A method for evaluating the equivalent total dose of fan angle for micro-nano satellites

Abstract

The invention discloses a fan-angle equivalent total dose evaluation method for a micro-nano satellite, and relates to the field of satellite radiation protection in a space environment. The method takes a to-be-tested point in a satellite as the center, calculates vector rays by a space grid division method, sequentially calculates the intersection length and material density between each vector ray and a satellite model, obtains the equivalent intersection length in each vector ray direction, combines a dose-depth curve obtained by simulating a space radiation environment, obtains the dose value in each ray direction, and performs weighted addition to obtain the three-dimensional total dose at the to-be-tested point, thereby improving the calculation efficiency and accuracy of the total dose evaluation, clearly reflecting the weak direction of the total dose protection, and providing accurate guidance for the reinforcement of the total dose protection.

Description

Technical Field

[0001] The present invention relates to a total dose calculation and evaluation in a micro-nano satellite in a space environment, and in particular to a method for calculating and evaluating the accumulated total dose value at a sensitive chip in a micro-nano satellite based on a regular icosahedron space grid division method. Background Art

[0002] The space environment is full of radiation from the universe, the sun, and the Earth's capture belt, which directly threatens the normal operation of micro-nano satellites. Among them, the total dose effect directly affects the service life of on-orbit electronic devices. Charged particles such as electrons and protons from the Earth's radiation belt and protons from solar cosmic rays enter the oxide interface layer of semiconductor devices, affecting carrier mobility, or enter the gate oxide layer, generating positive charges in it, thereby changing the voltage threshold value and causing the electrical performance of the device to deteriorate, thereby reducing the reliability of micro-nano satellites. Studies have shown that the total dose effect has caused many serious consequences for the hazards of onboard devices. In order to ensure the safe and reliable operation of micro-nano satellites, it is necessary to conduct accurate and efficient total dose evaluation and protection design around sensitive chips.

[0003] The total dose value accumulated on the device is directly related to the satellite shielding material and its thickness. The total dose protection is mainly achieved by optimizing the satellite structure or thickening the baffle. However, the total dose protection requires a highly accurate total dose assessment for effective and clear guidance, especially for micro-nano satellites that are constrained by cost and quality. Therefore, accurately assessing the dose value at the sensitive chip in the micro-nano satellite is a prerequisite and an important basis for protection.

[0004] At present, the methods for evaluating the total dose of satellites mainly include on-orbit flight tests, ground simulation tests and simulation evaluation. On-orbit flight tests are conducted directly in space, with long cycles, few opportunities and high costs; ground simulation tests use emission sources to simulate radiation effects, which is difficult to simulate the real space environment. The simulation evaluation method combines the space radiation model and the satellite model to conduct on-orbit total dose evaluation of sensitive devices in the satellite, which has the advantages of short evaluation cycle and low cost. Summary of the invention

[0005] The present invention proposes a fan angle equivalent total dose evaluation method for micro-nano satellites, which is used to evaluate the total dose value of sensitive positions in the micro-nano satellite, improve the calculation accuracy and efficiency of the total dose, analyze the weak direction of the total dose protection, and thus solve the bottleneck of the total dose system-level protection.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A method for evaluating the equivalent total dose of a micro-nano satellite based on a fan angle includes the following steps:

[0008] S1, based on the satellite's preset flight orbit and mission duration, uses the space radiation environment model to analyze the space radiation flux during the satellite's in-orbit operation.

[0009] S2, based on the SHIELDOSE-2 model and the spatial radiation flux analyzed in S1, calculates the total dose shielding effect of the simple geometric model, calculates the relationship between the absorbed dose and the specified material thickness, and obtains the dose-depth curve under the specified material geometric shielding.

[0010] S3, obtain the simplified shape model of the satellite by equivalently simplifying the satellite three-dimensional model, and obtain the coordinates of the total dose test point according to the position of the sensitive chip in the satellite three-dimensional model.

[0011] S4, taking the coordinates of the point to be measured obtained based on S3 as the center point, obtaining vector rays emitted from the center point through a space grid division method, and dividing the space grid based on these vector rays.

[0012] S5, based on the satellite shape model constructed in S3 and the vector ray set constructed in S4, performs intersection analysis, calculates the intersection thickness between each vector ray and the satellite shape model component module one by one, and combines the material density of the module to equate the thickness of each intersection material in the direction of each vector ray to the thickness of the aluminum material.

[0013] S6, based on the dose-depth curve calculated by S2 and the equivalent aluminum material thickness in each vector ray direction calculated by S5, convert the equivalent aluminum material thickness in each direction into the dose value in each vector ray direction to obtain the three-dimensional distribution of the dose, and perform weighted summation to obtain the total dose value of the test point.

[0014] S7, based on the three-dimensional distribution of the dose at the test point and the total dose value obtained in S6, perform total dose protection analysis. Compare the total dose value with the total dose threshold of the device at the test point to determine whether the total dose protection requirements are met. If not, reinforce and optimize the satellite model based on the three-dimensional distribution data of the dose, and return to S3 to recalculate the total dose. If satisfied, end the total dose assessment.

[0015] As a preferred solution of the present invention, the space grid division method in step S4 uses an equal angle division method, an equal solid angle division method or an icosahedron space grid division method. Different numbers of vector rays are output according to the number of divisions. The vector rays calculated and output are evenly distributed.

[0016] In the comparison of simple models, compared with angle division, the vector ray distribution from the icosahedron division is more uniform, which is more conducive to the stability and accuracy of the total dose calculation, and also helps to improve efficiency. As a further preferred embodiment of the present invention, the space grid division method in step S4 uses the icosahedron space grid division method, and the specific steps include:

[0017] S41, place the regular icosahedron symmetrically in the coordinate system, with its vertices all on the coordinate plane, and the coordinates of its center point are (0, 0, 0). Assuming the radius of its circumscribed sphere is r, the coordinates of each vertex are (±m, 0, ±n), (0, ±n, ±m), (±n, ±m, 0), where the calculation formulas for m and n are (1):

[0018]

[0019] Take the 7 faces that intersect with the first and second hexagrams for analysis, and construct an array to sequentially store the vertex coordinates of the 7 faces.

[0020] S42, based on the array obtained in S41, calculates the coordinates of the endpoints of each ray vector in a single face. Since there are common edges between faces, the ray vector endpoints on the AC edge are not calculated in the single face calculation, which avoids repeated calculations, improves calculation efficiency, and facilitates programming implementation. The calculation formulas are (2) and (3):

[0021]

[0022]

[0023] In formulas (2) and (3), A, C, and B are the vertices of the face, n is the number of iterative divisions in the regular icosahedron division, and u is The unit vector of The unit vector, i is the length in the u direction, j is the length in the v direction, (x ij ,y ij , z ij ) represents the coordinates of the endpoint of the ray vector with vector coordinates (i, j) on the surface, and iu represents (x ij ,y ij , z ij ) is the vector value in the u direction, jv represents (x ij ,y ij , z ij ) is the vector value in the v direction.

[0024] It is particularly important to note that due to the existence of common edges between faces, the ray vector endpoints on the AC edge are not calculated in the single-face calculation, which avoids repeated calculations, improves calculation efficiency, and improves the calculation efficiency of vector rays, which is conducive to programming implementation.

[0025] S43, repeating steps S41-S42, substituting the coordinates of the seven-face vertices stored in the array into A, B, and C in turn to calculate the ray vector endpoints, and obtaining the ray vector endpoints of the seven faces.

[0026] S44, using the symmetry properties of the regular icosahedron to traverse the coordinate points obtained above, and divide the points into three groups during the copying process, wherein each point in the first and second hexagrams needs to be symmetrically copied three times to obtain four points; the points on the oxy and oxz coordinate planes need to be symmetrically copied once to obtain two points; the remaining points are discarded.

[0027] If the point is within the first and second quadrants, it is replicated three times symmetrically, and the formula is (4):

[0028]

[0029] In formula (4), (x, y, z) represents the coordinates of a point in the first and second quadrants, where y>0, z>0, p and q are the replication coefficients of y and z respectively, (x pq ,y pq , z pq ) represents the coordinates of the vertex after three symmetric copies. pq is the formula expression of "each point in the first and second hexagrams needs to be symmetrically copied three times" in this copying process. The value of pq is in the set {-1, 1}, and there are four cases in total. When pq is 1, the coordinate xyz remains unchanged. When p is -1, it means that the coordinate xyz is copied to the coordinate x-yz with oxz as the symmetry plane; similarly, the copied coordinates xy-z and xyz are obtained.

[0030] If the point is on the oxy coordinate plane, it is copied symmetrically once, and the formula is (5):

[0031]

[0032] In formula (5), (x, y, 0) represents the coordinates of a point on the oxy coordinate plane, where y>0, (x p ,y p , 0) represents the vertex coordinates after symmetrical replication once.

[0033] If the point is on the oxz coordinate plane, it is copied symmetrically once, and formula (6) is:

[0034]

[0035] In formula (6), (x, 0, z) represents the coordinates of a point on the oxz coordinate plane, where z>0, (x q , 0, z q ) represents the vertex coordinates after symmetrical replication.

[0036] S45, obtain the endpoints of the ray vector of the regular icosahedron divided by n iterations. Since the coordinates of the center point of the regular icosahedron are (0, 0, 0), the ray vector is obtained by the endpoints of the ray vector and the origin of the coordinates, and finally the ray vector of the regular icosahedron divided by n iterations is obtained.

[0037] Preferably, the equivalent aluminum material thickness calculation in step S5 is expressed as formula (7):

[0038]

[0039] In formula (7): t o is the equivalent thickness in the direction of the vector ray, k is the number of intersection segments in the direction of the vector ray, t x is the thickness of the x-th intersection segment in the direction of the vector ray, ρ x is the density of the xth intersection segment in the direction of the vector ray, ρ Al is the density of aluminum alloy material, D tatal is the total dose of the test point, D sphere is the dose deposition function at the center of a solid sphere with radius as a variable.

[0040] Preferably, the total dose value of the test point in step S6 is calculated as follows:

[0041]

[0042] In formula (8): D tatal is the total dose of the test point, δ o is the solid angle in the direction of the vector ray, m is the number of vector rays, D sphere is the dose deposition function at the center of a solid sphere with radius as a variable.

[0043] As a preferred solution of the present invention, the calculation of the equivalent intersection thickness in step S5 is specifically to calculate the intersection thickness of each vector ray and the satellite shape model component module, and combine the material density of the module to make the thickness of each intersection material in the direction of each vector ray equivalent to the thickness of the material specified in step S2.

[0044] As a preferred solution of the present invention, the calculated total dose distribution can reflect the total dose distribution values ​​in all directions at the test point, and can provide a basis for total dose protection.

[0045] As a preferred solution of the present invention, step S7 also includes: if the total dose protection requirement of the device is not met, the satellite model is reinforced and optimized according to the three-dimensional distribution of the dose, and the total dose is recalculated again in step S3. As a preferred solution of the present invention, the specified material is the main material of the satellite; aluminum, titanium alloy or other materials can be selected.

[0046] Compared with the prior art, the present invention has the following beneficial effects:

[0047] (1) The present invention is based on a simulation evaluation method, combined with a satellite three-dimensional model and space radiation environment simulation. On the basis of obtaining the structural thickness and material of the satellite, the total dose value and dose distribution of the test points within the satellite for a specified orbit and mission period are evaluated, which is in line with engineering practice.

[0048] (2) The present invention can perform grid division based on regular icosahedron division, the number of calculated vector rays is adjustable, the vector rays are evenly distributed, and the shapes of the divided spatial grids are consistent, which can improve the accuracy of total dose assessment.

[0049] (3) The present invention completes the total dose evaluation based on the equivalent sector angle of the regular icosahedron division. The obtained total dose distribution diagram can more accurately reflect the dose values ​​in various directions at the test point, and can clearly reflect the weak direction of the total dose protection, provide accurate guidance for the total dose protection reinforcement, and improve the total dose protection efficiency and accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is a flow chart of the equivalent total dose evaluation of the fan angle based on the regular icosahedron division of the present invention;

[0051] Figure 2 is the simplified shape model of the satellite used in Example 1;

[0052] Figure 3 is the distribution of the total dose calculated in Example 1;

[0053] Figure 4 It is a schematic diagram of the division of a regular icosahedron;

[0054] Figure 5 This is a schematic diagram of the calculation of the endpoints of a single-sided ray vector when dividing a regular icosahedron. DETAILED DESCRIPTION

[0055] In order to make the purpose, technical solution and advantages of the present invention more clear, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific implementation methods described here are only used to explain the present invention and do not limit the present invention.

[0056] Example 1

[0057] like Figure 1 As shown in FIG. 1 , a method for evaluating the equivalent total dose of a fan angle for a micro-nano satellite includes the following steps:

[0058] S1, according to the satellite's scheduled flight orbit and mission duration, the space radiation environment model is used to analyze the space radiation flux during the satellite's life. The mission orbit is set to geosynchronous orbit, the launch time is 2021, and the on-orbit life is one year. The radiation belt proton model AP8 min, the radiation belt electron model AE8 min, and the solar proton long-term statistical model ESP are used to analyze the space environment radiation dose of the satellite during the flight mission.

[0059] S2, based on the SHIELDOSE-2 model and the spatial radiation flux analyzed in S1, calculates the shielding effect of the simple geometric model, calculates the relationship between the absorbed dose and the thickness of the aluminum material, and obtains the dose-depth curve under the geometric shielding of the aluminum material.

[0060] S3, using SolidWorks software to perform equivalent simplification on the satellite 3D model to obtain a simplified satellite shape model, such as Figure 2 The center point in the model is selected as the point to be measured.

[0061] S4, taking the coordinates of the point to be measured obtained based on S3 as the center point, performing space grid division by the icosahedron division method, obtaining vector rays emitted from the center, and evenly dividing the space grid based on these vector rays.

[0062] S5, based on the simplified satellite model constructed in S3 and the vector rays constructed in S4, performs intersection analysis on the satellite model and the vector rays, calculates the intersection thickness between each vector ray and the satellite component module one by one, and combines the material density of the module to equate the thickness of each intersection material in the direction of each vector ray to the thickness of the aluminum material.

[0063] S6, based on the dose-depth curve calculated by S2 and the equivalent aluminum material thickness in each vector ray direction calculated by S5, the equivalent aluminum material thickness in each direction is converted into the dose value on the grid unit, and finally the weighted sum is performed to obtain the dose distribution and total dose value of the final test point. The evaluation results show that with the increase in the number of vector rays, the total dose calculation result tends to be stable. The calculation result of the icosahedron partitioning method is 13.76krad when the number of vector rays is 42; as the number of vector rays increases, the calculated value continues to decrease, and when the number of vector rays is 162, the calculation result is 13.46krad; when the number of vector rays is 642, it tends to be stable, and the calculation result is 13.14krad; when the number of vector rays is 2562, the calculation result is 13.18krad.

[0064] S7, based on the dose distribution of the final test point obtained in S6, the total dose protection analysis is performed. Taking the three-dimensional total dose calculation result of the icosahedron partitioning method when the number of vector rays is 642 as an example, the total dose distribution value of each azimuth angle is obtained, such as Figure 3The characteristics of total dose distribution are analyzed by combining the three-dimensional shape model. The weak directions of total dose at the measurement points in the three-dimensional dose distribution reaction shape model are the upper and lower planes and the front and back planes. The analysis results can provide accurate guidance for targeted total dose reinforcement.

[0065] Preferably, the specific steps of regular icosahedron space grid division in step S4 include:

[0066] S41, place the regular icosahedron symmetrically in the coordinate system, with its vertices all on the coordinate axis, and the coordinates of its center point are (0, 0, 0). Assuming the radius of its circumscribed sphere is r, the coordinates of each vertex are (±m, 0, ±n), (0, ±n, ±m), (±n, ±m, 0), where the calculation formulas for m and n are (1):

[0067]

[0068] like Figure 4 As shown, the 7 faces intersecting with the first and second hexagrams are analyzed. The seven faces are s1, s2, s3, s4, s5, s6, and s7. The vertices of the 7 faces are numbered, and an array is constructed to sequentially store the vertex coordinates of the 7 faces, which are stored as an array {(M, P1, P2), (M, P2, P3), (M, P3, P4), (M, P4, P5), (M, P5, P1), (P5, P1, P6), (P3, P2, P7)}.

[0069] S42, based on the array obtained in S41, calculates the coordinates of the endpoints of each ray vector in a single face. Since there are common edges between faces, the ray vector endpoints on the AC edge are not calculated in the single face calculation, which avoids repeated calculations and improves calculation efficiency. Improving the calculation efficiency of vector rays is conducive to programming implementation, such as Figure 5 As shown, only the coordinates of the endpoints within the dotted box are calculated. The calculation formulas are (2) and (3):

[0070]

[0071]

[0072] In formulas (2) and (3), A, C, and B are the vertices of the face, n is the number of iterative divisions in the regular icosahedron division, and u is The unit vector of The unit vector, i is the length in the u direction, j is the length in the v direction, (x ij ,y ij , z ij ) represents the coordinates of the endpoint of the ray vector with vector coordinates (i, j) on the surface, and iu represents (x ij ,y ij , z ij) is the vector value in the u direction, jv represents (x ij ,y ij , z ij ) is the vector value in the v direction.

[0073] S43, repeat the operations of steps S41-S42, and substitute the vertex coordinates of the second to seventh faces stored in the array into A, B, and C in turn to calculate the ray vector endpoints, so as to obtain the ray vector endpoints of the seven faces.

[0074] S44, using the symmetry properties of the regular icosahedron to traverse the coordinate points obtained above, and divide the points into three groups during the copying process, wherein each point in the first and second hexagrams needs to be symmetrically copied three times to obtain four points; the points on the oxy and oxz coordinate planes need to be symmetrically copied once to obtain two points; the remaining points are discarded.

[0075] If the point is within the first and second quadrants, it is replicated three times symmetrically, and the formula is (4):

[0076]

[0077] In formula (4), (x, y, z) represents the coordinates of a point in the first and second quadrants, where y>0, z>0, p and q are the replication coefficients of y and z respectively, (x pq ,y pq , z pq ) represents the coordinates of the vertex after three symmetric copies. pq is the formula expression of "each point in the first and second hexagrams needs to be symmetrically copied three times" in this copying process. The value of pq is in the set {-1, 1}, and there are four cases in total. When pq is 1, the coordinate xyz remains unchanged. When p is -1, it means that the coordinate xyz is copied to the coordinate x-yz with oxz as the symmetry plane; similarly, the copied coordinates xy-z and xyz are obtained.

[0078] If the point is on the oxy coordinate plane, it is copied symmetrically once, and the formula is (5):

[0079]

[0080] In formula (5), (x, y, 0) represents the coordinates of a point on the oxy coordinate plane, where y>0, (x p ,y p , 0) represents the vertex coordinates after symmetrical replication once.

[0081] If the point is on the oxz coordinate plane, it is copied symmetrically once, and formula (6) is:

[0082]

[0083] In formula (6), (x, 0, z) represents the coordinates of a point on the oxz coordinate plane, where z>0, (x q , 0, z q ) represents the vertex coordinates after symmetrical replication.

[0084] S45, obtain the endpoints of the ray vector of the regular icosahedron divided by n iterations. Since the coordinates of the center point of the regular icosahedron are (0, 0, 0), the ray vector is obtained by the endpoints of the ray vector and the origin of the coordinates, and finally the ray vector of the regular icosahedron divided by n iterations is obtained.

[0085] Preferably, the equivalent aluminum material thickness calculation in step S5 is expressed as formula (7):

[0086]

[0087] In formula (7): t o is the equivalent thickness in the direction of the vector ray, k is the number of intersection segments in the direction of the vector ray, t x is the thickness of the x-th intersection segment in the direction of the vector ray, ρ x is the density of the xth intersection segment in the direction of the vector ray, ρ Al is the density of aluminum alloy material, D tatal is the total dose of the test point, D sphere is the dose deposition function at the center of a solid sphere with radius as a variable.

[0088] Preferably, the equivalent aluminum material thickness calculation in step S5 is expressed as formula (2):

[0089]

[0090] In formula (8): D tatal is the total dose of the test point, δ o is the solid angle in the direction of the vector ray, m is the number of rays, D sphere is the dose deposition function at the center of a solid sphere with radius as a variable.

[0091] As a preferred embodiment of the present invention, various materials existing in the satellite can be selected in step S2. In order to facilitate unified calculation, the material generally specified is aluminum, but it can also be replaced with other materials. No matter what material it is, only the thickness is concerned, and the conversion relationship between materials depends on the density.

[0092] As a preferred embodiment of the present invention, step S7 also includes: if the total dose protection requirement of the device is not met, the satellite model is reinforced and optimized according to the three-dimensional distribution of the dose, and the total dose is recalculated in step S3.

[0093] The above-mentioned embodiments only express several implementation methods of the present invention, and the description is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. For ordinary technicians in this field, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for evaluating the equivalent total dose of a micro-nano satellite based on a fan angle, characterized in that: The following steps are involved: S1, based on the satellite's preset flight orbit and mission duration, uses the space radiation environment model to analyze the space radiation flux received by the satellite during its lifespan; S2, calculating the shielding effect of the simple geometric model based on the SHIELDOSE-2 model and the spatial radiation flux analyzed in step S1, and obtaining a dose-depth curve under the geometric shielding of the specified material; S3, obtain the simplified shape model of the satellite by using the equivalent simplified satellite three-dimensional model, and obtain the coordinates of the total dose test point according to the position of the sensitive chip in the satellite three-dimensional model; S4, taking the coordinates of the point to be measured obtained based on step S3 as the center point, obtaining vector rays emitted from the center point by a space grid division method, and dividing the space grid based on these vector rays; S5, performing intersection analysis based on the satellite shape model constructed in step S3 and the vector ray set constructed in step S4, and calculating an equivalent intersection thickness; S6, based on the dose-depth curve calculated in step S2 and the equivalent intersection thickness in each vector ray direction calculated in step S5, the dose value in each vector ray direction is calculated to obtain the three-dimensional distribution of the dose, and the total dose value of the test point is obtained by weighted summation; S7, performing total dose protection analysis based on the three-dimensional distribution of the dose at the test point and the total dose value obtained in step S6; comparing the total dose value with the total dose threshold of the device at the test point to determine whether the total dose protection requirement is met.

2. A method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 1, characterized in that The space grid division method in step S4 is selected from the equal angle division method, the equal solid angle division method or the regular icosahedron space grid division method.

3. A method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 2, characterized in that Outputs different numbers of vector rays depending on the number of divisions.

4. A method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 2, characterized in that The vector rays output by the calculation are evenly distributed.

5. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 1, characterized in that The calculation of equivalent intersection thickness in step S5 specifically calculates the intersection thickness of each ray and the satellite shape model component module, and combines the material density of the module to make each intersection material thickness in each vector ray direction equivalent to the thickness of the material specified in step S2.

6. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 1, characterized in that The calculated total dose distribution reflects the total dose distribution values ​​in all directions at the test point, which can provide a basis for total dose protection.

7. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 1, characterized in that Step S7 also includes: if the total dose protection requirement of the device is not met, the satellite model is reinforced and optimized according to the three-dimensional distribution of the dose, and the total dose is recalculated in step S3 again.

8. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 2, characterized in that The depth in the dose-depth curve is the thickness of the specified material.

9. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 2, characterized in that The designated material is the main material of the satellite, and is selected from aluminum or titanium alloy.

10. The method for evaluating the equivalent total dose of a micro-nano satellite fan angle according to claim 2, characterized in that The icosahedron space grid division method specifically comprises the following steps: S41, place the regular icosahedron symmetrically in the coordinate system, with its vertices all on the coordinate plane, and the coordinates of its center point are (0, 0, 0). Assuming the radius of its circumscribed sphere is r, the coordinates of each vertex are (±m, 0, ±n), (0, ±n, ±m), (±n, ±m, 0), where the calculation formulas for m and n are (1): Take the 7 faces that intersect with the first and second hexagrams for analysis, and construct an array to sequentially store the vertex coordinates of the 7 faces; S42, based on the array obtained in S41, calculates the coordinates of the endpoints of each ray vector in a single face. Since there are common edges between faces, the ray vector endpoints on the AC edge are not calculated in the single face calculation, which avoids repeated calculations, improves calculation efficiency, and is conducive to programming implementation. The calculation formulas are (2) and (3): In formulas (2) and (3), A, C, and B are the vertices of the face, t is the number of iterations in the icosahedron partitioning, and u is The unit vector of The unit vector, i is the length in the u direction, j is the length in the v direction, (x ij ,y ij , z ij ) represents the coordinates of the endpoint of the ray vector with vector coordinates (i, j) on the surface, and iu represents (x ij ,y ij , z ij ) is the vector value in the u direction, jv represents (x ij ,y ij , z ij )The vector value in the v direction; S43, repeating steps S41-S42, substituting the coordinates of the seven-face vertices stored in the array into A, B, and C in turn to calculate the ray vector endpoints, and obtaining the ray vector endpoints of the seven faces; S44, using the symmetry property of the regular icosahedron to traverse the coordinate points obtained above, and in the process of copying, the points are divided into three groups, wherein each point in the first and second hexagrams needs to be symmetrically copied three times to obtain four points; the points on the oxy and oxz coordinate planes need to be symmetrically copied once to obtain two points; and the remaining points are discarded; If the point is within the first and second quadrants, it is replicated three times symmetrically, and the formula is (4): In formula (4), (x, y, z) represents the coordinates of a point in the first and second quadrants, where y>0, z>0, p and q are the replication coefficients of y and z respectively, (x pq ,y pq , z pq ) represents the coordinates of the vertex after three symmetrical copies; pq is the formula expression of "each point in the first and second hexagrams needs to be copied symmetrically three times" in this copying process. The value of pq is in the set {-1, 1}, and there are four cases in total. When pq is 1, the coordinate xyz remains unchanged. When p is -1, it means that the coordinate xyz is copied to the coordinate x-yz with oxz as the symmetry plane; similarly, the copied coordinates xy-z and xyz are obtained; If the point is on the oxy coordinate plane, it is copied symmetrically once, and the formula is (5): In formula (5), (x, y, 0) represents the coordinates of a point on the oxy coordinate plane, where y>0, (x p ,y p , 0) represents the vertex coordinates after symmetrical replication once; If the point is on the oxz coordinate plane, it is copied symmetrically once, and formula (6) is: In formula (6), (x, 0, z) represents the coordinates of a point on the oxz coordinate plane, where z>0, (x q , 0, z q ) represents the vertex coordinates after symmetrical replication; S45, obtain the endpoints of the ray vector of the regular icosahedron divided t times by iteration. Since the coordinates of the center point of the regular icosahedron are (0, 0, 0), the ray vector is obtained by the endpoints of the ray vector and the origin of the coordinates, and finally the ray vector of the regular icosahedron divided t times by iteration is obtained.

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