into the lower cylindrical part [14]. However, the fabrication of typical ellipsoids is still challenging due to the unequally distributed positive Gaussian curvature of these forms [15,16].
For example, Wang and coinvestigators explored the effects of polygon shape and number on the results of hydrobulging large spherical shells from football-shaped preforms. They also investigated the effects of petal shape and number on the results of hydrobulging large spherical shells from basketball-shaped preforms. They reported that satisfactory spheres can be obtained with both methods in accordance with the physical and geometrical adjustment principle [20][21][22][23][24]. Similar investigations have also performed by Hashemi for single-and five-layered spheres [25][26][27].
On the basis of these studies, Wang guided his doctoral student Yuan to explore the hydrobulging of toroidal shells that were intended for use in pipe elbow fabrication. They observed that the polygonal cross-section of the innermost toroidal preform required the use of more sides than the outermost toroidal preform did [28][29][30][31][32]. The stresses on the innermost toroid were greater than those on the outermost toroid [33].
Subsequently, Yuan guided his doctoral student Zhang to explore the effect of petal geometry and number on the hydrobulging of ellipsoidal shells from petal-shaped preforms. Double generating lines were used to design the petal-shaped preform to avoid insufficient hydrobulging at its two ends [34][35][36][37]. However, the resulting hydrobulged ellipsoidal shells were not compared with the perfect target geometry due to the poor fabrication accuracy of the method. Similar findings were obtained from recent studies on hydrobulging egg-shaped shells from a petal-shaped preform [38].
To increase the fabrication accuracy and facilitate preform fabrication, Zhang and coinvestigators explored the hydrobulging and buckling performance of egg-shaped shells from cone-segmented preforms. The egg-shaped geometry is promising for application as pressure hulls in submersibles. Their exploration revealed that cone-segmented preforms were more effective than petal-shaped preforms because bending and welding petals were difficult [39,40].
Moreover, the proper hydrobulging in the metallic plates has been widely studied. A new sheet hydroforming was proposed and applied for forming of two industrial parts that currently are produced in industry by conventional deep drawing and stamping in several stages [41]. The effect of load path on thickness distribution and product geometry in the tube hydroforming process was studied by Elyasi et al. [42]. They found that if pressure reached to maximum faster, bulge value and thinning of the part would be more and wrinkling value would be less. In addition, they carried out a series of researches on the hydroforming of cylindrical stepped tubes and metallic bellows [43][44][45][46][47].
However, a comparison of the hydrobulged and perfect geometries revealed that the hydrobulging of the two ends was insufficient. This was attributed to the constant thickness but variable curvature distributions of the cone-segmented preforms.
In this study, the integrated hydrobulging of prolate ellipsoids with two stepwise thicknesses under internal pressure was investigated analytically, numerically, and experimentally. The remainder of the paper is structured as follows. Section 2 details a theoretical study of the hydrobulging of prolate ellipsoids with two stepwise thicknesses under internal pressure. In Section 3, two nominally identical ellipsoidal preforms are fabricated, measured, and hydrobulged to confirm the theoretical predictions. Section 4 briefly concludes the work. The results revealed that varying the preform thickness is an efficient method of overcoming the insufficient hydrobulging of the ends of prolate ellipsoids in other methods.

Theoretical analysis
This section presents a theoretical study of the hydrobulging of prolate ellipsoids with two stepwise thicknesses with internal pressurization, including a geometric definition, analytical formulation, finite element modeling, and theoretical analyses.
Nevertheless, the more the segment number, the higher the hydroforming accuracy and the less the hydroforming pressure. The external surfaces of the perfect prolate ellipsoid and assumed ellipsoidal preform are mathematically described in Eqs. (1) and (2), respectively, as follows: and where a is the semiminor axis and b is the semimajor axis. As can be seen from Fig. 1, for the aforementioned equations, eight discrete conical segments (S1-S8) are inscribed inside the prolate ellipsoid. The segments are connected by the welding process in actual manufacturing. But the welding influence is not considered. It is assumed that the weld seams do not change before and after hydrobulging. In the actual manufacturing process, the thickness of weld seams is about two times the shell thickness. Also, there exists geometric discontinuity at each weld seam, leading to the stiffness increase. The detailed geometrical dimensions of the perfect prolate ellipsoid and assumed ellipsoidal preform are listed in Table 1.
The lower end of the ellipsoidal preform is closed by a thick, circular closure. The upper end of the ellipsoidal preform is closed by a thick, circular closure with an inlet hole. The inlet hole is used to fill the preform with water to apply internal pressurization during hydrobulging. If the hydrobulged ellipsoid were to be used as an underwater pressure hull for a manned vessel, these thick closures could be replaced by heavy flanges to install windows or an access port [5]. Two stepwise thicknesses are defined for the ellipsoidal preform. Four conical segments (S1, S2, S7, and S8) near the lower and upper ends were assumed to have thin walls with uniform thickness of t 1 = 0.67 mm; the remaining four conical elements (S3, S4, S5, S6) in the middle were assumed to have thicker walls with uniform thickness of t 2 = 0.83 mm. These thickness values were selected based on the standard thicknesses of stainless-steel plates available on the market and in accordance with the analytically derived stress distribution of the perfect prolate ellipsoid. Based on the principle of equal strength conception, the continuous variable thickness is transferred into the discrete thickness [5]. The detailed calculation is given by following Eqs. (1)-(13).

3
The stress distributions of a prolate ellipsoid and ellipsoidal preform can be obtained analytically based on the membrane theory of shells of revolution under uniform internal pressure [48]. The longitudinal stress (σ φ1 ), latitudinal stress (σ θ1 ), and equivalent stresses (σ e1 ) of perfect prolate ellipsoid can be expressed as follows: and where p is the applied internal pressure and t is the wall thickness of the prolate ellipsoid. R 1 and R 2 are the meridional and circumferential principal radii of the prolate ellipsoidal shell, respectively. These can be expressed as follows: and Under the assumption that the equivalent stress is equal to the material yielding point (σ e1 = σ y ), the yielding load (P y1 ) of a perfect prolate ellipsoid can be obtained by combining Eqs. (3)-(7) as follows: The longitudinal stress (σ φ2 ), latitudinal stress (σ θ2 ), and equivalent stresses (σ e2 ) of a segmented ellipsoidal preform can be expressed, respectively, as follows: where t is the wall thickness of the ellipsoidal preform and α i is the angle between the linear meridian of the ith segment and the axis of revolution, which can be determined as follows: Assuming that the equivalent stress is equal to the material yield point (σ e2 = σ y ), the yield load P y2 of the segmented preform can be obtained by combining Eqs. (9)-(12) as follows: To obtain the material properties of the ellipsoidal preform, tensile tests were performed in accordance with GB/T 228.1-2010 [49] and ISO 6892-1:2009 [50]. Two coupons, namely, a 0.83 and a 0.67 mm stainless-steel coupon, were cut from two parent stainless plates purchased from a market supplier and used to fabricate the ellipsoidal preforms. Figure 2 reveals that the material stress-strain  5) and (11), the equivalent stress profiles of the perfect prolate ellipsoid and the ellipsoidal preform under 1 MPa of internal pressure were obtained analytically (Fig. 3). After Eqs. (8) and (13) were applied, the yielding load profiles of the perfect prolate ellipsoid and ellipsoidal preform were obtained analytically (Fig. 4).
According to the principle of constant material volume, the average thickness (t i1 ) of the ith hydrobulged segment can be determined as follows: where the subscript i is the segment number, the subscript 0 represents the ellipsoidal preform, and the subscript 1 represents the hydrobulged prolate ellipsoid.
The surface area of the ith segment before and after hydrobulging can be calculated as follows: where h i is the frustum height of the ith segment.
The average thicknesses of the eight conical segments obtained using Eqs.

Finite element modeling
To numerically examine the results of hydrobulging prolate ellipsoids, geometrical and material nonlinear analyses were performed with the Newton-Raphson method available in ABAQUS/Standard software. A finite element model of a hydrobulged prolate ellipsoid with two stepwise thicknesses  is displayed in Fig. 6. The blue, green, and red areas indicate the shells with thickness t 1 = 0.67 mm, the shell with thickness t 2 = 0.83 mm, and the closures with thickness T = 16 mm. The thickness was defined as the offset from the mid-surface of each shell. This definition was facilitated by generating finite shell elements on the external surfaces of the blue and green shells as well as on the internal surfaces of the red closures. These external and internal surfaces were considered the top and bottom surfaces, respectively, in the finite element model. The top and bottom surfaces of conventional shell elements are those in the positive and negative normal directions, respectively. As can be seen from the inset of Fig. 6, the positive normal is given by the right-hand rule in accordance with the order of the specification of the nodes in the element definition.
The blue, green, and red areas share nodes at their boundaries, enabling the three different shell thicknesses to be assigned. The four-node general-purpose shell element with finite membrane strains (S4) was selected to optimize the accuracy of the numerical model. A mesh convergence analysis revealed that the 0.67 mm blue shell had 11,600 element nodes and 11,200 shell elements, the 0.83 mm green shell had 16,200 nodes and 16,000 shell elements, and the 16-mm red closure had 10,802 nodes and 10,600 shell elements.
To eliminate rigid body movement, three-point constraint boundary conditions were defined on the basis of previous research on eggs [5,6] as well as the guidelines of the China Classification Society [51] as follows. The degrees of freedom in the x, y, and z directions (in which the x-axis is the axis of revolution) were denoted as U x , U y , and U z , respectively. At the points U x = U z = 0, U z = U y = 0, and U x = U y = 0, the support forces were set to zero during the hydrobulging process.
Two loading cases were defined in the analysis: hydrobulging and springback. In hydrobulging, uniform internal pressure is gradually applied to the ellipsoidal preform until a prolate ellipsoid is obtained. The pressure value is determined in accordance with the yielding loads of the prolate ellipsoid and ellipsoidal preform. In the springback case, the internal pressure is gradually reduced to zero. The solving parameters for these cases were set as follows: an initial time increment of 0.02 s, a time period per step of 1 s, a minimum time increment of 10 −8 s, and a maximum time increment of 0.05 s. The numerically obtained results are presented in Figs. 7, 8, 9, and 10.

Analytical analysis
To predict the hydrobulging pressure range, the equivalent stresses of the perfect prolate ellipsoid and ellipsoidal preform under uniform internal pressure were determined based on the membrane theory of shells of revolution. The first yielding loads of the perfect prolate ellipsoid and ellipsoidal preform were obtained based on linear elastic mechanics. The initial hydrobulging pressure should approach the yielding load of the ellipsoidal preform, and the final hydrobulging pressure should approach the yielding load of the perfect prolate ellipsoid. Prolate ellipsoids with both constant and stepwise thickness were analyzed for comparison.
The equivalent stresses of ellipsoids and preforms with stepwise and constant thicknesses were uniformly distributed in the latitudinal direction due to the axially symmetric geometric, loading, and boundary conditions. Thus, the equivalent stresses in the longitudinal direction were analytically determined (Fig. 3), which reveals that the equivalent stress of perfect prolate ellipsoid with constant thickness  increases smoothly from the extrema to the middle. This increasing stress was primarily attributed to the increasing principal radii, as has been observed for ellipsoidal and eggshaped shells under uniform external pressure [2,3]. Small stresses near the two ends can result in a high yield load (Fig. 4). For the constant-thickness shells, the yield load was 3.921-6.320 MPa for the ellipsoid but 2.510-4.580 MPa for the preform. Because these ranges overlap, the two ends of the ellipsoid are insufficiently hydrobulged when the other areas are maximally hydrobulged. Similar results were determined for the integral hydrobulging of prolate ellipsoids and eggs from petal-shaped preforms [34,35,38].
Reducing the yielding loads of two ends by reducing their thickness can efficiently overcome this insufficient hydrobulging. As can be inferred from Eqs. 5 and 11, the equivalent stress increases linearly as the thickness decreases. As can be observed in Fig. 3, if the end thickness of the ellipsoid and preform is reduced from 0.83 to 0.67 mm, the end stresses of the ellipsoid and preform increase by approximately 20%, causing the end yielding load of the ellipsoid and preform to decrease by approximately 20%. Therefore, with stepwise thickness, the yielding load is 3.638-4.859 MPa for the ellipsoid and 2.096-3.564 MPa for the preform. The equivalent stresses and yielding loads of the prolate ellipsoids with stepwise thickness are nearly uniformly distributed during hydrobulging, and insufficient hydrobulging of the ellipsoid ends can be avoided.
From a geometrical perspective, the average thickness tends to decrease from the end segment to the middle segment. The analytically obtained thickness profile of hydrobulged prolate ellipsoid is presented in Fig. 5. The average thickness reduction of each hydrobulged segment can be obtained based on this profile and the initially defined nominal thicknesses. The average thickness reduction is 0.006 mm for the first and eighth hydrobulged segments, 0.008 mm for the second and seventh hydrobulged segments, 0.001 mm for the third and sixth hydrobulged segments, and 0.011 mm for the third and fourth hydrobulged segments. These thickness reduction differences are strongly associated with surface area increases (see Eq. 14).

Numerical analysis
On the basis of the analytical yielding load profiles of Fig. 4, hydrobulging pressures of 3.0, 3.5, 4.0, 4.1, 4.2, 4.3, 4.4, 4.5, and 4.6 MPa were applied to ellipsoidal preforms with stepwise or constant thicknesses. The obtained geometries and shape deviations from perfect geometry of the prolate ellipsoids hydrobulged under various internal pressures ( Fig. 1) are presented in Figs. 7 and 8, respectively. Positive (red) and negative (blue) values correspond to excessive and insufficient hydrobulging, respectively. Figures 7 and 8 reveal that as hydrobulging pressure increases, the meridian of each conical segment changes from a straight line to a smooth arc. The ellipsoidal preform becomes a nearly perfect prolate ellipsoid at a hydrobulging pressure of 4.2 MPa. At this pressure, hydrobulging is only slightly insufficient for the third and sixth conical segments in the stepwise case. However, insufficient hydrobulging was observed for not only the third and sixth but also the second and seventh segments in the constant case due to their high yield loads. The results suggest that a perfect ellipsoid can be hydrobulged by varying the wall thickness to ensure a uniform stress distribution.
At pressure higher than 4.2 MPa, the middle four segments are excessively hydrobulged. The shape deviations of hydrobulged prolate ellipsoids with both stepwise and constant thickness increase substantially as the pressure increases, indicating instability. This instability can be clearly observed in Fig. 9, which presents charts of the axial height, closed capacity, surface area, and radial displacement of hydrobulged prolate ellipsoids versus applied pressure. The results revealed that hydrobulging instability can be effectively monitored by measuring the geometric dimensions of axial height and displacement ( Fig. 9a and d). Figure 9 reveals that the geometric parameters of the hydrobulged prolate ellipsoids with stepwise thickness are nearly constant at an internal pressure of approximately 3 MPa but increase steadily between pressure of 3 and 4 MPa. At higher pressure, the axial height decreases and the remaining geometric parameters increase rapidly. Internal pressures of 3 and 4 MPa correspond with the yield loads of the preform and perfect prolate ellipsoid, respectively (Fig. 4). A comparison of Figs. 4, 7, and 8 reveals that the optimum hydrobulging loads of prolate ellipsoids with stepwise thickness are approximately equal to the average yielding loads of a perfect prolate ellipsoid.
Internal pressure can be considered as the combination of the axial compression on the two thick closures and the lateral pressure on the eight conical segments. Axial compression may result in a steady increase in axial height, whereas lateral pressure may result in a sudden decrease in axial height. Figure 9a indicates that the axial deformation of the prolate ellipsoid transitions from stretching to shrinking as the primary factor affecting axial height transitions from axial compression to lateral pressure. Therefore, axial height is the most effective indicator of the completion of the hydrobulging process, and hydrobulging can be halted when the axial height decreases to its original magnitude. Figures 7 and 9d reveal that the radial deformations of the segmented boundaries (P1-P5) can be differentiated by their stiffnesses. Radial deformation is smaller further away from the rigid heavy closure and can be neglected at internal pressure less than 4 MPa. At 4.2 MPa, the maximum radial deformation is less than 1 mm at the mid-height of the hydrobulged ellipsoid. However, conical segments are often welded together in practical hydrobulging, and weld seams can increase the stiffness of the segmented boundaries [38][39][40]. Therefore, the actual radial deformation may be substantially less than the numerical predictions. These findings further suggest that an inscribed polyhedron can be used as a segmented ellipsoidal preform for a perfect prolate ellipsoid.
The wall thicknesses, residual stresses, and plastic strains of hydrobulged prolate ellipsoids under satisfactory internal pressure (4.2 MPa) are presented in Fig. 10. As can be shown from Figs. 7 and 10a, changes in thickness are strongly associated with changes in shape. Greater outbulging was associated with a lower wall thickness. For eight conical segments, the reduction in thickness increases from two ends to the middle, which is in agreement with the analytical findings in Fig. 5. For each conical segment, the thickness also increases from the two ends to its middle area. As can be observed in Fig. 10b and c, hydrobulged prolate ellipsoids had high residual stresses and plastic strains. The segmented boundaries were subjected to high stress and strain concentrations due to the bending effect and geometric discontinuities. The tensile residual stress caused by internal pressure may be beneficial for underwater applications. Underwater, ellipsoids are subjected to external pressure, resulting in a compressive

Experimental analysis
The bulging performance levels of prolate ellipsoids with stepwise thicknesses under internal pressure were investigated experimentally through sample fabrication, geometric measurements, and hydrobulging experiments.
The experimental data were compared with numerical evaluations.

Materials and methods
Two nominally identical ellipsoidal preforms were fabricated according to the geometric dimensions listed in Fig. 1b and Table 1. The fabricated preforms were quasistatically hydrobulged into nearly perfect prolate ellipsoids in accordance with the theoretical findings of Section 2.3. The geometric Fig. 11 Fabrication of the ellipsoidal preform. a Blanking, b bending, c assembling, d welding, and e closing properties of the fabricated preforms were measured ultrasonically and optically both before and after hydrobulging.
Fabrication was performed through blanking with a computer numerical control laser cutting machine, bending with a two-roller plate bending machine, assembly with a spot welding machine, welding with a tungsten inert gas welding machine, and closing with two heavy closures. Images of the results of each fabrication process are presented in Fig. 11. The fabrication was performed at Jiangsu Provincial Key Laboratory of Advanced Welding Technology, Jiangsu University of Science and Technology. No stress relief was performed because the ellipsoidal preform structures had thin walls. After fabrication, the external geometries and wall thicknesses of two fabricated ellipsoidal preforms were measured to ensure that fabrication was accurate.
The external geometry of the fabricated ellipsoidal preforms was optically measured using an optical 3D scanner (Cronos 3D; Open Technologies, Italy) with an error of less than 0.02 mm. As presented in Fig. 12a, a thin-layered contrast aid (FC-5, Hyperd NDT Material Corporation, China) was sprayed on the surface to prevent glare. Circular marking stickers were pasted on the sprayed surface to ensure accurate picture stitching. The in-house software Optical RevEng was used to generate computer-aided design models Fig. 12 Experimental procedures for integrated hydrobulging of prolate ellipsoids with stepwise thicknesses. a Preform optical scanning, b ultrasonic preform measurements, c hydrobulging, d ultrasonic ellipsoid measurements, e ellipsoid optical scanning. Numbers are as follows: 1, ellipsoidal test article; 2, optical scanner; 3, ultrasonic thickness meter; 4, digital pressure transducer; 5, hand-operated pump; and 6, data acquisition system of the fabricated ellipsoidal preforms from the optically measured data.
After geometry scanning, all marking stickers were removed manually. The contrast aid film was carefully washed off with tap water. A total of 528 thickness measurement points were drawn on each fabricated ellipsoidal preform using a marker pen. The measuring points were the intersections of 24 equally distributed meridional lines and 22 equally distributed circumferential lines. The thicknesses of the fabricated ellipsoidal preforms were ultrasonically measured using a nondestructive thickness meter (DAKOTA/PX-7; Sonatest Corporation, USA).
Subsequently, hydrobulging tests were performed by applying quasistatic internal pressure. The ellipsoidal preforms were filled with tap water through the inlet tube. The inlet tube was connected to a hand-operated water pump (SRB-30X; Zhenhuan Hydraulic Apparatus, China) with a pressure hose. The water pump had a maximum working pressure of 30 MPa, and a single stroke had a maximum flow of 38 mL. The internal pressure was slowly increased to achieve quasistatic loading. The pressure values were recorded using a digital pressure transducer (SUP-P3000; United Test Automation, China) connected to the data acquisition system (DH5902; Donghua Test Technology, China). The recording frequency was 50 Hz. Hydrobulging was ceased when a nearly perfect prolate ellipsoid was obtained.
After hydrobulging, the thicknesses of the hydrobulged prolate ellipsoids were ultrasonically measured at the same points as those of fabricated ellipsoidal preforms to enable a point-to-point comparison of changes in thickness due to hydrobulging. After ultrasonic measurements, the marked points were cleaned using ethanol (CH 3 CH 2 OH, > 99.7%). The external geometries of the two hydrobulged prolate ellipsoids were optically measured using the same method as that used for the two fabricated ellipsoidal preforms to enable a quantitative comparison of the geometry of a nominal and hydrobulged prolate ellipsoid.
The applied pressure values recorded during the hydrobulging tests are presented in Fig. 13. Photographs of the geometric evolution of the ellipsoidal preform #2 are displayed in Fig. 14. The measured external surfaces of the two fabricated ellipsoidal preforms before and after hydrobulging are presented in Figs. 15 and 16, respectively. The measured thicknesses of the two fabricated ellipsoidal preforms before and after hydrobulging are displayed in Fig. 17.

Testing analysis
The hydrobulging tests applied to two fabricated ellipsoidal preforms were nearly hydrostatic; however, the loading profiles clearly differed. This difference was ascribed to the manual internal loading. Figure 13 reveals that each recorded pressure profile comprises three stages: slow loading, steady loading, and sudden unloading. The total hydrobulging time was similar for ellipsoidal preforms #1 and #2 at approximately 62 and 74 s, respectively; however, the steady loading time differed substantially at 27 and 54 s, respectively. The average loading rate of the steady loading stage was approximately 0.115 and 0.066 MPa/s for ellipsoidal preforms #1 and #2, respectively. These small loading rates were consistent with hydrostatic loading.
The final hydrobulging pressure corresponds to the maximum applied pressure in Fig. 13. At these two pressure values, the heights of the two hydrobulged ellipsoids are nearly equal to those of the two fabricated preforms (230 mm). The final hydrobulging pressure values were 4.705 and 4.604 MPa for the ellipsoidal preforms #1 and #2. The difference between them, at 0.101 MPa, was small. The average hydrobulging pressure was 4.655 MPa, which is approximately 1.11 greater higher than the numerical predictions. The numerical hydrobulging Fig. 13 Recorded pressure values as a function of time during hydrobulging pressure was considered to be in good agreement with the experimental value.
Photographs of a hydrobulged prolate ellipsoid at typical internal pressures are presented in Fig. 14, which reveals nearly no deformation at internal pressures less than 2.6 MPa because these pressures are in the elastic range of the ellipsoidal preform. Higher pressures are in the elastic-plastic range of the ellipsoidal preform, which is gradually hydrobulged into a nearly perfect prolate ellipsoid as internal pressure increases. After unloading, the water is poured out of the ellipsoids to obtain hydrobulged prolate ellipsoids after the springback stage. Thickness and geometry measurements were conducted to examine the performance of the hydrobulging process.

Measurement analysis
The results for the two fabricated ellipsoidal preforms revealed that the proposed process was accurate and repeatable. Figure 15 reveals that the deviations of the two fabricated ellipsoidal preforms from a perfect ellipsoid are approximately normally distributed with a mean value of zero in the range of − 0.75 to 0.75 mm. Serval large deviations were observed for the two end and serval weld seams due to assembly and welding deformations. However, these large deviations had little effect on the hydrobulging results because uniformly distributed internal pressure could reduce these geometric imperfections [13,[17][18][19].
The optically measured geometry indicates that the closed capacities of the two fabricated ellipsoidal preforms were 4,057,451 and 4,077,336 mm 3 . The average value was 4,067,393.5 mm 3 , which is nearly equal to the nominal value of 4,057,420 mm 3 . The external surface areas of the two fabricated ellipsoidal preforms were 117,775 and 118,238 mm 2 . The average value was 118,006.5 mm 2 , which is nearly equal to the nominal value of 117,839 mm 2 . The difference between the fabricated and nominal geometries was 0.25% for a closed capacity and 0.14% for an external surface area.
Reductions in imperfections were confirmed by observing the deviations from the nominal geometry of the two hydrobulged prolate ellipsoids. Figure 16 reveals that the hydrobulged prolate ellipsoids are slightly excessively bulged; however, their geometry is nearly identical to the nominal geometry. Relatively large deviations were observed for the two end and serval weld seams due to preform deviations. The results suggest that accurate fabrication of the ellipsoid preform is key for optimal hydrobulging.
The optically measured geometries indicate that the closed capacities of two hydrobulged prolate ellipsoids were 4,118,285 and 4,144,900 mm 3 . The average value was 4,131,592.5 mm 3 , which is nearly equal to the nominal value of 4,125,052 mm 3 . The external surface areas of the two hydrobulged prolate ellipsoids were 118,702 and 119,219 mm 2 with an average value of 118,960.5 mm 2 , which is nearly equal to the nominal value of 118,847 mm 2 . The difference between the hydrobulged and nominal geometries was 0.16% for closed capacity and 0.01% for external surface area.  The measured thicknesses of the fabricated ellipsoidal preforms and hydrobulged prolate ellipsoids had a nearly uniform distribution. As can be observed in Table 2 and Fig. 17, the thickness of the four end segments varied from 0.656 to 0.672 mm for the fabricated ellipsoidal preform #1 and from 0.664 to 0.676 mm for fabricated ellipsoidal preform #2. The thickness of the four middle segments varies from 0.826 to 0.841 mm for fabricated ellipsoidal preform #1 and from 0.824 to 0.838 mm for fabricated ellipsoidal preform #2. These values are nearly equal to the nominal values in Table 1.
The thickness of the hydrobulged prolate ellipsoids was slightly reduced due to the increase in surface area. The thickness of the four end segments varied from 0.647 to 0.668 mm for hydrobulged prolate ellipsoid #1 and from 0.661 to 0.670 mm for hydrobulged prolate ellipsoid #2. The thickness of the four middle segments varied from 0.801 to 0.821 mm for hydrobulged prolate ellipsoid #1 and from 0.792 to 0.816 mm for hydrobulged prolate ellipsoid #2. These values are nearly equal to the nominal values in Table 1.
The average thickness reductions in the four end segments were only 0.005 and 0.004 mm for fabricated ellipsoidal preforms #1 and #2, respectively. The average thickness reductions in the four middle segments were 0.018 and 0.028 mm for fabricated ellipsoidal preforms #1 and #2, respectively. The clear thinning of the middle segments is consistent with both analytical and numerical predictions.

Conclusion
An analytical, numerical, and experimental study of hydrobulging of prolate ellipsoids with stepwise thickness under internal pressure was performed. The main conclusions are as follows: (1) Reduction of the yield loads of the two ends is an efficient method of overcoming insufficient hydrobulging. This can be achieved by reducing the thickness because lower thickness results in higher equivalent stress. The yielding load for the preforms with stepwise thickness was obtained analytically as 3.638-4.859 MPa for the ellipsoid and 2.096-3.564 MPa for the preform; these calculations were used as a theoretical basis for a hydrobulging design. (2) Numerical predictions revealed that the ellipsoidal preform becomes a nearly perfect prolate ellipsoid at a hydrobulging pressure of 4.2 MPa. At higher pressures, the middle four segments bulge excessively, indicating instability. This instability can be effectively monitored by measuring the axial height of the ellipsoid. (3) High residual stresses and plastic strains were observed in the hydrobulged prolate ellipsoids. This tensile residual stress due to internal pressure may be beneficial for underwater applications; however, this phenomenon requires further investigation. The segmented boundaries are subjected to stress and strain concentrations because of the bending effect and geometric discontinuities. (4) The numerical predictions were compared with experimental data. A nearly hydrostatic process was used for hydrobulging tests of two fabricated ellipsoidal preforms. The experimental hydrobulging pressures were 4.705 and 4.604 MPa for the two ellipsoidal preforms. The average hydrobulging pressure was approximately 1.11 times the numerical prediction. (5) The fabrication of the ellipsoidal preforms and hydrobulging of the prolate ellipsoids were reasonably accurate and repeatable. The prolate ellipsoids were slightly outbulged, but the deviations from nominal geometry were small. Relatively large deviations were observed at the ends and weld seams of the prolate ellipsoids; these deviations were attributed to imperfections in the preforms. The thickness distributions of the fabricated ellipsoidal preforms and hydrobulged prolate ellipsoids were nearly uniform.
To the best of our knowledge, this is the first study to carry out the research on the hydrobulging of ellipsoidal shells with variable thickness. The variable thickness could alleviate the insufficient deformation at both ends of the shells. Importantly, the analytical equations of prolate ellipsoids with multiple thicknesses were deduced theoretically.
In future works, more conical segments with various thicknesses could be used to fabricate ellipsoidal preforms to increase the accuracy of the hydrobulged ellipsoids.