A convection-reaction-diffusion model construction method and system in the peridynamics framework

Abstract

The present invention belongs to the field of peridynamic technology, and discloses a method and system for constructing a convection-reaction-diffusion (ARD) model under the peridynamic framework. The present invention first proposes a PD-ARD equation (Peridynamic ARD Equation) suitable for non-uniform flow conditions, which effectively solves the problem of inaccurate description of ARD problems in complex flow fields in the prior art. In order to achieve more accurate coupling calculations, the present invention adopts the PD control equation to solve the velocity field of the fluid under the peridynamic (PD) framework, and forms an overall model by coupling the PD equation of viscous flow with the PD-ARD equation under non-uniform flow. The present invention can accurately simulate the convection-diffusion-reaction problem under viscous flow conditions, is suitable for modeling complex environments with multi-scale and multi-physical field coupling, and is widely used in industrial design, environmental monitoring, scientific research and other fields.

Description

Technical Field

[0001] The present invention belongs to the technical field of peridynamics, and in particular relates to a method and system for constructing a convection-reaction-diffusion model within a peridynamics framework. Background Art

[0002] Convection-reaction-diffusion (ARD) processes are common in various systems, such as the transport of drugs in the human body, the diffusion of pollutants in groundwater, and the spread of infectious diseases in complex networks. Although analytical solutions exist for some special cases of the classical ARD equation, most problems require numerical approximations. However, classical models of the ARD problem have two major limitations. First, classical models of steady-state ARD problems have difficulty accurately tracking interfacial changes in complex situations. Although phase-field models can address this issue, they typically rely on predefined assumptions about the interfacial process. Second, the classical ARD equation does not take into account more general long-range interactions (such as anomalous diffusion observed in heterogeneous environments and long-range inhibition in biological pattern formation).

[0003] Peridynamics (PD) is a non-local theory that introduces the concept of "horizon" and replaces spatial derivatives with integral operators. It does not require the smoothness of the unknown function, so it is very suitable for dealing with spontaneous changes at continuous points. In addition, PD can also solve the singularity problems caused by point contact and separation of materials in classical theory. Therefore, the PD method effectively makes up for the two shortcomings of the classical ARD model. First, the PD framework can naturally handle the interface change process and can accurately predict it without relying on prior assumptions. Second, as a non-local method, PD can naturally deal with behaviors with long-range effects. However, the current models for simulating ARD problems in the PD framework are limited to conditions of uniform flow velocity, but the physical scenario is generally under non-uniform flow fields.

[0004] 1. The problem of phase change is difficult to deal with in existing technologies

[0005] When dealing with phase change problems, traditional convection-reaction-diffusion models either find it difficult to capture the actual phase change process or require the pre-set laws of the phase change process.

[0006] 2. Problems of low calculation accuracy and insufficient numerical stability of existing models

[0007] Existing numerical solution methods are prone to numerical oscillations when dealing with viscous flow, reaction-diffusion and material transfer, especially in high-gradient regions, resulting in unstable calculation results and large errors.

[0008] 3. Difficulty in coupling reaction dynamics and fluid fields in existing technologies

[0009] Under complex fluid motion, material diffusion and chemical reaction rates are difficult to dynamically couple with the fluid field, and traditional models cannot effectively describe the material transfer and transformation process.

[0010] 4. The problem of poor intuitiveness and real-time output of existing technology results

[0011] The calculation results of existing models are mainly presented in the form of numerical data, lacking intuitive data visualization methods and insufficient real-time monitoring capabilities, making it difficult to meet the needs of actual engineering applications. Summary of the Invention

[0012] To address the challenges of existing technologies, this paper provides a method for modeling convection-reaction-diffusion within a peridynamic framework. This method effectively addresses existing technical issues, such as the difficulty in handling phase change, low computational accuracy, difficulty coupling reaction-diffusion with the fluid field, and insufficiently intuitive results. Through a system architecture combining software and hardware, this method significantly improves the computational accuracy, numerical stability, data visualization, and real-time application capabilities of convection-reaction-diffusion models.

[0013] The present invention is implemented as follows: a method for constructing a convection-reaction-diffusion model within a peridynamics framework includes:

[0014] Step 1, initialization;

[0015] Initialize velocity, pressure, density, and concentration fields;

[0016] Step 2, viscous flow solution;

[0017] Step 3, solution of solid dissolution / bimolecular reaction.

[0018] Furthermore, the solution of the solid dissolution / bimolecular reaction includes:

[0019] The advection term in the PD-ARD equation is replaced by the PD continuity equation in the middle Euler form, resulting in the following equation:

[0020]

[0021] Where C p (x, t) represents the concentration of substance p at point x at time t; represents the near field of x; represents other nodes in the near field of x;

[0022] In the PD model, each material point x∈Ω interacts with other points in its near field. In two dimensions, the near field is usually a circle with a radius of δ; yes The area covered; the velocity of the material point x at time t is represented by v(x,t), and It is x to A unit vector in the direction of

[0023] R pq (x, t) represents the reaction rate between substances p and q; substance p and M p The substances react, which can be expressed as q∈[1,M p ];

[0024] Depending on the role of the substance in a particular reaction, the corresponding reaction rate can be positive or negative; the α constant establishes the connection between macroscopic and microscopic speeds, and its value is equal to the dimension;

[0025] Set the speed The projection in the direction is defined as For the microscopic diffusion function Generally speaking, the relationship between the macroscopic diffusion coefficient and the microscopic diffusion coefficient is definite;

[0026] The two common forms of this function are "constant form" and "trigonometric form", which are: and

[0027] According to the matching between the PD model and the classical model when the concentration distribution of substances in the solution is linear, it can be concluded that in the two-dimensional case, d0=4D / πδ 2 and d1=12D / πδ 2 , where D is the diffusion coefficient; in formula (1), set

[0028] For the advection part in formula (1), three schemes are considered: the center scheme, the upwind scheme, and the downwind scheme. The center scheme has higher accuracy than the upwind scheme, but its numerical stability is not as good as the upwind scheme. The characteristics of the downwind scheme are similar to those of the upwind scheme. The advection term in formula (2) adopts a hybrid scheme, which combines the center scheme and the downwind scheme as follows:

[0029]

[0030] in β is the coefficient connecting the macroscopic velocity and the microscopic velocity in the downwind scheme, which is considered a constant. When the flow field is uniform, the PD-ARD equation is compared to obtain β = 2α. Here, w = 0.8.

[0031] The PD-ARD equation for inhomogeneous flow has been established;

[0032] Furthermore, the viscous flow solving module:

[0033] The PD control equation for viscous flow will experience numerical oscillation. This paper adds a density diffusion term to the mass conservation equation to reduce this phenomenon:

[0034]

[0035] Where, The abbreviation is It's the material point The density at time t, It's a material point is the pressure at time t, b is the volume force at point x of the material at time t; μ is the fluid viscosity; α p is a constant whose value is equal to the dimension; α μ The value of is determined by a simple flow problem to ensure the linear consistency of the formula. For the 2D case, α μ =16 / 3; η is the correction coefficient, usually set to 0.1, is a density-dependent microdiffusion function, and the diffusion coefficient is D ρ =c0δ; Similar to what was mentioned above, There are also two types: "Constant Type" and "Triangle Type". To ensure consistency with the previous micro coefficients, set is a "constant type"

[0036] The PD governing equations for viscous flow include a pressure field, but the model lacks an explicit equation. For incompressible Newtonian fluids, directly solving the original incompressible equations is somewhat challenging. An artificial equation of state is used to express the pressure as an algebraic function of density:

[0037]

[0038] Where ρ0 is the initial density, ρ * is the predicted density for the current step, γ is the material constant, and the material constant for water is 7; c0 is the artificial sound velocity; the artificial sound velocity affects the choice of time step; the higher the artificial sound velocity, the smaller the time step, but the more stable the value; generally, the artificial sound velocity is set to 10 times the maximum velocity of the flow field, ensuring that the Mach number M = v / c0 is less than 0.1, so that the density change is less than 1% of the initial density;

[0039] Therefore, we first obtain the velocity field through equations (3)-(5), and then bring the velocity field into equation (1) to solve the ARD process.

[0040] Another object of the present invention is to provide a convection-reaction-diffusion model building system within a peridynamics framework, comprising:

[0041] Initialization module, used to initialize velocity, pressure, density and concentration fields;

[0042] Viscous flow solving module, used to solve viscous flow;

[0043] Solid dissolution / bimolecular reaction solving module is used to solve solid dissolution / bimolecular reactions.

[0044] Another object of the present invention is to provide a computer device, comprising a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor performs the steps of the method for constructing a convection-reaction-diffusion model under the peridynamics framework.

[0045] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method for constructing a convection-reaction-diffusion model under the peridynamics framework.

[0046] Another object of the present invention is to provide an information data processing terminal, which is used to implement the convection reaction diffusion model construction system under the peridynamics framework.

[0047] In combination with the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solutions to be protected by the present invention are as follows:

[0048] First, the present invention proposes the PD-ARD equation for nonuniform flow, which addresses the existing technical issues. To facilitate coupling, the velocity field is derived within the PD framework. Therefore, the present invention couples the PD governing equations for viscous flow with the PD-ARD equations for nonuniform flow to simulate the ARD problem in viscous flow.

[0049] This paper proposes a method for constructing a convection-reaction-diffusion model within a peridynamic framework. By introducing a dynamic initialization and refined solution process, it addresses the inadequate description of complex flow and concentration field evolution in existing peridynamic models. By coupling the solution of viscous flow with the dissolution / reaction process, the interaction between fluid motion and chemical reactions is accurately captured. Furthermore, the present invention employs an iterative algorithm to achieve stable calculations for non-uniform initial conditions and complex boundary conditions, significantly improving the stability and accuracy of numerical simulations.

[0050] The present invention has significant technological advancements in industrial applications: (1) It achieves high-precision dynamic simulation of fluid flow and chemical reaction processes, which is suitable for material transfer and reaction research in the fields of power transmission and transformation engineering, environmental monitoring, chemical engineering design, etc.; (2) By coupling convection, diffusion and reaction processes, it improves the ability to describe complex multi-scale problems and provides a new tool for optimizing industrial production processes.

[0051] This invention overcomes the technical bottlenecks of traditional models in dealing with complex convection-reaction-diffusion problems, providing a more efficient and reliable analytical tool for industry and scientific research. Its application not only promotes technological advancements in the fields of chemical engineering, energy, and the environment, but also lays a theoretical foundation for intelligent industrial design and efficient resource utilization. Furthermore, the modular design approach of this invention facilitates integration with existing industrial software or systems, offering broad application prospects and economic benefits.

[0052] Second, the technical solution of the present invention fills the technical gap in the industry at home and abroad:

[0053] While many models exist for describing the ARD process, most are derived within the framework of classical mechanics, with few studies examining the process within the framework of peridynamics. Furthermore, the few models that describe the ARD process within the framework of peridynamics are limited to uniform flow conditions. The present invention proposes a model within the framework of peridynamics that can simulate the ARD process under non-uniform flow conditions.

[0054] The technical solution of the present invention solves a technical problem that people have long been eager to solve but have never been successful: although traditional partial differential equation models have been used to numerically solve the flow of one-dimensional bimolecular reaction transport flow in isotropic homogeneous media and compared with experimental results. However, these types of models, when extended to higher dimensions and applied to complex fields, have difficulty tracking the evolving interface. Although the phase field model of the reaction-diffusion problem has greater flexibility in tracking the changing interface, it still has to adopt preset functional changes at the interface, which usually does not correspond to the actual physics / chemistry. In contrast, the damage or phase change model based on peridynamics does not have this limitation and has been proven to accurately predict the evolution of the moving interface. However, the model for simulating the ARD process in the peridynamic framework is limited to uniform flow conditions. Here, the present invention introduces a new PD model for transient ARD problems that is universal under non-uniform flow conditions to solve the above problems.

[0055] Third, current models of convection-diffusion-reaction systems mostly rely on a classical mechanics framework, but lack sufficient accuracy when addressing phase transitions in multidimensional scenarios involving strong coupling of mass transfer, fluid dynamics, and chemical reactions. This limitation makes it difficult to effectively predict microscopic dynamic behavior and macroscopic effects in industrial applications involving complex multi-physics interactions (such as solute diffusion, solid dissolution, and chemical reaction kinetics). Furthermore, traditional methods struggle to capture some nonlocal behaviors, such as heterogeneous diffusion in heterogeneous environments and long-range inhibition during the differentiation and growth of biological cells. Finally, existing ARD models within the peridynamic framework struggle to describe ARD processes under nonuniform flow rates. Based on the peridynamic framework, this paper provides a model construction method that is more suitable for phase transition processes involving nonlocal interactions and multi-physics coupling. Within this framework, by incorporating kinetic descriptions from peridynamics, the model can more accurately simulate the flow characteristics, solute diffusion behavior, and chemical reaction rates of fluids undergoing phase transitions during ARD processes. Furthermore, by combining the convection, diffusion, and reaction equations, nonlocal behaviors that are difficult to resolve with traditional methods are incorporated into the solution, thus addressing the shortcomings of existing models. Finally, this model is constructed within the framework of peridynamics and can describe the ARD process under non-uniform flow rates.

[0056] This invention demonstrates significant technological advancements in a wide range of industrial applications. For example, in the chemical industry, it can more accurately predict the rate of solid dissolution and the distribution of reactant concentrations within a reactor, supporting improved reaction efficiency and reduced raw material costs. In environmental engineering, it can be used to simulate the diffusion and degradation of pollutants and optimize water purification processes. In the pharmaceutical industry, it can accurately simulate drug release and diffusion processes, providing technical support for drug design and formulation development. By accurately modeling these complex processes, this invention provides an important basis for improving industrial production efficiency and optimizing process design.

[0057] By efficiently utilizing the peridynamics framework, this paper proposes a comprehensive methodology, from initialization to fluid dynamics solutions and then to solid dissolution / chemical reaction simulation. Compared with traditional models, this method significantly improves the accuracy and stability of solutions in complex situations. This systematic approach provides a comprehensive solution for the study and application of complex convection-diffusion-reaction systems, filling a technological gap in coupled microscale and macroscale computations and promoting innovation and development in related industries. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is a flow chart of a method for constructing a convection reaction diffusion model within a peridynamics framework provided by an embodiment of the present invention.

[0059] Figure 2This is a structural block diagram of a convection-reaction-diffusion model building system within a peridynamics framework provided by an embodiment of the present invention.

[0060] Figure 3 This is a calculation flow chart of the PD-ARD model for viscous flow in bimolecular reactions and solid dissolution provided by an embodiment of the present invention.

[0061] Figure 4 It is an interaction diagram between a material point provided by an embodiment of the present invention and other material points in its near field.

[0062] Figure 5 This is a diagram of the central format, reverse-wind format, and forward-wind format provided by an embodiment of the present invention.

[0063] Figure 6 1 is a diagram of initial conditions and boundary conditions for a numerical verification example provided by an embodiment of the present invention.

[0064] Figure 7 It is a cloud diagram of the x-direction velocity of the PD solution and the COMSOL solution in a homogeneous medium example provided by an embodiment of the present invention.

[0065] Figure 8 It is a velocity comparison diagram of the PD solution and the COMSOL solution for the cylinder flow example provided by an embodiment of the present invention.

[0066] Figure 9 This is a comparison chart of the product AB concentration in a homogeneous medium using the PD-ARD model and COMSOL provided in an embodiment of the present invention.

[0067] Figure 10 3 is a comparison chart of the concentration of each substance along the horizontal line obtained by PD COMSOL at 200 s (a) and 500 s (b) in a calculation example of a homogeneous medium provided by an embodiment of the present invention.

[0068] Figure 11 3 is a comparison chart of AB concentrations obtained in a flow around a cylinder using the PD-ARD model and COMSOL provided in an embodiment of the present invention.

[0069] Figure 12 ARD example of flow around a cylinder provided by an embodiment of the present invention, a comparison of the concentrations of substance AB obtained along the horizontal line by PD and COMSOL.

[0070] Figure 13 1 and 2. (a), (b) and (c) provided in an embodiment of the present invention respectively represent three groups of figures showing the dissolution of solid A at .

[0071] Figure 143 is a graph of the concentration distribution of the dissolved product AB at 75 s for different Reynolds numbers provided by an embodiment of the present invention (wherein it represents the concentration of the node AB normalized by the saturation concentration of AB).

[0072] Figure 15 Graphs showing the dissolution rates on the left and right sides of the cylinder under three conditions are provided in accordance with an embodiment of the present invention. DETAILED DESCRIPTION

[0073] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0074] The convection-reaction-diffusion modeling system within the peridynamics framework first acquires initial state data for the fluid, concentration, pressure, and density fields through a data acquisition module. Specifically, the system can utilize satellite remote sensing equipment, laboratory fluid monitoring equipment, or high-precision sensors to collect real-time data, covering information such as temporal evolution and spatial distribution. The collected data is then imported into the computing device via a transmission interface, providing accurate and complete initial conditions for subsequent model calculations. Furthermore, users can input boundary conditions and initial state parameters through a graphical interface, ensuring flexibility and accuracy in model initialization.

[0075] In the data processing module, the system first eliminates invalid data and classifies and processes the data according to its characteristics, distinguishing different types of fluid field or concentration field information. Subsequently, the system solves the viscous flow, material diffusion, and reaction processes based on the principle of peridynamics (PD). The PD model calculates the interaction between material points and other points in the near field domain, and accurately describes the motion characteristics of the fluid, the diffusion behavior of the solute, and the reaction process by solving the momentum conservation equation, mass conservation equation, and reaction-diffusion equation of viscous flow. In this process, numerical calculation algorithms are used to improve calculation accuracy and ensure numerical stability.

[0076] The system performs core numerical calculation tasks through the computing unit, which involves the collaborative work of hardware processors and memory. The computing unit couples the solution of the viscous flow field with the material transfer and reaction processes, and dynamically updates the fluid field and concentration field. The system gradually optimizes the reconstructed data based on an iterative algorithm and evaluates the reconstruction error (such as the normalized root mean square error NRMSE) at each time step. When the error meets the preset accuracy requirements, the system stops iterating and outputs the final calculation results. This process ensures the accuracy and efficiency of the calculation, and can provide stable solution results even in complex fluid scenarios.

[0077] Finally, the system displays and stores the calculation results through the result output module. This module includes a data visualization unit that presents the evolution of the fluid and concentration fields as dynamic two-dimensional images, allowing users to intuitively understand the entire process of fluid motion and reaction diffusion. Furthermore, the system supports the export of calculation data reports and key parameters for further analysis and application. If users require real-time monitoring, the system can display the model's operating status and updated results in real time on a display device, meeting the needs of engineering decision-making and scientific research.

[0078] Through four core modules, namely data acquisition, data processing, iterative calculation and result visualization output, this system realizes the precise construction and efficient solution of the convection-reaction-diffusion model under the framework of peridynamics. It has the advantages of comprehensive data acquisition, high calculation accuracy and intuitive dynamic display.

[0079] like Figure 1 、 Figure 3 As shown, an embodiment of the present invention provides a method for constructing a convection reaction diffusion model in a peridynamics framework, comprising the following steps:

[0080] S101, initialization;

[0081] Initialize velocity, pressure, density, and concentration fields;

[0082] S102, viscous flow solution;

[0083] S103, Solution of solid dissolution / bimolecular reactions.

[0084] Within the framework of peridynamics, the initialization phase aims to establish the computational foundation for the model. By defining the initial conditions for the velocity, pressure, density, and concentration fields, the initial state of the simulated object is accurately described. These field variables can be obtained through experimental measurements, known boundary conditions, or theoretical calculations. Specifically, the velocity field is used to describe the flow characteristics of the fluid, the pressure field is used to describe the pressure distribution in the fluid, the density field reflects the spatial distribution of substances in the fluid, and the concentration field describes the distribution of solutes or reactants. The accuracy of initialization directly affects the stability and accuracy of subsequent calculations.

[0085] At this stage, the model uses the basic equations of peridynamics (such as the momentum conservation equation and the mass conservation equation) to solve the flow characteristics of viscous fluids. By considering the effects of viscosity, fluid inertia, and external forces (such as gravity or shear force), the dynamic behavior of small regions in the fluid is simulated. During the solution process, Gaussian orthogonality is used to approximate the integral term. For the time derivative, the forward Euler method is selected, and an iterative algorithm is used to obtain a steady-state or transient solution. The solution of viscous flow provides the basis for the transport of solutes or reactants, while also affecting the spatial distribution and speed of solid dissolution or chemical reactions.

[0086] Solid dissolution or bimolecular reactions are solved using the convection-reaction-diffusion equations within a peridynamic framework, describing the transport and transformation of species. In this process, the convection term reflects the movement of solutes or reactants with the fluid, the diffusion term describes the diffusion of species due to concentration gradients, and the reaction term describes the rate and mechanism of the chemical reaction. By coupling convection, diffusion, and reaction, the model simulates the dynamics of solid dissolution rates or chemical reactions and accurately predicts the evolution of the concentration field.

[0087] At each time step, the viscous flow solution is coupled with the dissolution / reaction process to dynamically update the fluid and concentration fields. Through an iterative algorithm, the model can track changes in the fluid-solid interface, the consumption of reactants, and the distribution of products in real time. Ultimately, the model outputs complete flow and concentration field evolution data, revealing the complex dynamic behavior of the convection-diffusion-reaction system. This process can simulate practical engineering problems involving multi-scale and multi-physics coupling, providing important reference for industrial design, environmental monitoring, and scientific research.

[0088] Solution of solid dissolution / bimolecular reaction provided by the embodiment of the present invention:

[0089] The advection term in the PD-ARD equation is replaced by the PD continuity equation in the middle Euler form, resulting in the following equation:

[0090]

[0091] Where C p (x, t) represents the concentration of substance p at point x at time t; represents the near field of x;

[0092] In the PD model, each material point x∈Ω interacts with other points in its near field. In two dimensions, the near field is usually a circle with a radius of δ; yes The area covered; the velocity of the material point x at time t is represented by v(x,t), and It is x to A unit vector in the direction of

[0093] R pq (x, t) represents the reaction rate between substances p and q; substance p and M p The substances react, which can be expressed as q∈[1,M p ];

[0094] Depending on the role of the substance in a particular reaction, the corresponding reaction rate can be positive or negative; the α constant establishes the connection between macroscopic and microscopic speeds, and its value is equal to the dimension;

[0095] Set the speed The projection in the direction is defined as For the microscopic diffusion function Generally speaking, the relationship between the macroscopic diffusion coefficient and the microscopic diffusion coefficient is definite;

[0096] The two common forms of this function are "constant form" and "trigonometric form", which are: and

[0097] According to the matching between the PD model and the classical model when the concentration distribution of substances in the solution is linear, it can be concluded that in the two-dimensional case, d0=4D / πδ 2 and d1=12D / πδ 2 , where D is the diffusion coefficient; in formula (1), set

[0098] For the advection part in formula (1), three schemes are considered: the center scheme, the upwind scheme, and the downwind scheme. The center scheme has higher accuracy than the upwind scheme, but its numerical stability is not as good as the upwind scheme. The characteristics of the downwind scheme are similar to those of the upwind scheme. The advection term in formula (2) adopts a hybrid scheme, which combines the center scheme and the downwind scheme as follows:

[0099]

[0100] in β is the coefficient connecting the macroscopic velocity and the microscopic velocity in the downwind scheme, which is considered a constant. When the flow field is uniform, the PD-ARD equation is compared to obtain β = 2α. Here, w = 0.8.

[0101] The PD-ARD equation for inhomogeneous flow has been established;

[0102] The viscous flow solution provided by the embodiment of the present invention:

[0103] The PD control equation for viscous flow will experience numerical oscillation. This paper adds a density diffusion term to the mass conservation equation to reduce this phenomenon:

[0104]

[0105] Where, The abbreviation is It's a material point The density at time t, It's a material point is the pressure at time t, b is the volume force at point x of the material at time t; μ is the fluid viscosity; α p is a constant whose value is equal to the dimension; α μThe value of is determined by a simple flow problem to ensure the linear consistency of the formula. For the 2D case, α μ =16 / 3; η is the correction coefficient, usually set to 0.1, is a density-dependent microdiffusion function, and the diffusion coefficient is D ρ =c0δ; Similar to what was mentioned above, There are also two types: "Constant Type" and "Triangle Type". To ensure consistency with the previous micro coefficients, set is a "constant type"

[0106] The PD governing equations for viscous flow include a pressure field, but the model lacks an explicit equation. For incompressible Newtonian fluids, directly solving the original incompressible equations is somewhat challenging. An artificial equation of state is used to express the pressure as an algebraic function of density:

[0107]

[0108] Where ρ0 is the initial density, ρ * is the predicted density for the current step, γ is the material constant, and the material constant for water is 7; c0 is the artificial sound velocity; the artificial sound velocity affects the choice of time step; the higher the artificial sound velocity, the smaller the time step, but the more stable the value; generally, the artificial sound velocity is set to 10 times the maximum velocity of the flow field, ensuring that the Mach number M = v / c0 is less than 0.1, so that the density change is less than 1% of the initial density;

[0109] Therefore, we first obtain the velocity field through equations (3)-(5), and then bring the velocity field into equation (1) to solve the ARD process.

[0110] like Figure 2 As shown, an embodiment of the present invention provides a convection reaction diffusion model construction system in a peridynamics framework, including:

[0111] Initialization module, used to initialize velocity, pressure, density and concentration fields;

[0112] Viscous flow solving module, used to solve viscous flow;

[0113] Solid dissolution / bimolecular reaction solving module is used to solve solid dissolution / bimolecular reactions.

[0114] Another object of the present invention is to provide a computer device, comprising a memory and a processor, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor performs the steps of the method for constructing a convection-reaction-diffusion model under the peridynamics framework.

[0115] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method for constructing a convection-reaction-diffusion model under the peridynamics framework.

[0116] Another object of the present invention is to provide an information data processing terminal, which is used to implement the convection reaction diffusion model construction system under the peridynamics framework.

[0117] The present invention is specifically implemented:

[0118] The advection term in the PD-ARD equation is replaced by the PD continuity equation in the middle Euler form, and the following equation can be obtained:

[0119]

[0120] like Figure 4 , where C p (x, t) represents the concentration of substance p at point x at time t; represents the near field of x; in the PD model, each material point x∈Ω interacts with other points in its near field. In two dimensions, the near field is usually a circle with a radius of δ; yes The area covered (area in two dimensions); the velocity of a material point x at time t is represented by v(x,t), and It is x to The unit vector in the direction of R pq (x, t) represents the reaction rate between substances p and q; substance p and M p The substances react, which can be expressed as q∈[1,M p ]; Depending on the role of a substance in a particular reaction (e.g., as a product or reactant), the corresponding reaction rate can be positive or negative; the α constant establishes the connection between macroscopic and microscopic speeds, and its value is equal to the dimension (e.g., α = 2 in two dimensions); the speed is expressed in The projection in the direction is defined as For the microscopic diffusion function Generally speaking, the relationship between the macroscopic diffusion coefficient and the microscopic diffusion coefficient is definite; the two common forms of this function are "constant type" and "triangular type", respectively: and According to the matching between the PD model and the classical model when the concentration distribution of substances in the solution is linear, it can be concluded that in the two-dimensional case, d0=4D / πδ 2 and d1=12D / πδ 2 , where D is the diffusion coefficient; in formula (1), set

[0121] For the advection part in formula (1), three schemes are considered (see Figure 5 ): center scheme, upwind scheme and downwind scheme; the center scheme has higher accuracy than the upwind scheme, but its numerical stability is not as good as the upwind scheme; the characteristics of the downwind scheme are similar to those of the upwind scheme; the advection term in formula (2) adopts a hybrid scheme, combining the center scheme and the downwind scheme as follows:

[0122]

[0123] in β is the coefficient connecting the macroscopic velocity and the microscopic velocity in the downwind scheme. For simplicity, it is considered a constant. When the flow field is uniform, β = 2α is obtained by comparison with the PD-ARD equation. Here, w = 0.8 is taken.

[0124] So far, the PD-ARD equation under non-uniform flow has been established;

[0125] Since numerical oscillations may occur when using the proposed PD governing equation for viscous flow, the present invention adds a density diffusion term to the mass conservation equation to mitigate this phenomenon:

[0126]

[0127] Where, The abbreviation is It's a material point The density at time t, It's a material point is the pressure at time t, b is the volume force at point x of the material at time t; μ is the fluid viscosity; α p is a constant whose value is equal to the dimension (because this article discusses two-dimensional cases, so α p =2); α μ The value of is determined by a simple flow problem to ensure the linear consistency of the formula. For the 2D case, α μ =16 / 3

[0128] ; η is the correction factor, usually set to 0.1, is a density-dependent microdiffusion function, and the diffusion coefficient is D ρ =c0δ; Similar to what was mentioned above, There are also two types: "Constant Type" and "Triangle Type". To ensure consistency with the previous micro coefficients, set is a "constant type"

[0129] The PD equation for viscous fluids includes a pressure field, but the model lacks an explicit equation. For incompressible Newtonian fluids, it is difficult to directly solve the original incompressible equation. Therefore, an artificial state equation is used here to express the pressure as an algebraic function of density:

[0130]

[0131] Where ρ0 is the initial density, ρ * is the predicted density for the current step, γ is the material constant, and the material constant for water is 7; c0 is the artificial sound velocity; the artificial sound velocity affects the choice of time step; the higher the artificial sound velocity, the smaller the time step, but the more stable the value; generally, the artificial sound velocity is set to 10 times the maximum velocity of the flow field, ensuring that the Mach number M = v / c0 is less than 0.1, so that the density change is less than 1% of the initial density;

[0132] Therefore, we first obtain the velocity field through equations (3)-(5), and then bring the velocity field into equation (1) to solve the ARD process;

[0133] Assume that the reaction between the purifier and the pollutant is an irreversible bimolecular reaction A+B→AB. Assume that all substances are water-soluble. Assume that the diffusion coefficient of each substance is equal to D = 0.0017 cm 2 / s. Therefore, formula (1) can be rewritten accordingly:

[0134]

[0135] Where C A , C B and C AB Represents the concentration of substances A, B and AB respectively. AB is the reaction rate of the bimolecular reaction, here set as R AB =4.1L / (mol·s).

[0136] Consider the two-dimensional ARD problem, such as Figure 6 Specifically, Figure 6 (a) describes the purification example without obstacles (ARD problem in homogeneous medium). Figure 6 (b) illustrates a purification example with an obstacle (ARD problem for flow around a cylinder). These examples are used to verify the proposed coupled PD-ARD model for viscous flow. The density of the fluid is ρ = 1000 kg / m 3 , viscosity is μ=10 -3 kg·m -1 This value is used in all the examples. In addition, the artificial sound velocity is set to c0 = 0.01 m / s. The inlet velocity is defined as v0 = 1 × 10 -4m / s, initial concentration C0 = 0.02 mol / L, and the outlet maintains zero pressure.

[0137] Figure 7 and Figure 8 The paper presents the velocity field convergence results obtained using the modified PD governing equations for viscous flow and COMSOL simulations. The results show that the velocity field predicted by the PD model is in good agreement with the velocity field observed by COMSOL.

[0138] Figure 9 and Figure 11 The concentration distribution of product AB at different times is shown. It can be seen that the results of the PD model are very consistent with the results of COMSOL. For further analysis, we compared Figure 6 (a) The concentrations of the three substances A, B, and AB in PD and COSMOL solutions at the horizontal line y = 0 (see Figure 10 ). Similarly, in Figure 6 In the example shown in (b), the concentrations of three substances A, B, and AB in the PD solution and the COSMOL solution at the horizontal line y = 0.015 m are compared (see Figure 12 ). This shows that the results of the PD-ARD coupling model under viscous flow are highly consistent with the COMSOL output results.

[0139] consider Figure 6 In the example in (b), the cylinder representing substance A is assumed to be soluble, and the substance dissolved in water is AB. This dissolution process is assumed to be an irreversible chemical reaction: A → AB. The PD-ARD equation governing this process is as follows:

[0140]

[0141] Among them, R A represents the dissolution rate of substance A in the solution. Assume that the dissolution rate is constant at R A =4.2mol / (L·s). The solid concentration of substance A is C A,soild =100mol / L. Assume that the saturation concentration of AB is C sat = 20 mol / L. During solid dissolution, dissolution stops once the concentration of AB reaches saturation. Dissolution resumes when the concentration of AB drops below saturation.

[0142] The diffusion rate of substance AB in the solution is set to D = 1×10 -6 m 2 / s. This example uses Figure 6 (b) The same configuration, the only modification is the solubility of the cylinder and different inlet velocities. Three flow cases are considered, with inlet velocities v0 = 1 × 10 -4m / s, 5×10 -4 m / s, 25×10 -4 m / s, and the corresponding Reynolds numbers are Re = 2, 10, and 50. The relevant artificial sound speeds in these cases are c0 = 0.01 m / s, 0.05 m / s, and 0.25 m / s.

[0143] Figure 13 The dissolution of solid substance A at Re = 2, 10, and 50 is shown. As shown in the figure, increasing the Reynolds number not only accelerates the dissolution rate of the cylinder, but also causes asymmetry in the remaining part of the solid. For example, a higher Reynolds number will enhance the fluid scouring effect on the left side and the upper and lower sides of the cylinder, resulting in faster dissolution in these areas. This left-right asymmetry can be quantified by calculating the amount of substance A on both sides of the cylinder (see Figure 15 ), as shown below: represents half of the total amount of substance A of all nodes in the domain at the initial moment, and n represents the total number of nodes. In addition, and Respectively represent the sum of the amount of substance A at all nodes on the left and right sides of the cylinder at time t.

[0144] Further analysis, Figure 14 and Figure 15 The results show that during the initial stages of dissolution, the dissolution rates on the left and right sides of the cylinder are nearly identical. This is because diffusion primarily influences the dissolution rate during this phase, and the diffusion rate remains consistent under all three conditions. However, as dissolution progresses, substance AB flows from the left to the right. Therefore, at a given Reynolds number, the dissolution rate on the left side of the cylinder always exceeds that on the right. This effect becomes more pronounced at higher Reynolds numbers, leading to a more severe asymmetry.

[0145] Example 1: Optimization of high-efficiency reactors in the chemical industry

[0146] In the chemical industry, dissolution, diffusion, and reaction processes play a central role in reactor design and optimization. For example, optimizing the dissolution and mass transfer of industrial catalysts requires precise control of reaction conditions to improve production efficiency. This convection-reaction-diffusion model, constructed within a peridynamic framework, can accurately simulate the coupled relationship between catalyst particle dissolution, solute diffusion, and reaction rate.

[0147] Specific applications:

[0148] In a multiphase reactor, this model can be used to simulate the concentration distribution, diffusion path, and reaction rate of dissolved catalyst particles.

[0149] Optimize the fluid dynamic conditions (such as pressure, temperature and flow rate) in the reactor to maximize the reaction efficiency and feedstock conversion rate.

[0150] Example 2: Diffusion and degradation of water pollutants in environmental engineering

[0151] In the field of environmental protection, the diffusion and degradation of pollutants in water bodies are key to designing effective treatment solutions. This invention, by coupling modeling the diffusion, flow, and degradation processes of pollutants, can simulate the dynamic behavior of pollutants in flowing water bodies and predict concentration changes during the degradation process.

[0152] Specific applications:

[0153] Regarding the diffusion behavior of harmful chemicals in rivers (such as heavy metals or organic pollutants), the model can combine water flow velocity and pressure distribution to predict the spatial distribution and diffusion path of pollutant concentrations.

[0154] Combining convection and reaction kinetics, the degradation process of pollutants after the addition of chemical agents is simulated to optimize the dosage and delivery strategy of the agents.

[0155] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware portion can be implemented using dedicated logic; the software portion can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. Those skilled in the art will appreciate that the above-mentioned devices and methods can be implemented using computer-executable instructions and / or contained in processor control code, for example, such as a carrier medium such as a disk, CD or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuits such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, or can be implemented by a combination of the above-mentioned hardware circuits and software, such as firmware.

[0156] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with this technical field within the technical scope disclosed by the present invention and within the spirit and principles of the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for constructing a convection-reaction-diffusion model in a peridynamics framework, characterized by: The following steps are involved: Step 1, initialization: Initialize the velocity field, pressure field, density field, and concentration field in the model. Determine the initial state through experimental measurement, boundary conditions, or theoretical calculations to provide a basis for subsequent calculations. Step 2: Viscous flow solution: Based on the basic equations of peridynamics, including the momentum conservation equation and the mass conservation equation, combined with the viscosity effect, fluid inertia and external force influence, the flow characteristics of the fluid are solved using numerical methods to generate a viscous flow field. Step 3: Solving solid dissolution / bimolecular reactions: Based on the convection-diffusion equation and combined with the reaction kinetics equation, a mass transfer and transformation model is established. The convection term describes the solute's movement with the fluid, the diffusion term describes the mass diffusion process caused by the concentration gradient, and the reaction term describes the rate and mechanism of the chemical reaction. Step 4: Comprehensive calculation and iterative solution: Based on each time step, the viscous flow solution results are coupled with the solution results of the dissolution / reaction process. The fluid field and concentration field are dynamically updated through an iterative algorithm, and the complete flow field and concentration field evolution data are finally output; The solution of the solid dissolution / bimolecular reaction includes: The advection term in the PD-ARD equation is replaced by the PD continuity equation in the middle Euler form, resulting in the following equation: Where C p (x, t) represents the concentration of substance p at point x at time t; represents the near field of x; In the PD model, each material point x∈Ω interacts with other points in its near field. In two dimensions, the near field is a circle with a radius of δ; yes The area covered; the velocity of the material point x at time t is represented by v(x,t), and It is x to A unit vector in the direction of R pq (x, t) represents the reaction rate between substances p and q; substance p and M p The substances react, which can be expressed as q∈[1,M p ]; Depending on the role of the substance in a particular reaction, the corresponding reaction rate can be positive or negative; the α constant establishes the connection between macroscopic and microscopic speeds, and its value is equal to the dimension; Set the speed The projection in the direction is defined as For the microscopic diffusion function Generally speaking, the relationship between the macroscopic diffusion coefficient and the microscopic diffusion coefficient is definite; The two common forms of this function are "constant form" and "trigonometric form", which are: and According to the matching between the PD model and the classical model when the concentration distribution of substances in the solution is linear, it can be concluded that in the two-dimensional case, d0=4D / πδ 2 and d1=12D / πδ 2 , where D is the diffusion coefficient; in formula (1), set For the advection part in formula (1), three schemes are considered: the center scheme, the upwind scheme, and the downwind scheme. The center scheme has higher accuracy than the upwind scheme, but its numerical stability is not as good as the upwind scheme. The characteristics of the downwind scheme are similar to those of the upwind scheme. The advection term in formula (2) adopts a hybrid scheme, which combines the center scheme and the downwind scheme as follows: in β is the coefficient connecting the macroscopic velocity and the microscopic velocity in the downwind scheme, which is considered a constant. When the flow field is uniform, the PD-ARD equation is compared to obtain β = 2α. Here, w = 0.

8. The PD-ARD equation for inhomogeneous flow has been established; The PD governing equation for viscous flow will experience numerical oscillations, which can be mitigated by adding a density diffusion term to the mass conservation equation: Where, The abbreviation is It's a material point The density at time t, It's a material point is the pressure at time t, b is the volume force at point x of the material at time t; μ is the fluid viscosity; α p is a constant whose value is equal to the dimension; μ The value of is determined by a simple flow problem to ensure the linear consistency of the formula. For the 2D case, μ =16 / 3; η is the correction coefficient, set to 0.1, is a density-dependent microdiffusion function, and the diffusion coefficient is D ρ =c0δ, c0 is the artificial sound speed; similar to the above, There are also two types: "Constant" and "Triangle"; to ensure consistency with the previous micro coefficients, set "Constant type" 2. The method for constructing a convection-reaction-diffusion model in a peridynamics framework according to claim 1, wherein: The PD equation for viscous flow includes a pressure field, but the model lacks an explicit equation. For incompressible Newtonian fluids, it is difficult to directly solve the original incompressible equation. An artificial state equation is used to express the pressure as an algebraic function of density: Where ρ0 is the initial density, ρ * is the predicted density for the current step, γ is the material constant, and the material constant for water is 7; c0 is the artificial sound velocity; the artificial sound velocity affects the choice of time step; the higher the artificial sound velocity, the smaller the time step, but the more stable the value; the artificial sound velocity is set to 10 times the maximum velocity of the flow field, ensuring that the Mach number M = v / c0 is less than 0.1, so that the density change is less than 1% of the initial density; Therefore, we first obtain the velocity field through equations (3)-(5), and then bring the velocity field into equation (1) to solve the ARD process.

3. A system for constructing a convection-reaction-diffusion model under a peridynamics framework, which implements the method for constructing a convection-reaction-diffusion model under a peridynamics framework as claimed in any one of claims 1 to 2, characterized in that: The convection-reaction-diffusion model building system in the peridynamics framework includes: Initialization module, used to initialize velocity, pressure, density and concentration fields; Viscous flow solving module, used to solve viscous flow; Solid dissolution / bimolecular reaction solving module is used to solve solid dissolution / bimolecular reactions.

4. A computer device, characterized in that: The computer device includes a memory and a processor, the memory stores a computer program, and when the computer program is executed by the processor, the processor performs the steps of the method for constructing a convection reaction diffusion model under the peridynamics framework as described in any one of claims 1 to 2.

5. A computer-readable storage medium storing a computer program, wherein when the computer program is executed by a processor, the processor performs the steps of the method for constructing a convection-reaction-diffusion model in the peridynamics framework as described in any one of claims 1 to 2.

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