Aged martensitic steel component prediction sampling method
By combining epitaxial and interpolation sampling techniques in the aging-state martensite steel composition prediction sampling method, the problem of inefficiency in traditional methods in material design and optimization is solved, and efficient and highly diverse sampling point generation is achieved.
Patent Information
- Application Number
- CN202510186061.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-19
- Publication Date
- 2025-05-30
AI Technical Summary
Traditional thermodynamic phase space calculation methods are inefficient and resource-consuming in the development of new materials and optimized performance of existing materials, making it difficult to meet the needs of rapid iteration and high accuracy.
A method for predicting and sampling of components of aging martensite steel is proposed. By obtaining the initial sampling point, determining the core sampling point, performing epitaxial and interpolated sampling, and performing alternatingly until the preset sampling number is reached.
The effective expansion of epitaxial and interpolated sampling points is achieved, the diversity and effectiveness of sampling points are improved, repeated experiments are avoided, and sampling efficiency is significantly improved.
Smart Images
Figure CN120068439A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of material calculation, and particularly to a method for predicting sampling of the composition of aged martensitic steel. Background Art
[0002] Traditional thermodynamic phase space calculation methods usually rely on a large number of manual iterations and experimental data verification, which makes the material design and optimization process time-consuming and inefficient. In industrial and scientific research, when facing complex systems, this method often fails to meet the requirements for rapid iteration and high accuracy. Especially in the scenarios of new material development and existing material property optimization, the traditional method is limited due to its low efficiency and high resource consumption. To solve this problem, in recent years, machine learning and computer-aided design technologies have been widely applied in materials science, especially in predicting material properties and optimizing material compositions. However, most of these methods focus on model training and still have deficiencies in exploring new phase space regions and sample collection efficiency. Therefore, a method capable of dynamically adjusting the sampling strategy in the phase space is needed to improve the sampling efficiency and effectiveness and reduce unnecessary repeated experiments. Summary of the Invention
[0003] In order to improve the sampling efficiency and effectiveness, the present invention proposes a method for predicting sampling of the composition of aged martensitic steel, and the method includes: Step S1, obtaining initial sampling points formed by the known composition of aged martensitic steel; Step S2, projecting the molar fraction of each sampling point as a vector into a low-dimensional phase space to determine the core sampling points among the sampling points, and performing extrapolation sampling based on the core sampling points; Step S3, performing interpolation sampling on the basis of the extrapolation sampling and dynamically adjusting the interpolation sampling points by calculating the driving force between the interpolation sampling points and the already generated sampling points; Step S4, alternately performing extrapolation sampling and interpolation sampling until the number of sampling points reaches a preset sampling quantity.
[0004] In one or more embodiments, projecting the molar fraction of each sampling point as a vector into a low-dimensional phase space to determine the core sampling points among the sampling points and performing extrapolation sampling based on the core sampling points includes: determining the molar fraction of the initial sampling point at a preset aging temperature; projecting the molar fraction of each sampling point as a vector into a low-dimensional phase space to obtain projection points, and calculating the center point of the projection points; calculating the projection point closest to the center point to determine the core sampling point, and using the convex hull algorithm to determine the boundary set of the core sampling point; extracting boundary sampling points from the boundary set according to a preset ratio, and generating new extrapolation sampling points respectively based on each boundary sampling point and the core sampling point.
[0005] In one or more embodiments, interpolation sampling is performed based on epitaxial sampling, and the interpolation sampling points are dynamically adjusted by calculating the driving force between the interpolation sampling points and the generated sampling points, including: randomly generating intermediate sampling points different from the components of the generated sampling points according to a preset martensitic steel composition range; calculating the mole fraction of the intermediate sampling points at the preset aging temperature according to a preset thermodynamic model and projecting it into the low-dimensional phase space to obtain intermediate sampling projection points; calculating the Euclidean distance between the intermediate sampling projection points and each generated sampling point with the mole fraction of the intermediate sampling projection points as vectors to determine the driving force between the intermediate sampling projection points and each generated sampling point; moving the intermediate sampling projection points based on the resultant driving force of the generated sampling points on the intermediate sampling projection points to determine the final interpolation sampling points.
[0006] In one or more embodiments, the mole fraction of each sampling point is used as a vector and projected into the low-dimensional phase space to obtain projection points, and the center point of the projection points is calculated, including: using the mole fraction of each sampling point as a vector and projecting it into a two-dimensional or three-dimensional space to obtain projection points; calculating the average value of the projection coordinates of the projection points in each dimension in the two-dimensional or three-dimensional space to determine the center point of the projection points.
[0007] In one or more embodiments, boundary sampling points are extracted from the boundary set according to a preset ratio, and new epitaxial sampling points are generated based on each of the boundary sampling points and the core sampling points, including calculating the epitaxial sampling points using the following formula: ; where, is the new epitaxial sampling point, is the boundary sampling point, λ is a random number and λ is a positive number less than 1, is the core sampling point.
[0008] In one or more embodiments, the preset ratio is determined by the number of boundary points in the boundary set and the number of generated sampling points.
[0009] In one or more embodiments, the Euclidean distance between the intermediate sampling projection points and each generated sampling point is calculated with the mole fraction of the intermediate sampling projection points as vectors to determine the driving force between the intermediate sampling projection points and each generated sampling point, including calculating the driving force using the following formula:
[0010] where, F is the driving force, in the low-dimensional phase space is the jth generated sampling point, is the intermediate sampling projection point, is the Euclidean distance between the intermediate sampling projection point and the generated sampling point j, and k is the number of generated sampling points.
[0011] In one or more embodiments, the aging martensitic steel composition prediction sampling method of the present invention further includes: configuring a boundary driving force for the intermediate sampling projection point, and configuring the boundary driving force to be much greater than the driving force between the intermediate sampling projection point and any generated sampling point; wherein, the boundary driving force is generated when any phase content in the intermediate sampling projection point approaches its preset upper limit or preset lower limit under the corresponding metal phase, and the direction of the boundary driving force is towards the direction of reducing the corresponding phase content.
[0012] In one or more embodiments, the aging martensitic steel composition prediction sampling method of the present invention further includes: configuring a maximum number of movements for the intermediate sampling projection point.
[0013] In one or more embodiments, moving the intermediate sampling projection point based on the resultant driving force of the generated sampling points on the intermediate sampling projection point to determine the final interpolation sampling point includes: moving the intermediate sampling projection point to the steady state position or the position where the maximum number of movements is reached based on the resultant driving force of the generated sampling points on the intermediate sampling projection point; selecting a preset number of generated sampling points closest to the intermediate sampling projection point; and averaging the intermediate sampling projection point and the preset number of generated sampling points to determine the final interpolation sampling point.
[0014] The beneficial effects of the present invention include: The present invention realizes the expansion of sampling points by combining extrapolation and interpolation, and the alternation of the two can promote and adjust each other, thereby effectively realizing the predictive sampling of sampling points with new components and new component ratios, thereby enhancing the diversity and effectiveness of sampling points, and being able to avoid repeated sampling, thus effectively improving the sampling efficiency. BRIEF DESCRIPTION OF THE DRAWINGS
[0015] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other embodiments can be obtained based on these drawings without creative efforts.
[0016] Figure 1 is the working flow chart of the aging martensitic steel composition prediction sampling method according to the embodiment of the present invention; Figure 2 is the sampling distribution schematic diagram using the method of the present invention and taking the aging martensitic steel of Ni3Ti phase, LAVES phase and austenite phase as the initial sampling points. DETAILED DESCRIPTION OF THE INVENTION
[0017] To make the objectives, technical solutions and advantages of the present invention more clear and understandable, the following further elaborates on the embodiments of the present invention in detail with reference to specific embodiments and the accompanying drawings.
[0018] It should be noted that all the expressions using "first" and "second" in the embodiments of the present invention are for distinguishing two entities or parameters with the same name but different identities. It can be seen that "first" and "second" are only for the convenience of expression and should not be construed as a limitation on the embodiments of the present invention. This will not be elaborated one by one in the subsequent embodiments.
[0019] In order to improve the sampling efficiency and the effectiveness of sampling to reduce unnecessary repeated experiments in the later stage, in an embodiment of the present invention, a sampling method for predicting the composition of age-hardened martensitic steel is proposed. As Figure 1 shown, the method includes: Step S1, obtaining the initial sampling points formed by the known compositions of age-hardened martensitic steel; Step S2, projecting the mole fraction of each sampling point as a vector into a low-dimensional phase space to determine the core sampling points among the sampling points, and performing extrapolation sampling based on the core sampling points; Step S3, performing interpolation sampling on the basis of the extrapolation sampling and dynamically adjusting the interpolation sampling points by calculating the driving force between the interpolation sampling points and the already generated sampling points; Step S4, alternately performing extrapolation sampling and interpolation sampling until the number of sampling points reaches the preset sampling quantity.
[0020] Specifically, maraging steel (or age-hardened martensitic steel) is a super-high-strength steel with a carbon-free or micro-martensite matrix that can produce precipitation hardening of intermetallic compounds during aging. Different from traditional high-strength steels, it is strengthened by the dispersion precipitation of intermetallic compounds instead of carbon, which endows it with some unique properties: high strength and toughness, low hardening index, good formability, simple heat treatment process, almost no deformation during aging, and excellent weldability. Each initial sampling point formed in Step S1 represents the composition of an age-hardened martensitic steel (including various elemental particles), and the vector representation form is as follows:
[0021] wherein, represents the i-th sampling point, represents the d-th constituent element in the i-th sampling point, that is, the composition; wherein, represents the lower limit of the component, represents the upper limit of the component.
[0022] In one embodiment, the mole fraction of each sampling point is used as a vector and projected into a low-dimensional phase space to determine the core sampling points among the sampling points, and extrapolation sampling is performed based on the core sampling points, including: determining the mole fraction of the initial sampling point at a preset aging temperature; using the mole fraction of each sampling point as a vector and projecting it into the low-dimensional phase space to obtain projection points, and calculating the center point of the projection points; calculating the projection point closest to the center point to determine the core sampling points, and using the convex hull algorithm to determine the boundary set of the core sampling points; extracting boundary sampling points from the boundary set according to a preset ratio, and generating new extrapolation sampling points based on each boundary sampling point and the core sampling points respectively.
[0023] Specifically, since the initial sampling points are known materials, the mass fraction of each component can be determined, and the mole fraction of each component can be determined through the conversion between mass and molar mass. Among them, the mole fraction represents the ratio of the amount of substance of a substance to the sum of the amounts of substance of each component. In this embodiment, the expansion of the sampling points is realized through the extrapolation method, and the core sampling points among the initial sampling points are found by projecting the mole fractions of each initial sampling point as vectors, so that other initial sampling points can be regarded as the extrapolation sampling of the core sampling. Thus, the extrapolation expansion is carried out with the core sampling point as the center, realizing the sampling with predictive properties, and effectively improving the effectiveness of the extrapolation sampling points. Among them, the effectiveness refers to the possibility that the sampling component can form the aging martensite with the corresponding metal phase in the experiment. More specifically, the extrapolation sampling is mainly used to expand the mole fraction of the generated sampling points, aiming to find more possible mole fractions, and it will not add new components. For this reason, in another embodiment, the present invention also proposes to perform interpolation sampling on the basis of the extrapolation sampling to realize the addition of new components. The specific steps are as follows: In one embodiment, interpolation sampling is performed on the basis of the extrapolation sampling, and the interpolation sampling points are dynamically adjusted by calculating the driving force between the interpolation sampling points and the generated sampling points, including: randomly generating intermediate sampling points different from the components of the generated sampling points according to the preset martensitic steel composition range; calculating the mole fraction of the intermediate sampling points at the preset aging temperature according to the preset thermodynamic model and projecting it into the low-dimensional phase space to obtain intermediate sampling projection points; using the mole fraction of the intermediate sampling projection points as vectors to calculate the Euclidean distance between the intermediate sampling projection points and each generated sampling point to determine the driving force between the intermediate sampling projection points and each generated sampling point; moving the intermediate sampling projection points based on the resultant driving force of the generated sampling points on the intermediate sampling projection points to determine the final interpolation sampling points.
[0024] Specifically, the preset martensitic steel composition range in this embodiment includes all the constituent elements of known martensitic steels with different metal phases and aging states at different aging temperatures, as well as some predicted elements that may be used as aging martensitic steels, and upper and lower limits of the content are set for each element. In one embodiment, the upper and lower limits are expressed in the form of element ratios. That is, the upper limit is 1, indicating that only one element is contained, and the lower limit is 0, indicating that a certain element is not contained at all. In actual use, the content of the corresponding element can only approach the upper and lower limits infinitely but cannot reach them. The main function of the upper and lower limits is to generate a boundary driving force in the subsequent process. This embodiment can form a molar fraction and / or constituent elements different from known aging martensitic steels; in addition, in this embodiment, the Euclidean distance between it and other sampling points is calculated by converting it into a vector of molar fractions, so as to adjust the molar fraction of the interpolation sampling point to generate the final interpolation sampling point. The determination of the molar fraction of the interpolation sampling point makes use of the experience of known sampling points to a certain extent. By alternately performing extrapolation sampling and interpolation sampling, the extrapolation sampling points and interpolation sampling points can promote and adjust each other, thereby improving the effectiveness of the sampling points.
[0025] In one embodiment, for the molar fraction of the interpolation sampling point, it can be optionally predicted by existing thermodynamic models and computer simulations.
[0026] In one embodiment, in step S2, the molar fraction of each sampling point is used as a vector to project into a low-dimensional phase space to obtain projection points, and the center point of the projection points is calculated, including: using the molar fraction of each sampling point as a vector to project into a two-dimensional or three-dimensional space to obtain projection points; calculating the average value of the projection coordinates of the projection points in each dimension in the two-dimensional or three-dimensional space to determine the center point of the projection points.
[0027] Specifically, since the element compositions between the sampling points are not completely the same, it is difficult to determine the center of the sampling points by using conventional processing methods. Therefore, the present invention proposes to analyze the sampling points from the perspective of the element composition ratio with the molar fraction of the sampling points as a vector, and determine the center point and thus the core sampling point in this way; and, in order to improve the calculation efficiency, this embodiment proposes to perform dimensionality reduction processing on the sampling points with the molar fraction as a vector, so as to quickly determine the center point. In one embodiment, it can be selected to project each sampling point to a two-dimensional or three-dimensional space for dimensionality reduction.
[0028] In one embodiment, in step S4, boundary sampling points are extracted from the boundary set according to a preset ratio, and new extrapolation sampling points are generated based on each boundary sampling point and the core sampling point, including calculating the extrapolation sampling points using the following formula: ; Among them, is the new extrapolation sampling point, is a boundary sampling point, λ is a random number and λ is a positive number less than 1, is a core sampling point.
[0029] In one embodiment, the preset ratio is determined by the number of boundary points in the boundary set and the number of generated sampling points. Combined with the subsequent driving force adjustment, this embodiment is used to ensure that the newly generated epitaxial sampling points conform to the distribution law of the initial sampling points, thereby improving the effectiveness of the sampling points.
[0030] In one embodiment, taking the mole fraction as a vector, calculate the Euclidean distance between the intermediate sampling projection point and each generated sampling point to determine the driving force between the intermediate sampling projection point and each generated sampling point, including calculating the driving force using the following formula:
[0031] where F is the driving force (vector), is the j-th generated sampling point in the low-dimensional phase space, is the intermediate sampling projection point, is the Euclidean distance between the intermediate sampling projection point and the generated sampling point j, and k is the number of generated sampling points. Specifically, in this embodiment Its expanded representation form is:
[0032] where, represents the mole fraction of the j-th sampling point in the low-dimensional phase space on the m-th phase. When projecting in a two-dimensional or three-dimensional phase space, the maximum value of m is 2 or 3 respectively.
[0033] In one embodiment, the sampling method for predicting the composition of age-hardened martensitic steel of the present invention further includes: configuring a boundary driving force for the intermediate sampling projection point, and configuring the boundary driving force to be much greater than the driving force between the intermediate sampling projection point and any generated sampling point; wherein, the boundary driving force is generated when any phase content in the intermediate sampling projection point approaches its preset upper limit or preset lower limit under the corresponding metal phase, and the direction of the boundary driving force is towards the direction of reducing the corresponding phase content. Specifically, the purpose of setting the boundary driving force is to ensure the effectiveness of the mole fraction of the randomly generated intermediate sampling projection point or during the subsequent movement process, and to avoid sampling points that do not conform to the actual situation. Among them, the phase content includes the mole fraction of each phase of the sampling point in the phase space.
[0034] In one embodiment, the sampling method for predicting the composition of age-hardened martensitic steel of the present invention further includes: configuring a maximum number of movements for the intermediate sampling projection point. Specifically, the purpose of configuring the maximum number of movements is to avoid the movement falling into an infinite loop. In one embodiment, the maximum number of movements can be configured to be 30 - 60 times.
[0035] In one embodiment, the intermediate sampling point is moved based on the driving force of the generated sampling point on the intermediate sampling point to determine the final interpolated sampling point, including: moving the intermediate sampling projection point to a steady-state position or a position at which the maximum number of moves is reached based on the driving force of the generated sampling point on the intermediate sampling projection point; selecting a preset number of generated sampling points closest to the intermediate sampling projection point; averaging the intermediate sampling projection point and the preset number of generated sampling points to determine the final interpolated sampling point. Specifically, by selecting a preset number of generated sampling points closest to the intermediate sampling projection point; this implementation realizes the correction of the interpolated sampling point by averaging, and can realize the addition of new components through the interpolated sampling points, which is helpful to explore new effective components.
[0036] Through the above embodiments, the present invention realizes the expansion of sampling points by extension and / or interpolation, and the alternation of the two can effectively realize the predictive sampling of new components and new component ratios, thereby improving the diversity and effectiveness of the sampling points.
[0037] Example 1 Using the above steps, a phase having a Ni3Ti phase, a LAVES phase and an austenite phase (i.e. Figure 2 The aged martensitic steel with FCC phase in the steel was taken as the initial sampling point, and the composition sampling was carried out at 500℃. The sampling results are as follows Figure 2 As shown. Figure 2 It can be seen that the sampling points are symmetrically distributed rather than randomly distributed, with a certain regularity, which can avoid the generation of a large number of invalid sampling points; at the same time, in view of the addition of driving force, in the process of generating sampling points by the method of the present invention, the distribution of sampling points in the phase space can also be monitored and adjusted in real time, so as to explore and utilize the phase space more efficiently. And avoid excessive concentration of dynamic sampling. The method of the present invention not only improves the comprehensiveness and balance of sample collection, but also significantly reduces the waste of computing resources. This method is particularly suitable for material systems with large parameter spaces and strong interdependence, such as alloy design, composite materials, and high-performance ceramics. In addition, by combining this technology with existing high-throughput computing methods, the speed and accuracy of material design can be further improved, opening up a new model of material research and development, and effectively supporting the needs of rapid material innovation and performance optimization.
[0038] The above are exemplary embodiments disclosed in the present invention, but it should be noted that various changes and modifications may be made without departing from the scope of the embodiments disclosed in the claims. The functions, steps and / or actions of the method claims according to the disclosed embodiments described herein do not need to be performed in any particular order. The serial numbers of the embodiments disclosed in the above embodiments of the present invention are for description only and do not represent the advantages and disadvantages of the embodiments.
[0039] Those of ordinary skill in the art should understand that the discussion of any of the above embodiments is only exemplary and is not intended to imply that the scope (including the claims) disclosed by the embodiments of the present invention is limited to these examples; under the concept of the embodiments of the present invention, the technical features in the above embodiments or different embodiments can also be combined, and there are many other variations in different aspects of the embodiments of the present invention as above, which are not provided in detail for the sake of brevity. Therefore, any omission, modification, equivalent replacement, improvement, etc. made within the spirit and principle of the embodiments of the present invention shall be included in the protection scope of the embodiments of the present invention.
Claims
1. A method for predicting and sampling the composition of aged martensitic steel, characterized in that: The method comprises: Obtaining the initial sampling point of the formation of the known aged martensitic steel composition; Projecting the mole fraction of each sampling point as a vector into a low-dimensional phase space to determine a core sampling point among the sampling points, and performing epitaxial sampling based on the core sampling point; Performing interpolation sampling based on the epitaxial sampling and dynamically adjusting the interpolation sampling point by calculating the driving force between the interpolation sampling point and the generated sampling point; The extrapolation sampling and the interpolation sampling are performed alternately until the sampling points reach the preset sampling number.
2. The composition prediction sampling method for aged martensitic steel according to claim 1, characterized in that: The mole fraction of each sampling point is projected as a vector into a low-dimensional phase space to determine a core sampling point among the sampling points, and an epitaxial sampling is performed based on the core sampling point, including: Determining the mole fraction of the initial sampling point at a preset aging temperature; Projecting the mole fraction of each sampling point as a vector into the low-dimensional phase space to obtain a projection point, and calculating the center point of the projection point; Calculate the projection point closest to the center point to determine the core sampling point, and use the convex hull algorithm to determine the boundary set of the core sampling point; Boundary sampling points are extracted from the boundary set according to a preset ratio, and new extension sampling points are generated based on each of the boundary sampling points and the core sampling points.
3. The composition prediction sampling method for aged martensitic steel according to claim 1 or 2, characterized in that: Performing interpolation sampling based on the epitaxial sampling and dynamically adjusting the interpolation sampling point by calculating the driving force between the interpolation sampling point and the generated sampling point includes: Randomly generate intermediate sampling points with different compositions from the generated sampling points according to a preset martensitic steel composition interval; Calculate the mole fraction of the intermediate sampling point at the preset aging temperature according to a preset thermodynamic model and project it into the low-dimensional phase space to obtain an intermediate sampling projection point; Calculating the Euclidean distance between the intermediate sampling projection point and each generated sampling point using the mole fraction of the intermediate sampling projection point as a vector to determine the driving force between the intermediate sampling projection point and each generated sampling point; The intermediate sampling projection point is moved in the low-dimensional phase space based on the driving force resultant of the generated sampling point on the intermediate sampling projection point to determine a final interpolated sampling point.
4. The composition prediction sampling method for aged martensitic steel according to claim 2, characterized in that: The mole fraction of each sampling point is projected as a vector into the low-dimensional phase space to obtain a projection point, and the center point of the projection point is calculated, including: The mole fraction of each sampling point is projected into a two-dimensional or three-dimensional space as a vector to obtain a projection point; The average value of the projection coordinates of the projection point in each dimension is calculated in a two-dimensional or three-dimensional space to determine the center point of the projection point.
5. The composition prediction sampling method for aged martensitic steel according to claim 2, characterized in that: Extracting boundary sampling points from the boundary set according to a preset ratio, and generating new extension sampling points based on each boundary sampling point and the core sampling point, including calculating the extension sampling points using the following formula: ； in, is the new epitaxial sampling point, is the boundary sampling point, λ is a random number and λ is a positive number less than 1, The core sampling point.
6. The composition prediction sampling method for aged martensitic steel according to claim 2 or 5, characterized in that: The preset ratio is determined by the number of boundary points in the boundary set and the number of generated sampling points.
7. The composition prediction sampling method for aged martensitic steel according to claim 3, characterized in that: The mole fraction of the intermediate sampling projection point is used as a vector to calculate the Euclidean distance between the intermediate sampling projection point and each generated sampling point to determine the driving force between the intermediate sampling projection point and each generated sampling point, including using the following formula to calculate the driving force: ； Where F is the driving force, is the jth generated sampling point in the low-dimensional phase space, is the middle sampling projection point, is the Euclidean distance between the intermediate sampling projection point and the generated sampling point j, and k is the number of generated sampling points.
8. The composition prediction sampling method for aged martensitic steel according to claim 7, characterized in that: The method further comprises: configuring a boundary driving force for the intermediate sampling projection point, and configuring the boundary driving force to be much greater than a driving force between the intermediate sampling projection point and any generated sampling point; The boundary driving force is generated when the content of any phase in the intermediate sampling projection point approaches a preset upper limit or a preset lower limit of the corresponding metal phase, and the direction of the boundary driving force is toward reducing the content of the corresponding phase.
9. The composition prediction sampling method for aged martensitic steel according to claim 3, characterized in that: The method further comprises: A maximum number of moves is configured for the intermediate sampling projection point.
10. The composition prediction sampling method for aged martensitic steel according to claim 9, characterized in that: The method further comprises: moving the intermediate sampling projection point based on the driving force of the generated sampling point on the intermediate sampling projection point to determine a final interpolation sampling point, comprising: Based on the driving force of the generated sampling points on the intermediate sampling projection point, the intermediate sampling projection point is moved to a steady-state position or a position when the maximum number of moves is reached; Selecting a preset number of generated sampling points closest to the intermediate sampling projection point; The intermediate sampling projection point and the preset number of generated sampling points are averaged to determine a final interpolated sampling point.