Conformal advection unified calculation method and system suitable for any polygonal mesh

Abstract

The invention belongs to the field of advection atmospheric mode algorithm development, and discloses a shape-preserving advection unified calculation method and system suitable for any polygon mesh, and the method comprises the following steps: carrying out the mesh division of a calculation domain of a standard example in an advection equation algorithm; standardized grid topology information is constructed for the grid units of the computational domain; initializing a standard example in the algorithm of the advection equation; and performing discretization solution on the advection equation through standardized grid topology information to complete shape-preserving advection unified calculation. According to the method, a general advection solution with calculation precision and stability can be provided for a complex grid numerical model in the fields of atmosphere simulation, ocean circulation and the like, and the development period of an atmosphere advection mode can be remarkably shortened.

Description

Technical Field

[0001] The present invention belongs to the field of advection atmospheric model algorithm development, and specifically relates to a conformal advection unified calculation method and system applicable to arbitrary polygonal grids, which can be used to simulate atmospheric transport processes. Background Art

[0002] Atmospheric advection models, the foundation of other global atmospheric models, employ numerical methods to solve the governing equations describing atmospheric advection motion. Their algorithmic architecture is crucial for the physical conservation and simulation accuracy of complex models such as shallow water wave models and non-hydrostatic dynamic frameworks. The implementation of advection algorithms requires an adaptive grid system. Currently, the most commonly used grids in atmospheric advection models are quadrilateral-based latitude and longitude grids, cubic sphere grids, yin-yang grids, and hexagonal-based icosahedral grids. The development and implementation of existing advection algorithms heavily rely on this single grid system, which creates technical difficulties for algorithm versatility and the development of different limiters within the algorithm, significantly increasing development, debugging, and maintenance costs. Therefore, developing a unified advection algorithm applicable to arbitrary polygonal grids has important practical application value for the development of atmospheric models.

[0003] Driven by the rapid development of high-performance computers in recent years, atmospheric numerical simulations are rapidly migrating toward heterogeneous many-core parallel architectures. This trend poses new theoretical challenges and technical demands for the scalability of numerical algorithms. Currently, the most widely used computational methods are the discontinuous Galerkin method (DGMM), the spectral element method (SEM), and the multi-moment constrained finite volume method (MFVM) based on local reconstruction. These methods can achieve the required computational accuracy through high-order local reconstruction and exhibit strong scalability. However, the DGMM and SSMM are subject to strict CFL conditions. The CFL number is a convergence criterion in computational fluid dynamics (CFD). Specifically, the dependent domain of a difference equation must contain the dependent domain of the corresponding differential equation. Most simply, it means that the time-marching solution must be faster than the propagation speed of the physical perturbation. Only in this way can all physical perturbations be captured. As numerical simulation grid resolution increases, improvements in computational efficiency are significantly limited. In comparison, the MFVM exhibits a more relaxed CFL condition while maintaining computational accuracy and stability, making it a key technical approach to breaking through the bottleneck of ultra-large-scale parallel computing.

[0004] However, when dealing with discontinuous regions in the flow field, high-order multi-moment constrained finite volume methods often encounter the problem of non-physical numerical oscillations, which is also a technical difficulty faced by other high-precision algorithms. This phenomenon not only violates the actual physical process, but also leads to nonlinear accumulation of simulation errors in long-term integration, resulting in unpredictable computational problems. To solve this problem, scholars have considered adding limiters to the numerical algorithm: such as TVD-type monotonic limiters, WENO-based maximum value flux limiters, slope limiters, etc. However, the above-mentioned limiter strategies are often limited by their basic numerical algorithms and are heavily dependent on the shape of the computational grid, making them difficult to implement in multi-moment constrained finite volume methods based on unstructured grids.

[0005] To address the above challenges, there is an urgent need for a new algorithmic framework for efficient conformal advection based on the multi-moment constrained finite volume method, which is decoupled from grid geometry and supports arbitrary polygonal units, to provide theoretical reference and technical support for the development of a new generation of atmospheric advection models. Summary of the Invention

[0006] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide a unified calculation method and system for conformal advection applicable to arbitrary polygonal grids. This method is independent of the shape of the calculation grid and has the characteristics of high precision, positive definiteness and conformal preservation.

[0007] The purpose of the present invention is to be achieved through the following technical solutions:

[0008] A unified calculation method for conformal advection applicable to arbitrary polygonal meshes includes the following steps:

[0009] Meshing the computational domain of the standard example in the algorithm for the advection equation;

[0010] Construct standardized grid topology information for grid cells in the computational domain;

[0011] Initialize the standard example of the algorithm for the advection equation;

[0012] Through the standardized grid topology information, the advection equation is discretized and solved to complete the unified calculation of conformal advection.

[0013] Preferably, constructing standardized grid topology information for grid cells in the computational domain includes the following process:

[0014] Encode all vertices, all edges, and the centers of all grid cells in the computational domain;

[0015] For each grid cell, according to the code of the grid cell, index the set of all vertices and the set of all edges belonging to the grid cell;

[0016] For each vertex, according to the code of the vertex, index the set of all grid cells to which the vertex belongs;

[0017] For each edge, according to the encoding of the edge, index the set of all grid cells to which the edge belongs.

[0018] Preferably, the initialization is performed for a standard example in the algorithm of the advection equation, specifically including:

[0019] According to the coding of the grid cells of the standard case, the grid cell integral average value is defined at the center of the grid cell of the standard case;

[0020] According to the codes of the vertices in the grid cells, point values ​​are defined at the vertices of the grid cells;

[0021] The initial wind field and variable distribution of the standard example are set; wherein the variables in the variable distribution are advection substances in the atmosphere, and the advection substances in the atmosphere include atmospheric density and water vapor.

[0022] Preferably, the process of discretizing and solving the advection equation using standardized grid topology information includes:

[0023] Calculate smoothness indicator parameters based on variables in the advection equation;

[0024] The maximum and minimum values ​​of the integrated mean values ​​of all grid cells are calculated using the smooth indicator parameters;

[0025] Based on the maximum and minimum integral average values ​​of each grid cell, the vertex values ​​and edge midpoint values ​​of the grid cell are corrected; the edge midpoint value is calculated by substituting the edge midpoint value coordinates into the reconstruction function of the corresponding grid cell; the integral average value of the grid cell is corrected by adding the integral average conservation correction process;

[0026] After the vertex values ​​and edge midpoint values ​​of the grid cells are corrected, the advection equation is spatially discretized to obtain the discretized advection equation;

[0027] The time integration of the discretized advection equation is performed to complete the solution of the advection equation.

[0028] Preferably, for any grid cell i, the smooth indicator parameter λ of any grid cell i is i The calculation method is as follows:

[0029]

[0030] Among them, β i (x i ,y i ) is the center coordinate (x) of grid cell i based on grid cell i i ,yi ) smoothing factor; β s (x i ,y i ) is the coordinate of all adjacent grid cells of grid cell i based on the center coordinate of grid cell i (x i ,y i ) smoothing factor; ε is a very small positive number to prevent the denominator from being 0, and its value is not greater than 10 -16 ; s is the code of the grid unit, the variable n represents the number of adjacent grid units; the smoothing factor β i (x i ,y i ) is calculated based on the variable q in the advection equation and the concept of weighted essential non-oscillation;

[0031] The maximum value V of the integral average value of any grid cell i and the minimum value of the integral mean v i as follows:

[0032]

[0033] Where, V q s is the integrated average of the adjacent grid cells of grid cell i, e is the maximum adjustment parameter. In order not to affect the calculation accuracy, e is a positive number before the last time integration step and is 0 at the last time integration step; max() is the maximum value function; min() is the minimum value function.

[0034] Preferably, the correction of the vertex value and the edge midpoint value of the grid unit includes the correction of the maximum value of the vertex value, the correction of the minimum value of the vertex value, the correction of the maximum value of the edge midpoint value and the correction of the minimum value of the edge midpoint value;

[0035] The vertex value maximum correction process includes the following steps:

[0036] Constructing auxiliary variables p H k =min(V j ),j=1,m, where p q k is an arbitrary vertex value, m is the total number of cells sharing point k, j is all grid cells sharing point k, and shared point k is a vertex shared by several grid cells;

[0037] For any grid cell j that shares point k, calculate the correction coefficient for grid cell j Among them, M j and M o Through auxiliary variables The calculated mass within the grid cell and the maximum mass that can leave the grid cell;

[0038]

[0039] Calculate the maximum correction coefficient of shared point k Among them, δf j To pass auxiliary variables The calculated flux from shared point k to grid cell j is If the correction coefficient of grid unit j is used, grid unit j is marked as j1;

[0040] Correction of auxiliary variables:

[0041] Calculate the corrected vertex value: in, The vertex value to satisfy the maximum correction constraint;

[0042] The process of correcting the vertex value minimum is the same as the process of correcting the vertex value maximum. In the process of correcting the vertex value minimum, the auxiliary variable constructed is for: p L k =max(v j ),j=1,m; the vertex value that satisfies the minimum constraint is: Correct the vertex value to satisfy the minimum constraint;

[0043] The process of correcting the maximum value of the edge midpoint value is the same as that of correcting the maximum value of the vertex value;

[0044] The process of correcting the minimum value of the edge midpoint value is the same as that of correcting the minimum value of the vertex value.

[0045] Preferably, the integral mean value conservation correction process is performed on the integral mean value of the grid unit, specifically comprising the following steps:

[0046] Vertex values ​​based on satisfying minimum correction constraints Use the vertex value maximum correction process to calculate the point value that meets the vertex value maximum constraint condition The grid cell to which the correction coefficient is applied is marked as j3;

[0047] pass and The extra mass δM in grid cell j3 is calculated by the difference between j3 ;

[0048] Utilization mass δM j3 Perform the compensatory correction for the conservation of the integral mean value according to the following formula:

[0049]

[0050] in, V q j3 and V q j2 is the original integral average value of grid unit j3 and grid unit j2, ( V q j3 ) m and( V q j2 ) m is the corrected integral average value of grid unit j3 and grid unit j2, A j3 is the area of ​​grid cell j3.

[0051] Preferably, the shape of the grid unit is an arbitrary polygon, and the shape of the arbitrary polygon includes a triangle, a quadrilateral or a hexagon;

[0052] The standard calculation examples are complex wave calculation examples, stepped cylinder calculation examples, or deformed flow calculation examples.

[0053] The present invention further provides a unified calculation system for conformal advection applicable to arbitrary polygonal meshes, which is used to implement the unified calculation method for conformal advection applicable to arbitrary polygonal meshes as described above, comprising:

[0054] Meshing module: used to mesh the computational domain of the standard examples in the advection equation algorithm;

[0055] Grid topology information construction module: used to construct standardized grid topology information for grid cells in the computational domain;

[0056] Initialization module: used to initialize the standard examples in the algorithm of the advection equation;

[0057] Computational module: used to discretize and solve the advection equation through standardized grid topology information to complete the unified calculation of conformal advection.

[0058] The present invention has the following beneficial effects:

[0059] The proposed unified conformal advection calculation method, applicable to arbitrary polygonal grids, is independent of grid shape and can provide a more competitive theoretical basis for global atmospheric advection models based on different grid types. The proposed method effectively suppresses unphysical numerical oscillations while ensuring that the numerical solution is conservative and positively definite, possessing significant practical application value for atmospheric advection simulations. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 The figure is a flow chart of the unified calculation method of conformal advection applicable to arbitrary polygonal grids of the present invention.

[0061] Figure 2 This is a schematic diagram of grid data storage in the present invention.

[0062] Figure 3 This is a waveform diagram of complex wave calculation according to an embodiment of the present invention.

[0063] Figure 4 This is a diagram of the relative error of the maximum value calculated for complex waves according to an embodiment of the present invention.

[0064] Figure 5 This is a diagram of the minimum relative error of complex wave calculation in an embodiment of the present invention. DETAILED DESCRIPTION

[0065] The present invention will be described clearly and completely below with reference to the accompanying drawings and embodiments of the present invention. The described embodiments are only a part of the embodiments of the present invention, not all embodiments.

[0066] See also Figure 1 The present invention is applicable to a unified calculation method of conformal advection of arbitrary polygonal grids, comprising the following steps:

[0067] S1, meshing the computational domain of a standard example in the algorithm for the advection equation; wherein the mesh cells may be in the shape of any polygon, specifically, a triangle, a quadrilateral, or a hexagon; the standard example may be a complex wave example, a stepped cylinder example, or a deformed flow example;

[0068] S2. Construct standardized grid topology information for the grid cells in the computational domain, including the following process:

[0069] For all vertices in the computational domain (refer to Figure 2 , specifically refers to the vertex of the grid unit, the common vertex of adjacent grid units is counted as one vertex), all edges (refer to Figure 2 , specifically referring to the edges of the grid cells, the edges shared by adjacent grid cells are counted as one edge) and the centers of all grid cells are encoded;

[0070] For each grid cell, according to the code of the grid cell, index the set of all vertices and the set of all edges belonging to the grid cell;

[0071] For each vertex, according to the code of the vertex, index the set of all grid cells to which the vertex belongs;

[0072] For each edge, according to the encoding of the edge, index the set of all grid cells to which the edge belongs;

[0073] S3. Initializing the standard example in the algorithm of the advection equation, specifically including: defining the volume integrated-average value (VIA) at the center of the grid cell of the standard example according to the code of the grid cell of the standard example; defining the point value (PV) at the vertex of the grid cell according to the code of the vertex in the grid cell, and then setting the initial wind field and variable distribution of the standard example; wherein the variable q corresponding to the variable distribution can be density, water vapor, or other advective substances in the atmosphere;

[0074] S4. Discretize and solve the advection equation using standardized grid topology information to complete the conformal advection unified calculation. This step specifically includes the following steps:

[0075] S401, calculate the smooth indicator parameter λ based on the variable q in the advection equation. For any grid cell i, the smooth indicator parameter λ of any grid cell i is i The calculation method is as follows:

[0076]

[0077] Among them, β i (x i ,y i ) is the center coordinate (x) of grid cell i based on grid cell i i ,y i ) smoothing factor; β s (x i ,y i ) is the number of adjacent grid cells of grid cell i (the variable n represents the number of adjacent cells) based on the center coordinates of grid cell i (x i ,y i ) smoothing factor; ε is a very small positive number to prevent the denominator from being 0, and is generally not greater than 10 -16 ; s is the code of the grid unit; smoothing factor β i (x i ,y i ) is calculated based on the variable q and the weighted essential non-oscillation concept;

[0078] S402, calculate the maximum value V of the integrated average value of grid cell i i and the minimum value of the integral mean v i ; Take the maximum value of the integral average value V i For example, if λ i If the value is greater than or equal to the set threshold Λ, the grid cell i is marked as a problem cell and the maximum value V of the integral average value is calculated using the integral average value of the adjacent grid cells of the grid cell i. i , if λ iIf it is less than Λ, the global initial maximum value V0 is used to calculate V i :

[0079]

[0080] Where, V q s is the integrated average of the adjacent grid cells of grid cell i, e is the maximum adjustment parameter. To not affect the calculation accuracy, e is a positive number before the last time integration step and is 0 at the last time integration step; max() is the maximum value function; min() is the minimum value function;

[0081] S403, based on the maximum value V of the integrated average value of each grid unit i and the minimum value of the integral mean v i , the vertex and edge midpoint values ​​of the grid cells are corrected to ensure that the numerical solution of the advection equation is positive and non-oscillatory. The edge midpoint values ​​are calculated by substituting the edge midpoint coordinates into the reconstruction function of the corresponding grid cell. The physical meaning of this correction process is to ensure that the mass leaving the target grid cell is no greater than the mass already inside the grid cell.

[0082] Specifically, based on the maximum value V of the integrated average value of each grid cell i and the minimum value of the integral mean v i , the vertex values ​​and edge midpoint values ​​of the mesh cells are corrected, including vertex value maximum correction, vertex value minimum correction, edge midpoint value maximum correction, and edge midpoint value minimum correction. Because each vertex is shared by multiple mesh cells (more than two), the vertex value maximum and minimum correction processes will affect each other. Therefore, it is also necessary to add the integral average value conservation correction process to the mesh cell integral average value. The vertex value is used as an example here to introduce it.

[0083] The vertex value maximum correction process includes the following steps: p q k For example:

[0084] (1) Auxiliary variable construction: introducing auxiliary variables

[0085]

[0086] Where m is the total number of cells that share point k, and j is the number of all grid cells that share point k.

[0087] (2) Flux correction coefficient calculation: For any grid cell j that shares point k, calculate the correction coefficient of grid cell j

[0088]

[0089] Among them, M j and M o Through auxiliary variables The calculated mass within the grid cell and the maximum mass that can leave the grid cell;

[0090] (3) Calculate the maximum correction coefficient of the shared point k

[0091]

[0092] Among them, δf j To pass auxiliary variables The calculated flux from shared point k to grid cell j is If the correction coefficient of grid cell j is used, the grid cell is marked as j1;

[0093] (4) Modify auxiliary variables:

[0094]

[0095] (5) Calculate the corrected vertex value:

[0096]

[0097] in, The vertex value that satisfies the maximum correction constraint.

[0098] The difference between the minimum correction process of the vertex value and the maximum correction process is the different auxiliary variable construction methods. The other processes can directly refer to the maximum correction process. The auxiliary variables constructed in the minimum correction process of the vertex value are:

[0099]

[0100] The vertex values ​​that satisfy the minimum constraint are:

[0101]

[0102] in, The calculation method and the correction process of the maximum value The calculation method is consistent with that of , and the unit to which the correction coefficient used belongs is marked as j2.

[0103] The process of correcting the maximum value of the edge midpoint value is exactly the same as the process of correcting the maximum value of the vertex value, and the process of correcting the minimum value of the edge midpoint value is exactly the same as the process of correcting the minimum value of the vertex value. Different algebraic symbols can be used to represent the corresponding values.

[0104] The vertex values ​​used in the spatial discretization process are the values ​​corrected by the maximum and minimum values. As mentioned above, for vertex values, It cannot strictly guarantee that the maximum condition constraint is met, and an additional integral average value conservation correction process needs to be added to the integral average value of the grid unit.

[0105] The integral mean value of the grid cells is corrected for the conservation of the integral mean value. The specific steps include:

[0106] (1) Based on Use the vertex value maximum correction process to calculate the point value that meets the vertex value maximum constraint condition The grid cell to which the correction coefficient is applied is marked as j3;

[0107] (2) Pass and The extra mass δM in grid cell j3 is calculated by the difference between j3 ;

[0108] (3) Utilization mass δM j3 Perform compensatory correction for conservation of integral mean value:

[0109]

[0110] in, V q j3 and V q j2 is the original integral average value of grid unit j3 and grid unit j2, ( V q j3 ) m and( V q j2 ) m is the corrected integral average value of grid unit j3 and grid unit j2, A j3 is the area of ​​grid cell j3.

[0111] S404, spatially discretizing the advection equation to obtain a discretized advection equation;

[0112] The spatial discretization of advection equation includes the spatial discretization of point value and the spatial discretization of integral average value;

[0113] The spatial discretization of point values ​​is achieved by calculating the derivatives of the reconstruction function of the grid cells. Specifically, the reconstruction function is constructed using the corrected vertex values, and the coefficients of the reconstruction function are solved by the known point values ​​and the integral average value (the integral average value of the grid cells defined in S3) in the grid cells; the derivatives of the vertex values ​​are calculated based on the upwind numerical method.

[0114] The spatial discretization of the integral average is performed in the form of flux to ensure the conservation of the unified calculation method of conformal advection of the present invention, specifically including: calculating the flux through each edge in the grid cell in the form of line integral, and summing all the edge fluxes of the grid cell to obtain the total flux through the grid cell;

[0115] S405: Time-integrate the discretized advection equation to complete the solution of the advection equation. When performing the time integration, a TVD-fidelity Runge-Kutta integration method may be used.

[0116] Example

[0117] Reference Figure 1 , the advection equation involved in this implementation is a two-dimensional advection equation, and its specific form is as follows:

[0118]

[0119] Where q is the input variable, e(q) is the flux in the x-direction, and f(q) is the flux in the y-direction.

[0120] Specifically, the unified calculation method for conformal advection of arbitrary polygonal grids in this embodiment includes the following steps:

[0121] (1) Meshing the computational domain of the standard example in the algorithm of the advection equation; wherein the mesh cells are hexagonal in shape, and the standard example is a complex wave example;

[0122] (2) Construct standardized grid topology information for the grid cells in the computational domain:

[0123] All vertices (specifically, vertices of grid cells, vertices shared by adjacent grid cells are counted as one vertex), all edges (specifically, edges of grid cells, edges shared by adjacent grid cells are counted as one edge) and centers of all grid cells in the computational domain are encoded.

[0124] For each grid cell, according to the code of the grid cell, the set of all vertices and the set of all edges to which the grid cell belongs are indexed. Figure 2 Any grid cell C shown i For example, grid cell C i The set of all vertices of i1 ~P i6 , grid cell C i The set of all edges of i1 ~l i6 .

[0125] For each vertex, according to the code of the vertex, index the set of all grid cells to which the vertex belongs. Figure 2 Any vertex P showni1 For example, vertex P i1 The set of all grid cells is C i 、C j 、C k .

[0126] For each edge, according to the encoding of the edge, index the set of all grid cells to which the edge belongs. Figure 2 Any edge l shown i1 For example, edge l i1 The set of all grid cells is C i 、C k .

[0127] (3) Initialize the standard example in the algorithm of the advection equation and define the integral average value at the center of the unit V q, defines the point value at the vertex P q. Set initial speed Set the initial variable distribution to 2D complex wave.

[0128] (4) Through the standardized grid topology information, the advection equation is discretized and solved to complete the uniform calculation of conformal advection. i Take this as an example to introduce the specific steps:

[0129] (4.1) Using formula (1) to calculate the grid cell C i The smooth indicator parameter λ i ;

[0130] (4.2) Using formula (2) to calculate the grid cell C i The maximum value of the integral average value V i and the minimum value of the integral mean v i .

[0131] (4.3) Based on grid cell C i The maximum value of the integral average value V i and the minimum value of the integral mean v i , for grid cell C i The vertex value and edge midpoint value of the grid cell C are used to correct the i Vertex P i1 First, let's take P i1 Point value p q i1 Performing maximum value correction includes the following steps:

[0132] (4.3.1) Use formula (3) to construct auxiliary variables The parameter m in formula (3) is 3, representing the shared P i1 Grid cell C of the point i、C j 、C k .

[0133] (4.3.2) Calculate the grid cell C using formula (4) i 、C j 、C k Correction factor of

[0134] (4.3.3) Calculate using formula (5) p q i1 The maximum correction coefficient of ;

[0135] (4.3.4) The modified auxiliary variable is obtained by formula (6):

[0136] (4.3.5) The point value that satisfies the maximum correction constraint is obtained through formula (7)

[0137] (5) Yes p q i1 Perform minimum value correction. Construct the auxiliary variable shown in formula (8) Complete the calculation process from steps (4.3.2) to (4.3.4) in sequence, and finally obtain the point value that satisfies the minimum correction constraint according to formula (9): And mark the unit to which the correction coefficient used belongs, assuming it is Figure 2 The grid cell C shown in j ;

[0138] (6) Based on Use the vertex value maximum correction process to calculate the point value that meets the vertex value maximum constraint condition And mark the grid unit to which the correction coefficient used belongs, assuming it is Figure 2 The grid cell C shown in i ;

[0139] (7) Pass and The difference calculation grid cell C i The extra mass δM i ;

[0140] (8) Using formula (10) to calculate the grid cell C i and C j Perform conservation correction of the integral average value;

[0141] (9) Spatial discretization of the advection equation, including spatial discretization of point values ​​and spatial discretization of integral average values;

[0142] (10) The time integration of the discretized advection equation is performed to complete the solution of the advection equation.

[0143] Figure 3 is the waveform of the numerical solution of the complex wave. It can be seen that no non-physical numerical oscillation occurs in the numerical solution, which proves that the proposed unified calculation method for conformal advection applicable to arbitrary polygonal grids has conformal characteristics. Figure 4 The maximum relative error of the numerical solution changes with time. It can be seen that the maximum relative error is always negative, indicating that the numerical solution is always within the maximum value range of the variable integral average value. Figure 5 The curve of the minimum relative error of the numerical solution over time shows that the minimum relative error is always 0, indicating that the numerical solution is always within the minimum range of the variable integral average. These results demonstrate the conformal positive definiteness of the unified calculation method for conformal advection applicable to arbitrary polygonal meshes.

[0144] In addition, an embodiment of the present invention further provides a system for implementing the above-mentioned unified calculation method of conformal advection applicable to arbitrary polygonal grids. The system includes:

[0145] Meshing module: used to mesh the computational domain of the standard examples in the advection equation algorithm;

[0146] Grid topology information construction module: used to construct standardized grid topology information for grid cells in the computational domain;

[0147] Initialization module: used to initialize the standard examples in the algorithm of the advection equation;

[0148] Computational module: used to discretize and solve the advection equation through standardized grid topology information to complete the unified calculation of conformal advection.

[0149] The present invention can provide a universal advection solution with both computational accuracy and stability for complex grid numerical models in the fields of atmospheric simulation, ocean circulation, etc., and can significantly shorten the development cycle of atmospheric advection models.

[0150] Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0151] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention should be covered by the scope of protection of the claims of the present invention.

Claims

1. A unified calculation method for conformal advection of arbitrary polygonal grids, characterized by: The process includes the following: Meshing the computational domain of the standard example in the algorithm for the advection equation; Construct standardized grid topology information for grid cells in the computational domain; Initialize the standard example of the algorithm for the advection equation; Through the standardized grid topology information, the advection equation is discretized and solved to complete the unified calculation of conformal advection.

2. A unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 1, characterized in that: Constructing standardized grid topology information for the grid cells in the computational domain includes the following steps: Encode all vertices, all edges, and the centers of all grid cells in the computational domain; For each grid cell, according to the code of the grid cell, index the set of all vertices and the set of all edges belonging to the grid cell; For each vertex, according to the code of the vertex, index the set of all grid cells to which the vertex belongs; For each edge, according to the encoding of the edge, index the set of all grid cells to which the edge belongs.

3. A unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 2, characterized in that: Initialize the standard example of the advection equation algorithm, including: According to the coding of the grid cells of the standard case, the grid cell integral average value is defined at the center of the grid cell of the standard case; According to the codes of the vertices in the grid cells, point values ​​are defined at the vertices of the grid cells; The initial wind field and variable distribution of the standard example are set; wherein the variables in the variable distribution are advection substances in the atmosphere, and the advection substances in the atmosphere include atmospheric density and water vapor.

4. A unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 2, characterized in that: The process of discretizing and solving the advection equation using standardized grid topology information includes: Calculate smoothness indicator parameters based on variables in the advection equation; The maximum and minimum values ​​of the integrated mean values ​​of all grid cells are calculated using the smooth indicator parameters; Based on the maximum and minimum integral average values ​​of each grid cell, the vertex values ​​and edge midpoint values ​​of the grid cell are corrected; the edge midpoint value is calculated by substituting the edge midpoint value coordinates into the reconstruction function of the corresponding grid cell; the integral average value of the grid cell is corrected by adding the integral average conservation correction process; After the vertex values ​​and edge midpoint values ​​of the grid cells are corrected, the advection equation is spatially discretized to obtain the discretized advection equation; The time integration of the discretized advection equation is performed to complete the solution of the advection equation.

5. A unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 4, characterized in that: For any grid cell i, the smooth indicator parameter λ of any grid cell i i The calculation method is as follows: Among them, β i (x i ,y i ) is the center coordinate (x) of grid cell i based on grid cell i i ,y i ) smoothing factor; β s (x i ,y i ) is the coordinate of all adjacent grid cells of grid cell i based on the center coordinate of grid cell i (x i ,y i ) smoothing factor; ε is a very small positive number to prevent the denominator from being 0, and its value is not greater than 10 -16 ; s is the code of the grid unit, the variable n represents the number of adjacent grid units; the smoothing factor β i (x i ,y i ) is calculated based on the variable q in the advection equation and the concept of weighted essential non-oscillation; The maximum value V of the integral average value of any grid cell i and the minimum value of the integral mean v i as follows: Where, V q s is the integrated average of the adjacent grid cells of grid cell i, e is the maximum adjustment parameter. In order not to affect the calculation accuracy, e is a positive number before the last time integration step and is 0 at the last time integration step; max() is the maximum value function; min() is the minimum value function.

6. A unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 5, characterized in that: Correcting the vertex value and edge midpoint value of the grid unit includes vertex value maximum correction, vertex value minimum correction, edge midpoint value maximum correction and edge midpoint value minimum correction; The vertex value maximum correction process includes the following steps: Constructing auxiliary variables in, p q k is an arbitrary vertex value, m is the total number of cells sharing point k, j is all grid cells sharing point k, and shared point k is a vertex shared by several grid cells; For any grid cell j that shares point k, calculate the correction coefficient for grid cell j Among them, M j and M o Through auxiliary variables The calculated mass within the grid cell and the maximum mass that can leave the grid cell; Calculate the maximum correction coefficient of shared point k Among them, δf j To pass auxiliary variables The calculated flux from shared point k to grid cell j is If the correction coefficient of grid unit j is used, grid unit j is marked as j1; Correction of auxiliary variables: Calculate the corrected vertex value: in, The vertex value to satisfy the maximum correction constraint; The process of correcting the vertex value minimum is the same as the process of correcting the vertex value maximum. In the process of correcting the vertex value minimum, the auxiliary variable constructed is for: The vertex values ​​that satisfy the minimum constraint are: Correct the vertex value to satisfy the minimum constraint; The process of correcting the maximum value of the edge midpoint value is the same as that of correcting the maximum value of the vertex value; The process of correcting the minimum value of the edge midpoint value is the same as that of correcting the minimum value of the vertex value.

7. The unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 5, characterized in that: The integral mean value conservation correction process for the grid cell integral mean value includes the following steps: Vertex values ​​based on satisfying minimum correction constraints Use the vertex value maximum correction process to calculate the point value that meets the vertex value maximum constraint condition The grid cell to which the correction coefficient is applied is marked as j3; pass and The extra mass δM in grid cell j3 is calculated by the difference between j3 ; Utilization mass δM j3 Perform the compensatory correction for the conservation of the integral mean value according to the following formula: in, V q j3 and V q j2 is the original integral average value of grid unit j3 and grid unit j2, ( V q j3 ) m and( V q j2 ) m is the corrected integral average value of grid unit j3 and grid unit j2, A j3 is the area of ​​grid cell j3.

8. The unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 1, characterized in that: The spatial discretization of advection equation includes the spatial discretization of point value and the spatial discretization of integral average value; The spatial discretization of point values ​​is achieved by calculating the derivative of the reconstruction function of the grid cell, which includes: constructing the reconstruction function with the corrected vertex value, solving the coefficient of the reconstruction function by the known point value and the integral average value in the grid cell; calculating the derivative of the vertex value based on the upwind numerical method; The spatial discretization of the integral average is performed in the form of flux, specifically including: calculating the flux through each edge in the grid cell in the form of line integral, and adding all the edge fluxes of the grid cell to obtain the total flux through the grid cell.

9. The unified calculation method for conformal advection applicable to arbitrary polygonal grids according to claim 1, characterized in that: The shape of the grid unit is an arbitrary polygon, and the shape of the arbitrary polygon includes a triangle, a quadrilateral or a hexagon; The standard calculation examples are complex wave calculation examples, stepped cylinder calculation examples, or deformed flow calculation examples.

10. A unified calculation system for conformal advection of arbitrary polygonal meshes, used to implement the unified calculation method for conformal advection of arbitrary polygonal meshes according to any one of claims 1 to 9, characterized in that: include: Meshing module: used to mesh the computational domain of the standard examples in the advection equation algorithm; Grid topology information construction module: used to construct standardized grid topology information for grid cells in the computational domain; Initialization module: used to initialize the standard examples in the algorithm of the advection equation; Computational module: used to discretize and solve the advection equation through standardized grid topology information to complete the unified calculation of conformal advection.

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