High-precision numerical simulation method for fluid based on least square finite volume particle method

Abstract

This invention discloses a high-precision numerical simulation method for fluids based on the least-squares finite-volume particle method. The method includes the following steps: S1: Reading the geometric model and fluid particles; S2: Determining free-surface particles using an angle scan method, calculating the unit normal vector at the free surface, and generating virtual particles outside the fluid along the normal direction; S3: Constructing a Type-B least-squares moving particle semi-implicit operator model; S4: Transforming the partial differential equations of fluid dynamics into an integral form based on the particle control volume, and applying the Gaussian divergence theorem to reduce the dimension of the volume integral to a flux surface integral along the closed boundary of the control volume; S5: Constructing a composite particle displacement correction vector and correcting the phase space position of the fluid particles; S6: Completing the fluid evolution calculation for the current time step. This invention enables high-precision and high-stability numerical simulation of complex free-surface flows with low computational cost.

Description

Technical Field

[0001] This invention relates to the fields of computational fluid dynamics and numerical simulation technology, specifically to a high-precision numerical simulation method for fluids based on the least squares finite volume particle method. Background Technology

[0002] In modern scientific computing and cutting-edge industrial engineering simulations, numerical simulations of fluid dynamics involving large deformations, complex geometric moving boundaries, and free surface tracking have always been extremely challenging frontier topics. Industry has long relied on traditional meshing methods based on the Euler perspective to solve the Navier-Stokes equations and various partial differential equations. Although traditional meshing methods possess rigorous mathematical theoretical support and conservation properties, they require enormous computational resources and manual effort for mesh generation and topology quality optimization when dealing with complex geometries.

[0003] To fundamentally overcome the physical constraints imposed on computation by mesh topology, meshless methods have emerged with their pure Lagrangian perspective. Meshless methods utilize a series of discrete particles carrying physical properties to characterize the continuous medium, eliminating the need to generate and maintain complex mesh topological connections. This demonstrates a significant natural advantage in solving extreme deformations and free-surface flows. Typical classical meshless particle methods include Smoothed Particle Hydrodynamics (SPH), Moving Particle Semi-Implicit (MPS), Generalized Finite Difference Method (GFDM), Radial Basis Function Method, and Regenerative Kernel Method. However, the spatial discretization accuracy and numerical stability of classical SPH and MPS methods are highly dependent on the spatial uniformity of particle distribution. Particle distribution often inevitably exhibits a highly irregular state, leading to a sharp amplification of the approximation error of classical kernel functions and their derivatives. This not only degrades the accuracy of spatial derivative calculations but also induces strong, non-physical pressure high-frequency oscillations.

[0004] To improve computational accuracy on irregular point clouds, academia and industry have developed various higher-order gradient and Laplacian operator models, such as the correction matrix method, pairwise relaxation method, and generalized finite difference method and least squares moving particle semi-implicit method (LSMPS) based on higher-order Taylor series expansions. The LSMPS method theoretically enables arbitrary higher-order reconstruction of partial differential operators on irregularly distributed nodes by performing Taylor series expansions on the scalar variables of neighboring particles at fixed positions and minimizing the error functional using weighted least squares techniques. LSMPS is specifically divided into two schemes: Type-A schemes assume that the function value of the central particle at a fixed position is known, and the system only solves for the spatial derivatives of each order as unknowns; Type-B schemes treat the function value of the central particle and the derivatives of each order as unknowns for joint solution.

[0005] While strong-form direct discretization methods such as LSMPS and GFDM significantly improve theoretical accuracy, they introduce catastrophic computational complexity. To obtain strictly second-order accurate second-order partial differential operators in strong form, the Taylor series polynomials must be forcibly expanded to the third order or higher. Inverting higher-order matrices not only requires extremely large neighborhood particle support domains to ensure matrix nonsingularity, greatly increasing memory bandwidth pressure and cache miss rates, but also involves massive floating-point operations, resulting in extremely high practical computational costs. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, the present invention aims to provide a high-precision numerical simulation method for fluids based on the least squares finite volume particle method. This method transforms the fluid partial differential equation into an integral form based on the particle control volume and solves it by reducing the dimension using the Gaussian divergence theorem. Combined with the Type-B least squares operator and particle displacement correction, it can achieve high-precision and high-stability numerical simulation of complex free surface flows with low computational cost.

[0007] To achieve the objective of this invention, the following solution is adopted: A high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method includes the following steps: S1: Read the geometric model and fluid particles, and model each discrete fluid particle as a square finite control volume with characteristic length in two-dimensional space to obtain a set of fluid particles with square control volumes. S2: Based on the fluid particle set, the free surface particles are determined by the angle scanning method, the unit normal vector at the free surface is calculated, and virtual particles are generated outside the fluid along the normal direction to obtain a virtual particle set used to maintain the integrity of the boundary support domain. S3: Based on the fluid particle set and the virtual particle set, construct a Type-B least squares moving particle semi-implicit operator model to obtain a local low-order reconstruction operator for reconstructing the function value and derivative at the face center; S4: The partial differential equations of fluid dynamics are transformed into integral forms based on particle control volume, and the Gaussian divergence theorem is applied to reduce the dimension of the volume integral to the flux surface integral along the closed boundary of the control volume. The surface normal flux is estimated using the Type-B least squares moving particle semi-implicit operator model to obtain the discretized fluid dynamics algebraic equations. S5: Based on fluid continuity and particle number density distribution, construct a composite particle displacement correction vector and correct the phase space position of fluid particles to obtain the corrected uniformly distributed fluid particle positions. S6: Solve the discretized fluid dynamics algebraic equations and use the corrected uniformly distributed fluid particle positions to update the velocity and position of the fluid particles, thus completing the fluid evolution calculation for the current time step.

[0008] Furthermore, in step S4, the Gaussian divergence theorem is applied to transform the volume integral into a flux surface integral along the closed boundary of the control volume, and the transformed form is as follows: The volume integral that originally dealt with the second-order partial derivative is transformed into the flux integral along the closed surface of the control volume, thus converting the solution of the second-order partial derivative into the calculation of the first-order derivative in the direction of the outward normal of the surface.

[0009] Furthermore, in step S3, the constructed Type-B least squares moving particle semi-implicit operator model is used to uniformly assume the spatial function values ​​and spatial derivatives of all orders as unknown variables for weighted least squares solution. Based solely on the known physical state quantities of the particle at the center of the control volume and the state quantities of the neighboring particles in its supporting domain, the unknown function values ​​and their gradient information at any face center position can be accurately extracted and reconstructed.

[0010] Furthermore, in step S4, after estimating the surface normal flux using the Type-B least squares moving particle semi-implicit operator model, the Gaussian numerical integration method is also used to numerically integrate the nonlinear continuous flux distribution passing through the particle surface to ensure that the overall calculation format achieves second-order spatial accuracy.

[0011] Furthermore, in step S2, the generated virtual particles are forcibly incorporated into the least squares support domain of the fluid particles, and jointly participate in the moment matrix construction of the Type-B least squares moving particle semi-implicit operator model to correct the discrete format of the boundary truncation region.

[0012] Furthermore, in step S5, the constructed composite particle displacement correction vector includes a physical repulsion displacement based on the particle number density gradient and a volume correction displacement based on the density error gradient. The physical repulsion displacement is used to push away particles that get too close, and the volume correction displacement is used to compensate for macroscopic volume conservation.

[0013] Furthermore, before step S1 or during step S6, a dynamic time step control step is also included, specifically: At the beginning of each time integration cycle, the entire field is automatically scanned to obtain the maximum characteristic convection rate and the physical field viscous diffusion coefficient. Following the Courant-Friedrich-Lyuvi condition and the von Neumann diffusion stability limit criterion, the optimal time step is automatically selected and output.

[0014] Furthermore, in step S6, during the collision detection of the moving boundary, a rebound recovery coefficient based on relative velocity is introduced, and the normal velocity of the dynamic wall is directly injected into the source term on the right-hand side of the pressure Poisson equation, and fluid-structure interaction is achieved by combining the velocity rebound logic of the relative moving boundary.

[0015] Furthermore, the method is used for fluid dam failure evolution and impact simulation, specifically as follows: Initialize rectangular water column particles on one side of the rectangular computational domain, set the initial velocity to zero, release the fluid so that it collapses under the action of gravity and impacts the wall in front; Within the dynamic time step, zero-pressure boundary conditions are applied by generating virtual particles, and non-physical aggregation of droplets is avoided by correcting the phase space position of fluid particles.

[0016] Furthermore, the method is applied to numerical simulations of hydrostatic pressure tests or dynamic boundary tank sloshing problems: In hydrostatic pressure testing, long-term numerical integration evolution is performed on uniformly distributed fluid particles in a closed static water tank to calculate the numerical density and volume conservation terms of the fluid. In the dynamic boundary water tank sloshing problem, horizontal harmonic excitation displacement and velocity are applied to the solid wall boundary of the two-dimensional water tank, and the normal velocity of the dynamic wall is injected into the right-hand source term of the pressure Poisson equation.

[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention models fluid particles as square control volumes with characteristic lengths and transforms the partial differential equations of fluid dynamics into integral forms based on these control volumes. Then, it applies the Gaussian divergence theorem to reduce the dimension of the volume integral to the flux surface integral along the closed boundary of the control volume. This completely avoids the complex process of constructing and inverting high-order polynomial matrices to maintain second-order accuracy in traditional high-order meshless methods. Flux reconstruction can be completed with only a low-order polynomial basis, which greatly reduces the floating-point operations and memory addressing overhead per time step, thereby significantly improving computational efficiency.

[0018] 2. This invention utilizes a Type-B least squares moving particle semi-implicit operator model to perform high-precision estimation of the normal flux of the control volume surface. This operator treats the function value and partial derivative at the face center as unknowns for local weighted least squares solution. At the same time, it combines a composite particle displacement correction based on fluid continuity and particle number density distribution to ensure the uniformity of particle distribution. This enables the method to achieve third-order accuracy in approximating function values ​​and second-order accuracy in estimating first-order derivatives, achieving strict second-order spatial calculation accuracy under the global error norm.

[0019] 3. This invention uses an angle scanning method to determine free surface particles and generate virtual particles along the normal direction to maintain the integrity of the support domain at the boundary. At the same time, when constructing the Type-B operator model, the unknown function value and gradient information of the face center position can be accurately reconstructed only by the known state variables of the particle at the control volume center and the state variables of its neighboring particles. This mechanism makes the boundary flux calculation without introducing additional unknown global degrees of freedom at the face center, while ensuring the high-precision implementation of the free surface zero-pressure boundary condition and the solid wall boundary condition, thereby achieving the accuracy of boundary processing without increasing the global degrees of freedom.

[0020] 4. This invention constructs a composite particle displacement correction vector based on fluid continuity and particle number density distribution, adaptively correcting the phase space position of fluid particles. This effectively pushes away excessively close particles and compensates for macroscopic volume conservation, avoiding extreme distortion of particle distribution and singularity of least square moment matrix caused by severe deformation or long-term evolution. Furthermore, by solving the discretized fluid dynamics algebraic equations and updating particle velocity and position, the method can maintain the normalized density stable around 1.0 and without non-physical pressure oscillations in long-term, large-deformation, and strongly nonlinear flow simulations, thus endowing the calculation process with extremely strong stability and long-term simulation capability. Attached Figure Description

[0021] Figure 1 This is a flowchart of a high-precision numerical simulation method for fluid based on the least squares finite volume particle method in an embodiment of the present invention; Figure 2 This is a schematic diagram illustrating the concept of finite volume particles based on least squares and the definition of surface flux in an embodiment of the present invention. Figure 3 This is a flowchart illustrating the overall execution process of the LSFVP fluid numerical simulation algorithm in this embodiment of the invention. Figure 4 This is a cloud map showing the evolution of the liquid surface and pressure field during large deformation of the dam-break flow in an embodiment of the present invention. Figure 5 This is a comparison curve of the dam-break flow impact experiment verification in the embodiments of the present invention; Figure 6 This is a pressure profile and density history diagram of the hydrostatic pressure balance test in an embodiment of the present invention; Figure 7 This is a physical topological distribution diagram of the liquid surface under extreme deformation of the container during sloshing, as shown in an embodiment of the present invention. Figure 8 This is a graph showing the evolution of the container swaying response and density curves in an embodiment of the present invention. Detailed Implementation

[0022] The present invention will now be further described in conjunction with the accompanying drawings and specific embodiments. It should be noted that, without conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments.

[0023] like Figure 1-8 As shown, this embodiment of the invention provides a high-precision numerical simulation method for fluids based on the least squares finite volume particle method, including the following steps: S1: Read the geometric model and fluid particles, and model each discrete fluid particle as a square finite control volume with characteristic length in two-dimensional space to obtain a set of fluid particles with square control volumes.

[0024] S2: Based on the fluid particle set, determine the free surface particles by angle scanning method, calculate the unit normal vector at the free surface, and generate virtual particles outside the fluid along the normal to obtain a virtual particle set used to maintain the integrity of the boundary support domain.

[0025] S3: Based on the fluid particle set and the virtual particle set, construct a Type-B least squares moving particle semi-implicit operator model to obtain a local low-order reconstruction operator for reconstructing the function value and derivative at the face center.

[0026] S4: The partial differential equations of fluid dynamics are transformed into integral forms based on particle control volume, and the Gaussian divergence theorem is applied to reduce the dimension of the volume integral to the flux surface integral along the closed boundary of the control volume. The surface normal flux is estimated using the Type-B least squares moving particle semi-implicit operator model to obtain the discretized algebraic equations of fluid dynamics.

[0027] S5: Based on fluid continuity and particle number density distribution, construct a composite particle displacement correction vector and correct the phase space position of fluid particles to obtain the corrected uniformly distributed fluid particle positions.

[0028] S6: Solve the discretized fluid dynamics algebraic equations and use the corrected uniformly distributed fluid particle positions to update the velocity and position of the fluid particles, thus completing the fluid evolution calculation for the current time step.

[0029] The following is a more detailed description of the high-precision numerical simulation method for fluid based on the least squares finite volume particle method according to embodiments of the present invention.

[0030] This invention presents a least-squares finite-volume particle method that combines high precision, high efficiency, and strong stability. This method is a high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method. It overcomes the difficulties faced by existing high-order meshless methods in reconstructing second derivatives, such as excessively high matrix dimension, low computational efficiency, and unstable boundaries.

[0031] This invention's high-precision numerical simulation method for fluid dynamics based on the least squares finite volume particle method innovatively introduces the particle control volume assumption and a dimension reduction technique using the integral form of partial differential equations. Unlike traditional GFDM or LSMPS, which directly discretize partial differential operators using strong forms, the LSFVP framework proposed in this invention is based on a rigorous integral form expansion of partial differential equations. In the two-dimensional computational space, this scheme conceptually models each discrete meshless particle as a square control volume occupying a defined geometric space. Assuming the initial particle spacing is the characteristic length, the particle control volume is a square space with sides equal to this characteristic length. Under this basic framework, the evolutionary control equations of physical quantities are integrally integrated over the entire domain within the control volume of each particle. For the volume second-order differential term in the control equations, which is extremely computationally expensive, this invention introduces the classical Gaussian divergence theorem, rigorously and equivalently transforming the volume integral, originally for the second-order partial derivative, into a flux integral along the closed geometric surface of the control volume. Through this dimensionality reduction transformation, the system conceptually transforms the second-order partial derivatives of the physical field, which originally required explicit solution, into the first-order derivatives calculated along the outward normal direction of the surface. This transformation completely avoids the theoretical dilemma of directly reconstructing high-order differential operators in a meshless framework, enabling the algorithm to easily obtain high-precision gradient information of physical quantities with minimal computational cost, provided that it approximates with lower-order polynomials.

[0032] This invention presents a high-precision numerical simulation method for fluid dynamics based on the least-squares finite-volume particle method. This method utilizes a Type-B least-squares moving particle semi-implicit operator to achieve high-precision local reconstruction of surface flux. At the four boundaries of the particle-shaped control volume, it is essential to accurately estimate the normal gradient of physical quantities and the face-centered function value of the nonlinear convection term. If the traditional Type-A LSMPS operator is used to estimate the gradient at the edge midpoints, the physical variables at each edge midpoint will inevitably be introduced as entirely new unknown global degrees of freedom into the sparse equations due to the known preset function values ​​in the Type-A scheme, leading to an expansion of the entire system's degrees of freedom. Therefore, this invention creatively introduces a Type-B LSMPS operator into the flux calculation of a finite-volume surface. Since the underlying Taylor series expansion model of the Type-B scheme uniformly assumes that the spatial function values ​​and spatial derivatives are unknown variables for weighted least-squares solution, the algorithm can accurately extract and reconstruct the unknown function values ​​and gradient information at any face-centered position using only the known physical state quantities of the particle at the volume center and the state quantities of its neighboring particles within its support domain, through a pre-calculated low-order matrix. More importantly, this face-centered flux reconstruction strategy perfectly maintains the degree of freedom scale of the original system. All unknowns to be solved are strictly limited to the center of the particle control volume, and no redundant variables need to be added at the mesh edge.

[0033] This invention's high-precision numerical simulation method for fluid dynamics based on the least squares finite volume particle method integrates Gaussian numerical integration to ensure absolute second-order spatial accuracy of the overall computational scheme. In the theoretical derivation of LSFVP, after the first-order gradient of the face-centered flux is accurately recovered by Type-B LSMPS, surface integration is still required on each surface of the control volume to summarize the net local physical flux value. This invention explicitly employs Gaussian numerical integration to numerically integrate the nonlinear continuous flux distribution passing through the particle surface. Theoretical analysis and rigorous numerical verification show that when Gaussian numerical integration is used to evaluate the surface flux integral reconstructed by the first-order gradient model, the algorithm can ensure perfect second-order spatial accuracy of the overall partial differential equation solution under global error norm measurement. This accuracy guarantee mechanism means that this invention does not need to invert the computationally intensive high-order large matrix as in traditional LSMPS methods; it only uses low-order fundamental matrices with minimal computation and extremely fast inversion to surpass the theoretical limits of strongly formal algorithms under the same computational resources.

[0034] This invention presents a high-precision numerical simulation method for fluids based on the least-squares finite-volume particle method, which integrates virtual particle boundary theory to improve computational efficiency. When dealing with complex industrial fluid geometric boundaries, the technical solution of this invention includes a geometric scanning algorithm at the free surface to accurately identify free surface modules. Through this module, the system calculates and marks free surface particles in real time and generates virtual particles with symmetrical physical properties along the fluid surface normal. These virtual particles are incorporated into the least-squares support domain for joint correction of position and physical quantities, thereby ensuring a consistent high-precision discretization format is maintained even in boundary truncation regions.

[0035] The high-precision numerical simulation method for fluids based on the least-squares finite-volume particle method in this invention provides the system with ultimate stability through a dual-constraint dynamic time-step control and particle displacement correction mechanism. For complex fluid evolution processes with strong convection and transient large deformation characteristics, this method introduces a unified dynamic time-step adaptive control system. At the beginning of each time integration cycle, the program automatically scans the entire field to obtain the maximum characteristic convection velocity and the physical field viscous diffusion coefficient, and strictly follows the Courant-Friedrich-Lyuvy condition and the von Neumann diffusion stability limit criterion. It automatically selects and outputs the optimal time step that balances convection cutoff and diffusion stability, thereby eliminating the accumulation of cutoff errors during the time integration stage. Furthermore, to address the risk of extreme disorder in the distribution of scattered particles caused by violent fluid collisions, compression, or centrifugal stretching, this invention incorporates an independent particle displacement correction technique after discretizing the momentum equation. This technique evaluates the local particle number density gradient and applies a small geometric correction displacement to repel and push away excessively close particles, completely eliminating the singularity of the least square moment matrix caused by extreme distortion in the neighborhood, thus ensuring the robustness and stability of long-term time-domain simulation processes.

[0036] The specific process of the high-precision numerical simulation method for fluid based on the least squares finite volume particle method in this invention is as follows: 1. Read and initialize the complex geometric mesh and fluid particles. In two-dimensional space, the discrete fluid particles are conceptually modeled as finite control volumes of squares with characteristic lengths.

[0037] 2. Free surfaces are determined for fluid particles using the angle scan method. The unit normal vector at the free surface is calculated, and virtual particles are generated outside the fluid along the normal vector. These virtual particles are used to maintain the integrity of the kernel function support domain at the free surface boundary and to apply zero-pressure boundary conditions.

[0038] 3. Based on fluid particles and virtual particles, a Type-B least-squares moving particle semi-implicit operator model is constructed. The Type-B scheme unifies the function values ​​and partial derivatives at the face center as unknowns for local low-order reconstruction. The expression for its local matrix is ​​as follows: in, For the central particle The moment matrix, For kernel function weights, These are local polynomial basis vectors. This is a scaling matrix.

[0039] 4. Transform the partial differential equations of fluid dynamics into integral forms based on the particle control volume. Applying the Gaussian divergence theorem, the volume integral for solving the second spatial derivative is reduced in dimension to a flux surface integral along the closed geometric boundary of the control volume: High-precision estimation of surface normal flux using the Type-B operator built in S3 This avoids the process of inverting the higher-order matrix required for directly discretizing the second derivative.

[0040] 5. Based on fluid continuity and particle number density distribution, a composite particle displacement correction vector is constructed to correct the phase space positions of fluid particles, thereby maintaining volume conservation and point cloud uniformity under strong convection conditions. (Particle displacement vector) Including repulsive displacement and volume correction displacement : in Given the current particle number density, For reference particle number density, This is a local compression error. To conservatively exclude displacement coefficients and prevent oscillations caused by correction overshoot, the default safety threshold of this system is set to [value missing]. , The volume conservation correction factor is set to [value] in this system. .

[0041] 6. Solve the discretized fluid dynamics equations to update the velocity and position of fluid particles. In moving boundary collision detection, a rebound recovery coefficient based on relative velocity is introduced to complete the calculation for the current time step.

[0042] The following example, using fluid dam-break evolution and impact simulation, illustrates the high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method in this invention: 1. For example Figure 4 As shown, a rectangular water column particle is initialized on one side of the rectangular computational domain, with an initial velocity of zero. The fluid is released to cause it to collapse under the action of gravity and impact the wall in front.

[0043] 2. Within the dynamic time step, virtual particles are generated by normal extrapolation to apply zero-pressure boundary conditions; the pressure is solved using the LSFVP hybrid operator, and particle displacement correction technology is used to avoid non-physical aggregation of droplets.

[0044] 3. For example Figure 5 As shown, the method of the present invention plots the dimensionless wavefront displacement time evolution and sidewall impact pressure time history curves based on the probe data tracked by the calculation program. The calculated curves are highly consistent with the black scatter points representing classical experimental measurements, which fully demonstrates the extremely high accuracy of the algorithm in capturing strong nonlinear impact rupture and fluid loads.

[0045] The following uses hydrostatic pressure testing as an example to illustrate the high-precision numerical simulation method for fluid based on the least squares finite volume particle method in this invention: 1. Initialize uniformly distributed fluid particles inside a closed, static water tank, where the fluid is only subject to gravity.

[0046] 2. The algorithm continuously calculates the numerical density of the fluid by performing long-term numerical integral evolution and uses a high-order modified moving particle semi-implicit operator to calculate the volume conservation term as a position correction.

[0047] 3. For example Figure 6 As shown, under long-term simulation, the pressure history curve at the bottom of the tank calculated by LSFVP remains stable, and the average calculated pressure is extremely close to the rigorous theoretical analytical solution. Furthermore, the normalized numerical density of the fluid remains stable around 1.0, without numerical dissipation or volume collapse.

[0048] The following example, using the sloshing problem of a moving boundary water tank, illustrates the high-precision numerical simulation method for fluid dynamics based on the least squares finite volume particle method according to an embodiment of the present invention: 1. Create a two-dimensional water tank containing fluid particles at a certain depth, and apply horizontal harmonic excitation displacement and velocity to the solid boundary of the water tank.

[0049] 2. When constructing the finite volume surface flux integral of LSFVP, the normal velocity of the dynamic wall is directly injected into the source term of the pressure Poisson equation on the right side; at the same time, fluid-structure interaction is achieved by combining the velocity bounce logic relative to the moving boundary.

[0050] 3. For example Figure 7 As shown, under continuous periodic excitation, the water body experiences violent shaking and climbing, such as... Figure 8 As shown, the free surface elevation history curve recorded by this invention accurately tracks high-fidelity experimental data, with both phase and peak values ​​showing high consistency. Even during prolonged and severe swaying deformation, the normalized density inside the fluid remains stable and fluctuates around 1.0, demonstrating the extremely strong robustness of this framework in handling complex dynamic boundary fluid-structure interaction problems.

[0051] This invention employs a partial differential equation (PDE) solution framework based on finite volume integral reduction. In this embodiment, an equivalent square or cubic control volume is established for each randomly distributed, meshless fluid particle. Second-order PDEs such as Laplace's equations are integrated globally over this control volume, and the volume integral of the second derivative is rigorously transformed into a flux surface integral along the closed geometric surface of the control volume using the Gaussian divergence theorem. This architecture completely avoids the inversion calculations of high-dimensional, massive matrices, enabling the algorithm to achieve globally rigorous second-order spatial convergence accuracy using only a low-order polynomial basis.

[0052] This invention introduces a Type-B least squares operator collaborative technique in flux reconstruction. When calculating the normal gradient flux of each surface of the control volume, a Type-B least squares operator is used. This operator treats the function value and partial derivative at the face center as unknowns for local weighted least squares solution. It can accurately interpolate the normal gradient at the face center position using only the known state variables of the central particle and its neighboring particles in the support domain. This mechanism enables high-precision flux calculation without introducing any additional unknown global degrees of freedom at the face center, maintaining the minimal scale of the sparse matrix.

[0053] This invention proposes a dynamic virtual particle boundary enhancement technique based on real geometric normals. When tracking a highly nonlinear free liquid surface, a geometric sector scanning method is used to determine free surface particles in real time and accurately calculate their unit outward normal vectors. Subsequently, virtual particles with corresponding physical properties are dynamically generated by extending this normal vector outwards by a distance equal to the distance between characteristic particles. These virtual particles are forcibly incorporated into the least-squares kernel function support domain of the fluid particles, jointly participating in the construction of the operator matrix. This solves the error amplification problem caused by kernel function truncation at the interface in traditional particle methods.

[0054] This invention employs a dual particle displacement correction and stabilization mechanism that couples volume conservation and physical repulsion. Addressing the objective physical phenomenon that the hybrid framework cannot guarantee strict flux conservation during particle topological dynamic evolution, a post-processing position fine-tuning mechanism is introduced. A composite shift vector is constructed: one is a physical repulsion displacement based on the particle number density gradient, used to push away excessively close particles; the other is a pull-back displacement based on a higher-order correction of the semi-implicit density error gradient of the moving particles. Furthermore, for particles located on free surfaces, strict normal displacement filtering is applied, allowing them to slide only tangentially. This mechanism can enforce volume conservation on a macroscopic scale, completely eliminating high-frequency oscillations caused by uneven particle distribution during violent collisions or prolonged shaking, thus endowing the solver with extreme stability.

[0055] Compared with existing finite mesh technology and various classical and high-order meshless particle computation methods, the complex geometric fluid simulation technology based on LSFVP provided in this invention, namely, a high-precision numerical simulation method for fluids based on the least squares finite volume particle method, has the following advantages: 1. This invention achieves a leapfrog improvement in computational efficiency. Through an innovative dimensionality reduction operation using integral forms of partial differential equations, this invention mathematically eliminates the dependence on directly reconstructing second-order or even higher-order spatial derivatives. The algorithm abandons the high-order (up to fifth- or sixth-order) matrix inversion problem that must be faced in traditional strong-form LSFVPs to maintain second-order accuracy, and fully reduces dimensionality using low-order tensor basis functions to meet flux interpolation requirements. By avoiding the high floating-point operations and memory addressing overhead caused by high-dimensional matrix operations, the algorithm's execution speed per unit time step is greatly improved. 2. This invention establishes global second-order spatial computational accuracy. By strictly performing flux computation on a square control volume, this invention successfully overcomes the problem of rapid degradation of traditional meshless computational accuracy on unstructured, irregular point clouds. Error testing and analysis of the Type-B operator on the perturbation node set according to this invention show that the operator itself achieves extremely high third-order accuracy for approximating function values ​​and clear second-order accuracy for estimating the first derivative. Furthermore, after coupling with the Gaussian surface integral rule, the LSFVP algorithm exhibits excellent error convergence characteristics for all fundamental physical partial differential equations. 3. The boundary treatment of this invention is more accurate. The Type-B operator makes the treatment of boundary flux and in-plane gradient completely consistent, avoiding the introduction of additional global degrees of freedom. Simultaneously, the dynamically generated virtual particles allow for high-precision realization of free surface pressure boundary conditions and solid wall boundaries.

[0056] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A high-precision numerical simulation method for fluid based on the least squares finite volume particle method, characterized in that, Includes the following steps: S1: Read the geometric model and fluid particles, and model each discrete fluid particle as a square finite control volume with characteristic length in two-dimensional space to obtain a set of fluid particles with square control volumes. S2: Based on the fluid particle set, the free surface particles are determined by the angle scanning method, the unit normal vector at the free surface is calculated, and virtual particles are generated outside the fluid along the normal direction to obtain a virtual particle set used to maintain the integrity of the boundary support domain. S3: Based on the fluid particle set and the virtual particle set, construct a Type-B least squares moving particle semi-implicit operator model to obtain a local low-order reconstruction operator for reconstructing the function value and derivative at the face center; S4: The partial differential equations of fluid dynamics are transformed into integral forms based on particle control volume, and the Gaussian divergence theorem is applied to reduce the dimension of the volume integral to the flux surface integral along the closed boundary of the control volume. The surface normal flux is estimated using the Type-B least squares moving particle semi-implicit operator model to obtain the discretized fluid dynamics algebraic equations. S5: Based on fluid continuity and particle number density distribution, construct a composite particle displacement correction vector and correct the phase space position of fluid particles to obtain the corrected uniformly distributed fluid particle positions. S6: Solve the discretized fluid dynamics algebraic equations and use the corrected uniformly distributed fluid particle positions to update the velocity and position of the fluid particles, thus completing the fluid evolution calculation for the current time step.

2. The high-precision numerical simulation method for fluid based on the least squares finite volume particle method according to claim 1, characterized in that, In step S4, the Gaussian divergence theorem is applied to transform the volume integral into a flux surface integral along the closed boundary of the control volume. The transformed form is as follows: The volume integral that originally dealt with the second-order partial derivative is transformed into the flux integral along the closed surface of the control volume, thus converting the solution of the second-order partial derivative into the calculation of the first-order derivative in the direction of the outward normal of the surface.

3. The high-precision numerical simulation method for fluid based on the least squares finite volume particle method according to claim 1, characterized in that, In step S3, the constructed Type-B least squares moving particle semi-implicit operator model is used to uniformly assume the spatial function values ​​and spatial derivatives of all orders as unknown variables for weighted least squares solution. Based solely on the known physical state quantities of the particle at the center of the control volume and the state quantities of the neighboring particles in its supporting domain, the unknown function values ​​and their gradient information at any face center position can be accurately extracted and reconstructed.

4. The high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method according to claim 1, characterized in that, In step S4, after estimating the surface normal flux using the Type-B least squares moving particle semi-implicit operator model, the Gaussian numerical integration method is used to numerically integrate the nonlinear continuous flux distribution passing through the particle surface to ensure that the overall calculation format achieves second-order spatial accuracy.

5. The high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method according to claim 1, characterized in that, In step S2, the generated virtual particles are forcibly incorporated into the least squares support domain of the fluid particles, and jointly participate in the moment matrix construction of the Type-B least squares moving particle semi-implicit operator model to correct the discrete format of the boundary truncation region.

6. The high-precision numerical simulation method for fluid based on the least squares finite volume particle method according to claim 1, characterized in that, In step S5, the constructed composite particle displacement correction vector includes a physical repulsion displacement based on the particle number density gradient and a volume correction displacement based on the density error gradient. The physical repulsion displacement is used to push away particles that get too close, and the volume correction displacement is used to compensate for macroscopic volume conservation.

7. The high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method according to claim 1, characterized in that, Before step S1 or during step S6, a dynamic time step control step is also included, specifically: At the beginning of each time integration cycle, the entire field is automatically scanned to obtain the maximum characteristic convection rate and the physical field viscous diffusion coefficient. Following the Courant-Friedrich-Lyuvi condition and the von Neumann diffusion stability limit criterion, the optimal time step is automatically selected and output.

8. The high-precision numerical simulation method for fluid based on the least squares finite volume particle method according to claim 1, characterized in that, In step S6, during the collision detection of the moving boundary, a rebound recovery coefficient based on relative velocity is introduced, and the normal velocity of the dynamic wall is directly injected into the source term on the right side of the pressure Poisson equation. Combined with the velocity rebound logic of the relative moving boundary, fluid-structure interaction is achieved.

9. The high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method according to claim 1, characterized in that, The method is used for fluid dam failure evolution and impact simulation, specifically as follows: Initialize rectangular water column particles on one side of the rectangular computational domain, set the initial velocity to zero, release the fluid so that it collapses under the action of gravity and impacts the wall in front; Within the dynamic time step, zero-pressure boundary conditions are applied by generating virtual particles, and non-physical aggregation of droplets is avoided by correcting the phase space position of fluid particles.

10. The high-precision numerical simulation method for fluid based on the least-squares finite-volume particle method according to claim 1, characterized in that, The method is applied to numerical simulations of hydrostatic pressure tests or dynamic boundary tank sloshing problems. In hydrostatic pressure testing, long-term numerical integration evolution is performed on uniformly distributed fluid particles in a closed static water tank to calculate the numerical density and volume conservation terms of the fluid. In the dynamic boundary water tank sloshing problem, horizontal harmonic excitation displacement and velocity are applied to the solid wall boundary of the two-dimensional water tank, and the normal velocity of the dynamic wall is injected into the right-hand source term of the pressure Poisson equation.

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