Nonlinear liquid sloshing proportion boundary finite element numerical simulation method

Abstract

The invention is suitable for the cross technical field of computational fluid mechanics and structural engineering, and relates to a nonlinear liquid sloshing proportion boundary finite element numerical simulation method, which comprises the following steps of: S10, tracking the motion of a liquid free surface by adopting a semi-Lagrange method, and deducing a nonlinear sloshing motion equation under pitching excitation by utilizing speed potential; s20, creating a nonlinear liquid sloshing model under a proportional coordinate system and establishing a proportional boundary finite element system equation; s30, performing time domain solution by using a second-order Runze-Kutta algorithm to ensure accurate time update of the physical variables and gradients thereof; and S40, simplifying the control equation and the boundary condition into a second-order ordinary differential equation by using a weighted residual method, obtaining the free liquid level elevation change by solving the equation, and calculating the hydrodynamic pressure on the liquid container by using a Bernoulli equation. The method is simple in process, the calculation process is convenient and efficient, and the prediction precision and efficiency of numerical simulation under the swing excitation working condition are effectively improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of computational fluid dynamics and structural engineering, and particularly relates to a nonlinear liquid sloshing proportional boundary finite element numerical simulation method. BACKGROUND

[0002] Liquid sloshing, as a typical nonlinear fluid-structure interaction phenomenon, widely exists in civil engineering liquid storage structures, water conservancy engineering water guide pipe networks and transportation engineering transport containers. When such structures are subjected to swing excitation such as seismic waves, vehicle braking or wave loads, the free surface will excite large gravity waves, local splashing and even impact loads, resulting in a 40%-60% increase in the dynamic stress amplitude of the structure.

[0003] Traditional numerical analysis methods face significant bottlenecks when dealing with swing excitation working conditions: the finite element method (FEM) requires global mesh dynamic reconstruction. For a typical two-dimensional container (aspect ratio 3:1), a single 0.5Hz excitation simulation takes 6.2 hours (ANSYS Fluent 2022R2, mesh number 120k); the boundary element method (BEM) reduces the dimensionality requirement, but it is difficult to handle the strong nonlinear term of the free surface velocity potential, resulting in a wave height prediction deviation of more than 20%; existing commercial software (such as ADINA, ABAQUS) has insufficient adaptability to time-varying boundary conditions, making it difficult to accurately simulate the subharmonic resonance phenomenon caused by sudden changes in excitation frequency (Δf≥0.2Hz). The above methods have engineering applicability in specific working conditions, but their limitations are more pronounced when subjected to swing excitation, the calculation process is inconvenient, the prediction accuracy and efficiency are low, and the engineering applicability is limited. Patent No. CN117150868B provides a finite element numerical simulation model generation method and system for damage assessment, including the following steps: calibrating the finite element numerical simulation model of the warhead and the equivalent finite element numerical simulation model of the target based on the actual damage element distribution in the damage process; generating a new finite element numerical simulation model of the warhead damage target based on the calibrated finite element numerical simulation model of the warhead and the equivalent finite element numerical simulation model of the target; calibrating the new finite element numerical simulation model of the warhead damage target based on the actual damage element distribution in the damage process to obtain a finite element numerical simulation model for damage assessment. In this patent, numerical simulation is also realized by finite element method, which has the same disadvantages as existing technology.

[0004] Therefore, how to provide a numerical simulation method that is convenient and efficient in calculation process, and has high prediction accuracy and efficiency under swing excitation working conditions, is a problem that needs to be solved by technical personnel in this field. SUMMARY

[0005] In view of the defects of the prior art, the purpose of the present application is to provide a nonlinear liquid sloshing proportional boundary finite element numerical simulation method to solve the problems of the prior art that the numerical simulation method is inconvenient to calculate under the rolling excitation working condition, and the prediction accuracy and efficiency are low.

[0006] In order to solve the above technical problems, the technical scheme adopted by the present application is as follows:

[0007] The present application provides a nonlinear liquid sloshing proportional boundary finite element numerical simulation method, comprising the following steps:

[0008] S10, a semi-Lagrangian method is used to track the motion of the liquid free surface, and a nonlinear sloshing motion equation under the pitch excitation is derived by using the velocity potential;

[0009] S20, a nonlinear liquid sloshing model in a proportional coordinate system is created and a proportional boundary finite element system equation is established;

[0010] S30, a second-order Runge-Kutta algorithm is used for time domain solving to ensure accurate time updating of physical variables and their gradients;

[0011] S40, the weighted residual method is used to simplify the control equation and the boundary condition into a second-order ordinary differential equation, the free surface elevation change is obtained by solving the equation, and the fluid dynamic pressure on the liquid container is calculated by using the Bernoulli equation.

[0012] Further, in the step S10, the rectangular container in the rolling oscillation is subjected to the rolling excitation defined by the angular velocity ω(t) motion, and the rolling excitation is determined by .

[0013] Wherein, θ represents the included angle between the rectangular container and the vertical direction, and the clockwise rotation is the positive direction; t represents time, in the case that there is no viscosity and compressibility in the fluid, it is assumed that the fluid does not rotate, and the control equation is defined as the Laplace equation:

[0014]

[0015] Wherein, represents the velocity potential of the liquid in the domain Ω, represents a two-dimensional Laplace operator.

[0016] Further, in the step S10, under the assumption condition that the atmospheric pressure is 0, the pressure of the free surface is equal to the atmospheric pressure, and the following dynamic boundary condition is obtained:

[0017]

[0018] Wherein, g represents the acceleration of gravity, Γ S1 represents the boundary formed by the free surface;

[0019] According to the assumption that the fluid particles reaching the free surface still stay on the free surface in the subsequent movement, the dynamic boundary condition on the free surface is expressed as:

[0020]

[0021] where n y and n x are the direction cosines of the normal vector and the outward projection along the boundary, respectively, denotes the differential along the normal direction;

[0022] The remaining boundary conditions on the inner wall of the container wetted by the liquid are:

[0023]

[0024] Further, the step S20 specifically comprises:

[0025] A scaled coordinate system is introduced, and the geometric shape of the problem domain boundary is defined by the scaled coordinates;

[0026] The boundary conditions are discretized by using the weighted residual method, and the SBFEM system equation in the scaled coordinate system is derived, in which the following relationship is obtained through the control equation and the boundary conditions:

[0027]

[0028] where v n is the normal liquid velocity outside the domain boundary, w is an arbitrary weighted function, and the spatial derivative in the equation is expressed as:

[0029] dΩ = ξ |J b |dξds

[0030] where ξ and s are the scaled boundary coordinates, and the weighted function w of the liquid potential energy is expressed according to the Galerkin method as:

[0031] w(ξ, s) = [N(s)]{w(ξ)} = {w(ξ)} T [N(s)] T

[0032] The integral and expansion of the domain integral with respect to the radial coordinate are further realized by applying the Green's theorem.

[0033] Further, the weighted function w is substituted into the spatial derivative to obtain:

[0034]

[0035] The integral and expansion are obtained as:

[0036]

[0037] wherein [E0] = ∫ s [B1(s)] T [B1(s)]|J b |ds, [E1] = ∫ s [B2(s)] T [B1(s)]|J b |ds, [E2] = ∫ s [B2(s)] T [B2(s)]|J b |ds.

[0038] Further, in the step S30, the motion and dynamic boundary conditions are expressed by the midpoint method as follows:

[0039]

[0040]

[0041] wherein n and n+1 represent the values of the physical quantity at t n and t n+1 , and t n =t0+n△t, c1, c2 and c3 satisfy c1+c2=1, c2c3=1 / 2.

[0042] Further, in the step S40, the hydrodynamic pressure applied to the liquid container is calculated by the Bernoulli equation:

[0043]

[0044] The nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided by the application has at least the following beneficial effects compared with the prior art:

[0045] The numerical analysis method in the prior art has more obvious limitations when facing the rolling excitation, the calculation process is inconvenient, the prediction accuracy and efficiency are low, and the engineering applicability is limited. The application has a simple process and is convenient to operate, adopts the semi-Lagrangian method to track the motion of the liquid free surface, derives the nonlinear sloshing motion equation under the pitch excitation by using the velocity potential, creates a nonlinear liquid sloshing model in the proportional coordinate system and solves the proportional boundary finite element system equation, uses the second-order Runge-Kutta algorithm for time domain solving, ensures the accurate time update of the physical variables and their gradients, simplifies the control equation and the boundary conditions into second-order ordinary differential equations by using the weighted residual method, obtains the free surface elevation change by solving the equations, calculates the hydrodynamic pressure on the liquid container by using the Bernoulli equation, and further reduces the calculation cost of the liquid motion characteristics under the rolling action, effectively improves the prediction accuracy and calculation efficiency of the numerical simulation. BRIEF DESCRIPTION OF DRAWINGS

[0046] To more clearly illustrate the solution of the present invention, a brief introduction will be given to the drawings used in the description of the embodiments below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0047] Figure 1 A flowchart of a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in an embodiment of the present invention;

[0048] Figure 2 This is a motion diagram illustrating a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in an embodiment of the present invention.

[0049] Figure 3 for Figure 2 The rectangular container shown is a schematic diagram of mesh discretization in a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in an embodiment of the present invention;

[0050] Figure 4 A comparison diagram of the liquid surface elevation with the reference solution at different time steps is provided for a nonlinear liquid sloshing proportional boundary finite element numerical simulation method in an embodiment of the present invention.

[0051] Figure 5 Time history diagrams of liquid surface elevations at different heights corresponding to the rotation centers in a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in an embodiment of the present invention.

[0052] Figure 6 The free liquid surface elevation time history diagram of a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in this embodiment of the invention;

[0053] Figure 7 for Figure 6 The free surface elevation time history diagram shown is a slice of the liquid surface at time 1Time(s).

[0054] Figure 8 for Figure 6 The free liquid surface elevation time history diagram shown is a slice of the liquid surface at time 3Time(s);

[0055] Figure 9 for Figure 6 The free liquid surface elevation time history diagram shown is a slice of the liquid surface at time 5 (Time(s)).

[0056] Figure 10 for Figure 6 The free liquid surface elevation time history diagram shown is a slice of the liquid surface at time 7 (Time(s)).

[0057] Figure 11 A schematic diagram of the proportional boundary finite element method mesh discretization of a rectangular liquid storage structure with rounded corners in a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided in an embodiment of the present invention;

[0058] Figure 12 In a nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided for an embodiment of the present invention, a time history diagram of the free liquid surface elevation of a container with a round bottom radius under swaying is introduced.

[0059] Figure 13 for Figure 11 The time history diagram shows the pressure exerted by the liquid surface sloshing at the bottom of the container. Detailed Implementation

[0060] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0061] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0062] This invention provides a nonlinear liquid sloshing scaled boundary finite element numerical simulation method, which is applied to the dynamic response prediction and safety assessment of liquid storage structures such as hydrogen energy storage and transportation equipment and large aqueducts. The nonlinear liquid sloshing scaled boundary finite element numerical simulation method includes the following steps:

[0063] S10. The motion of the free surface of the liquid is tracked using the semi-Lagrange method, and the nonlinear swaying motion equation under pitch excitation is derived using the velocity potential; S20. A nonlinear liquid swaying model is created in a scaled coordinate system, and the scaled boundary finite element system equations are established; S30. The second-order Lonzo-Kutta algorithm is used for time-domain solution to ensure accurate time updates of physical variables and their gradients; S4. The control equations and boundary conditions are simplified to second-order ordinary differential equations using the weighted residual method. The elevation change of the free liquid surface is obtained by solving the equations, and the hydrodynamic pressure on the liquid container is calculated using the Bernoulli equation.

[0064] The present invention has a simple process and a convenient and efficient calculation process, which effectively improves the prediction accuracy and efficiency of numerical simulation under swing excitation conditions.

[0065] In order for the person skilled in the art to better understand the technical scheme of the present application, the technical scheme in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings.

[0066] The present application provides a nonlinear liquid sloshing proportional boundary finite element numerical simulation method, which is applied to the dynamic response prediction and safety evaluation of liquid storage structures such as hydrogen energy storage and transportation equipment and large aqueducts, and combines Figure 2 With Figure 3 In this embodiment, Figure 2 The motion schematic diagram of the nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided by the present application, Figure 3 For Figure 2 The proportional boundary finite element method grid discrete schematic diagram of the rectangular container shown in the figure, in Figure 3 The model is established and discretized, and the operation object of the model establishment and discretization process is a rectangular container, and the detailed method specifically includes the following steps:

[0067] S10, the semi-Lagrangian method is used to track the motion of the liquid free surface, and the nonlinear sloshing motion equation under the pitch excitation is derived by using the velocity potential.

[0068] Specifically, as shown in Figure 2 , a multi-reference frame kinematic model is established, the coordinate system is defined as a fixed inertial system O-XY: the origin O is located at the geometric center of the container, the X axis is horizontal to the right, and the Y axis is vertical upward; the motion system o-xy: the container makes a swing motion around the O point, and the x axis is always coincident with the container bottom. The rectangular container in swing oscillation is subjected to angular velocity motion Determined;

[0069] In the case that there is no viscosity and compressibility in the fluid, it is assumed that the fluid does not rotate, and the control equation can be defined as the Laplace equation: Under the assumption condition that the atmospheric pressure is 0, the pressure of the free surface is equal to the atmospheric pressure to obtain the dynamic boundary condition:

[0070]

[0071] Wherein, g represents the acceleration of gravity, Γ S1 Indicates the boundary formed by the free surface;

[0072] According to the assumption that the fluid particles reaching the free surface still stay on the free surface in the subsequent motion, the dynamic boundary condition on the free surface can be expressed as:

[0073]

[0074] Wherein, n y And n xare the direction cosines of the normal vector and the outward projection along the boundary, respectively, denotes the differential along the normal direction;

[0075] The remaining boundary conditions on the inner wall of the container wetted by the liquid are:

[0076]

[0077] S20, create a nonlinear liquid sloshing model in a scaled coordinate system and establish a scaled boundary finite element system (SBFEM) equation.

[0078] Specifically, as shown in Figure 3 , a scaled boundary coordinate system (ξ, η) is introduced, the radial coordinate ξ ∈ [0, 1]: the scaling ratio of the domain, ξ = 1 corresponds to the physical boundary; the circumferential coordinate η ∈ [-1, 1]: the dimensionless parameter along the boundary, the geometry of the problem domain boundary is defined by the scaled boundary coordinate.

[0079] The boundary conditions are discretized using the weighted residual method, and the SBFEM system equation in the scaled coordinate system is derived, where through the control equation and the boundary condition, the following relationship can be obtained:

[0080]

[0081] where, v n The normal liquid velocity outside the domain boundary is shown, and w is an arbitrary weighting function. The spatial derivative in the equation is expressed as:

[0082] dΩ = ξ |J b |dξds

[0083] where,

[0084]

[0085]

[0086] where, ξ, s are scaled boundary coordinates, and according to the Galerkin method, the weighted function w of the liquid potential energy can be expressed as:

[0087] w(ξ, s) = [N(s)]{w(ξ)} = {w(ξ)} T [N(s)] T

[0088] Substituting the above equation into the boundary condition equation, we get:

[0089]

[0090] By applying Green's theorem, the integral and expansion of the domain integral with respect to the radial coordinate are realized, and the SBFEM system equation is obtained:

[0091]

[0092] wherein,

[0093] [E0] = ∫ s [B1(s)] T [B1(s)]|J b |ds

[0094] [E1] = ∫ s [B2(s)] T [B1(s)]|J b |ds

[0095] [E2] = ∫ s [B2(s)] T [B2(s)]|J b |ds.

[0096] S30, time domain solving is performed using the second-order Runge-Kutta algorithm (RK2), ensuring accurate time updating of physical variables and their gradients.

[0097] Specifically, the improved midpoint method format is adopted to represent the motion and dynamic boundary conditions,

[0098]

[0099]

[0100] wherein, n and n+1 represent the values of physical quantities at t n and t n+1 , and t n =t0+n△t, c1, c2 and c3 satisfy c1+c2=1, c2c3=1 / 2.

[0101] S40, the control equation and the boundary condition are simplified into second-order ordinary differential equations using the weighted residual method, the free surface elevation change is obtained by solving the equations, and the fluid dynamic pressure on the liquid container is calculated using the Bernoulli equation.

[0102] Specifically, in view of the arbitrariness of the weighting function, [E0], [E1] and [E2] must satisfy the following relationship:

[0103]

[0104] The second-order ordinary differential equation is solved by introducing a dual variable vector to convert it into a more simplified form:

[0105]

[0106] wherein,

[0107]

[0108] When ξ = ξ1, the nodal values of {q(ξ)} on the right side of the equation can be calculated by the following formula:

[0109]

[0110] where, can be expressed as: then get ξ{X(ξ)}, ξ = [Z] {X(ξ)}, where the coefficient matrix [Z] is determined as:

[0111]

[0112] Substituting the above formula gives: λ{Φ} = [Z] {Φ}, whose eigenvalue decomposition gives:

[0113] [Z] [Φ] = [Φ]<λ>

[0114] [Φ] = [{Φ1} {Φ2} … {Φ i} … {Φ 2n}]

[0115] <λ> = diag(λ1 λ2 … λ i … λ 2n )

[0116] then:

[0117]

[0118] And because can be deduced:

[0119]

[0120] If <λ b > = diag(λ1,λ2,…,λ i ,…,λ n )

[0121]

[0122] get where {c} = [c1,c2,…,c n ] T represent the integral constant vector.

[0123] Finally, the liquid velocity on the domain boundary is calculated as

[0124]

[0125] The fluid dynamic pressure applied to the liquid container is calculated by Bernoulli's equation The field variables at any point in the liquid domain are calculated. Specifically, the information is brought into the equation solution to obtain the liquid surface elevation, and the result data is imported into the prepared matlab visualization program for post-processing, i.e. drawing the liquid surface elevation variation graph, the container side wall pressure variation curve or the bottom pressure variation curve, and then the movement characteristics of the liquid in the container under the swing action can be determined intuitively.

[0126] Further, in this embodiment, in order to verify the applicability, accuracy and efficiency of the nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided by the embodiment of the present application in the numerical simulation of liquid motion under the swing action, the calculation performance of the present application is investigated from different angles, such as Figure 2 As shown in the figure, the swing motion of the liquid in the two-dimensional rectangular container is established, the distance between the undisturbed free surface in the rectangular container and the x-axis is represented by e, the positive value of which represents that the free liquid surface is located above the x-axis, the center of rotation o is located at the center of the x-axis and on the center line of the container, at the same time, the distance from the undisturbed free surface to the actual free surface is represented by η, as shown in the figure Figure 4 As shown in the figure, the height of the two-dimensional rectangular container is set to 0.6 meters and the width is set to 0.9 meters, the height of the free liquid surface on the left side of the rectangular liquid storage structure is selected as the verification index, the swing angular velocity of the container is ω, wherein the initial angle between the rectangular container and the vertical direction is 0.8°, the angular velocity of rotation is ω, the time is t, and the time marching scheme adopts multiple time steps, including 0.01, 0.005 and 0.001, the model proposed by the present application adopts a smaller number of elements, a lower element order and a smaller degree of freedom, but has a high consistency with the reference solution, thereby proving that the model proposed in this paper is suitable for simulating the nonlinear liquid sloshing phenomenon of the liquid storage structure under the swing excitation.

[0127] Further, as shown in the figure Figure 5 When a first-order frequency swing motion is applied, the liquid sloshing phenomenon will intensify over time, the wave height fluctuation amplitude of the free liquid surface gradually increases, and the change period gradually shortens, it is worth noting that the influence of the first-order natural frequency external excitation on the liquid sloshing phenomenon decreases as the center of rotation rises. In addition, under the same frequency excitation, the simulation results of the liquid with different filling amounts under the swing excitation are also tested, from Figure 6 It can be seen that under the same frequency of pitch excitation, the liquid sloshing of different heights shows different periods and amplitudes, as the filling height decreases, the period will be extended, and the amplitude will be increased. In Figures 7 to 10 The cross-sectional images of the moment when the liquid sloshing phenomenon in the container reaches the most severe moment in each period are obtained, it can be seen that for the container with higher loading capacity, the free liquid surface shape also shows strong nonlinearity.

[0128] Further, in order to prove the applicability of the nonlinear liquid sloshing proportional boundary finite element numerical simulation method provided by the embodiment of the present application, the model is improved to a rectangular liquid storage structure with rounded corners, as shown in Figure 11 Figure 11 The SBFEM grid of the rectangular liquid storage structure with rounded corners is shown, and in this embodiment, 19 cubic units are used, containing 57 nodes, as shown in Figure 12 As shown in the figure, increasing the radius of the rounded corner at the bottom of the rectangular container slightly prolongs the total vibration force period of the liquid in the x direction during free vibration, while the amplitude remains unchanged, as shown in Figure 13 As shown in the figure, the increase of the rounded corner at the bottom of the rectangular container has little effect on the total sloshing force of the liquid in the y direction during free vibration, but it will cause the period of the sloshing force acting on the container bottom to be prolonged.

[0129] Compared with the prior art, the nonlinear liquid sloshing proportional boundary finite element numerical simulation method described in the above embodiments has the following advantages: the prior art numerical analysis method has the following limitations when facing a rolling excitation: the calculation process is inconvenient, the prediction accuracy and efficiency are low, and the engineering applicability is limited. The process of the present application is simple and convenient to operate. The semi-Lagrangian method is used to track the motion of the liquid free surface, the nonlinear sloshing motion equation under the pitch excitation is derived using the velocity potential, the nonlinear liquid sloshing model in the proportional coordinate system is created and the proportional boundary finite element system equation is solved, the second-order Runge-Kutta algorithm is used for time domain solving to ensure accurate time updating of physical variables and their gradients, the control equation and boundary conditions are simplified into second-order ordinary differential equations using the weighted residual method, the free surface elevation change is obtained by solving the equation, and the fluid dynamic pressure on the liquid container is calculated using Bernoulli's equation, thereby reducing the calculation cost of the liquid motion characteristics under the rolling action, and effectively improving the prediction accuracy and calculation efficiency of the numerical simulation.

[0130] Obviously, the above-described embodiments are only preferred embodiments of the present application, not all embodiments, and the preferred embodiments of the present application are shown in the drawings, but do not limit the patent scope of the present application. The present application can be realized in many different forms, and conversely, the purpose of providing these embodiments is to make the disclosure of the present application more thorough and comprehensive. Although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments, or make equivalent substitutions for some of the technical features. Any equivalent structure made by using the contents of the present application specification and drawings, directly or indirectly applied to other related technical fields, is also within the scope of the patent protection of the present application.​

Claims

1. A nonlinear liquid sloshing proportional boundary finite element numerical simulation method, characterized in that, The method comprises the following steps: S10, a semi-Lagrangian method is used to track the movement of the liquid free surface, and a nonlinear sloshing motion equation under the pitch excitation is derived by using the velocity potential; S20, a nonlinear liquid sloshing model in a scaled coordinate system is created, and a scaled boundary finite element system equation is established; S30, a second-order Runge-Kutta algorithm is used for time domain solving to ensure accurate time updating of physical variables and their gradients; S40, the control equation and the boundary condition are simplified into second-order ordinary differential equations by using a weighted residual method, the free surface elevation change is obtained by solving the equations, and the fluid dynamic pressure on the liquid container is calculated by using the Bernoulli equation.

2. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 1, characterized in that, In the step S10, the angular velocity ω(t) of the rectangular container in the rolling oscillation is defined by the rolling excitation received by the rectangular container in the rolling oscillation determined; Wherein, θ represents the angle between the rectangular container and the vertical direction, and the clockwise rotation is the positive direction; t represents time, in the case that there is no viscosity and compressibility in the fluid, it is assumed that the fluid does not rotate, and the control equation is defined as the Laplace equation: wherein, denotes the velocity potential of the liquid within the domain Ω, denotes the two-dimensional Laplace operator.

3. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 2, characterized in that, In the step S10, the pressure of the free surface under the assumption condition that the atmospheric pressure is 0 is equal to the atmospheric pressure, and the following dynamic boundary condition is obtained: where g denotes the acceleration due to gravity, Γ S1 denotes the boundary at which the free surface is formed; According to the assumption that the fluid particles reaching the free surface still stay on the free surface in the subsequent movement, the dynamic boundary condition on the free surface is expressed as: where n y and n x are the directional cosines of the normal vector and the outward projection along the boundary, respectively, denotes the differential along the normal direction; The remaining boundary condition on the inner wall of the container wetted by the liquid is:

4. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 2, characterized in that, The step S20 specifically comprises: A scaled coordinate system is introduced, and the geometric shape of the problem domain boundary is defined by the scaled coordinates; The boundary condition is discretized by using a weighted residual method, and the SBFEM system equation in the scaled coordinate system is derived, wherein the following relationship is obtained by the control equation and the boundary condition: where v n The display domain boundary outside normal liquid velocity, w is any weighted function, the spatial derivative in the equation is expressed as: dΩ = ξ | J b |dξds Wherein, ξ and s are scaled boundary coordinates, according to the Galerkin method, the weighted function w of the liquid potential energy is expressed as: w(ξ, s) = [N(s)] {w(ξ)} = {w(ξ)} T [N(s)] T And then, the integral and expansion of the domain integral with respect to the radial coordinate are realized by applying the Green theorem.

5. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 4, characterized in that, The weighted function w is substituted into the spatial derivative to obtain: The integral and expansion are obtained as follows: where [E0] = ∫ s [B1(s)] T [B1(s)]|J b |ds, [E1] = ∫ s [B2(s)] T [B1(s)]|J b |ds, [E2] = ∫ s [B2(s)] T [B2(s)]|J b |ds.

6. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 2, characterized in that, In the step S30, the midpoint method is used to represent the motion and the dynamic boundary condition as follows: where n and n+1 represent the values of the physical quantity at t n and t n+1 , respectively, and t n = t0+ n△t, and c1, c2 and c3 satisfy c1+c2=1 and c2c3=1 / 2.

7. The nonlinear liquid sloshing proportional boundary finite element numerical simulation method according to claim 2, characterized in that, In the step S40, the fluid dynamic pressure applied to the liquid container is calculated by the Bernoulli equation:

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