Overdetermined equation parallel random solving method and system for aircraft structure grid

Abstract

The invention discloses an aircraft structure grid-oriented overdetermined equation parallel random solving method and system, and relates to the technical field of grid generation, and the method comprises the steps: obtaining a numerical aircraft parameter model; generating a grid by adopting a variational harmonic partial differential equation on the basis of an aircraft parameter model, discretizing the variational harmonic partial differential equation, and establishing an overdetermined linear equation set taking a grid vertex coordinate value as an unknown number after discretization; performing low-rank approximate processing on the overdetermined linear equation set to obtain an overdetermined linear equation set after dimension reduction; a greedy random hybrid Kaczmarz algorithm is adopted to carry out parallel solution on the overdetermined linear equation set after dimension reduction, and an optimal solution is obtained; and mapping the optimal solution to a physical space to generate a hybrid grid, and importing the hybrid grid into an aircraft simulation platform for heat flow analysis. According to the method, parallel optimization is carried out on the solution of an overdetermined equation set generated by a grid, and rapid solution of an overdetermined linear equation set is realized through a greedy random hybrid Kaczmarz algorithm, so that the purpose of improving the grid quality is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of grid generation, and in particular to an overdetermined equation parallel random solution method and system for aircraft structure grids. Background Technique

[0002] The statements in this part only provide background technical information related to the present disclosure and do not necessarily constitute prior art.

[0003] The thermal protection system of hypersonic aircraft poses extremely high requirements for heat flux simulation, and it is necessary to accurately capture heat flux details with wavelengths in the centimeter to millimeter range, while the overall size of the aircraft usually reaches dozens of meters. To meet the needs of fine heat flux analysis, the scale of the simulation grid can be extended to tens of billions, hundreds of billions, or even trillions. Facing the processing requirements of such large-scale and high-quality simulations, traditional serial grid generation methods are no longer sufficient. In view of the characteristics of structured grids required in aerospace aircraft design, it is very important to efficiently generate high-quality structured grids with good orthogonality, regular shapes, and smooth transitions.

[0004] The Variational Harmonic Partial Differential Equation (VH-PDE) is a method that transforms a PDE into a functional extremum problem through the variational method. VH-PDE has the ability of adaptive grid refinement and can dynamically adjust the grid density according to the gradient change of the physical field or the requirements of specific regions, so as to generate high-quality grids, especially showing superiority at complex geometric boundaries. Therefore, the VH-PDE method is particularly suitable for scenarios that require accurate capture of physical phenomena and environments with complex heat flux distributions.

[0005] In the process of generating grids in VH-PDE, first, it is necessary to map the physical domain to the computational domain through a mapping function, and then use numerical methods to discretize the PDE into a system of equations. This system of equations is usually overdetermined, that is, the number of equations is much larger than the number of unknowns. Then, the least squares method or other methods need to be used to solve this overdetermined system of equations. Finally, structured grids are generated according to the solution results. This problem ultimately boils down to the solution of a large-scale linear system. Therefore, the solution of the overdetermined linear system is a key step in the VH-PDE method because it directly affects the quality, stability, and computational efficiency of grid generation. However, the solution of the overdetermined linear system consumes a large amount of time in the VH-PDE process, accounting for more than 70% of the total time. Therefore, how to effectively and quickly solve the overdetermined linear system becomes very important.

[0006] Since an overdetermined linear equation system has the characteristic that the number of equations is greater than the number of unknowns, an exact solution will not be generated. In actual use, the optimal approximate solution of the overdetermined linear equation system is usually used as the solution of the overdetermined linear equation system. The most common form of the optimal solution is the least squares solution, which finds the optimal solution by minimizing the Euclidean norm of the residuals.

[0007] Traditional methods for solving overdetermined linear equation systems include the least squares method, QR decomposition method, etc. However, when the data scale increases day by day, continuing to use the least squares method to solve the overdetermined linear equation system will face problems such as high computational complexity and memory limitations. Therefore, the least squares method is no longer applicable to the solution of large-scale overdetermined equation systems. Adding a randomization method to the solution process of the overdetermined linear equation system can improve the algorithm stability, reduce the computational complexity, and explore a wider solution space. Therefore, the randomized Kaczmarz algorithm (Random kaczmarz, RK) is a simple and effective iterative method for solving overdetermined linear equation systems. It accelerates the iterative process by randomly selecting hyperplanes, improves the convergence speed, and is applicable to the solution of large-scale sparse matrices. However, in the randomized Kaczmarz algorithm, if the selection probability is not adjusted according to the row norm, its convergence rate will be affected, and only single-row information is used in each iteration process, resulting in the convergence speed not reaching the ideal effect. Summary of the Invention

[0008] To overcome the deficiencies of the above-mentioned prior art, the present invention provides a parallel random solution method and system for overdetermined equations for aircraft structural grids, which performs parallel optimization on the solution of the overdetermined equation system generated by grid generation, and realizes the rapid solution of the overdetermined linear equation system through the greedy random hybrid Kaczmarz algorithm, so as to achieve the purpose of improving the grid quality and enhancing the efficiency and accuracy of aircraft heat flux analysis and calculation.

[0009] To achieve the above object, one or more embodiments of the present invention provide the following technical solutions: In a first aspect, the present invention provides a parallel random solution method for overdetermined equations for aircraft structural grids, including: Obtain a numerical aircraft parameter model; Based on the aircraft parameter model, generate a grid using the variational harmonic partial differential equation, and discretize the variational harmonic partial differential equation. After discretization, establish an overdetermined linear equation system with the grid vertex coordinate values as unknowns; Perform low-rank approximation processing on the overdetermined linear equation system to obtain a reduced-dimensional overdetermined linear equation system; Use the greedy random hybrid Kaczmarz algorithm to perform parallel solution on the reduced-dimensional overdetermined linear equation system to obtain the optimal solution; Map the optimal solution to the physical space to generate a hybrid grid, and import the hybrid grid into the aircraft simulation platform for heat flux analysis.

[0010] In a further technical solution, a random rangefinder is used to perform low-rank approximation processing on the overdetermined linear equations.

[0011] In a further technical solution, the parallel optimization of the solution process of the dimension-reduced overdetermined linear equations is as follows: The overdetermined linear equations are divided into multiple blocks by rows and distributed to multiple processors for joint execution. Each processor respectively executes the greedy random hybrid Kaczmarz algorithm for iterative solution.

[0012] In a further technical solution, the greedy random hybrid Kaczmarz algorithm is used to solve the dimension-reduced overdetermined linear equations to obtain the optimal solution. The specific steps are as follows: The total processor process collects the local solution vectors from each processor process to obtain the global solution vector; Each processor process uses the greedy random hybrid model to select the hyperplane and uses the caching method to update the local solution vector to obtain a new local solution vector; The total processor process collects the new local solution vectors from each processor process to obtain a new global solution vector; The Monte Carlo error estimation method is used for error estimation to obtain the error estimation value of the solution vector; Judge whether the error estimation value reaches the threshold. If so, calculate the true error value. If not, broadcast the new global solution vector to each processor process to continue the iteration; Judge whether the true error value reaches the threshold. If so, obtain the optimal solution. If not, broadcast the new global solution vector to each processor process to continue the iteration.

[0013] In a further technical solution, the specific method of using the greedy random hybrid model to select the hyperplane is as follows: Set the proportional value of the greedy method; Randomly generate a positive number less than or equal to 1; If the positive number is less than or equal to the set value, use the greedy method to select the hyperplane. If the positive number is greater than the set value, use the random method to select the hyperplane.

[0014] In a further technical solution, the caching method to update the local solution vector is as follows: There is no need to repeat the calculation of the product. Directly use the value in the cache. After the solution vector is updated, only multiply the data of the current row by the solution vector, and update the cache of the current row. Perform a full cache update every set number of iterations.

[0015] For a further technical solution, the Monte Carlo error estimation method is adopted for error estimation, and the error estimation value of the solution vector is specifically obtained as follows: randomly select multiple sample rows, calculate the residual of each sample, accumulate the squares of the residuals, and obtain the error estimation value of the solution vector by calculating the mean of the sum of the squared residuals and taking the square root.

[0016] In a second aspect, the present invention provides an overdetermined equation parallel random solution system for an aircraft structural grid, including: A model acquisition module, which is configured to: acquire a numerical aircraft parameter model; An equation system establishment module, which is configured to: generate a grid based on the aircraft parameter model by using a variational harmonic partial differential equation, discretize the variational harmonic partial differential equation, and establish an overdetermined linear equation system with the grid vertex coordinate values as unknowns; An equation system dimension reduction module, which is configured to: perform a low-rank approximation process on the overdetermined linear equation system to obtain a dimension-reduced overdetermined linear equation system; An equation system random solution module, which is configured to: perform parallel solution on the dimension-reduced overdetermined linear equation system by using a greedy randomized Kaczmarz algorithm to obtain an optimal solution; A grid generation module, which is configured to: map the optimal solution to the physical space to generate a hybrid grid, and import the hybrid grid into an aircraft simulation platform for heat flux analysis.

[0017] In a third aspect, the present invention provides a computer-readable storage medium, on which a computer program is stored, and when the program is executed by a processor, the steps in the overdetermined equation parallel random solution method for an aircraft structural grid as described in the first aspect are implemented.

[0018] In a fourth aspect, the present invention provides a computer device, including a memory, a processor, and a computer program stored on the memory and executable on the processor, and when the processor executes the program, the steps in the overdetermined equation parallel random solution method for an aircraft structural grid as described in the first aspect are implemented.

[0019] The above one or more technical solutions have the following beneficial effects: The present invention proposes a parallel optimization method for the greedy random Kaczmarz algorithm based on low-rank approximation. By improving the VH-PDE grid generation technology, the efficiency and accuracy of aircraft heat flux analysis are effectively enhanced. This method accelerates the solution process through parallel computing, significantly shortening the generation time of high-quality grids and enhancing the ability to quickly respond to changes in the flight environment; the generated grids are smoother and can accurately fit the complex aerodynamic shape of the aircraft, ensuring accurate modeling of the key structural boundaries in heat flux analysis; at the same time, this method reduces the consumption of computing resources, making it possible to generate high-precision grids on embedded platforms, thus expanding the application scope of aircraft heat flux simulation.

[0020] The present invention provides a parallel optimization method for the greedy random Kaczmarz algorithm based on low-rank approximation. This method aims to improve the computational efficiency of the randomized Kaczmarz algorithm and the convergence rate of the randomized Kaczmarz algorithm. This method uses methods such as data reuse and error estimation to accelerate the convergence rate, and adopts a way of combining processes and threads to effectively parallelize the randomized Kaczmarz method based on low-rank approximation, thereby improving the performance of the RK algorithm, giving full play to its high efficiency, and providing strong support for the efficient operation of the VH-PDE method.

[0021] When calculating the error in the present invention, the true value of the error is not directly calculated, but the Monte Carlo error estimation method is used to replace part of the error calculation, effectively reducing the calculation time. First, the error estimate value is calculated. When the error estimate value reaches the threshold, the true value of the error is calculated. The effective combination of the Monte Carlo error estimation method and the Kaczmarz method significantly reduces its calculation time and calculation amount, and further improves the algorithm efficiency.

[0022] The present invention effectively combines the greedy method and the randomized method by setting a flexible parameter, combines the advantages of both, and applies the two to the Kaczmarz algorithm, improving the algorithm efficiency and laying a good foundation for generating high-quality grids. And the present invention proposes a caching method, which can effectively reduce the calculation time during residual calculation and solution vector update, achieve the purpose of data reuse, reduce the calculation amount for the hybrid Kaczmarz algorithm, and improve its convergence rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] The specification drawings constituting a part of the present invention are used to provide a further understanding of the present invention. The schematic embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation to the present invention.

[0024] Figure 1 is a flowchart of the parallel random solution method for overdetermined equations for aircraft structural grids in an embodiment of the present invention; Figure 2 is the flowchart of the greedy random hybrid Kaczmarz parallel method in the embodiment of the present invention; Figure 3 is the flowchart of Algorithm 1 in the embodiment of the present invention; Figure 4 is the experimental result graph in the embodiment of the present invention. Detailed implementation manners

[0025] It should be noted that the following detailed description is exemplary and is intended to provide further illustration of the present invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present invention belongs.

[0026] It should be noted that the terms used herein are only for describing specific implementation manners and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0027] In the case of no conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0028] Embodiment 1 As Figure 1 shown, this embodiment discloses an overdetermined equation parallel random solution method for aircraft structural grids, and the method includes the following steps: S1: Obtain a numerical aircraft parameter model; In this embodiment, a heat flux analysis is performed on the thermal protection system of a hypersonic aircraft to predict the temperature distribution and heat flux density on the aircraft surface at high Mach numbers and evaluate the performance of thermal protection materials. A parametric model of the aircraft is established, including components such as the fuselage, wings, and control surfaces.

[0029] Modeling is carried out for the target aircraft, defining its geometric shape, covering key components such as the fuselage, wings, and control surfaces, so as to obtain a complete aircraft parameter model; at the same time, the external environment in which the aircraft is located is modeled, and an environmental parameter model including terrain, obstacles, atmospheric disturbances, etc. is constructed to obtain an environmental parameter model for heat flux analysis.

[0030] S2: Generate grids based on the aircraft parameter model by using variational harmonic partial differential equations, and discretize the variational harmonic partial differential equations. After discretization, an overdetermined linear equation system with the grid vertex coordinate values as unknowns is established; In this embodiment, based on the mapping PDE model, the orthogonality and uniformity metrics of grid cells are introduced into the harmonic mapping PDE. By combining the harmonic mapping term with the energy function, a new objective function mathematical model for structured grid generation is constructed, enabling the grid generation problem to be ultimately transformed into the solution of a large-scale overdetermined linear equation system.

[0031] By solving the harmonic partial differential equation (PDE), a grid with an adaptive density distribution is generated to accurately fit the boundary of a complex aircraft. Specifically, the variational harmonic partial differential equation (VH-PDE) method is used to generate high-quality adaptive grids that can closely fit the boundary of a complex aircraft (such as the wing boundary). By introducing the harmonic mapping theory, a coordinate transformation is performed on the entire aircraft operation space, an elliptic partial differential equation is constructed, and the PDE is solved using numerical methods, thereby automatically generating a continuous and smooth grid distribution in the parametric space. By flexibly setting the boundary conditions and control functions of the PDE, the density and directionality of the grid can be significantly adjusted - high-density division is achieved in key areas such as the aircraft edge and control surface to optimize the overall computing resources.

[0032] In the VH-PDE method, grid generation is a dynamic adaptive optimization process. The entire process starts with the initialization of a coarse grid within the computational domain, usually using uniform rectangular or triangular grids, and boundary conditions are set to ensure geometric matching. Subsequently, high-error regions are identified through error estimation, and these regions are locally refined, while the grids in low-error regions are appropriately coarsened to optimize the allocation of computing resources. Next, high-order interpolation methods are used to smooth the grid data to reduce numerical errors that may be caused by irregular grids.

[0033] The collocation method is used to discretize the harmonic PDE; the continuous PDE model is transformed into a linear algebra system to facilitate the solution of the grid node distribution within the computational domain. The surface heat flux density of the aircraft is taken as the unknown variable, and the "heat flux needs to satisfy the physical boundary conditions and avoid material limit constraints" is represented by linear constraints. At the same time, dynamic restrictions (such as the change in heat flux caused by flight speed and angle of attack changes) are embedded to form a complex linear constraint system to model the heat flux distribution of the aircraft. After the variational harmonic partial differential equation is discretized, the number of constraint equations usually far exceeds the number of free variables, resulting in an overdetermined linear equation system.

[0034] In the VH-PDE method, mesh generation is first transformed into solving a set of partial differential equations that describe the distribution of mesh points within a specified region, usually taking into account constraints such as the relative positions, densities, and boundary conditions between mesh points. Then, by discretizing these partial differential equations, the continuous mesh generation problem is transformed into a discrete system of linear equations. During this process, the introduced boundary conditions and geometric constraints usually result in the number of equations exceeding the number of mesh points, thus forming an overdetermined system of linear equations , where represents the matrix, represents the solution vector, represents the vector, that is, the number of equations is greater than the number of unknowns. Through the matrix and the vector the unknown solution is obtained

[0035] The partial differential equations are transformed into the weak form by weighted integration, and the physical quantities are approximated at the mesh nodes using basis functions. Then, through discretization and integration operations, the continuous partial differential equations are transformed into algebraic equations. Since the number of equations is usually more than the number of unknowns, an overdetermined system of linear equations is finally formed

[0036] S3: Perform low-rank approximation processing on the overdetermined system of linear equations to obtain a reduced-dimensional overdetermined system of linear equations

[0037] In this embodiment, the low-rank approximation method is effectively combined with the hybrid Kaczmarz parallel algorithm. The low-rank approximation method is used to effectively reduce the dimension of the system of equations, providing a good foundation for solving the overdetermined system of linear equations and laying a solid foundation for the effective execution of the hybrid Kaczmarz parallel algorithm

[0038] Since the number of iteration steps of the random Kaczmarz algorithm is highly related to the number of equations in the system, in order to accelerate the convergence rate of the algorithm, the present invention uses a random rangefinder to perform low-rank approximation on the original system of equations, reducing the dimension of the overdetermined system of linear equations and achieving effective reduction of the dimension of the overdetermined system of linear equations, thereby improving the algorithm convergence rate

[0039] The specific steps are as follows Randomly generate a matrix , expressed as

[0040] where the dimension of the matrix A is rows columns, and is much larger than ; represents the rank

[0041] Multiply matrix by matrix to obtain matrix , which is expressed as:

[0042] Orthogonalize the columns of matrix to obtain matrix , which is expressed as:

[0043] Multiply matrix by matrix to obtain the low-rank approximation matrix , which is expressed as:

[0044] Perform the same operation on vector to obtain: .

[0045] S4: Use the greedy randomized hybrid Kaczmarz algorithm to parallelly solve the overdetermined linear equations after dimensionality reduction to obtain the optimal solution; Since the randomized Kaczmarz algorithm depends on the previous solution result when solving overdetermined linear equations, there are difficulties in effectively parallelizing the randomized Kaczmarz algorithm. The present invention uses a method combining threads and processes to effectively parallelize it, and uses methods such as the greedy randomized hybrid selection of hyperplanes method, data reuse method, Monte Carlo error estimation, etc. to effectively improve the performance of the randomized Kaczmarz method. When solving partial differential equations, the error is re-evaluated in each calculation step to determine whether further adjustment of the grid is required. If necessary, the grid is dynamically updated to ensure the accuracy and stability of the calculation. Finally, when the calculation reaches the convergence criterion, the grid is optimized, including merging redundant grid points, adjusting the grid distribution, etc., to reduce the calculation cost and improve the numerical stability. The whole process is a cyclic iterative optimization process, and the grid is continuously adjusted as the PDE solution progresses to adapt to the local characteristics of the problem, thereby improving the calculation efficiency and accuracy. The local characteristics refer to the specific requirements for grid generation. In some regions, the solution changes violently or the error is large, and a finer grid is required; while in other regions, the solution is relatively smooth or the error is very small, and a coarser grid can be used. These regions where the change is violent are called local characteristic regions.

[0046] Meanwhile, perform two-level parallel processing on the greedy randomized hybrid Kaczmarz method based on low-rank approximation. Instead of simply using processes for parallelization, the present invention divides the equations into blocks, performs parallel processing at the process level between each block, and performs two-level parallel optimization using threads within each block. The threads are mainly in the initialization Cache, calculate the residual, update the solution vector within the block, and perform a comprehensive update Parallel optimization is carried out in aspects such as caching and calculating the true value of the global error. Through the combination of process-level and thread-level optimization, the solution speed is further improved.

[0047] Furthermore, parallel optimization means that after dividing the system of equations into blocks, parallel calculations are performed among the blocks using processes, and then initialization within each block Cache, calculate the residual, update the solution vector within the block, and perform a comprehensive update Parallel calculations are also carried out during the processes of caching and calculating the true value of the global error. That is to say, parallel processing is performed both between and within the blocks.

[0048] In this embodiment, as Figure 2 shown, the overdetermined linear equations are divided into multiple blocks by rows and distributed to multiple processors for joint execution. Each processor separately performs the greedy random hybrid Kaczmarz algorithm for iterative solution.

[0049] The entire system of equations is divided into blocks and jointly executed by processors, with each processor processing one of the blocks. During the block division process, the present invention performs a remainder operation on the rows that cannot be evenly distributed to each process, so as to achieve the purpose of load balancing. Each process initially receives the same number of rows, and then compares the remainder with the process number. If the remainder is greater than this process number, then this process needs to process one more row of data. By this method, the number of data rows processed among each process can be made uniform, avoiding the situation of excessive workload on a single process, thereby effectively achieving load balancing.

[0050] In this embodiment, during the parallel iterative calculation process, the optimal update direction and step size of the current approximate solution (greedily and randomly select the hyperplane) are dynamically searched for each iteration, and a new approximate solution is calculated accordingly. The calculation will continue until the set termination condition is met or the maximum number of iterations is reached. Subsequently, all calculation results are merged to the main processor through the reduction interface, and finally the optimal solution is selected as the grid vertex coordinate value of the overdetermined system of equations, thereby constructing a structured grid.

[0051] Specifically, the steps for solving the overdetermined linear equations after dimensionality reduction using the greedy random hybrid Kaczmarz algorithm to obtain the optimal solution are as follows: The total processor process collects the local solution vectors from each processor process to obtain the global solution vector; Each processor process selects the hyperplane using the greedy random hybrid model and updates the local solution vector using the cache method to obtain a new local solution vector; The total processor process collects new local solution vectors from each processor process to obtain a new global solution vector; The Monte Carlo error estimation method is used for error estimation to obtain an error estimation value of the new global solution vector; Determine whether the error estimation value reaches a threshold. If so, calculate the true error value. If not, broadcast the new global solution vector to each processor process for continued iteration; Determine whether the true error value reaches a threshold. If so, obtain the optimal solution. If not, broadcast the new global solution vector to each processor process for continued iteration.

[0052] On the basis of the randomized Kaczmarz algorithm, a method of greedily randomly mixing and selecting hyperplanes is added, which avoids the drawback that some important hyperplanes are not selected when only using the randomized method, thus slowing down the convergence rate. And because the residual needs to be calculated every time the greedy method is used, each residual calculation will multiply the matrix with , and the product of the matrix also needs to be repeatedly calculated during the iterative update. This causes serious waste of resources. Therefore, the present invention proposes a caching method. Each time the residual is calculated, only the product of the corresponding row needs to be taken out, without having to recalculate the product of the matrix and , thereby improving the calculation efficiency. When the vector is updated, only the value of the selected row this time needs to be locally updated. In order to avoid errors caused by untimely cache update, which affects the convergence rate of the algorithm, the present invention adopts a method of comprehensively updating the cache at fixed time intervals to avoid the drawback of slow convergence speed caused by untimely cache update. cache update. cache update.

[0053] The solution steps are specifically as follows: (1) After each processor process obtains the corresponding data, a hyperplane is selected by using a greedy random hybrid model to project and update the solution vector ( ), and a flexible parameter is introduced into this model. This parameter allows the relative proportion of the greedy algorithm and the random algorithm to be dynamically adjusted according to the actual matrix . When the matrix When there are significant differences in the row performance, the weight of the greedy method can be increased. Conversely, when the row differences are small, increasing the weight of the randomization method can produce better results. First, the present invention sets the proportion of the greedy method to 40%, and then randomly generates a positive number less than or equal to 1. If the randomly generated value is less than or equal to the parameter (set to 0.4 in this embodiment), the greedy method is used to select the hyperplane projection. On the contrary, the random method is used to select the hyperplane projection for solution update.

[0054] Parameter needs to be judged according to the conditions of the matrix . If the row differences of the matrix are large, the weight of greed is increased, that is, increasing the value can achieve an acceleration effect. If the row differences are not large or there is noise, the random weight is increased, that is, reducing the value can achieve an acceleration effect. The specific process is as Figure 3 shown in Algorithm 1.

[0055] After the step of selecting the hyperplane is completed, the update of the solution vector starts. At this time, when updating the solution vector , there is no need to repeat the calculation of the product. The present invention can directly use the value in the cache. The expression for updating the solution vector is:

[0056] Among them, represents the current approximate solution, represents the -th element of the vector , represents the column vector corresponding to the -th row of the matrix , represents the product of the column vector corresponding to the -th row of the matrix and the current approximate solution.

[0057] After the solution vector is updated, only the data of this row is multiplied by the vector to update the cache corresponding to the corresponding row. Since the differences between multiple solution vectors are too small, and the calculation amount of frequently updating the entire cache is too large, so a full Cache update. That is to say, after partitioning the system of equations, each process only processes its own small piece of data and performs updates to the local solution vector. After a certain number of iterations, it is necessary to recycle the local solution vectors within each block and add them to obtain the average value to get the global solution vector.

[0058] In parallel computing, since the communication overhead is large and the significance of frequent global projection is small, the present invention performs global communication after each process reaches a certain number of iterations, so as to reduce the additional overhead brought by communication. Since the cache needs to be comprehensively updated at regular intervals, the comprehensive update step is placed after the global communication. It is found in the experiment that the error calculation takes a large amount of time, resulting in a performance bottleneck. Therefore, the Monte Carlo error estimation method is used to reduce the overhead caused by the calculation error, thereby further improving the performance.

[0059] The specific implementation steps for calculating the error are as follows: (1) Use multiple cores for parallel computing. After the number of iterations within each core reaches the specified number, global communication is performed. Use MPI_Allreduce to collect the local vectors of each process back to the main process, add the updated local solution vectors of each process, and then take their average value to obtain the global solution vector.

[0060] (2) After obtaining the latest global solution vector, instead of directly calculating its true error, the present invention first uses the Monte Carlo error estimation method for error estimation. The Monte Carlo error estimation method randomly selects multiple sample rows, calculates the residuals of each sample, and accumulates the squares of the residuals. Finally, by calculating the mean value of the sum of the squared residuals and taking the square root, the error estimation value of the solution vector is obtained. Using the Monte Carlo error estimation method can efficiently estimate the error and is especially suitable for large-scale problems.

[0061] The specific error estimation process is as follows: First, the error of the overdetermined system of linear equations is:

[0062] Among them, represents the error, represents the number of rows of matrix (i.e., the number of equations), represents the number of columns of matrix (i.e., the number of unknowns), represents the row, represents the column, represents matrix The element in the th row and th column, represents the th element in the unknown vector, and represents the th element in the constant vector

[0063] Using the Monte Carlo method, randomly select rows, calculate the sum of squared residuals for each row, and then take their average to estimate the overall estimation error:

[0064] where, represents the overall estimation error, represents the sample index, represents the th group and the coefficient in the th group and th equation,

[0065] To estimate this value (error estimate), the Monte Carlo method relies on randomly selecting rows. Selecting each row with equal probability, the expected value of the sum of squared residuals of randomly selecting a row is:

[0066] where, represents the residual of the th row (i.e., the th equation) in the linear equation system. The average sum of squared residuals is used to measure the error in the linear system.

[0067] In this way, we can obtain:

[0068] where, represents the index number of the selected row in the th iteration.

[0069] Since the expectations of the sum of squared residuals of all rows are the same, we can obtain: .

[0070] Thus, the effectiveness of the Monte Carlo method can be proved, and the Monte Carlo error estimate can effectively represent the error value. And using this method can further improve the calculation efficiency. The present invention adopts the Monte Carlo error estimation method, which can greatly reduce the calculation amount and greatly shorten the error calculation time.

[0071] If the error estimation value is less than the threshold, the true error is calculated. If the true error value is less than the threshold at this time, the program execution is completed, and the optimal result has been obtained. If the error is greater than the threshold at this time, it turns to step (3). If the error estimation is greater than the threshold, there is no need to calculate the true error and it directly turns to step (3) to continue.

[0072] (3) If the error value does not meet the predetermined requirements, use MPI_Bcast to broadcast the latest global solution vector to each process, and each process continues to iterate based on the latest global solution vector. Broadcasting the latest global solution vector to each process keeps the update directions of all processes generally consistent, avoiding the solution vector from having too large an error.

[0073] Efficient solution of overdetermined linear equations Plays an important role in the mesh generation of VH-PDE. First, through fast solution and error analysis, local error sources can be identified to guide adaptive mesh refinement, thereby refining the mesh in necessary regions and reducing the computational cost. Second, regularization or optimized iterative methods can improve the condition number of matrix A, enhance numerical stability, and optimize the mesh distribution. In addition, the analysis of the properties of the solution during the solution process can detect redundant information, reduce unnecessary grid points, and optimize storage and computational efficiency. At the same time, by analyzing numerical instability, the mesh can be smoothed and adjusted, the mesh structure can be optimized, the discretization error can be reduced, and the PDE calculation accuracy can be improved. Therefore, efficiently solving the overdetermined system can not only improve the numerical stability and computational efficiency of the VH-PDE method, but also reverse-optimize the mesh generation to make it more adaptable to the problem requirements, thereby improving the overall computational performance.

[0074] S5: Map the optimal solution to the physical space to generate a hybrid mesh, and import the hybrid mesh into the aircraft simulation platform for heat flux analysis.

[0075] Map the obtained optimal solution to the physical space to generate a hexahedron-dominated hybrid mesh, taking into account both computational efficiency and accuracy, and dynamically update the mesh to adapt to environmental changes (such as moving obstacles); import the mesh into the aircraft simulation platform for heat flux analysis, use a sparse solver or GPU-accelerated algorithm to calculate the trajectory in real time, and evaluate the heat flux distribution and temperature distribution on the surface of the aircraft at different flight stages. For example, import the mesh into ANSYS·Mechanical, define the material properties, and call the sparse direct solver to calculate the temperature field. Through the VH-PDE mesh generation technology, a high-precision environmental representation is constructed to ensure that the aircraft structure and system can work properly and not be damaged under extreme thermal loads.

[0076] In this embodiment, during the process from solving the overdetermined linear equations to mesh generation, first, after solving the overdetermined linear equations, error analysis and optimization of the solution are required to ensure the accuracy of the solution. Next, post-processing operations are performed to calculate physical quantities and visualize the solution, such as drawing contour lines, cloud maps, etc., in order to understand the results from a physical perspective. In the verification stage, the numerical solution needs to be compared with the theoretical solution or experimental data to verify the accuracy of the solution. After that, if further analysis of the problem is required, multi-scale methods or adaptive mesh techniques need to be introduced to provide more refined calculations in local areas. Finally, based on the above processing and analysis results, mesh generation and adjustment can be carried out to improve the accuracy and adapt to the complexity of the problem in subsequent calculations. The whole process ensures the efficiency and accuracy from the solution of the overdetermined equations to the final mesh generation. Experiment Description and Results Experimental Environment: Shanhe Supercomputer.

[0077] Experimental Parameters: A set of data generated after discretizing the partial differential equation for mesh generation: 80000×8000 dimensions.

[0078] A set of tests were conducted using an overdetermined system of equations with dimensions of 80000×8000. The test scale of the present invention ranges from 1 processor to 64 processors, and the solution time and computational acceleration ratio were recorded respectively. The experimental results are as Figure 4 shown. It can be seen in Figure 4 the performance of the algorithm of the present invention when using 1 to 64 cores. When using 64 cores, a speedup ratio of 21.9 times can be achieved, and when using 64 cores, the algorithm of the present invention can achieve an acceleration of more than 582 times compared to the randomized Kaczmarz algorithm.

[0079] The present invention uses the low-rank approximation method to effectively reduce the dimension of the system of equations, improve the convergence rate of the randomized Kaczmarz, and experiments prove that the randomized Kaczmarz method can never obtain an optimal solution with an error less than 1, while the randomized Kaczmarz based on low-rank approximation can reach the optimal solution within the error bound, thus proving that low-rank approximation can not only improve the convergence rate but also provide a higher-quality approximate solution.

[0080] The present invention uses the Monte Carlo error estimation method to replace part of the error calculation, effectively reducing the calculation time. When using 32 cores, the time using the error estimation method can be shortened by more than 19.4 times compared to not using this method, and when using 64 cores, the time can be shortened by more than 21.9 times, thus proving the efficiency of this method.

[0081] Compared with the randomized Kaczmarz method, the greedy randomized hybrid Kaczmarz algorithm based on low-rank approximation proposed by the present invention can achieve an acceleration ratio of more than 582 times. This result shows that the Kaczmarz method implemented by the present invention has excellent performance advantages, can greatly shorten the calculation time compared with the original randomized Kaczmarz method, and demonstrates its superiority and efficiency.

[0082] Example Two This example discloses an overdetermined equation parallel random solution system for an aircraft structural grid, including: A model acquisition module, configured to: acquire a numerical aircraft parameter model; An equation system establishment module, configured to: generate a grid based on the aircraft parameter model by using a variational harmonic partial differential equation, and discretize the variational harmonic partial differential equation, and establish an overdetermined linear equation system with the grid vertex coordinate values as unknowns after discretization; An equation system dimension reduction module, configured to: perform low-rank approximation processing on the overdetermined linear equation system to obtain a reduced-dimension overdetermined linear equation system; An equation system random solution module, configured to: perform parallel solution on the reduced-dimension overdetermined linear equation system by using the greedy randomized hybrid Kaczmarz algorithm to obtain an optimal solution; A grid generation module, configured to: map the optimal solution to the physical space to generate a hybrid grid, and import the hybrid grid into an aircraft simulation platform for heat flux analysis.

[0083] Example Three The purpose of this example is to provide a computing device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the steps of the method in Example One are implemented.

[0084] Example Four The purpose of this example is to provide a computer-readable storage medium. A computer-readable storage medium stores a computer program, and when the program is executed by a processor, the steps of the method in Example One are executed.

[0085] The steps involved in the devices in Examples Three and Four above correspond to those in Method Example One, and the specific implementation manners can refer to the relevant description part of Example One. The term "computer-readable storage medium" should be understood to include a single medium or multiple media including one or more instruction sets; it should also be understood to include any medium that can store, encode, or carry an instruction set for execution by a processor and enable the processor to execute any method in the present invention.

[0086] Those skilled in the art should understand that the various modules or steps of the present invention described above can be implemented by a general-purpose computer device. Optionally, they can be implemented by program codes executable by a computing device, so that they can be stored in a storage device and executed by the computing device, or they can be separately fabricated into individual integrated circuit modules, or multiple modules or steps among them can be fabricated into a single integrated circuit module for implementation. The present invention is not limited to any specific combination of hardware and software.

[0087] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

[0088] Although the specific implementation manners of the present invention have been described above in conjunction with the accompanying drawings, it is not a limitation to the protection scope of the present invention. Those skilled in the art should understand that, based on the technical solution of the present invention, various modifications or deformations that can be made by those skilled in the art without creative efforts are still within the protection scope of the present invention.

Claims

1. A parallel stochastic solution method for overdetermined equations for aircraft structural grids, characterized in that: include: Obtaining numerical aircraft parameter models; Based on the aircraft parameter model, a grid is generated by using a variational harmonic partial differential equation, and the variational harmonic partial differential equation is discretized, and after discretization, an overdetermined linear equation group is established with the coordinate values ​​of the grid vertices as unknowns; Performing low-rank approximation processing on the overdetermined linear equations to obtain a reduced-dimensional overdetermined linear equations; The greedy random hybrid Kaczmarz algorithm is used to solve the overdetermined linear equations after dimension reduction in parallel to obtain the optimal solution. The optimal solution is mapped to the physical space to generate a hybrid grid, and the hybrid grid is imported into an aircraft simulation platform for thermal flow analysis.

2. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 1, characterized in that: A random range finder is used to perform low-rank approximation processing on the overdetermined linear equations.

3. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 1, characterized in that: The parallel optimization of the solution process of the overdetermined linear equations after dimensionality reduction is specifically as follows: the overdetermined linear equations are divided into multiple blocks by row and assigned to multiple processors for joint execution. Each processor executes the greedy random hybrid Kaczmarz algorithm for iterative solution.

4. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 3, characterized in that: The greedy random hybrid Kaczmarz algorithm is used to solve the overdetermined linear equations after dimensionality reduction, and the specific steps to obtain the optimal solution are as follows: The general processor process collects local solution vectors from each processor process to obtain a global solution vector; Each processor process uses a greedy random mixing model to select a hyperplane and uses The cache method updates the local solution vector to obtain a new local solution vector; The general processor process collects new local solution vectors from each processor process to obtain a new global solution vector; The Monte Carlo error estimation method is used to estimate the error and obtain the error estimation value of the solution vector; Determine whether the error estimate reaches a threshold value, if so, calculate the true value of the error, if not, broadcast the new global solution vector to each processor process to continue iteration; It is determined whether the true value of the error reaches a threshold value. If so, the optimal solution is obtained. If not, the new global solution vector is broadcast to each processor process to continue iteration.

5. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 4, characterized in that: The greedy random mixture model is used to select the hyperplane as follows: Set the ratio value of the greedy method; Randomly generate a positive number less than or equal to 1; If the positive number is less than or equal to the set value, a greedy method is used to select a hyperplane; if the positive number is greater than the set value, a random method is used to select a hyperplane.

6. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 4, characterized in that: use The cache method updates the local solution vector as follows: No need to repeat calculation Product, use directly After the solution vector is updated, only the data of the current row is multiplied by the solution vector. The cache is updated and a full update is performed after every set number of iterations. Cache updates.

7. The method for solving overdetermined equations in parallel and stochastically for aircraft structural grids according to claim 4, characterized in that: The Monte Carlo error estimation method is used for error estimation to obtain the error estimate value of the solution vector as follows: randomly select multiple sample rows, calculate the residual of each sample, and accumulate the squares of the residuals. The error estimate value of the solution vector is obtained by finding the mean of the sum of squared residuals and taking the square root.

8. Parallel stochastic solution system for overdetermined equations for aircraft structural grids, characterized by: include: A model acquisition module is configured to: acquire a numerical aircraft parameter model; An equation group establishment module is configured to: generate a grid based on the aircraft parameter model using a variational harmonic partial differential equation, discretize the variational harmonic partial differential equation, and establish an overdetermined linear equation group with grid vertex coordinate values ​​as unknowns after discretization; An equation group dimensionality reduction module is configured to: perform low-rank approximation processing on the overdetermined linear equation group to obtain an overdetermined linear equation group after dimensionality reduction; The random solution module of the equation system is configured to: solve the overdetermined linear equation system after dimension reduction in parallel by using the greedy random hybrid Kaczmarz algorithm to obtain the optimal solution; The grid generation module is configured to: map the optimal solution to the physical space to generate a hybrid grid, and import the hybrid grid into the aircraft simulation platform for thermal flow analysis.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps in the parallel stochastic solution method for overdetermined equations for aircraft structural grids as described in any one of claims 1 to 7 are implemented.

10. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the program, the steps in the parallel stochastic solution method for overdetermined equations for aircraft structural grids as described in any one of claims 1 to 7 are implemented.

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