A method for analyzing the overall stress of a small celestial body

Abstract

The application discloses a kind of small celestial body overall stress analysis method, comprising the following steps: (1) for small celestial body, its boundary is, a spherical surface is set to its complete encirclement outside small celestial body, the three-dimensional space surrounded by spherical surface is recorded as;(2) the spherical surface is discretized, generates triangular grid, the area and unit normal vector of each triangle element are calculated, and are recorded as and;(3) the gravity vector of triangle element center is obtained, and is recorded as;(4) is substituted into formula (), and the resultant force that small celestial body is received is calculated.

Description

Technical Field

[0001] This invention belongs to the field of aerospace dynamics and space gravitational field calculation, and specifically relates to a method for analyzing the overall forces acting on a small celestial body. Background Technology

[0002] Modeling the dynamic environment of small celestial bodies (such as asteroids and comets) is a core prerequisite for the safe implementation of deep space exploration missions. Accurately understanding the overall forces acting on small celestial bodies in space is crucial for orbit prediction, attitude control, and collision risk assessment.

[0003] Currently, the mainstream methods for calculating the forces acting on small celestial bodies heavily rely on their internal density. Specifically, traditional Newtonian mechanics methods theoretically require knowledge of the density distribution of the small celestial body, using volume integrals to calculate its gravitational field and forces. Another common method is orbital dynamics inversion, which uses precise tracking data from the orbits of probes flying around or past small celestial bodies to infer their total mass. While this method avoids the difficulty of directly measuring density, it still requires assumptions about the internal density distribution after obtaining the total mass, and the resulting error is not negligible. Neither theoretical methods relying on internal density integrals nor orbital dynamics inversion methods solve the core problem of "directly obtaining the overall forces without prior internal information." Therefore, a new method is urgently needed that can directly calculate the overall forces acting on a small celestial body using the forces acting on its external gravitational field. Summary of the Invention

[0004] The purpose of this invention is to overcome the limitations of traditional methods that rely on the internal density distribution information of small celestial bodies, and to provide a method for calculating the overall force on a small celestial body using the gravitational vector of the external gravitational field. This method achieves accurate and efficient calculation of the overall force on irregularly shaped small celestial bodies by directly obtaining the gravitational vector at the boundary of the gravitational field.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for analyzing the forces acting on a small celestial body as a whole includes the following steps:

[0007] S1: Obtain the three-dimensional geometric model of the target small celestial body, and construct a smooth surface (such as a sphere) that completely surrounds the geometric model of the target small celestial body in space. Let the center of the sphere be located inside the small celestial body, and the radius of the sphere be... Three times the average size of small celestial bodies;

[0008] S2: The spherical surface described in S1 Discretize to generate a triangular grid. Calculate each triangular element area and unit normal vector And record and ;

[0009] S3: Obtain the triangle element gravitational vector on ,remember It reflects the combined effect of the gravitational pull generated by the target small celestial body's own mass and the external gravitational environment;

[0010] S4: Will Substitute into formula (A) to calculate the overall force. .

[0011] Furthermore, the specific steps in S1 for constructing and discretizing the sphere that completely surrounds the geometric model of the target small celestial body are as follows:

[0012] Step (11) defines a closed surface that completely surrounds the target celestial body in space, and discretizes the closed surface into a continuous triangular grid, denoted as... ;

[0013] Step (12) uses the spatial coordinates of the boundary triangular grid generated in step (11) as the theoretical position reference for setting up gravity measurement points or performing field calculations.

[0014] Furthermore, in step S2, the area of ​​each triangular element is calculated. With unit normal vector The specific steps are as follows:

[0015] Step (21) based on the triangular element Given the spatial coordinates of the three vertices, calculate its area. ;

[0016] Step (22) based on the triangular element The normal vector of the two edge vectors is obtained by performing a cross product operation, and then normalized to obtain the unit normal vector of the cell. ;

[0017] Step (23) Traverse all grid cells to obtain the area of ​​each triangular element. With unit normal vector .

[0018] Furthermore, the specific steps for obtaining the gravitational vector g in S3 are as follows:

[0019] Step (31) Obtain the triangular element The actual gravitational vector at that point is denoted as . ;

[0020] Step (32) assumes a parallel gravitational field with uniform direction and intensity acting on the outside of the small celestial body, whose vector is denoted as . ;

[0021] Step (33) calculates the triangular grid. The gravitational vector at that location ,Right now .

[0022] Step (34) traverses all grid cells to obtain the gravity vector. .

[0023] Furthermore, the specific steps for solving the overall force on the small celestial body in S4 are as follows:

[0024] Step (41) According to Gauss's theorem, the divergence of the gravitational field outside the small celestial body satisfies the Laplace equation, that is:

[0025] (1)

[0026] Meanwhile, inside the small celestial body, the gravitational field satisfies the Poisson equation:

[0027] (2)

[0028] in The gravitational constant is... This refers to the internal density of a small celestial body.

[0029] Step (42) is based on the identity:

[0030] (3)

[0031] For equation (3) in the volume that completely surrounds the small celestial body Integrating the above, we get:

[0032] (4)

[0033] Step (43) Applying the divergence theorem to the above equation, the first term on the left side of equation (4) can be simplified to:

[0034] (5)

[0035] Step (44) Substituting the Poisson equation into the volume integral of the first term on the right side of equation (4) yields:

[0036] (6)

[0037] Step (45) The volume integral of the second term on the right side of equation (4) can be converted into a surface integral:

[0038] (7)

[0039] Step (46) Further, equations (5), (6), and (7) can be substituted into equation (2) to obtain the overall force on the small celestial body. The calculation formula is as follows:

[0040] (A)

[0041] in,

[0042] Let be the overall force vector of the small celestial body in its external gravitational field;

[0043] It is the gravitational constant;

[0044] This is the gravitational vector at the boundary grid.

[0045] The unit normal vector of the boundary grid node;

[0046] This represents the area of ​​the boundary grid.

[0047] Area of ​​corresponding triangular elements calculated by integrating S2 and its unit normal vector The gravitational vector at the boundary grid obtained by S3 The corresponding formulas are ( ) within Substitute this into the calculation of the overall force on the small celestial body.

[0048] The above technical solution can achieve the following beneficial effects:

[0049] This invention defines the boundary of the external gravitational field of a small celestial body and applies the gravitational field integral formula to transfer the calculation domain of its overall force from the highly complex interior of the celestial body to the external space. This makes the final calculation result completely independent of the internal density distribution and structure of the small celestial body, thus eliminating the key error caused by the inaccuracy of the internal model in principle.

[0050] This invention reduces computational complexity from three dimensions to two dimensions by integrating the surface on a two-dimensional discrete grid, compared with traditional methods, thus significantly improving computational efficiency.

[0051] The discretized grid processing method proposed in this invention can flexibly adapt to the complex geometric characteristics of irregularly shaped small celestial bodies. It can directly derive the overall force based on the force conditions at the gravitational field boundaries, demonstrating significant engineering application value for small celestial bodies with unknown internal structures. Attached Figure Description

[0052] Figure 1 This is a schematic diagram of the small celestial body in the method described in this invention and a derivation diagram of the core formula (A);

[0053] Figure 2This is a flowchart of a simulation experiment using the asteroid Itokawa as an example.

[0054] Figure 3 This is a schematic diagram illustrating the results of a simulation experiment using the asteroid Itokawa as an example. Detailed Implementation

[0055] The invention will be further described below with reference to the accompanying drawings:

[0056] like Figure 1 As shown, the small celestial body in the method described in this invention is on the left. The diagram shows the derivation of the core formula (A) on the right.

[0057] like Figure 2 As shown, a method for calculating the overall force on a small celestial body using its external gravitational force includes the following steps:

[0058] Step (1) involves acquiring and processing the internal data of the small celestial body. The specific method for this step is as follows:

[0059] Step (11): A simulation experiment was conducted based on the Itokawa asteroid data published on NASA's official website, defining it as a celestial body. , preset For its actual quality The grid was discretized in three dimensions using Gmsh software according to the actual size, with the unit being standard meters. ;

[0060] Step (12): Calculate the coordinates of the centroid of each tetrahedral grid element. ( Represented as the first (each tetrahedral grid element), and the total mass Evenly distributed to each centroid point, defined as ;

[0061] Step (2) Construct the external gravity boundary spherical grid. The specific method for this step is as follows:

[0062] Step (21): Use Gmsh software to construct a sphere concentric with the geometric center of the asteroid to completely surround the celestial body. spherical surface Discretize it to generate a triangular grid. ;

[0063] Step (22), based on the triangular element Given the spatial coordinates of the three vertices, calculate its area. Coordinates of the centroid and unit normal vector ;

[0064] Step (23), traverse the sphere Given all the triangular elements, we can obtain the area. and unit normal vector ;

[0065] Step (3) Obtain the gravitational vector on the boundary surface. The specific method for this step is as follows;

[0066] Step (31), based on each tetrahedral grid unit described in step (1) The quality is Define the mass of the centroid of each triangular element in step (2) as a unit mass, and calculate its gravitational vector in sequence according to Newton's law of universal gravitation.

[0067] Step (32) involves superimposing the gravitational vectors generated by each tetrahedral grid element to obtain the result in the triangular element. The gravitational vector under the influence of the asteroid's overall mass ;

[0068] Step (4) involves superimposing an external parallel gravitational field. The specific method for this step is as follows:

[0069] Step (41), assume there is a field strength of external parallel gravitational field It applies to the entire computational region of the asteroid;

[0070] Step (42), the gravitational vector obtained in step (3) With gravitational field By performing vector superposition, we obtain triangular elements. The gravitational vector at that location ;

[0071] Step (43), traverse the sphere Repeat step (42) for all triangular elements to obtain the celestial body. On the sphere gravitational vector on ;

[0072] Step (5) is to solve for the overall force on the asteroid in the external parallel gravitational field. The specific method for this step is as follows:

[0073] Step (51), the boundary grid area obtained in step (2) With unit normal vector and the gravitational vector obtained in step (4) Substitute into the formula ( ), to obtain the overall force of the asteroid ;

[0074] Step (6) verifies the accuracy of the calculation method. The specific method for this step is as follows:

[0075] Step (61), according to Newton's second law, based on the total mass of the Itokawa asteroid... and gravitational field The theoretical value of the resultant force on it in the external field is calculated. And calculate The value, and compare it with the actual value. and gravitational field Compare;

[0076] like Figure 3 As shown, Figure (a) is a three-dimensional schematic diagram of the asteroid Itokawa, and Figure (b) is a schematic diagram of its cross-section calculation results. This is the final result of the calculation;

[0077] actual value (approximately) ) and theoretical value (for The two exhibit good consistency in both numerical and directional aspects. (approximately) ) and gravitational field (for The results show a high degree of agreement, verifying the accuracy of the method described in this invention.

[0078] This invention constructs a smooth, closed boundary around a small celestial body, discretizes it into a triangular lattice, and obtains the geometric properties (i.e., normal vector and area) of the triangular elements. Combining this with the gravitational vectors observed on these triangular elements, the overall force on the small celestial body is calculated. This method can flexibly adapt to the complex geometry of small celestial bodies, requiring no prior knowledge of the internal density distribution or any a priori assumptions about it. The overall force calculation can be achieved solely through the observable gravitational vectors on the closed surface of the small celestial body. Compared to the volume integral (triple integral) formula based on Newton's law of universal gravitation, this method uses a surface integral (double integral) formula with less computation, making the overall force calculation of small celestial bodies more efficient. It has significant potential engineering application value for small celestial bodies with unknown internal structures.

[0079] The above descriptions are all preferred embodiments of the present invention. For those skilled in the art, any modifications to the present invention in various equivalent forms without departing from the principle of the present invention shall fall within the protection scope of the appended claims.

Claims

1. A method for analyzing the forces acting on a small celestial body as a whole, characterized in that, The method is as follows: S1: Obtain the three-dimensional geometric model of the target celestial body, and construct a smooth surface, i.e., a sphere, that completely surrounds the geometric model of the target celestial body in space. The center of the sphere is located inside the small celestial body, and the radius of the sphere is... Three times the average size of small celestial bodies; S2: The spherical surface described in S1 Discretize to generate a triangular grid. Calculate each triangular element area and unit normal vector And record and ; S3: Obtain the triangle element gravitational vector on ,remember It reflects the combined effect of the gravitational pull generated by the target small celestial body's own mass and the external gravitational environment; S4: Will Substitute into formula (A) to calculate the overall force. .

2. The method for analyzing the overall force on a small celestial body according to claim 1, characterized in that: The method for constructing and discretizing the sphere that completely encloses the geometric model of the target small celestial body in S1 is as follows: Step (11) defines a closed sphere that completely surrounds the target celestial body in space, and discretizes the closed sphere into a continuous triangular grid, denoted as... ; Step (12) uses the midpoint of the boundary triangular grid generated in step (11) as the gravity measurement point, and extracts the theoretical position for surface integral calculation.

3. The method for analyzing the overall force on a small celestial body according to claim 1, characterized in that: The area of ​​each triangular element is calculated in S2. With unit normal vector The method is as follows: Step (21) for the discretized triangular elements Calculate the area of ​​this unit. And obtain its unit normal vector ; Step (21) Traverse all grid cells to obtain the area of ​​each triangular element. With unit normal vector .

4. The method for analyzing the overall force on a small celestial body according to claim 1, characterized in that: The gravity vector is obtained in S3. The method is as follows: Step (31) Obtain the triangular element The actual gravitational vector at that point is denoted as . ; Step (32) assumes there is an additional gravitational field acting outside the small celestial body, whose vector is denoted as . ; Step (33) involves the gravity vector described in step (31). With respect to the external gravitational field described in step (32) By superimposing the values, the triangular elements are calculated. The gravitational vector at that location ,Right now .

5. The method for analyzing the overall force on a small celestial body according to claim 1, characterized in that: The method for solving the overall force on the small celestial body in S4 is as follows: Step (41) According to Gauss's theorem, the divergence of the gravitational field outside the small celestial body satisfies the Laplace equation, that is: Meanwhile, inside the small celestial body, the gravitational field satisfies the Poisson equation: in The gravitational constant is... This refers to the internal density of a small celestial body. Step (42) is based on the identity: For equation (3) in the volume that completely surrounds the small celestial body Integrating the above, we get: Step (43) Applying the divergence theorem to the above equation, the first term on the left side of equation (4) can be simplified to: Step (44) Substituting the Poisson equation into the volume integral of the first term on the right side of equation (4) yields: Step (45) The volume integral of the second term on the right side of equation (4) can be converted into a surface integral: Step (46) involves substituting equations (5), (6), and (7) into equation (2) and rearranging the equations to obtain the overall force on the small celestial body. The calculation formula is as follows: in, Let be the overall force vector of the small celestial body in its external gravitational field; It is the gravitational constant; #imgpt45# represents the gravitational vector at the boundary grid. #imgpt46# is the unit normal vector of the boundary grid node; #imgpt47# represents the area of ​​the boundary grid. Integrate the corresponding triangular element area #imgpt48# and its unit normal vector #imgpt49# calculated by S2, and the gravitational vector #imgpt50# at the boundary grid obtained by S3, which correspond to #imgpt52# in formula (#imgpt51#), and substitute them into the calculation of the overall force on the small celestial body.

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