Method for validating spacecraft design based on structured sampling optimization of ugkwp method

Abstract

This invention discloses a method for verifying spacecraft design based on structured sampling optimization of the UGKWP method, relating to the field of spacecraft design. The method includes: S1, using the designed shape data of the spacecraft as input data for a three-dimensional unstructured UGKWP solver; S2, the three-dimensional unstructured UGKWP solver performs simulation calculations based on the input data to obtain theoretical values ​​of the external flow field of the spacecraft in a multi-scale environment; S3, verifying the spacecraft's performance based on the theoretical values ​​obtained in S2 to guide the next step of spacecraft shape design optimization; wherein, during the solving process of the three-dimensional unstructured UGKWP solver, the macroscopic quantities obtained at the current time step are resampled. P samp The pseudo-random numbers corresponding to the particles are replaced with sample sequences generated by structured sampling to reduce sample variance. This invention, targeting the characteristics of the 3D unstructured UGKWP solver, introduces a structured sampling method into the generation process of particle microscopic properties, reducing the overall variance of UGKWP and saving memory and computational resources.

Description

Technical Field

[0001] This invention relates to the field of spacecraft design. More specifically, this invention relates to a method for validating spacecraft designs based on the structured sampling optimization UGKWP method. Background Technology

[0002] As hypersonic vehicles become increasingly complex in shape, their external flow fields exhibit multi-scale and multi-velocity domain characteristics. To simultaneously simulate multi-scale and multi-velocity flow fields in rarefied environments, particle-based or semi-particle-based methods such as DSMC and UGKWP are necessary. These methods require sampling the velocity, position, and flight time of simulated particles using random numbers generated by pseudo-random number functions. This random sampling process inevitably introduces statistical noise, which typically needs to be reduced by increasing the scale of the simulated particles, but this undoubtedly leads to increased memory and computational resource consumption. Summary of the Invention

[0003] One object of the present invention is to solve at least the above-mentioned problems and / or defects, and to provide at least the advantages described below.

[0004] To achieve these objectives and other advantages of the present invention, a method for verifying spacecraft designs based on the structured sampling optimization UGKWP method is provided, characterized by comprising: S1. Use the spacecraft's designed external shape data as input data for the 3D unstructured UGKWP solver. S2, the three-dimensional unstructured UGKWP solver performs simulation calculations based on input data to obtain the theoretical values ​​of the external flow field of the spacecraft in a multi-scale environment; S3. Verify the spacecraft's performance based on the theoretical values ​​obtained in S2 to guide the next step of spacecraft shape design optimization; In the solution process of the 3D unstructured UGKWP solver, the macroscopic quantities obtained at the current time step are resampled. P samp The pseudo-random numbers corresponding to the particles are replaced with sample sequences generated by structured sampling to reduce sample variance.

[0005] Preferably, the sample sequence adopts the pseudo-Monte Carlo method. Halton This is achieved through a sequence, and in a d-dimensional space, the first d prime numbers are selected to form a radix matrix, thus creating multidimensional data points. Represented as: In the above formula, The inverted root of data point a under cardinum b; When the number of sample points is fixed, if the first dimension is uniformly distributed when generating a multidimensional low-discrepancy sequence, then... Hammersley Multidimensional data points in the sequence Represented as: In the above formula, N is the total number of samples in each dimension.

[0006] Preferably, the sample sequence is implemented using a Latin hypercube, and a random permutation sequence is generated for each dimension d. π d The Latin hypercube sampling formula is characterized by the following equation: In the above formula, express d The first dimension k A structured number, N The total number of samples for each dimension. U d,k These are uniformly distributed random numbers.

[0007] Preferably, in the unstructured solver, the geometric meshes of different shapes are split into several tetrahedral meshes, so as to perform calculations within the tetrahedral meshes. P samp The position of the particle is sampled; In the tetrahedral mesh ABCD, the particle positions in three-dimensional space are... It is characterized by the following formula: In the above formula, r 1. r 2. r 3 are three (0,1) intervals generated using quasi-Monte Carlo or Latin hypercube, and are uniformly distributed structured sequences. , , , These represent the locations of points A, B, C, and D of the tetrahedral mesh. P Let be any point in the three-dimensional space enclosed by tetrahedron ABCD.

[0008] Preferably, P samp The particle velocity satisfies the condition that the mean value is a macroscopic quantity of the grid. The variance follows a normal distribution. R The gas constant is... T For temperature, then for r 1. r 2. r3. The following transformation is used to obtain a structured sequence that satisfies a normal distribution. n i ,and i =1, 2, 3: .

[0009] This invention includes at least the following beneficial effects: This invention, targeting the characteristics of the 3D unstructured UGKWP solver, introduces a structured sampling method into some (…) P samp The generation of particle microscopic properties (velocity and position) reduces the overall variance of UGKWP, enabling the method to capture non-equilibrium properties of the flow field with fewer particles, thus saving memory and computational resources.

[0010] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0011] Figure 1 This is a schematic diagram of the mesh particle position sampling of the tetrahedron ABCD of the present invention; Figure 2 A schematic diagram of particle spatial distribution when using pseudo-random number sampling in existing technology; Figure 3 The invention employs Halton A schematic diagram of the spatial distribution of particles in a sequence subjected to structured sampling; Figure 4 The invention employs Hammersley A schematic diagram of the spatial distribution of particles in a sequence subjected to structured sampling; Figure 5 This is a schematic diagram of the spatial distribution of particles using Latin hypercube for structured sampling in this invention. Figure 6 In the verification example of this invention, pseudo-random numbers are used. Halton sequence, Hammersley A comparative diagram showing the variation of star-shaped dispersion with the number of samples in four methods: sequence, Latin hypercube, and stellar dispersion. Figure 7 In the verification examples of this invention, a standard distribution, a random sequence, and LHS sequence, Hammersley A diagram showing the probability density distribution comparison of 100 normally distributed data obtained from four different random sequences; Figure 8This is a schematic diagram of the macroscopic quantity distribution when the number of particles is 100 in 100 independent experiments. Figure 9 This is a schematic diagram of the macroscopic quantity distribution when the number of particles is 500 in 100 independent experiments. Figure 10 This is a schematic diagram of the macroscopic quantity distribution when the number of particles is 1000 in 100 independent experiments. Figure 11 In 100 independent experiments, Kn =0.001 Sod Comparison of shock tube density and velocity distribution; Figure 12 In 100 independent experiments, Kn =10.0 Sod Comparison of shock tube density and velocity distribution. Detailed Implementation

[0012] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0013] The method used by the three-dimensional unstructured UGKWP solver employed in this invention (referred to as the UGKWP method) involves two types of particles in its simulation process: those that did not collide in the previous time step. P left Particles and macroscopic quantities resampled based on the current time step P samp Particles, the former characterizing the non-equilibrium properties of the flow field, whose positions and spatial distributions cannot be predicted in advance; the latter obtained from macroscopic quantities sampling, whose positions and velocities both conform to specific distributions. Therefore, if it is possible to reduce [the flow] based on macroscopic quantities... P samp By adjusting the sampling variance of particles, the sampling size can be reduced while maintaining the overall randomness of the UGKWP method, thereby reducing memory requirements.

[0014] In practical applications, this invention, taking into account the characteristics of the 3D unstructured UGKWP solver, introduces a structured sampling method into some parts. P samp In the process of generating particle microscopic properties (velocity and position), the structured sampling method used to reduce the overall variance of UGKWP is implemented as follows: By generating a more uniform sample sequence according to certain pre-defined rules, statistical noise can be reduced and convergence speed improved without affecting the randomness of the Monte Carlo simulation. This sampling method is called structured sampling. In the 3D unstructured UGKWP solver, P sampThe distribution of particle positions and velocities both satisfy specific "pre-defined rules." The positions of these particles are uniformly distributed within the grid, while their velocities satisfy a mean value based on the macroscopic quantities of the grid. (R is the gas constant, T is the temperature) represents a normal distribution with variance. Therefore, the main work of this invention is to replace P with sample sequences generated by structured sampling. samp The pseudo-random numbers used in the particle position and velocity sampling process reduce the overall variance of the 3D unstructured UGKWP solver.

[0015] Quasi-Monte Carlo (QMC) and Latin hypercube sampling (LHS) both fall under the category of structured sampling. The QMC method uses low-discrepancy sequences instead of computer-generated pseudo-random numbers, allowing samples to cover the sampling space more evenly. The radial inverse function is the fundamental function for generating low-discrepancy sequences; its function is to transform an integer c into a decimal with a base b distributed in the interval (0,1). The radial inverse function of c with base b can be expressed as: (1) In the above formula, a i express c In base b The coefficients below satisfy 0 ≤ a i < b If for a i Taking the reciprocal will result in a decimal. : (2) In the above formula, for c In base b The roots are turned downwards.

[0016] If the expansion factor is a i If we consider it as a vector, the above expression can be transformed into the following form, and matrix M is called the generating matrix.

[0017] (3) To avoid periodic repetition of sequences across different dimensions and reduce the correlation of datasets, prime numbers are typically used as the cardinality of the inverse root function, and the generating matrix M is calculated based on the inverse root function. Taking the commonly used Halton sequence as an example, this sequence is constructed by using coprime cardinality across all dimensions and generating mutually independent, low-discrepancy sequences based on the inverse root function to achieve multidimensional sequence construction. For d 3D space, usually choosing the first dimensional space d A base matrix composed of prime numbers, multidimensional data points The form can be expressed as: In the above formula, For data points a The inverted root under the base b; If the number of sample points is fixed, when generating a multidimensional low-discrepancy sequence, the first dimension can be uniformly distributed, resulting in a multidimensional data point distribution. The form can be expressed as: In the above formula, N is the total number of samples in each dimension. This form of low-discrepancy sequence is... Hammersley Sequence. Since the data in the first dimension is uniformly distributed, there is no need to calculate the inverted root function, therefore Hammersley The efficiency of sequence generation is higher than Halton The sequence is higher, and it can also reduce the correlation between data of different dimensions when there are few dimensions.

[0018] Another sampling method that can reduce sample variance is stratified sampling. The most basic stratified sampling divides the entire data space evenly into several intervals based on the number of samples to be sampled, and then randomly samples a data point from each interval. This sampling method ensures that each layer is sampled, avoiding sample clustering, and that sampling from different intervals is independent, ensuring the randomness and comprehensiveness of the samples. Currently, a Latin hypercube sampling method based on the stratified sampling idea is suitable for high-dimensional problems. In this method, the data space is evenly divided into several equally probable intervals and randomly arranged, thus ensuring the non-correlation between dimensions and allowing the data to be evenly and independently distributed in the multidimensional space. Specifically, if the number of samples to be sampled for each dimension d is N, then each dimension will be divided into equally probable intervals as represented by the following formula. (4) For each dimension d, a random permutation sequence π also needs to be generated. dThis sequence contains the number of each sub-interval. For example, for a two-dimensional sampling problem with 5 samples, the two random permutations can be in the form of π1=(3,1,2,5,4) or π2=(4,2,3,5,1), where the data in the sequence are the interval numbers. The following formula (5) is the formula for Latin hypercube sampling, where U d,k Uniformly distributed random numbers. Randomly permuted sequence. π d The introduction of this feature eliminates the correlation between different dimensions, thus ensuring that sample points can be evenly distributed in multidimensional space.

[0019] (5) In the above formula, express d The first dimension k A structured number, N The total number of samples for each dimension. U d,k These are uniformly distributed random numbers.

[0020] The structured sampling method described in this invention can be used for position and velocity sampling in a 3D unstructured UGKWP solver. The mesh shapes involved in the unstructured solver include tetrahedrons, pyramids, triangular prisms, and hexahedrons. To achieve uniform position sampling in meshes of different shapes, the geometry needs to be divided into several tetrahedrons, and the sampling is performed within the tetrahedral mesh. P samp The position of the particles is sampled, and the number of samples for each sub-region needs to be determined based on the volume ratio during this splitting process.

[0021] In a tetrahedral mesh, the particle positions in three-dimensional space can be determined using the mesh vertex coordinates and a uniformly distributed structured sequence of three (0,1) intervals, such as... Figure 1 As shown. Assume point E is any point in the three-dimensional space enclosed by tetrahedron ABCD, O is the intersection of the extension of line segment AE with the base, B is the vertex of the base BDC of triangle, and S is the intersection of the extension of the base BO with side CD. Draw a plane through point E parallel to the base BDC, intersecting the three edges of the tetrahedron at points P, F, and G respectively; draw a straight line through point O parallel to side CD, intersecting the two sides of the base at points M and N respectively.

[0022] Based on the geometric similarity theorem, we can assume that the volume ratio between tetrahedron AFGP and tetrahedron ABCD is _____. r 1: (6) In the above formula, V AFGPLet be the volume of the tetrahedron AFGP. V ABCD Let be the volume of tetrahedron ABCD; Similarly, the area ratio between triangle BMN and triangle BDC can also be obtained. r 2. The length ratio between line segment CS and line segment CD r 3: (7) (8) In the above formula, S BMN Let BMN be the area of ​​triangle BMN. S BDC Let BDC be the area of ​​triangle BDC. l CS Let CS be the length of line segment CS. l CD Let CD be the length of line segment CD; Furthermore, the proportional relationship between the line segments can be obtained as follows: (9) In the above formula, l AE Let AP be the length of line segment AP. l AO Let be the length of line segment AO. l BO Let BO be the length of line segment BO; Given the coordinates of A, B, C, and D, based on the length relationship shown in equation (9), the general formula for calculating the coordinates of a random point in three-dimensional space can be obtained as follows: (10) In the above formula, Represents the spatial coordinates of each point. r 1. r 2. r 3 represents the structured sequence generated by the quasi-Monte Carlo or Latin hypercube methods.

[0023] Verification Example To test the performance of pseudo-random numbers and different structured sampling methods in position sampling of the 3D unstructured UGKWP solver, a method was designed as follows: Figures 2-5 The example shown illustrates this. The sampling region contains 80 sampled particles, and sampling the positions of simulated particles in the grid under two-dimensional conditions requires two uniformly distributed random numbers. From... Figure 2As can be seen, if pseudo-random numbers are used directly, a significant "clustering" phenomenon will occur during location sampling (the locations marked by red circles in the figure). Clustering causes sample points to be overly concentrated in certain areas, failing to fully cover the entire sample space, thus leading to problems such as increased variance and slower convergence. From... Figures 3-5 The structured sampling results clearly show that the sampled particles are more evenly distributed in the quadrilateral grid, and the clustering phenomenon is significantly reduced.

[0024] To quantitatively assess the uniformity of sample point coverage of the sampling space, the "star discrepancy" between the sample point distribution and the ideal uniform distribution is further calculated. Star Discrepancy It measures the data points within a unit hypercube [0,1]. d An important indicator of the properties of the distribution, its definition is: (11) In the above formula, d Represents the dimension of the data space. n For data points, x and x i These represent the first and second elements in the unit hypercube, respectively. i Data points, #{ i | x i ≤ x} indicates that the condition is satisfied. x i ≤ x The number of data points This indicates that the data is ideally and uniformly distributed in the data space [0,1]. d In the cumulative distribution function, sup represents the minimum upper bound of the dataset.

[0025] Furthermore, Figure 6 This paper illustrates the variation of the star-shaped discreteness of 3D data obtained by different sampling methods with the number of samples (100 discrete points in the space). The figure shows that when the number of sample points is small (10), the Latin hypercube sampling algorithm based on the hierarchical approach has the smallest discreteness. However, as the number of sample points increases (above 20), the discreteness using Halton sequences and... Hammersley The uniformity of the samples obtained from sequence computation also becomes increasingly better, and Hammersley The sample dispersion obtained from the sequence should be slightly smaller than Halton Sequence. From Figure 6 As can be seen, the star-shaped dispersion of the location distribution obtained by the structured sampling method is consistently superior to that of pseudo-random number sampling under different sample size conditions, which confirms... Figures 3-5 The conclusions are as follows.

[0026] P samp The particle velocity satisfies the condition that the mean value is a macroscopic quantity of the grid. (R is the gas constant, T is the temperature) represents a normal distribution with variance. P samp Particle sampling requires the use of normally distributed random numbers, but structured sampling methods cannot generate normally distributed sequences. Therefore, it is necessary to use the Box-Muller formula to explicitly generate normally distributed sequences from uniformly distributed sequences, as shown in equation (12). r i This represents a structured uniformly distributed sequence obtained through a structured sampling method. n i To satisfy a structured sequence that follows a normal distribution, and i =1, 2, 3. According to variance propagation, the variance of a normal sequence calculated using a structured uniformly distributed sequence will be smaller than the variance of a normal random number generated directly using uniformly distributed pseudo-random numbers.

[0027] Furthermore, Figure 7 This section presents a comparison of the probability density distributions of 100 normally distributed data points obtained from different random sequences. Figure 7 As can be seen, the mean and standard deviation of the normal random numbers calculated by the structured sequence are very close to the standard normal distribution, and the probability density distribution curve also matches the reference curve well. However, the probability density distribution of the normal random numbers generated by pseudo-random numbers has a very obvious difference from the reference curve.

[0028] (12) Furthermore, a two-dimensional uniform flow example demonstrates the introduction of structured sampling methods into... P samp The variation of the standard deviation of macroscopic quantities in the unit cell after sampling the particle position and velocity variables. The computational domain of this example is [0,1]×[0,0.01], uniformly discretized by 100×1 points. The initial condition is (ρ,u,v,T)=(1,1,0,1).

[0029] Furthermore, Figure 8 The statistical characteristics of the macroscopic quantity distribution of the unit cells after 1000 iterations under different particle numbers are shown. To demonstrate the stability characteristics of the macroscopic quantity distribution, Figures 8-10 This presents 100 independent calculation results obtained under three particle number conditions and different sampling methods. Figures 8-10 It can be observed that using Hammersley structured sequence pairs Psamp By sampling the spatial position and velocity of particles, the noise of the final macroscopic quantity distribution within the cell is significantly less than that of the pseudo-random number result. This conclusion can also be further extended to general examples with flow structures.

[0030] Furthermore, Figures 11-12 This demonstrates 100 sets of Knudsen numbers obtained using different sampling methods, with Knudsen numbers of 0.001 and 10.0. Sod Shock tube density and velocity distribution. From Figures 11-12 As can be seen, the distribution noise obtained using structured sampling is also lower than that obtained using pseudo-random numbers, and this difference is more pronounced at higher density locations. This is because in areas with relatively high density, P samp The proportion of particles is relatively large, so the effect of structured sampling is more significant.

[0031] In practical applications, the shape data of the spacecraft after design can be used as input data for the three-dimensional unstructured UGKWP solver. The three-dimensional unstructured UGKWP solver performs simulation calculations based on the input data to obtain the theoretical values ​​of the external flow field of the spacecraft in a multi-scale environment. Furthermore, the performance of the spacecraft is verified based on the obtained theoretical values ​​to guide the next step of spacecraft shape design optimization.

[0032] The above solution is merely an illustration of a preferred example and is not limited thereto. When implementing this invention, appropriate substitutions and / or modifications can be made according to the user's needs.

[0033] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. It can be applied to various fields suitable for the present invention. Other modifications can be readily made by those skilled in the art. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and examples shown and described herein.

Claims

1. A method for verifying spacecraft design based on the structured sampling optimization UGKWP method, characterized in that, include: S1. Use the spacecraft's designed external shape data as input data for the 3D unstructured UGKWP solver. S2, the three-dimensional unstructured UGKWP solver performs simulation calculations based on input data to obtain the theoretical values ​​of the external flow field of the spacecraft in a multi-scale environment; S3. Verify the spacecraft's performance based on the theoretical values ​​obtained in S2 to guide the next step of spacecraft shape design optimization; In the solution process of the 3D unstructured UGKWP solver, the macroscopic quantities obtained at the current time step are resampled. P samp The pseudo-random numbers corresponding to the particles are replaced with sample sequences generated by structured sampling to reduce sample variance.

2. The method for verifying spacecraft design based on the structured sampling optimization UGKWP method as described in claim 1, characterized in that, The sample sequences were obtained using pseudo-Monte Carlo methods. Halton This is achieved through a sequence, and in a d-dimensional space, the first d prime numbers are selected to form a radix matrix, thus creating multidimensional data points. Represented as: In the above formula, The root of data point a under cardinum b; When the number of sample points is fixed, if the first dimension is uniformly distributed when generating a multidimensional low-discrepancy sequence, then... Hammersley Multidimensional data points in the sequence Represented as: In the above formula, N is the total number of samples in each dimension.

3. The method for verifying spacecraft design based on the structured sampling optimization UGKWP method as described in claim 1, characterized in that, The sample sequence is implemented using a Latin hypercube, and a random permutation sequence is generated for each dimension d. π d The Latin hypercube sampling formula is characterized by the following equation: In the above formula, express d The first dimension k A structured number, N The total number of samples for each dimension. U d,k These are uniformly distributed random numbers.

4. The method for verifying spacecraft design based on the structured sampling optimization UGKWP method as described in claim 2 or 3, characterized in that, In the unstructured solver, geometric meshes of different shapes are split into several tetrahedral meshes to perform calculations within the tetrahedral meshes. P samp The position of the particle is sampled; In the tetrahedral mesh ABCD, the particle positions in three-dimensional space are... It is characterized by the following formula: In the above formula, r 1. r 2. r 3 are three (0,1) intervals generated using quasi-Monte Carlo or Latin hypercube, and are uniformly distributed structured sequences. , , , These represent the locations of points A, B, C, and D of the tetrahedral mesh. P Let be any point in the three-dimensional space enclosed by tetrahedron ABCD.

5. The method for verifying spacecraft design based on the structured sampling optimization UGKWP method as described in claim 4, characterized in that, P samp The particle velocity satisfies the condition that the mean value is a macroscopic quantity of the grid. The variance follows a normal distribution. R The gas constant is T For temperature, then for r 1. r 2. r 3. The following transformation is used to obtain a structured sequence that satisfies a normal distribution. n i ,and i =1, 2, 3: 。

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