Introduction

With the gradual depletion of global shallow energy resources and intensifying energy supply–demand contradictions1,2, the exploration and development of deep energy resources have become crucial for achieving strategic energy transition3,4,5. As a core technology for accurately acquiring reservoir physical parameters, pressure-preserved coring technology plays a significant role in enabling precise reservoir evaluation and enhancing exploration and development efficiency6,7. However, current pressure-preserved coring equipment generally faces technical limitations in maintaining adequate pressure-preservation thresholds when operating in deep complex geological environments, resulting in core samples struggling to maintain their in-situ preservation state. This leads to severe distortion of critical geological information such as reservoir pore structures and fluid saturation. Therefore, breaking through the ultra-high pressure preservation technology for deep pressure-preserved coring equipment has become an urgent requirement for high-precision exploration of strategic resources like deep-sea natural gas hydrates and deep shale gas. The technological breakthrough in this field will provide scientific basis and technical support for efficient development and sustainable utilization of deep resources.

The development of pressure-preserved coring technology dates back to the 1960s, with maximum pressure-preservation capacity being one of its core indicators, which determines the maximum operating depth at which a pressure-preserved coring device can function effectively. The earliest pressure-preserved sampler, the Pressure Core Barrel (PCB)8, developed during the Deep Sea Drilling Project (DSDP, 1968–1983), achieved a maximum pressure-preservation capacity of 34.4 MPa. In 1989, the Ocean Drilling Program (ODP) introduced the Pressure Core Sampler (PCS)9, with a maximum pressure-preservation capacity of 68.9 MPa, enabling exploration in deeper marine environments. Fugro pressure corer(FPC) and HYACE Rotary Corer(HRC)10, two coring systems developed in Germany for hard formations, demonstrated maximum pressure-preservation capacities of 25 MPa. The Pressure–Temperature Core Sampler (PTCS)11,12, a joint Japanese-American development, achieved simultaneous temperature and pressure preservation with a maximum pressure capacity of 24 MPa. Germany designed the MeBo Autoclave Corer (MAC) in 2002 (14 MPa maximum pressure) and the Dynamic Autoclave Piston Corer (DAPC)13,14 in 2003 (20 MPa maximum pressure). China’s pressure-preserved coring technology started relatively late. Zhejiang University developed the Pressure–Temperature Preserved Corer (PTPC)15,16 in 2005 with a 30 MPa pressure capacity. Zhu et al. later engineered the Pressure–Temperature Preserved Sampler (PTPS)17,18, achieving 40 MPa. Recent scientific and resource exploration efforts have progressively targeted greater depths, with oil and gas drilling reaching a record vertical depth of 12,869 m19 (formation pore pressure > 100 MPa), demanding higher pressure-preservation capacities. A comparative analysis of maximum pressure-preservation capacities across various technologies is presented in Fig. 1.

Fig. 1
figure 1

Comparison of maximum pressure holding capacity.

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Pressure-preserving controllers, as core components of pressure-preserved coring systems20, primarily include ball valves and flap valves. Ball valves are favored for their strong adaptability to harsh environments, excellent sealing reliability, and simplicity in open/close control. The pressure-bearing strength of ball valves and the sealing performance of their sealing surfaces collectively determine the maximum pressure-preservation capacity of coring systems, both of which are significantly influenced by the structural and dimensional parameters of the valves. Ruiming et al.21 optimized the valve seat structure of ball valves, effectively addressing sealing failure issues. Jiang et al.22 investigated the effects of friction coefficients, ball dimensions, and valve seat dimensions on sealing performance using ANSYS, revealing that friction coefficients predominantly influence the maximum sealing gap, while valve seat height critically impacts sealing performance. Li et al.23 explored the relationship between deep-sea ball valve seat structures and sealing performance, demonstrating that reducing the radial thickness of valve seats enhances pressure-preservation capacity. While these studies improved pressure-preservation performance through structural optimization, their optimization efficiency remained limited. Implementing mathematical methods and multi-objective optimization algorithms could enhance engineering optimization efficiency and further refine structural performance24,25,26. For instance, Li et al.27 optimized valve seat dimensions for ultra-low-temperature conditions using the Radial Basis Function (RBF) and NSGA-II, achieving a 43.45% reduction in maximum valve seat stress and improving pressure-bearing strength. Park et al.28 employed the Finite Element Method (FEM) and Response RSM to optimize CF8M ball valves, identifying their optimal dimensions. Zhang et al.29 applied RSM to study the impact of soft-seal valve seat dimensions on pressure-bearing strength under ultra-low temperatures. Despite advancements in optimization efficiency, challenges persist in proxy model accuracy and multi-objective trade-off capabilities.

Currently, scholars have conducted extensive research on structural dimension optimization of ball valves, yet most studies focus on low and medium pressure environments. The influence mechanisms of structural parameters on pressure retention capability under high-pressure and ultra-high-pressure conditions remain unclear, and quantitative indicators for evaluating the pressure retention capability of ball valves have not yet been established. This paper improves the existing pressure-bearing structure of pressure-retaining ball valves. Through numerical simulations and theoretical analysis, two evaluation metrics for pressure retention capability were determined. A regression model correlating ball valve structural dimensions with these metrics was established using CCD experimental design and response RSM. Multi-objective optimization of ball valve dimensions was performed via the NSGA-II algorithm to identify optimal dimensions under a medium pressure of 100 MPa. The theoretical model of pressure-bearing strength was validated against numerical simulations, confirming the accuracy of the numerical model. Further comparison between numerical results and prediction outcomes verified the reliability of the predictive model, with the overall optimization workflow illustrated in Fig. 2. The proposed optimization methodology integrating RSM and NSGA-II successfully enhances the ultimate pressure retention capability of ball valves, providing a novel methodological framework for structural optimization of ultra-high-pressure ball valves.

Fig. 2
figure 2

Overall flow chart of ball valve structure size optimization.

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Floating ball valves

Structure and working principle

This study optimized the existing pressure-bearing structure of ball valves and proposed a floating pressure-retaining ball valve that utilizes the valve seat to withstand high-pressure media. The valve seat’s structure, which is comprised of three parts-the sealing seat, the valve body, and the valve seat-is depicted in Fig. 3. Figure 4 illustrates the working principle of a floating ball valve. In Fig. 4a for the open state of the valve and Fig. 4b for the closed state, the valve is controlled by rotating the valve body via the reciprocating movement of a pneumatic cylinder.

Fig. 3
figure 3

Floating ball valve structure.

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Fig. 4
figure 4

Working principle of floating ball valve.

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Ball valve structure size

The structural parameters of the sealing seat, valve body, and valve seat are directly associated with the ball valve’s maximum pressure holding capability. Figure 5 identifies the relevant size parameters that influence the pressure retention capacity of the ball valve, The parameters associated with the valve seat number six, those related to the sealing seat total seven, and those pertaining to the valve body consist of two, Notably, the seat aperture (D7), the sealing seat internal diameter (D4), and the valve body internal diameter (D5) are all of equal dimensions. Ultimately, the initial values for each of the ball valve’s size parameters and their respective ranges have been determined based on engineering experience, as detailed in Table 1.

Fig. 5
figure 5

Ball valve structure dimensions.

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Table 1 Initial design variables.
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Theoretical model of compressive strength

The compressive strength of the floating ball valve body acts as a critical determinant for the maximum pressure-retaining capability of the ball valve. In this section, the body of the ball valve is simplified to a planar curved rod, with appropriate simplifications made to the boundary conditions. Drawing upon the principles of the theory of curved rod bending and the laws of energy, a theoretical model for the compressive strength of the floating ball valve body is proposed, to lay the theoretical foundation for subsequent numerical model validation.

Physical model and assumptions

Floating ball valve body at the upper end of the medium pressure \({P}_{1}\) and sealing seat pressure \({P}_{2}\), the lower end of the valve seat support force \({P}_{3}\), sealing seat and seat force on the valve body can be approximated as a uniform load, the force model shown in Fig. 6, from the figure can be seen, \({P}_{1}\), \({P}_{2}\), and \({P}_{3}\) on the Z axis symmetrical distribution of uniform load \({P}_{1}\), \({P}_{2}\), \({P}_{3}\) simplified as a centralized load \({F}_{1}\), \({F}_{2}\), \({F}_{3}\), it is easy to know that the direction of action of the three are With the Z-axis co-linear, and there are \({F}_{1}+{F}_{2}={F}_{3}\). Therefore, the force of the floating ball valve body can be simplified to the cross-sectional shape of Fig. 7a for the large curvature of the ring of the plane bending problem, the force schematic shown in Fig. 7b, the ring bore radius of \({\text{r}}_{1}\), the axis of the radius of R, the outer diameter of the \({\text{r}}_{2}\), in the analysis of the calculations before the following assumptions are made on the ring:

  1. (1)

    The material is assumed to be homogeneous, continuous, isotropic, and the stress–strain follows Hooke’s law.

  2. (2)

    The deformation in the Z-direction of the ring is much smaller than the radius R of the ring axis, and the small deformation assumption is used.

  3. (3)

    It is assumed that the stress at each point of the ring along the direction parallel to the x-axis is constant.

Fig. 6
figure 6

Valve body force model.

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Fig. 7
figure 7

Simplified force diagram of valve body.

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Valve body stress calculation

Using the symmetry of the ring and the load, it can be simplified as a quarter circle fixed at one end and subjected to a pressure \(F\) and a torque \({M}_{0}\) at the other end, as shown in Fig. 7c, there are two unknown parameters \(F\) and \({M}_{0}\) on the cross-section A, which are easy to know because the ring is in equilibrium:

$$F = \left( {F_{1} + F_{2} } \right)/2 = F_{3} /2$$
(1)

where F is the pressure on section A, and \({F}_{1}\), \({F}_{2}\), and \({F}_{3}\) are the sum of the vertical components of \({P}_{1}\), \({P}_{2}\), and \({P}_{3}\), respectively, on their respective action surfaces.where the radius of the projected surface of the \({P}_{1}\) action surface in the z-direction is \({\text{D}}_{4}\), there:

$$F_{1} = P_{{1}} \pi D_{4}^{2} /4$$
(2)

The projection surface of the \({\text{P}}_{2}\) action surface in the Z-direction of the ring has an inner diameter of \({\text{D}}_{4}\) and an outer diameter of D. The sealing seat has an inner diameter of \({\text{D}}_{4}\) and an outer diameter of \({\text{D}}_{1}\), which can be known:

$$F_{2} = P_{2} \pi \left( {D^{2} - D_{4}^{2} } \right)/4 = P_{{1}} \pi \left( {D_{1}^{2} - D_{4}^{2} } \right)/4$$
(3)

That is, there is:

$$F = \left( {F_{1} + F_{2} } \right)/2 = P_{{1}} \pi D_{1}^{2} /8$$
(4)

For \({M}_{0}\), we can first solve for the internal forces on any cross section m-m (as in Fig. 7c) as:

$$\left\{ {\begin{array}{*{20}l} {M = M_{0} - FR(1 - \cos \varphi )} \hfill \\ {F_{s} = - F\sin \varphi } \hfill \\ {F_{N} = - F\cos \varphi } \hfill \\ \end{array} } \right.$$
(5)

where M is the bending moment on section m-m, \({F}_{s}\) is the shear force on section m-m, and \({F}_{N}\) is the axial force on section m-m.

The expression for the strain energy of the crank rod is given by:

$$V_{\varepsilon } = \mathop \int \nolimits_{s} \left( {\frac{{M^{2} }}{{2ESR_{0} }} + \frac{{MF_{N} }}{{EAR_{0} }} + \frac{{F_{N}^{2} }}{2EA} + \frac{{kF_{S}^{2} }}{2GA}} \right){\text{d}}s$$
(6)

where: E is the modulus of elasticity of the material, S = Ae, A is the cross-sectional area, e is the distance between the centroid axis of the cross-section shape and the neutral axis, and \({R}_{0}\) is the radius of curvature of the axis.

Due to symmetry, the corner of section A should be equal to zero, so the deformation coordination condition is:

$$\theta_{A} = \frac{{\partial V_{e} }}{{\partial M_{0} }} = \int_{0}^{{\frac{\pi }{2}}} \left( {\frac{M}{ES}\frac{\partial M}{{\partial M_{0} }} + \frac{{F_{N} }}{EA}\frac{\partial M}{{\partial M_{0} }} + \frac{M}{EA}\frac{{\partial F_{N} }}{{\partial M_{0} }} + \frac{{F_{N} R}}{EA}\frac{{\partial F_{N} }}{{\partial M_{0} }} + \frac{{kF_{S} R}}{GA}\frac{{\partial F_{S} }}{{\partial M_{0} }}} \right)d\varphi = 0$$
(7)

where: \(G\) is the shear modulus, for isotropic materials, \(G=E/2(1+\mu )\), \(\mu\) is the Poisson’s ratio of the material.

Substituting the internal forces in Eq. (5) into the above coordination conditions, we obtain:

$$\frac{1}{ES}\int_{0}^{{\frac{\pi }{2}}} \left[ {M_{0} - \frac{FR}{2}(1 - \cos \varphi )} \right]{\text{d}}\varphi - \frac{F}{EA}\int_{0}^{{\frac{\pi }{2}}} \cos \varphi {\text{d}}\varphi = 0$$
(8)

Complete the integral and solve for:

$$M_{0} = FR\left( {1 - \frac{2}{\pi }} \right) + \frac{2FS}{{\pi A}}$$
(9)

Substituting Eqs. (4) and (9) into Eq. (5) yields the internal force on any section m-m as:

$$\left\{ {\begin{array}{*{20}l} {M = P_{1} \pi D_{1}^{2} R\cos \varphi /8 + P_{1} D_{1}^{2} S/4A - P_{1} D_{1}^{2} R/4} \hfill \\ {F_{s} = - \pi P_{1} D_{1}^{2} /8 \times \sin \varphi } \hfill \\ {F_{N} = - \pi P_{1} D_{1}^{2} /8 \times \cos \varphi } \hfill \\ \end{array} } \right.$$
(10)

The positive stress at any point on an arbitrary section m-m is the sum of the positive stress due to the bending moment \(M\) and the axial force \({F}_{N}\) as:

$$\sigma = \sigma_{M} + \sigma_{N} = My/S\rho + F_{N} /A$$
(11)

where \(y\) is the distance of each point on the cross-section from the central axis and \(\rho\) is the radius of curvature of the fiber layer parallel to the axis.

It is shown that for the shear stress in bending of a curved rod, it can be approximated by the formula for the shear stress of a straight rod, yielding the shear stress at any point on any cross-section m-m as:

$$\tau = \frac{{F_{S} S_{Z}^{*} }}{{I_{Z} b}}$$
(12)

where \(b\) is the width of the cross-section, \({I}_{Z}\) is the moment of inertia of the entire cross-section to the central axis, and \({S}_{Z}^{*}\) is the static distance of the area of the part of the cross-section below the transverse line \(\text{y}\) from the central axis to the central axis.

In summary, we can find out the positive stress \(\sigma\) and tangential stress \(\tau\) at any point of any cross-section m-m by Eqs. (11) and (12), around the point to the longitudinal and transverse six sections to take the infinitesimal length of the unit cell, the unit cell for the bidirectional stress state (as shown in Fig. 8), by Eq. (13), we can find out the maximum and minimum positive stresses at any point of any cross-section m-m:

$$\left. \begin{gathered} \sigma_{\max } \hfill \\ \sigma_{\min } \hfill \\ \end{gathered} \right\} = \frac{{\sigma_{x} }}{2} \pm \sqrt {\left( {\frac{{\sigma_{x} }}{2}} \right)^{2} + \tau_{xy}^{2} }$$
(13)

where \({\sigma }_{x}=\sigma\), is the positive stress on the face normal to the X-axis, and \({\tau }_{xy}=\tau\), is the tangential stress on the face normal to the X-axis.

Fig. 8
figure 8

Stress state of the monolith.

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Based on Eq. (13) and according to the fourth strength theory yield criterion, we can find the yield stress \({\sigma }_{s}\) at any point on any cross-section m-m as in Eq. (14):

$$\sigma_{s} = \sqrt {\frac{1}{2}\left[ {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} } \right]}$$
(14)

where \({\sigma }_{1}={\sigma }_{max}\), the first principal stress; \({\sigma }_{2}={\sigma }_{min}\), the second principal stress; and \({\sigma }_{3}=0\), the third principal stress.

Therefore, the floating ball valve body strength conditions are:

$$\sigma_{s} \le \left[ \sigma \right]$$
(15)

where, \([\upsigma ]\) is the material allowable stress.

Quantitative evaluation indicators for pressure retention capacity

Numerous investigations have demonstrated that contact pressure and sealing performance are connected30,31. In actuality, the issue of seal failure—which can arise for a number of reasons—limits the maximum pressure that ball valves can withstand. This part analyzes the pressure holding mechanism of ball valves using numerical simulation, and it suggests quantitative assessment indices for the valves’ pressure holding capability.

Numerical simulation

A three-dimensional model of the ball valve is established based on the initial values of the structural dimensions parameters listed in Table 1. The material of the ball valve is chosen as \(42GrMo\), with yield strength \({R}_{e}=930 \;\text{MPa}\) and tensile strength \({R}_{a}=1080 \; \text{MPa}\). Finite element software is utilized to conduct a simulation analysis of the ball valve’s pressure-retaining process. The entire ball valve is made of tetrahedral mesh, with mesh encryption at the contact surfaces of the sealing seat and the valve body and the valve body and the valve seat. The upper surface of the valve seat is subjected to full constraints, the sealing surface to medium pressure P perpendicular to the surface, and the sealing surface and the valve body adopt a Coulomb friction contact with a coefficient of friction set to 0.2. The sealing surface and the valve seat is idealized as a frictionless contact, with no vectorial force acting on it.

First, a mesh size analysis was conducted. With the medium pressure PP set at 50 MPa and under the aforementioned boundary conditions, simulations were performed to evaluate the maximum Von Mises stress across five distinct mesh sizes, as detailed in Table 2. The simulation results, illustrated in Fig. 9, demonstrate that the maximum Von Mises stress stabilized when refined to Mesh4, exhibiting negligible variation with further mesh refinement. Consequently, the mesh dimensions corresponding to Mesh4 were adopted in this study for subsequent investigations.

Table 2 Mesh size specification table.
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Fig. 9
figure 9

Maximum Von Mises stress versus mesh sizes.

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Examine the ball valve’s deformation characteristics and stress distribution at six different media pressures: 50, 60, 70, 80, 90, and 100 MPa. Refer to Fig. 10 for the von mises stress cloud diagram of the floating ball valve under each media pressure. The cloud diagram shows that the maximum von mises stress in each media pressure appears in the valve body hole wall, and that it gradually increases as the medium pressure rises, reaching 877.64 MPa at 100 MPa. Figure 11 displays the pressure distribution across the sealing surface (contact surface) of the valve body and sealing seat at varying media pressures. As Fig. 12a illustrates, the coordinate system is set up at the contact surface’s center, Formula (16) for the ball valve necessary specific pressure empirical formula:

$$q_{b} = km\left( {a + cP} \right)/\sqrt b$$
(16)

where \(K\) is the safety coefficient, \(m\) is the coefficient related to the nature of the fluid; \(a\), \(c\) is the coefficient related to the sealing surface material; \(P\) is the working pressure of the fluid (MPa); \(b\) is the projection width of the sealing surface perpendicular to the direction of fluid flow (mm), \(b=tcos\varphi\); \(t\) is the sealing surface width (mm).

Fig. 10
figure 10

Von Mises stress cloud of floating ball valve at different media pressures.

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Fig. 11
figure 11

Cloud diagram of pressure distribution of sealing surface of floating ball valve under different media pressure.

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Fig. 12
figure 12

Contact pressure on the monitoring curves varies with the load.

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The necessary sealing specific pressure values for the ball valve under various medium pressures are calculated using Eq. (16) and presented in Table 3, with these values set as the legend values in Fig. 11. The cloud diagram reveals that under various medium pressures, the weakest points of the sealing surface pressure values consistently occur along the x-axis. The sections of the valve where the pressure is greater than or equal to the necessary specific pressure of the ball valve (the middle part indicated by the red arrow) can achieve effective sealing. A measuring line for the sealing pressure value is established along the positive direction of the x-axis (as shown by the red dashed line in Fig. 10a). The contact pressure values along the measuring line under different medium pressures are depicted in Fig. 12. By employing an interpolation method, the effective sealing width of the ball valve’s sealing surface along the x-axis at various medium pressures is calculated and presented in Fig. 13. It is observed that as the medium pressure increases, the effective sealing width gradually expands, leading to a progressive enhancement in the sealing performance of the ball valve.

Table 3 The necessary sealing specific pressure of ball valves under different mediapressure.
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Fig. 13
figure 13

Effective sealing width of ball valve under each medium pressure.

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Evaluation indicators

Based on the aforementioned analysis, this study proposes the maximum von Mises stress and the effective sealing width of the seating surface as two quantitative evaluation indicators to assess the pressure maintenance capability of the ball valve.

The maximum von Mises stress serves as a critical indicator of the pressure-bearing capacity of a spherical valve. It refers to the peak von Mises stress value experienced across the entire structure of the spherical valve during its operation under high-pressure, and even super-high-pressure fluid media conditions. The absence of yielding failure under high-pressure and ultra-high-pressure media is fundamental to the pressure maintenance capability of a ball valve. According to the fourth strength theory, the maximum von Mises stress should satisfy the following condition:

$$\sigma_{\max } \le \sigma_{s} /{\text{K}}$$
(17)

where: \({\sigma }_{max}\) is the maximum von mises stress value, MPa; \({\sigma }_{s}\) is the material yield stress value, MPa; K is the safety factor.

The term ‘effective sealing width of the sealing surface’ refers to the minimum width of the region on the projection of the ball valve’s sealing surface, perpendicular to the direction of fluid flow, where the contact pressure value meets or exceeds the necessary sealing pressure. A larger effective sealing width indicates better sealing performance of the ball valve, whereas a narrow effective sealing width may lead to sealing failure. The conditions for evaluation are as follows:

$${\text{S}} > {\text{h}}$$
(18)

where: S is the effective sealing width of the sealing surface, mm; h is the minimum value of the effective sealing width of the sealing surface, affected by the medium pressure.

Response surface modeling

Sensitivity analysis

Sensitivity analysis can be employed to eliminate parameters with minimal impact on the optimization objective and constraint conditions, reduce the number of unnecessary sampling points, decrease the number of iterations, and enhance the efficiency of structural optimization design. Based on the data in Table 1, the ball valve structure is parametrically modeled using SolidWorks and ANSYS Workbench, and the maximum von mises stress value \({\sigma }_{max}\) and the effective sealing width S of the sealing surface are taken as the response variables, combined with the ANSYS DesignXplorer module to analyze the degree of influence of the structural dimensions of the ball valve on the response variables. The sensitivity analysis results are shown in Fig. 14. Figure 14a shows the sensitivity analysis diagram of structural dimensions on \({\sigma }_{max}\). From the diagram, it can be seen that the design variables that have a greater impact on \({\sigma }_{max}\) are \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\), with the size order of \({D}_{5}>{L}_{6}{>L}_{2}\), and Fig. 14b shows the sensitivity analysis diagram of structural dimensions on S, and it is found that the design variables that have a greater impact on S are \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\), with the size order of \({D}_{5}>{L}_{2}>{L}_{6}\), The finalized design variables of the ball valve are shown in Table 4.

Fig. 14
figure 14

Sensitivity analysis plot of structural dimensions to response variables: (a) \({\upsigma }_{{\max}}\), (b) \({\text{S}}\).

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Table 4 Effective design variables.
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Experimental design and response surface modeling

In this study, the RSM module in Design-Expert was applied to the experimental design and a three-factor, five-level response surface test was conducted using a CCD design with \({\sigma }_{max}\) and S as the response variables and \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\) as the influencing factors. The different levels of each factor are represented using − 2, − 1, 0, 1 and 2. Table 5 shows the correspondence between the coded and experimental values of each factor. Numerical simulations at 100 MPa media pressure were carried out based on the experimental matrix given in Table 6, and the \({\sigma }_{max}\) and S results corresponding to each parameter combination are listed.

Table 5 Design factors and levels.
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Table 6 Response surface test design and results.
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RSM is an optimization technique that integrates experimental design and mathematical modeling. It employs a limited number of scientifically designed experimental samples to construct a continuous response surface model within a multivariate variable space. Compared to conventional experimental approaches, RSM effectively controls experimental costs while precisely characterizing nonlinear effects and interaction mechanisms among variables. Its core advantage resides in statistically grounded model significance testing, where analysis of variance (ANOVA) validates the predictive accuracy of regression models, followed by parameter space optimization based on fitting results32,33. The methodology quantifies the functional relationships between response variables and independent variables through a quadratic polynomial regression equation, as shown in Eq. (19), with the mathematical formulation expressed as:

$$G = K_{0} + \mathop \sum \nolimits_{i = 1}^{3} K_{i} x_{i} + \mathop \sum \nolimits_{i = 1}^{3} K_{ii} x_{i}^{2} + \sum \mathop \sum \nolimits_{i < j} K_{ij} x_{i} x_{j} + \varepsilon$$
(19)

where \(G\) is the response function, in this study, these are represented by the maximum Von Mises stress \({\upsigma }_{{\max}}\) or the effective sealing width \({\text{S}}\). \({K}_{0}\), \({K}_{i}\), \({K}_{ii}\) and \({K}_{ij}\) are constant, linear, second order and second order interaction coefficients respectively. \({x}_{i}\) is the optimization parameter, In this study, these are represented by the critical parameters \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\). \(\varepsilon\) is the difference between the actual value and the approximate value.

The mathematical models between \({\sigma }_{max}\) and S and the three dimensional parameters \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\) are established by Eq. (19) as shown in Eqs. (20) and (21):

$$\begin{aligned} \sigma_{{{{\max}}}} = \; &2987.76 - 67.33D_{5} - 18.38L_{2} - 41.96L_{6} + 0.1D_{5} L_{2} \\ & + 0.39D_{5} L_{6} - 0.06L_{2} L_{6} + 0.61D_{5}^{2} + 0.29L_{2}^{2} + 0.39L_{6}^{2} \\ \end{aligned}$$
(20)
$$\begin{aligned} S = & - 30.581 + 4.8D_{5} + 2.164L_{2} - 5.607L_{6} - 0.036D_{5} L_{2} \\ & - 0.021D_{5} L_{6} + 0.006L_{2} L_{6} - 0.03D_{5}^{2} + 0.0007L_{2}^{2} + 0.077L_{6}^{2} \\ \end{aligned}$$
(21)

In the two equations: \({\sigma }_{max}\) is the maximum Von Mises stress value, MPa; S is the effective sealing width, mm; \({D}_{5}\) is the inner diameter of the valve body, mm; \({L}_{2}\) is the sealing surface height adjustment, mm; \({L}_{6}\) is the pressure surface height adjustment, mm.

The explicit mathematical relationships established through the response surface model not only effectively reduce computational complexity but, more critically, provide a reliable modeling foundation for subsequent implementation of the NSGA-II algorithm in multi-objective optimization. This ensures the optimization process maintains both computational efficiency and engineering feasibility.

ANOVA

The significance of the model can be checked by ANOVA. \({\upsigma }_{{\max}}\) and S ANOVA results are shown in Tables 7 and 8, where P-value is the probability, and the significance of the model and factors can be judged by the P-value, the model is extremely significant when P < 0.0001, highly significant when P < 0.01, significant when P ≤ 0.05, and non-significant when P > 0.05. F-value is the test statistic. The order of magnitude of the F-value can be used to determine the primary and secondary order of the influencing factors.

Table 7 ANOVA results of RSM-Quadratic model for \({\sigma }_{max}\).
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Table 8 ANOVA results of RSM-Quadratic model for S.
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As can be seen in Table 7, \({\sigma }_{max}\) model is extremely significant (P < 0.0001) and the model is highly credible. \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\) have extremely significant effect on \({\sigma}_{max}\) (P < 0.0001), \({D}_{5}{L}_{6}\) has highly significant effect on a (P < 0.01) and the rest of the factors do not have any significant effect on a (P > 0.05), and the factors are ranked according to F-value as \({{D}_{5}(2730.33)>L}_{6}(2016.43)>{L}_{2}(691.19)>{{D}_{2}L}_{2}(22.22)\), and the most critical factors are \({D}_{5}\) & \({L}_{6}\). At the same time, \({\upsigma }_{{\max}}\) model is tested, the model coefficient of determination \({\text{R}}^{2}\)= 0.998, which tends to be close to 1, indicating that the correlation of the regression model is good; \(\text{Adj }{\text{R}}^{2}-\text{Pred }{\text{R}}^{2}=0.032<0.2\), indicating that the regression model is able to adequately indicate the engineering problem under study; \(\text{CV\%}=0.66\text{\%}<10\text{\%}\), indicating that the model derived from the test is credible and accurate; \(\text{Adeq Precision}=116.73>4\), indicating that the model is reasonable, and the combination of the above analyses indicates that the \({\sigma }_{max}\) regression model conforms to the principle of the test, and it has a good adaptability.

As shown in Table 8, the S model is extremely significant (P < 0.0001) and the model is highly credible. The effect of \({D}_{5}\), \({L}_{2}\), \({L}_{6}\) and \({D}_{5}{L}_{2}\) on is highly significant (P < 0.0001), the effect of \({D}_{5}{L}_{6}\) and \({L}_{6}^{2}\) on S is highly significant (P < 0.01), the effect of \({L}_{2}{L}_{6}\) on S is significant (P \(\le\) 0.05) and the rest of the factors do not have a significant effect on a (P > 0.05), the order of the factors according to the F-value is \({{D}_{5}(924.57)>L}_{2}(82.27)>{D}_{5}{L}_{2}(81.95)>{L}_{6}(52.81)\), and the most critical factor is \({D}_{5}\). At the same time, the S model is tested, the model coefficient of determination \({\text{R}}^{2}\) = 0.994, which tends to be close to 1, indicating that the correlation of the regression model is good; \(\text{Adj }{\text{R}}^{2}-\text{Pred }{\text{R}}^{2}=0.021<0.2\), indicating that the regression model can adequately indicate the engineering problem under study; \(\text{CV\%}=9.11\text{\%}<10\text{\%}\), indicating that the model derived from the test is credible and accurate; \(\text{Adeq Precision}=44.21>4\), indicating that the model is reasonable, and the combination of the above analyses shows that the S regression model conforms to the principle of the test, and it has a good adaptability.

The simulated and predicted values of the response variables \({\sigma }_{max}\) and S are shown in Fig. 15a,b, where the points are the simulated values, and the fitting effect is judged by the degree of overlap between the points and the straight line, and the simulated and predicted values of the two targets of \({\sigma }_{max}\) and S are in good agreement with the predicted values, indicating that the response surface model is reliable.

Fig. 15
figure 15

Regression curves of actual and predicted values (a) \({\sigma }_{max}\) model, (b) S model.

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Figure 16 shows the three-dimensional response surface between any two of the three influencing factors of \({D}_{5}\), \({L}_{2}\) and \({L}_{6}\) and \({\upsigma }_{{\max}}\). As shown in Fig. 16a, when \({L}_{6}\) is at the center point, \({\upsigma }_{{\max}}\) increases with the increase of \({D}_{5}\) and \({L}_{2}\). Figure 16b,c show the same pattern, and the increase of \({D}_{5}\), \({L}_{2}\) and \({L}_{2}\), \({L}_{6}\) cause a significant growth of \({\upsigma }_{{\max}}\).

Fig. 16
figure 16

RSM 3D plot of different structural parameters for \({\upsigma }_{{\max}}\): (a) \({\text{D}}_{5}\) versus \({\text{L}}_{2}\), (b) \({\text{D}}_{5}\) versus \({\text{L}}_{6}\), (c) \({\text{L}}_{2}\) versus \({\text{L}}_{6}\).

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Figure 17 shows the three-dimensional response surface between any two of the three influences \({D}_{5}\), \({L}_{2}\), \({L}_{6}\) and S. Figure 17a shows the influence of \({D}_{5}\) and \({L}_{2}\) on S when \({L}_{6}\) is at the center point, and it is found that S decreases with increasing \({D}_{5}\). When \({D}_{5}\) is at a low level, S increases with increasing \({L}_{2}\), and when \({D}_{5}\) is at a high level, there is almost no influence of \({L}_{2}\) on S. From Fig. 17b it can be found that S decreases with the increase of \({D}_{5}\) and increases with the increase of \({L}_{6}\) first decreases and then increases. From Fig. 17c, it can be seen that when \({L}_{6}\) is at a low level, \({L}_{2}\) has almost no effect on S. When \({L}_{6}\) is at a high level, S increases with the increase of \({L}_{2}\). S decreases first and then increases as \({L}_{6}\) increases.

Fig. 17
figure 17

RSM 3D plot of different structural parameters for \(\text{S}\): (a) \({\text{D}}_{5}\) versus \({\text{L}}_{2}\), (b) \({\text{D}}_{5}\) versus \({\text{L}}_{6}\), (c) \({\text{L}}_{2}\) versus \({\text{L}}_{6}\).

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Optimization

Multi-objective optimization

According to \({\upsigma }_{{\max}}\), S response surface regression mathematical model, combined with Eqs. (17) and (18) evaluation index judgment conditions and Table 3 design variable value range, the establishment of the ball valve structure multi-objective optimization model as shown in Eq. (23):

$$\left\{ \begin{gathered} {\text{Min:}}\sigma_{\max } \hfill \\ {\text{Max:S}} \hfill \\ \sigma_{\max } \le [\sigma ]/k \hfill \\ {\text{S}} \ge h \hfill \\ D_{5\min } \le D_{5} \le D_{5\max } \hfill \\ L_{2\min } \le L_{2} \le L_{2\max } \hfill \\ L_{6\min } \le L_{6} \le L_{6\max } \hfill \\ \end{gathered} \right.$$
(23)

where: \({\upsigma }_{{\max}}\) is the maximum Von Mises stress value, MPa; \([\upsigma ]\) is the material yield stress value, MPa; K is the safety coefficient; S is the effective sealing width of the sealing surface, mm; h is the minimum value of the effective sealing width of the sealing surface, mm; \({D}_{5}\) is the inner diameter of the valve body, mm; \({L}_{2}\) is the amount of adjustment of the height of the sealing surface, mm; \({L}_{6}\) is the amount of adjustment of the height of the pressure surface, mm.

In this study, the ball valve is optimized using the NSGA-II algorithm based on the Pareto solution34,35, NSGA-II is an elite strategy non-dominated sorting genetic algorithm, which is based on the basic genetic algorithm and improves the selection regeneration method, each individual is stratified according to their dominance and non-dominance relationship, and then do the selection operation, which makes the algorithm in the solving of multi-objective optimization problems It has good results in solving multi-objective optimization problems. The optimization solution flow of NSGA-II algorithm is shown in Fig. 18, and the Pareto optimal solution obtained by NSGA-II algorithm is shown in Fig. 19, where the horizontal axis corresponds to the maximum Von Mises stress value \({\upsigma }_{{\max}}\), and the vertical axis corresponds to the effective sealing width of the sealing surface S. It can be found that the initial S increases rapidly and \({\upsigma }_{{\max}}\) does not change much, and the subsequent S does not change much, and \({\upsigma }_{{\max}}\) grows rapidly, the optimization objectives of the present study are to obtain the minimum \({\upsigma }_{{\max}}\) and maximum S, it is easy to know that the inflection point of the curve is the optimal solution. According to the optimization model, the optimal floating ball valve structure size is obtained as \({D}_{6}=60 \; \text{mm}\), \({L}_{2}=37 \; \text{mm}\), \({L}_{6}=25 \; \text{mm}\), At this time, the corresponding maximum Von Mises stress \({\upsigma }_{{\max}}\) is 806.67 MPa, which is reduced by 8.1% compared with the initial structure size, this indicates an improvement in the pressure-bearing strength of the ball valve and provides potential for enhancing its maximum pressure retention capability; and the effective sealing width of the sealing surface S is 11.02 mm, which is increased by 118.2% compared with the initial structure size, this significantly raises the leakage resistance of the sealing medium and enhances the sealing performance of the ball valve.

Fig. 18
figure 18

Optimized solution flow of NSGA-II algorithm.

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Fig. 19
figure 19

Pareto optimal solution.

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Analysis of optimization results

Model validation

In order to ensure the accuracy of the numerical model of this study, under the initial structural dimensions, the theoretical model of compressive strength and the numerical simulation model in section “Theoretical model of compressive strength” were used to analyze the values of the ball valves under the pressures of six media, such as 50 MPa, 60 MPa, 70 MPa, 80 MPa, 90 MPa, 100 MPa, etc. The theoretical formulas in section “Theoretical model of compressive strength” showed that the maximum Von Mises stress appeared in the cross-section A inside (shown in Fig. 7c), combined with the data in Table 1 to calculate \({\upsigma }_{{\max}}\), while extracting the value of ball valve \({\upsigma }_{{\max}}\) in the numerical simulation in section “Numerical simulation”, the results are shown in Fig. 20, from Fig. 20, the simulation results are in good agreement with the theoretical calculation results, and the maximum error is no more than 4.671%, indicating that the numerical model has high reliability and can be used for the verification of the subsequent optimization model.

Fig. 20
figure 20

Calculation results and error values of different models.

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Optimal solution verification

In order to verify the accuracy of the optimal solution obtained by the multi-objective optimization model, the size parameters \({D}_{6}=60 \; \text{mm}\), \({L}_{2}=37 \; \text{mm}\), \({L}_{6}=25 \; \text{mm}\) corresponding to the optimal solution are reconstructed into a three-dimensional model for numerical simulation, and the simulation results of the maximum Von Mises stress value \({\upsigma }_{{\max}}\) and effective sealing width S of the sealing surface are shown in Figs. 21 and 22, and it can be seen in Fig. 21, the optimal size parameter of the ball valve \({\upsigma }_{{\max}}\) is 836.19 MPa, and the prediction value error is 3.53%, Fig. 22a shows the pressure distribution diagram of the contact surface, and the contact pressure value on the extracted measurement line is shown in Fig. 22b, and the part of the diagram that is larger than the specific pressure that must be sealed can realize the effective sealing, and it can be seen that S is 10.26 mm, and the error with the predicted value is 6.9%, and the analytical results show that the prediction model is reliable.

Fig. 21
figure 21

Von Mises stress distribution at 100 MPa medium pressure.

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Fig. 22
figure 22

Contact pressure distribution at 100 MPa medium pressure.

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Conclusion

This study integrated the pressure-holding working process of floating ball valves and established a technical framework of “numerical simulation—sensitivity screening—response surface modeling—multi-objective optimization”, which determined the optimal structural dimensions of the ball valve and significantly enhanced its ultimate pressure-holding capacity. The main research conclusions are as follows:

  1. (1)

    A novel floating-type pressure-holding ball valve design scheme was proposed, which achieves significant enhancement in ultimate pressure-holding performance through optimization of the pressure-bearing structure. The research results demonstrate that the improved ball valve achieves a maximum increase of 44.9% in pressure-holding threshold compared to traditional structures. This innovative design strategy is extendible to the engineering optimization of pressure equipment such as pipeline valve systems and hydraulic components.

  2. (2)

    A theoretical model for pressure-bearing strength of ball valves was constructed based on curved beam bending theory and energy principles. By introducing the energy variational principle, two-dimensional stress field analytical equations were established. This theoretical framework enables precise quantitative determination of Von Mises yield stress at any position within the valve body structure, with maximum deviation between calculated and simulated values not exceeding 4.671%.

  3. (3)

    Based on numerical simulation and theoretical analysis, a composite failure criterion for pressure retention in ball valves incorporating maximum von Mises stress (\({\upsigma }_{{\max}}\)) and effective sealing width (S) was proposed, integrating both structural strength and sealing performance. This criterion provides a theoretical foundation for reliability studies of high-pressure sealing systems.

  4. (4)

    A response surface mathematical model was established based on CCD experimental methodology and quadratic polynomial regression, followed by multi-parameter collaborative optimization using the NSGA-II algorithm. This process yielded an optimal dimensional combination (\({D}_{6}=60 \; \text{mm}\), \({L}_{2}=37 \; \text{mm}\), \({L}_{6}=25 \; \text{mm}\)) under 100 MPa working conditions. Numerical verification revealed prediction errors of 3.53% for stress and 6.9% for sealing pressure, confirming the engineering applicability of the optimization model.