Iterative solving method and device for finite element equation set of complete machine structure

Abstract

The invention belongs to the technical field of structural finite element data processing, and particularly relates to an iterative solution method and device for a finite element equation set of a whole machine structure. The method comprises the steps of obtaining a complete machine structure sparse positive definite stiffness matrix and a load vector applied to a complete machine structure, and constructing a complete machine structure finite element equation set used for representing the relation among the complete machine structure sparse positive definite stiffness matrix, the load vector and a displacement vector; performing degree-of-freedom elimination on the sparse positive definite stiffness matrix of the whole machine structure to obtain an independent degree-of-freedom matrix; performing bandwidth optimization sorting on the independent degree-of-freedom matrix to obtain a sorted stiffness matrix; processing the sorted stiffness matrix based on an algebraic multi-grid preprocessor to obtain a preprocessed sub-matrix; and calculating a displacement vector by adopting a conjugate gradient iteration method according to the pre-processing sub-matrix. According to the method, the problem that the sparse matrix of the aviation structure is not converged in iterative solution can be effectively solved, and the solution efficiency is remarkably improved while the calculation precision is ensured.

Description

An iterative solution method and apparatus for finite element equations of a complete machine structure Technical Field

[0001] This application belongs to the field of structural finite element data processing technology, specifically relating to an iterative solution method and apparatus for a complete machine structure finite element equation system. Background Technology

[0002] In the structural design and analysis of aerospace equipment, finite element simulation is an indispensable core technology. By discretizing a continuous structure into a finite number of elements, a large system of linear equations Kx=b can be established to describe the mechanical behavior of the structure, where K is the overall stiffness matrix, x is the unknown displacement vector, and b is the load vector. With the increasing complexity and refinement of simulation analysis of aerospace structures, the number of degrees of freedom in the model has increased dramatically, often reaching millions or even tens of millions, resulting in an exceptionally large stiffness matrix K.

[0003] For solving large-scale linear equation systems, numerical methods are mainly divided into two categories: direct methods and iterative methods. While direct methods are general and stable, they have significant drawbacks when dealing with large-scale sparse matrices. Their decomposition process generates a large number of padding elements, destroying the sparsity of the matrix, leading to high memory consumption and increased computational complexity, which is often unacceptable for ultra-large-scale problems. In contrast, iterative methods only need to store the non-zero elements of the matrix. The core operations are sparse matrix and vector multiplication, and their time and space complexity are proportional to the number of non-zero elements. Therefore, they have significant advantages in terms of memory usage and potential computational efficiency, making them more suitable for large-scale sparse problems.

[0004] However, the widespread application of iterative methods is constrained by a key bottleneck: convergence and convergence speed. The convergence of iterative methods heavily depends on the spectral properties of the coefficient matrix K, especially its condition number. In finite element analysis of aerospace structures, due to the complexity of the structure geometry, the large differences in material properties, and the existence of multiple constraints, the generated global stiffness matrix K often exhibits high sparsity, and the distribution of non-zero elements is irregular, resulting in a large condition number and scattered eigenvalues. More specifically, such matrices often have weak diagonal dominance properties, or even fail to meet the strict diagonal dominance or symmetric positive definite conditions required by iterative methods, causing the standard iterative method to converge slowly or even diverge completely. Summary of the Invention

[0005] To address the aforementioned issues, this application provides an iterative solution method and apparatus for finite element equations of a complete aircraft structure, enabling rapid and accurate calculation of the displacement response of the structure under specific loads, and providing support for improving the performance of efficient solution algorithms in aerospace autonomous structural analysis software.

[0006] The first aspect of this application provides an iterative solution method for finite element equations of a complete machine structure, mainly including:

[0007] Step S1: Obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and construct the finite element equation system of the whole structure to characterize the relationship between the sparse positive definite stiffness matrix, load vector and displacement vector of the whole structure.

[0008] Step S2: Eliminate the degrees of freedom of the sparse positive definite stiffness matrix of the whole structure to obtain the independent degree of freedom matrix;

[0009] Step S3: Use the RCM algorithm to perform bandwidth optimization sorting on the independent degree-of-freedom matrix to obtain the sorted stiffness matrix;

[0010] Step S4: Process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix;

[0011] Step S5: Calculate the displacement vector using the conjugate gradient iteration method based on the preprocessed submatrix.

[0012] Preferably, in step S2, the independent degrees of freedom matrix is ​​calculated using the following formula. :

[0013] ;

[0014] in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

[0015] Preferably, step S2 further includes:

[0016] The independent degree-of-freedom matrix is ​​converted into compressed storage row or triplet format and then stored.

[0017] Preferably, in step S3, the sorted stiffness matrix is ​​obtained using the following formula. :

[0018] ;

[0019] in, This is the RCM permutation matrix.

[0020] Preferably, step S5 further includes:

[0021] Step S51: Calculate the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration value;

[0022] Step S52: Calculate the preprocessed residual iteration value based on the initial residual iteration value and the preprocessed submatrix;

[0023] Step S53: Update the displacement vector iteration value based on the preprocessed residual iteration value, and further update the initial residual iteration value and the preprocessed residual iteration value until the L2 norm of the initial residual iteration value is lower than the set value, and output the displacement vector iteration value under the current iteration step.

[0024] The second aspect of this application provides an iterative solution device for finite element equations of a complete machine structure, mainly comprising:

[0025] The whole structure finite element equation parameter acquisition module is used to obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and to construct the whole structure finite element equation system to characterize the relationship between the sparse positive definite stiffness matrix, load vector and displacement vector of the whole structure.

[0026] The independent degree-of-freedom matrix generation module is used to eliminate the degree of freedom of the sparse positive definite stiffness matrix of the whole structure and obtain the independent degree-of-freedom matrix.

[0027] The stiffness matrix sorting module is used to perform bandwidth optimization sorting on the independent degree-of-freedom matrix using the RCM algorithm to obtain the sorted stiffness matrix.

[0028] The preprocessing submatrix generation module is used to process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix.

[0029] The displacement vector calculation module is used to calculate the displacement vector based on the preprocessed submatrix using the conjugate gradient iteration method.

[0030] Preferably, in the independent degree-of-freedom matrix generation module, the independent degree-of-freedom matrix is ​​calculated using the following formula. :

[0031] ;

[0032] in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

[0033] Preferably, the device further includes:

[0034] The compressed storage module is used to convert the independent degree-of-freedom matrix into a compressed storage row or triplet format for storage.

[0035] Preferably, in the stiffness matrix sorting module, the sorted stiffness matrix is ​​obtained using the following formula. :

[0036] ;

[0037] in, This is the RCM permutation matrix.

[0038] Preferably, the displacement vector calculation module includes:

[0039] The initial residual iteration value calculation unit is used to calculate the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration values.

[0040] The preprocessing residual iteration value calculation unit is used to calculate the preprocessing residual iteration value based on the initial residual iteration value and the preprocessing submatrix.

[0041] The iterative control unit is used to update the displacement vector iterative value based on the preprocessed residual iterative value, and further update the initial residual iterative value and the preprocessed residual iterative value until the L2 norm of the initial residual iterative value is lower than the set value, and output the displacement vector iterative value under the current iteration step.

[0042] This application can effectively solve the problem of non-convergence of sparse matrices in aerospace structures during iterative solutions, and significantly improves the solution efficiency while ensuring computational accuracy. Attached Figure Description

[0043] Figure 1 is a flowchart of a preferred embodiment of the iterative solution method for the finite element equations of the whole structure according to this application. Detailed Implementation

[0044] To make the objectives, technical solutions, and advantages of this application clearer, the technical solutions in the embodiments of this application will be described in more detail below with reference to the accompanying drawings. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, not all, of the embodiments of this application. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application. The embodiments of this application will be described in detail below with reference to the accompanying drawings.

[0045] The first aspect of this application provides an iterative solution method for finite element equations of a complete machine structure, as shown in Figure 1, which mainly includes:

[0046] Step S1: Obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and construct the finite element equation system of the whole structure to characterize the relationship between the sparse positive definite stiffness matrix, load vector and displacement vector of the whole structure.

[0047] Step S2: Eliminate the degrees of freedom of the sparse positive definite stiffness matrix of the whole structure to obtain the independent degree of freedom matrix;

[0048] Step S3: Use the RCM algorithm to perform bandwidth optimization sorting on the independent degree-of-freedom matrix to obtain the sorted stiffness matrix;

[0049] Step S4: Process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix;

[0050] Step S5: Calculate the displacement vector using the conjugate gradient iteration method based on the preprocessed submatrix.

[0051] In step S1 of this application, a sparse positive definite stiffness matrix of the whole structure is generated based on the element stiffness matrix generation technology and the overall stiffness assembly technology of the CAE software SABRE for aerospace autonomous structural analysis. At the same time, the load vector output by the load application mechanism is obtained so as to construct a linear equation system between the stiffness matrix, the load vector and the displacement vector.

[0052] Next, in step S2, the relevant degrees of freedom are eliminated using single-point and multi-point constraint elimination techniques to obtain the set of independent degrees of freedom matrices. In some optional embodiments, in step S2, the independent degrees of freedom matrix is ​​calculated using the following formula. :

[0053] ;

[0054] in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

[0055] To reduce memory usage and improve computational efficiency, matrix format conversion techniques can be used to compress and store the independent degree-of-freedom matrix in the SABRE system's internal matrix storage format. For example, in some optional implementations, step S2 further includes:

[0056] The independent degree-of-freedom matrix is ​​converted into compressed storage row or triplet format and then stored.

[0057] In this embodiment, data in the Compressed Storage Row (CSR) is represented using row indices, column indices, and non-zero element values, while data in the triplet (COO) is represented using row indices, column indices, and non-zero element values.

[0058] Subsequently, in step S3, the sparse matrix is ​​sorted using the adaptive inverse Cuthill-McKee algorithm (RCM) with bandwidth optimization. The starting node is dynamically selected, and weighted degrees and dynamic weights are introduced. Simultaneously, the matrix bandwidth and profile are optimized, thereby improving the locality and iterative stability of the sparse matrix and enhancing the diagonal dominance property of the matrix elements. For example, in some optional embodiments, in step S3, the sorted stiffness matrix is ​​obtained using the following formula. :

[0059] ;

[0060] in, This is the RCM permutation matrix.

[0061] Next, in step S4, based on the sorting optimization, an algebraic multigrid preprocessor is constructed to reduce the system condition number through coarse-fine grid levels and smoothing operators, as shown in the formula:

[0062] ;

[0063] Here, AMG is the algebraic multigrid preprocessor (function), and M is the preprocessing submatrix, which is a sparse matrix with a small condition number, which is convenient for fast iterative solution and facilitates the iterative calculation of subsequent equation systems.

[0064] Finally, in step S5, the sorted matrix is ​​used to efficiently solve the linear equation system using the conjugate gradient iteration method under the AMG preprocessing, and the final result is output according to the relevant convergence criteria.

[0065] In some alternative implementations, step S5 further includes:

[0066] Step S51: Calculate the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration value;

[0067] Step S52: Calculate the preprocessed residual iteration value based on the initial residual iteration value and the preprocessed submatrix;

[0068] Step S53: Update the displacement vector iteration value based on the preprocessed residual iteration value, and further update the initial residual iteration value and the preprocessed residual iteration value until the L2 norm of the initial residual iteration value is lower than the set value, and output the displacement vector iteration value under the current iteration step.

[0069] In this embodiment, initial displacement vector iteration values ​​are first given. In step S51, the initial residual iteration value is calculated. ,in, This is the right-hand item after sorting, i.e., the sorted load vector; then the AMG preprocessor is called, and in step S52, through... Calculate the preprocessed residual iteration value Introducing intermediate variables ,when season .

[0070] After setting the initial conditions and parameters, execute loop control. For example, for the k-th loop, execute... In step S53, the displacement vector iteration value of the (k+1)th iteration is updated. Then update the iterative values ​​of the residual vector. Using convergence criteria Perform a convergence check. As a set value, usually Output upon convergence If it does not converge, it continues to call the AMG preprocessor, according to Calculate the (k+1)th iteration value of the preprocessing residual. And the intermediate variable updated in the (k+1)th iteration. .

[0071] Through comparative analysis, this application reduces the solution time and the number of iteration convergence steps by using the sorted matrix. At the same time, compared with the direct solution method, the computational accuracy of this application meets the requirements.

[0072] The second aspect of this application provides an iterative solution device for finite element equations of a complete machine structure, corresponding to the above method, mainly comprising:

[0073] The whole structure finite element equation parameter acquisition module is used to obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and to construct the whole structure finite element equation system to characterize the relationship between the sparse positive definite stiffness matrix, load vector and displacement vector of the whole structure.

[0074] The independent degree-of-freedom matrix generation module is used to eliminate the degree of freedom of the sparse positive definite stiffness matrix of the whole structure and obtain the independent degree-of-freedom matrix.

[0075] The stiffness matrix sorting module is used to perform bandwidth optimization sorting on the independent degree-of-freedom matrix using the RCM algorithm to obtain the sorted stiffness matrix.

[0076] The preprocessing submatrix generation module is used to process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix.

[0077] The displacement vector calculation module is used to calculate the displacement vector based on the preprocessed submatrix using the conjugate gradient iteration method.

[0078] In some optional implementations, the independent degrees of freedom matrix generation module calculates the independent degrees of freedom matrix using the following formula. :

[0079] ;

[0080] in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

[0081] In some alternative embodiments, the device further includes:

[0082] The compressed storage module is used to convert the independent degree-of-freedom matrix into a compressed storage row or triplet format for storage.

[0083] In some optional embodiments, the sorted stiffness matrix is ​​obtained in the stiffness matrix sorting module using the following formula. :

[0084] ;

[0085] in, This is the RCM permutation matrix.

[0086] In some optional implementations, the displacement vector calculation module includes:

[0087] The initial residual iteration value calculation unit is used to calculate the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration values.

[0088] The preprocessing residual iteration value calculation unit is used to calculate the preprocessing residual iteration value based on the initial residual iteration value and the preprocessing submatrix.

[0089] The iterative control unit is used to update the displacement vector iterative value based on the preprocessed residual iterative value, and further update the initial residual iterative value and the preprocessed residual iterative value until the L2 norm of the initial residual iterative value is lower than the set value, and output the displacement vector iterative value under the current iteration step.

[0090] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. An iterative solution method for finite element equations of a complete machine structure, characterized in that, include: Step S1: Obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and construct a set of finite element equations for the whole structure to characterize the relationship between the sparse positive definite stiffness matrix, load vector, and displacement vector; Step S2: Eliminate the degrees of freedom of the sparse positive definite stiffness matrix of the whole structure to obtain the independent degree-of-freedom matrix; Step S3: Use the RCM algorithm to perform bandwidth optimization sorting on the independent degree-of-freedom matrix to obtain the sorted stiffness matrix; Step S4: Process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix; Step S5: Calculate the displacement vector using the conjugate gradient iteration method based on the preprocessed submatrix.

2. The iterative solution method for finite element equations of a complete machine structure as described in claim 1, characterized in that, In step S2, the independent degrees of freedom matrix is ​​calculated using the following formula. ： ;in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

3. The iterative solution method for finite element equations of a complete machine structure as described in claim 1, characterized in that, Step S2 is followed by: converting the independent degree-of-freedom matrix into a compressed storage row or triplet format and then storing it.

4. The iterative solution method for finite element equations of a complete machine structure as described in claim 2, characterized in that, In step S3, the sorted stiffness matrix is ​​obtained using the following formula. ： ;in, This is the RCM permutation matrix.

5. The iterative solution method for finite element equations of a complete machine structure as described in claim 1, characterized in that, Step S5 further includes: Step S51, calculating the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration value; Step S52, calculating the preprocessed residual iteration value based on the initial residual iteration value and the preprocessed submatrix; Step S53, updating the displacement vector iteration value based on the preprocessed residual iteration value, and further updating the initial residual iteration value and the preprocessed residual iteration value until the L2 norm of the initial residual iteration value is lower than the set value, and outputting the displacement vector iteration value under the current iteration step.

6. An iterative solution device for finite element equations of a complete machine structure, characterized in that, include: The whole structure finite element equation parameter acquisition module is used to obtain the sparse positive definite stiffness matrix of the whole structure and the load vector applied to the whole structure, and to construct the whole structure finite element equation system to characterize the relationship between the sparse positive definite stiffness matrix, load vector and displacement vector of the whole structure. The independent degree-of-freedom matrix generation module is used to eliminate the degree of freedom of the sparse positive definite stiffness matrix of the whole structure to obtain the independent degree-of-freedom matrix; the stiffness matrix sorting module is used to perform bandwidth optimization sorting of the independent degree-of-freedom matrix using the RCM algorithm to obtain the sorted stiffness matrix. The preprocessing submatrix generation module is used to process the sorted stiffness matrix based on the algebraic multigrid preprocessor to obtain the preprocessed submatrix. The displacement vector calculation module is used to calculate the displacement vector based on the preprocessed submatrix using the conjugate gradient iteration method.

7. The iterative solution device for finite element equations of a complete machine structure as described in claim 6, characterized in that, In the independent degree-of-freedom matrix generation module, the independent degree-of-freedom matrix is ​​calculated using the following formula. ： ;in, The overall structure has a sparse positive definite stiffness matrix. For the constraint cancellation operator, T is the matrix transpose operation.

8. The iterative solution device for finite element equations of a complete machine structure as described in claim 6, characterized in that, The device also includes a compressed storage module for converting the independent degree-of-freedom matrix into a compressed storage row or triplet format for storage.

9. The iterative solution device for finite element equations of a complete machine structure as described in claim 7, characterized in that, In the stiffness matrix sorting module, the sorted stiffness matrix is ​​obtained using the following formula. ： ;in, This is the RCM permutation matrix.

10. The iterative solution device for finite element equations of a complete machine structure as described in claim 6, characterized in that, The displacement vector calculation module includes: an initial residual iteration value calculation unit, used to calculate the initial residual iteration value based on the sorted stiffness matrix and displacement vector iteration value; a preprocessing residual iteration value calculation unit, used to calculate the preprocessing residual iteration value based on the initial residual iteration value and the preprocessing submatrix; and an iteration control unit, used to update the displacement vector iteration value based on the preprocessing residual iteration value, and further update the initial residual iteration value and the preprocessing residual iteration value until the L2 norm of the initial residual iteration value is lower than a set value, and output the displacement vector iteration value under the current iteration step.

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