A method and system for locating a pole based on a genetic algorithm of a MCSC spherical bipolar orthogonal grid

Abstract

The present application belongs to the global ocean numerical simulation technology field of earth science, and discloses a MCSC spherical double-pole orthogonal grid pole positioning method and system based on a genetic algorithm. The present application aims to solve the problem that in the existing global ocean grid generation technology, there is a complex nonlinear mapping relationship between the center point of the middle domain and the north pole point of the spherical grid, and it is seriously dependent on manual trial and error setting, which causes the grid pole to be difficult to accurately and flexibly positioned to the target geographical location to realize automatic optimization. The main technical scheme includes: obtaining the boundary information of the geographical area to be processed to construct the conformal mapping to the spherical projection complex plane; accepting the target north pole point coordinates specified by the user; calling the genetic algorithm to iteratively search for the optimal conformal center point in the circular boundary domain, combining the inverse projection transformation to minimize the mapping position error; generating a slit mapping according to the center point and mapping the grid back to the sphere to generate a global ocean model orthogonal curve grid.

Description

Technical Field

[0001] This invention belongs to the field of global ocean numerical simulation technology in Earth science, and provides a method and system for locating poles in a spherical bipolar orthogonal grid based on a genetic algorithm. Background Technology

[0002] Global ocean circulation models are one of the core tools for studying physical ocean processes, air-sea interactions, and climate change prediction. Model grid generation is a prerequisite for ocean numerical simulations, and its quality directly affects the computational efficiency, stability, and accuracy of the simulations. Traditional latitude and longitude grids exhibit singularities at the North Pole, resulting in excessively small zonal grid spacing, which severely limits the time step for model integration. To address this issue, existing techniques often employ bipolar or tripolar grids, using conformal transformations to "move" the North Pole to land regions such as North America or Eurasia.

[0003] In recent years, a mesh generation method based on the Schwarz-Christoffel conformal mapping of multi-connected regions has been proposed (Xu et al., 2015). This method maps continental blocks on Earth as slits on the complex plane, enabling the generation of orthogonal curved meshes globally. The mesh lines offer the advantage of enhanced resolution along coastal areas, providing new possibilities for high-resolution, multi-scale ocean simulations. However, existing techniques treat major continental blocks as deleted regions, resulting in mesh poles being limited to scattered land areas such as Greenland that were not deleted. This may restrict the flexibility of mesh design.

[0004] In the process of generating a global mesh based on the MCSC mapping, a key step is to specify a center point in the intermediate domain of the conformal mapping, which will ultimately be mapped to the north pole of the spherical mesh. In existing technologies, the selection of this center point typically relies on manual setting. However, since the mapping from the circular boundary domain to the final spherical mesh is a complex nonlinear composite function involving both the MCSC mapping and polar projection, there is no clear correspondence between the manually set point and the final mesh's north pole. This makes it difficult for mesh designers to accurately and flexibly place the north pole in the desired geographical location. For example, if they want to place the pole in a specific location on the Eurasian continent to optimize the mesh, they often need to repeatedly try and fail, which is inefficient and makes it difficult to guarantee optimality. Furthermore, how to automatically optimize the pole position according to computational stability requirements is also an unresolved problem in existing technologies. Summary of the Invention

[0005] The purpose of this invention is to solve the problems in existing global grid generation technology based on MCSC conformal mapping, which is difficult to accurately and flexibly locate the grid poles to the target geographical location due to the complex nonlinear mapping relationship between the intermediate domain center point and the final spherical grid north pole point and the reliance on manual trial and error setting. This results in low design iteration efficiency and the inability to achieve automatic optimization by combining computational stability or grid quality indicators.

[0006] To achieve the above objectives, the present invention employs the following technical means:

[0007] This invention provides a method for locating poles in a spherical bipolar orthogonal grid based on a genetic algorithm, comprising the following steps:

[0008] Step 1: Obtain the boundary information of the geographic region to be processed and construct a conformal mapping from the circular boundary domain to the spherical projection complex plane;

[0009] Step 2: Accept the geographic coordinates of the target North Pole specified by the user;

[0010] Step 3: Call the genetic algorithm, taking the geographic coordinates of the target North Pole as the search target, and iteratively search for the optimal conformal center point in the circular boundary domain, so that the error between the spherical position corresponding to the optimal conformal center point after conformal mapping and inverse projection transformation and the geographic coordinates of the target North Pole is minimized;

[0011] Step 4: Based on the optimal conformal center point obtained from the search, generate a slit mapping from the circular boundary domain to the slit domain, and invert the mesh on the slit domain back to the sphere to generate the final global ocean model orthogonal curve mesh.

[0012] In the above scheme, step 1, obtaining the boundary information of the geographic area to be processed, specifically includes: selecting... Given a set of continental blocks to be deleted, obtain the set of polygon boundary vertices for each continental block. ,in Indicates the first The set of latitude and longitude coordinates of all vertices of a continental block to be deleted is represented as:

[0013]

[0014] Where λ and φ represent longitude and latitude, respectively. It is the total number of vertices of the i-th continental block to be deleted.

[0015] In the above scheme, step 1, which involves constructing a conformal mapping from the circular boundary region to the spherical projection complex plane, further includes using stereographic projection to map the latitude and longitude coordinates of the polygon boundary vertices. Convert to complex plane coordinates The conversion formula for the stereographic projection is:

[0016]

[0017] in, Represents longitude. Represents latitude, The imaginary unit, This represents the stereographic projection function.

[0018] In the above scheme, the spherical projection complex plane mentioned in step 1 and circular boundary region Defined as an unbounded, multi-connected region, denoted as:

[0019]

[0020]

[0021] in, It is the complex plane. It is the region enclosed by a polygon formed by the vertices of the transformed complex coordinates. It is a circular region on the complex plane. , and These are the coordinates of the center and the radius, respectively.

[0022] In the above scheme, the construction of conformal mapping in step 1 specifically refers to generating a mapping from the circular boundary domain. To the spherical projection complex plane conformal mapping .

[0023] In the above scheme, the geographical coordinates of the target North Pole in step 2 are represented as follows: ,in The longitude of the target North Pole The latitude of the target North Pole.

[0024] In the above scheme, the fitness function of the genetic algorithm in step 3 Used to evaluate candidate center points via conformal mapping and inverse polar equatorial projection Then, the latitude and longitude positions are inverted to the spherical surface. Latitude and longitude position of the target North Pole specified by the user The degree of closeness between them; the fitness function is defined as the degree of closeness between them. and The Euclidean distance between two points is expressed as:

[0025]

[0026] in, , The calculation formula is:

[0027]

[0028] in, Representing complex numbers The imaginary part, Representing complex numbers The real part, Representing complex numbers The model.

[0029] In the above scheme, step 4, generating the slit mapping from the circular boundary domain to the slit domain, specifically includes:

[0030] Generate the corresponding circular boundary domain To the narrow domain Slit mapping The slit mapping Map the optimal conformal center point to the origin, and define the circular boundary domain. The circular boundaries on the surface are mapped one by one to different radial slits;

[0031] Subsequently, based on resolution requirements, in the slit domain The corresponding polar coordinate grid is generated on top.

[0032] In the above scheme, step 4, which involves inverting and mapping the mesh on the slit domain back to the sphere, specifically includes:

[0033] narrow space The polar coordinate grid on the grid is mapped sequentially through the inverse slit. Conformal mapping and inverse polar projection The inversion back to the sphere yields the final orthogonal curve grid of the global ocean model, ensuring that the North Pole of this grid corresponds to the geographical location of the target North Pole specified by the user.

[0034] This invention also provides a genetic algorithm-based MCSC spherical bipolar orthogonal grid pole localization system for implementing the method, comprising:

[0035] The mapping construction module is configured to obtain the boundary information of the geographic region to be processed and construct a conformal mapping from the circular boundary domain to the spherical projection complex plane;

[0036] The genetic algorithm search module is configured to accept the geographic coordinates of the target North Pole specified by the user and call the genetic algorithm to iteratively search for the optimal conformal center point in the circular boundary domain.

[0037] The mesh generation module is configured to generate slit mappings based on the optimal conformal center points obtained from the search, and to invert and map the meshes on the slit domains back to the sphere to generate the final orthogonal curve meshes for the global ocean model.

[0038] Because the present invention employs the above-mentioned technical means, it has the following beneficial effects:

[0039] 1. By transforming the pole location problem into an optimization problem and introducing a genetic algorithm, this invention accurately solves the inverse problem of complex nonlinear composite functions from a circular boundary domain to the final spherical mesh, ensuring that the pole location is highly consistent with the design target, and solving the problem of the lack of a clear correspondence between manually set poles and the final poles in the prior art.

[0040] 2. This invention introduces a heuristic search algorithm into the Schwarz-Christoffel conformal mapping mesh generation program for multi-connected regions for the first time, realizing an improvement from manual setting to automatic positioning of the grid north pole. The module can accurately and quickly place the pole at any geographical location specified by the user (which must be located outside of continental blocks), effectively improving the flexibility and optimizability of mesh design.

[0041] 3. By precisely controlling the pole locations, this invention can effectively regulate the grid distribution in high-latitude regions, especially near the poles. Combined with grid quality optimization objectives (such as maximizing the minimum latitudinal grid spacing), a larger minimum latitudinal grid spacing can be obtained, thereby allowing the model to use a longer time integration step, improving computational efficiency while ensuring numerical stability.

[0042] 4. This invention combines a genetic algorithm with the Schwarz-Christoffel conformal mapping of multi-connected regions, achieving a technical effect that cannot be achieved by a single technique: the genetic algorithm can efficiently search for the optimal solution in a complex nonlinear mapping space, while the Schwarz-Christoffel conformal mapping of multi-connected regions provides an accurate mathematical description and inversion path for the pole location. The synergistic effect of the two makes the pole location accuracy converge stably with the number of iterations, solving the inefficiency problem of repeated trial and error in the prior art. Attached Figure Description

[0043] Figure 1 : A simplified flowchart of the invention;

[0044] Figure 2 The overall process corresponding to Example 1;

[0045] Figure 3 The process of using a genetic algorithm to search for the optimal conformal center point;

[0046] Figure 4: This represents the search duration and corresponding final fitness function value of the genetic algorithm for each run in the three control experiments. Detailed Implementation

[0047] The embodiments of the present invention will be described in detail below. Although the present invention will be described and illustrated in conjunction with some specific embodiments, it should be noted that the present invention is not limited to these embodiments. On the contrary, any modifications or equivalent substitutions made to the present invention should be covered within the scope of the claims of the present invention.

[0048] Furthermore, to better illustrate the present invention, numerous specific details are set forth in the following detailed embodiments. Those skilled in the art will understand that the present invention can be practiced without these specific details.

[0049] Example 1: Basic Implementation

[0050] like Figure 1 As shown, the present invention provides a method for locating poles in a spherical bipolar orthogonal grid based on a genetic algorithm, comprising the following steps:

[0051] Step 1: Select the polygon boundary information of n continental blocks to be deleted. ,in It is a set containing the latitude and longitude coordinates of all vertices of the i-th continental block to be deleted, which can be represented as:

[0052]

[0053] Where λ and φ represent longitude and latitude, respectively. This represents the total number of vertices of the i-th continental block to be deleted. The value of n is a positive integer and can be determined based on the specific application requirements; for example, in global ocean simulations, the value of n corresponding to the major continental blocks is typically chosen.

[0054] Step 2: Using stereographic projection The latitude and longitude coordinates of each vertex of the continental block to be deleted. Convert to complex plane coordinates . polar projection The conversion formula is as follows:

[0055]

[0056] in, It is the imaginary unit.

[0057] Step 3: Obtain the complex coordinate set of continental block vertices from the stereographic projection { Create a complex plane projected from a sphere. and circular boundary region .in, and Both are unbounded, multi-connected regions, and can be represented as follows:

[0058] Step 4: Based on the user-created spherical projection complex plane and circular boundary region , generate from arrive conformal mapping .

[0059] Step 5: Accept the user-specified latitude and longitude coordinates of the target North Pole. .

[0060] Step 6: Call the genetic algorithm with the user-specified North Pole latitude and longitude position as the search target, within the circular boundary region. Search for the optimal conformal center point .

[0061] like Figure 2 As shown, the specific implementation process of the genetic algorithm is as follows:

[0062] Population initialization: in a circular boundary region Internal random generation Candidate center points ,in Population size;

[0063] Fitness assessment: For each candidate centroid Calculate its conformal mapping and inverse polar equatorial projection The latitude and longitude positions were then inverted to the spherical surface. And calculate the fitness function value;

[0064] Genetic operations: Selecting, crossing over, and mutating the population to generate a new generation of candidate centroids;

[0065] Iteration Termination: Repeat the fitness evaluation and genetic operations until the preset maximum number of iterations is reached or the change in the fitness function is less than the preset threshold.

[0066] Fitness function of genetic algorithm Used to evaluate a candidate center point via conformal mapping and inverse polar equatorial projection Then, the degree of closeness between the latitude and longitude position retrieved to the sphere and the latitude and longitude position of the target North Pole specified by the user is defined as the Euclidean distance between the two points:

[0067]

[0068] Among them, the inverse polar equatorial projection The formula is:

[0069]

[0070] in, The coordinates of the North Pole of the target on the complex plane. Denotes the real part of the candidate center point. This represents the imaginary part of the candidate center point.

[0071] Step 7: Based on the found optimal conformal center point Generate the corresponding from To the narrow domain Slit mapping .in, The center point will Mapped to the origin, and The circular boundary on Each slit is mapped to a different radial slit. Then, based on resolution requirements (such as a given number of grid points in the longitudinal and latitudinal directions), in... The corresponding polar coordinate grid is generated on top.

[0072] Step 8: Generate the final spherical mesh.

[0073] Will The polar coordinate grid on the grid is mapped sequentially through the inverse slit. Conformal mapping and inverse polar projection The inversion back to the sphere yields the final orthogonal curve grid for the global ocean model. The North Pole of this grid will geographically correspond precisely to the target North Pole specified by the user.

[0074] Example 2: System Implementation

[0075] This invention relates to a genetic algorithm-based MCSC spherical bipolar orthogonal grid pole localization system, comprising:

[0076] Mapping Builder: Used to accept user-specified information about continental blocks to be deleted. The coordinates are converted to complex plane coordinates using stereographic projection, and then generated from... arrive conformal mapping .

[0077] Genetic Algorithm Search Module: Used to accept the latitude and longitude coordinates of the target North Pole specified by the user. The fitness function is the Euclidean distance between the candidate center point's position on the sphere obtained by conformal mapping and inverse stereographic projection and the target position. The genetic algorithm is then used to iteratively search for the optimal conformal center point.

[0078] Mesh generation module: Used to find the optimal conformal center points based on the genetic algorithm. The final spherical orthogonal curve mesh is generated through slit mapping and inverse stereographic projection.

[0079] The modules are connected via data interfaces, and the mapping construction module generates conformal mappings. Output to the genetic algorithm search module and the grid generation module.

[0080] test

[0081] Conditions: Using the same MCSC mapping, three target North Pole points are preset, located in the center of Eurasia, North America, and Greenland, respectively. Three control experiments are conducted, in which the genetic algorithm searches for each of the three target North Pole points 100 times, and then its optimization efficiency and accuracy are evaluated. All three control experiments are performed using a single CPU, and the genetic algorithm is set with the same population size, search range, and termination condition.

[0082] Comparison with existing technologies: In 2015, Xu Shiming et al. proposed a global ocean grid generation method based on MCSC mapping, combining MCSC mapping and equatorial polar projection. They expanded the functionality of the MCSC mapping generation toolbox to support the generation of MCSC mappings that mixed radial and circular slots. Most landmasses were mapped as slots (including Eurasia, North America, South America, and Africa). Therefore, the grid's North Pole was limited to high-latitude areas near scattered landmasses not mapped as slots (such as Greenland), resulting in a limited range of pole location selection. Subsequently, the grid was used for simulation and evaluation in ocean circulation models. However, Xu Shiming et al.'s research did not focus on providing a fully automated and flexible improvement scheme for the grid generation process. The selection of the grid's North Pole required extensive human trial and error, which would hinder the efficiency of subsequent grid improvement work.

[0083] Experimental data:

[0084] (1) The search duration and corresponding final fitness function values ​​for each run of the genetic algorithm in the three control experiments, as follows: Figure 4 As shown, the target points for the three sets of experiments were set to (77.67°N, 41.07°W), (59.55°N, 103.03°E), and (46.03°N, 102.07°W), respectively.

[0085] (2) The three sets of global grid data generated by the reverse equatorial polar projection have 180 meridional grid points and 360 latitudinal grid points. The North Pole of the three sets of grids is the same as the three target North Poles preset in the conditions. Since the equatorial polar projection has a clear analytical formula, the position of the North Pole can be accurately set and the generated grid can be used as a reference.

[0086] (3) Three sets of MCSC global grid data with 180 meridional grid points and 360 latitudinal grid points were generated using the conformal center points corresponding to the optimal final fitness function values ​​obtained in the three sets of control experiments. Each set of grids has the same deletion region (South America and Africa).

[0087] Conclusion: Experimental data show that using a genetic algorithm to locate the North Pole in an MCSC grid (currently, no related research has attempted to apply this type of heuristic optimization method to the North Pole location problem in MCSC grids), although the randomness of the heuristic algorithm introduces some fluctuations in the results, this is... Figure 4 The fitness function value and search duration fluctuate, but accuracy and duration can still be controlled within an ideal range (highest fitness function value 0.0061, longest duration 31.81s). On the other hand, from Figure 4 Overall, as the latitude of the target North Pole increases, the fluctuations in search duration and accuracy become more pronounced, which may be related to the search range settings. By transforming pole location into a numerical optimization problem and utilizing the global search capability of genetic algorithms, the grid North Pole can be accurately and quickly located at any user-specified geographical location. This significantly improves the automation level and design efficiency of grid generation, and further enhances the intelligence and practicality of MCSC grid generation.

Claims

1. A method for locating poles in a spherical bipolar orthogonal grid based on a genetic algorithm, characterized in that, Includes the following steps: Step 1: Obtain the boundary information of the geographic region to be processed and construct a conformal mapping from the circular boundary domain to the spherical projection complex plane; Step 2: Accept the geographic coordinates of the target North Pole specified by the user; Step 3: Call the genetic algorithm, taking the geographic coordinates of the target North Pole as the search target, and iteratively search for the optimal conformal center point in the circular boundary domain, so that the error between the spherical position corresponding to the optimal conformal center point after conformal mapping and inverse projection transformation and the geographic coordinates of the target North Pole is minimized; Step 4: Based on the optimal conformal center point obtained from the search, generate a slit mapping from the circular boundary domain to the slit domain, and invert the mesh on the slit domain back to the sphere to generate the final global ocean model orthogonal curve mesh.

2. The method according to claim 1, characterized in that, Step 1, which involves obtaining the boundary information of the geographic area to be processed, specifically includes: selecting... Given a set of continental blocks to be deleted, obtain the set of polygon boundary vertices for each continental block. ,in Indicates the first The set of latitude and longitude coordinates of all vertices of a continental block to be deleted is represented as: Where λ and φ represent longitude and latitude, respectively. It is the total number of vertices of the i-th continental block to be deleted.

3. The method according to claim 2, characterized in that, Step 1, which involves constructing a conformal mapping from the circular boundary region to the spherical projection complex plane, also includes using stereographic projection to map the latitude and longitude coordinates of the polygon boundary vertices. Convert to complex plane coordinates The conversion formula for the stereographic projection is: in, Represents longitude. Represents latitude, The imaginary unit, This represents the stereographic projection function.

4. The method according to claim 3, characterized in that, The spherical projection complex plane mentioned in step 1 and circular boundary region Defined as an unbounded, multi-connected region, denoted as: in, It is the complex plane. It is the region enclosed by a polygon formed by the vertices of the transformed complex coordinates. It is a circular region on the complex plane. , and These are the coordinates of the center and the radius, respectively.

5. The method according to claim 4, characterized in that, The construction of conformal mapping in step 1 specifically refers to generating a map from the circular boundary domain. To the spherical projection complex plane conformal mapping .

6. The method according to claim 1, characterized in that, The geographical coordinates of the target North Pole mentioned in step 2 are expressed as follows: ,in The longitude of the target North Pole The latitude of the target North Pole.

7. The method according to claim 5, characterized in that, The fitness function of the genetic algorithm described in step 3 Used to evaluate candidate center points via conformal mapping and inverse polar equatorial projection Then, the latitude and longitude positions are inverted to the spherical surface. Latitude and longitude position of the target North Pole specified by the user The degree of closeness between them; the fitness function is defined as the degree of closeness between them. and The Euclidean distance between two points is expressed as: in, , The calculation formula is: in, Representing complex numbers The imaginary part, Representing complex numbers The real part, Representing complex numbers The model.

8. The method according to claim 7, characterized in that, Step 4, which involves generating the slit mapping from the circular boundary domain to the slit domain, specifically includes: Generate the corresponding circular boundary domain To the narrow domain Slit mapping The slit mapping Map the optimal conformal center point to the origin, and define the circular boundary domain. The circular boundaries on the surface are mapped one by one to different radial slits; Subsequently, based on resolution requirements, in the slit domain The corresponding polar coordinate grid is generated on top.

9. The method according to claim 8, characterized in that, Step 4, which involves inverting and mapping the mesh on the slit domain back to the sphere, specifically includes: narrow space The polar coordinate grid on the grid is mapped sequentially through the inverse slit. Conformal mapping and inverse polar projection The inversion back to the sphere yields the final orthogonal curve grid of the global ocean model, ensuring that the North Pole of this grid corresponds to the geographical location of the target North Pole specified by the user.

10. A genetic algorithm-based MCSC spherical bipolar orthogonal grid pole localization system, used to implement the method of any one of claims 1 to 9, characterized in that, include: The mapping construction module is configured to obtain the boundary information of the geographic region to be processed and construct a conformal mapping from the circular boundary domain to the spherical projection complex plane; The genetic algorithm search module is configured to accept the geographic coordinates of the target North Pole specified by the user and call the genetic algorithm to iteratively search for the optimal conformal center point in the circular boundary domain. The mesh generation module is configured to generate slit mappings based on the optimal conformal center points obtained from the search, and to invert and map the meshes on the slit domains back to the sphere to generate the final orthogonal curve meshes for the global ocean model.

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