A Supernode Approximate Symbolic Factorization Method for the Structural Finite Element Stiffness Matrix

Abstract

This application belongs to the field of structural finite element stiffness analysis, and particularly relates to a supernode approximate symbolic factorization method for a structural finite element stiffness matrix. The method includes: Step 1, dividing supernodes on the structural finite element stiffness matrix; Step 2, obtaining the row index segments of the supernodes; Step 3, constructing a supernode adjacency matrix according to the row index segments, and constructing a supernode elimination tree according to the supernode adjacency matrix; Step 4, performing symbolic factorization with the supernodes and the row index segments as the operation objects according to the supernode elimination tree; Step 5, performing aircraft structural strength design according to the structurally finite element stiffness matrix after symbolic factorization. The supernode approximate symbolic factorization method for the structural finite element stiffness matrix of this application is guided by the idea of dimensionality reduction, greatly reducing the time for symbolic factorization.

Description

Technical Field

[0001] This application belongs to the field of structural finite element stiffness analysis, and particularly relates to a supernode approximate symbolic factorization method for structural finite element stiffness matrices. Background Art

[0002] The aircraft structural strength design aims to ensure that the aircraft can withstand the effects of various working environments such as loads, vibrations, and temperatures during flight, avoid structural failures, and maintain the ability to operate safely. Structural finite element stiffness analysis is widely used in the static strength analysis and dynamic strength analysis of aircraft structures.

[0003] The role of symbolic factorization of the structural finite element stiffness matrix is to predict the positions of fill elements, allocate sufficient memory space for numerical factorization, and provide a reference for planning the factorization process. Traditional symbolic factorization is guided by the elimination tree of the reordered matrix and operates on single columns and single row indices for matrix L pattern prediction, and the time complexity of the algorithm is O (| L |), where | L | is the number of non-zero elements in the matrix L . As the matrix size and the number of non-zero elements continue to increase, the time consumption of symbolic factorization will increase significantly.

[0004] Therefore, it is desirable to have a technical solution to overcome or at least mitigate at least one of the above-mentioned defects of the prior art. Summary of the Invention

[0005] The purpose of this application is to provide a supernode approximate symbolic factorization method for structural finite element stiffness matrices to solve at least one problem existing in the prior art.

[0006] The technical solution of this application is as follows:

[0007] A supernode approximate symbolic factorization method for structural finite element stiffness matrices, comprising:

[0008] Step 1, dividing supernodes on the structural finite element stiffness matrix;

[0009] Step 2, obtaining the row index segments of the supernodes;

[0010] Step 3, constructing a supernode adjacency matrix according to the row index segments, and constructing a supernode elimination tree according to the supernode adjacency matrix;

[0011] Step 4, performing symbolic factorization with the supernodes and the row index segments as the operation objects according to the supernode elimination tree;

[0012] Step 5, performing aircraft structural strength design according to the structural finite element stiffness matrix after symbolic factorization.

[0013] In at least one embodiment of the present application, in step one, partitioning supernodes on the structural finite element stiffness matrix includes:

[0014] Obtaining the structural finite element stiffness matrix after METIS reordering;

[0015] Partitioning supernodes on the structural finite element stiffness matrix according to the boundaries between matrix blocks.

[0016] In at least one embodiment of the present application, the boundaries between the matrix blocks are determined according to the following characteristics:

[0017] The row where the boundary is located has only one non-zero element on the diagonal, and the number of non-zero elements in the previous row is greater than 1, then the number of non-zero elements in this row minus the number of non-zero elements in the previous row is less than 0, that is , , RN j is the number of non-zero elements in the j th row, RN j-1 is the number of non-zero elements in the j th - 1 row;

[0018] Determine the threshold TH , if , then take the j th column as the column where the boundary is located.

[0019] In at least one embodiment of the present application, in step two, obtaining the row index segments of the supernodes includes:

[0020] Obtaining the row index segments of the non-zero elements of each supernode;

[0021] Mapping all the row index segments of each supernode to an array for array representation.

[0022] In at least one embodiment of the present application, in step three, constructing the supernode adjacency matrix according to the row index segments includes:

[0023] If the segment of supernode J and the diagonal square matrix of supernode K have the same row index segments, then there is an adjacency relationship between supernode J and supernode K . Construct the supernode adjacency matrix according to the adjacency relationship.

[0024] In at least one embodiment of the present application, the diagonal of the supernode adjacency matrix is filled with 1, the non - diagonal elements are represented by 1 for the existence of an adjacency relationship, and 0 for the non - existence of an adjacency relationship.

[0025] In at least one embodiment of the present application, in step four, when performing symbol decomposition, a fusion supernode fragment task pool and an addition supernode fragment task pool are respectively constructed, where the fusion supernode fragment task pool is used to store fusion type tasks, the addition supernode fragment task pool is used to store addition type tasks, and each thread preferentially retrieves the fusion type tasks in each loop.

[0026] The invention has at least the following beneficial technical effects:

[0027] The supernode approximate symbol decomposition method for the structural finite element stiffness matrix of the present application is guided by the idea of dimensionality reduction, greatly reducing the time of symbol decomposition and solving the problem that the time complexity of the traditional symbol decomposition method increases sharply with the increase of the matrix scale and the number of non-zero elements. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 is a flowchart of the supernode approximate symbol decomposition method for the structural finite element stiffness matrix according to an embodiment of the present application;

[0029] Figure 2 is a schematic diagram of the non-zero element distribution of the structural finite element stiffness matrix before and after METIS reordering according to an embodiment of the present application;

[0030] Figure 3 is a schematic diagram of the boundary between matrix blocks according to an embodiment of the present application;

[0031] Figure 4 is a schematic diagram of the row index fragment of the supernode according to an embodiment of the present application;

[0032] Figure 5 is a schematic diagram of the supernode adjacency matrix and the supernode elimination tree according to an embodiment of the present application. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0033] To make the purpose, technical solutions, and advantages of the implementation of the present application clearer, the technical solutions in the embodiments of the present application will be described in more detail below with reference to the accompanying drawings in the embodiments of the present application. 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 some, but not all, of the embodiments of the present application. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to explain the present application and should not be construed as limiting the present application. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present application without creative efforts shall fall within the scope of protection of the present application. The embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0034] In the description of the present application, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "lateral", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present application and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as limiting the protection scope of the present application.

[0035] The following will further elaborate on the present application with reference to the Figures 1 to 5 accompanying drawings.

[0036] The present application provides a method for supernode approximate symbolic factorization of a structural finite element stiffness matrix. As Figure 1 shown, it includes the following steps:

[0037] Step 1: Divide supernodes on the structural finite element stiffness matrix;

[0038] Step 2: Obtain the row index segments of the supernodes;

[0039] Step 3: Construct a supernode adjacency matrix based on the row index segments, and construct a supernode elimination tree based on the supernode adjacency matrix;

[0040] Step 4: Perform symbolic factorization with the supernodes and row index segments as the operation objects according to the supernode elimination tree;

[0041] Step 5: Perform aircraft structural strength design based on the structurally finite element stiffness matrix after symbolic factorization.

[0042] For the method of supernode approximate symbolic factorization of the structural finite element stiffness matrix of the present application, supernodes are first divided. In Step 1, dividing supernodes on the structural finite element stiffness matrix specifically includes:

[0043] Obtain the structurally finite element stiffness matrix after METIS reordering;

[0044] Divide supernodes on the structural finite element stiffness matrix according to the boundaries between matrix blocks.

[0045] Directly divide supernodes on the structurally finite element stiffness matrix after METIS reordering. The METIS reordering algorithm reorders the matrix based on the multi-level nested dissection algorithm, which can effectively reduce the matrix fill elements and improve the parallelism of the matrix. As Figure 2 shown, the non-zero element distribution of the matrix after METIS reordering has obvious block characteristics. Based on the reordered matrix, supernode division is performed according to the block characteristics of the matrix. As Figure 3As shown, the key is to find the boundaries between matrix blocks. These positions have the following characteristics, and the boundaries between matrix blocks are determined based on these characteristics:

[0046] For the row where the boundary is located, there is only one non - zero element on the diagonal, and the number of non - zero elements in the previous row is greater than 1. Then, the number of non - zero elements in this row minus the number of non - zero elements in the previous row is less than 0, that is , , RN j is the number of non - zero elements in the j row, RN j-1 is the number of non - zero elements in the j -1 row;

[0047] Determine the threshold TH . If , then take the j column as the column where the boundary is located.

[0048] Since only the lower triangular part of the matrix is saved during symbolic decomposition, there is only one non - zero element on the diagonal for the row where the boundary is located.

[0049] In the method for super - node approximate symbolic decomposition of the structural finite - element stiffness matrix in this application, in step two, obtaining the row - index segment of the super - node includes:

[0050] Obtaining the row - index segment of the non - zero elements of each super - node;

[0051] Mapping all the row - index segments of each super - node into an array for array representation.

[0052] After the super - nodes are divided, map all the row - index segments of the non - zero elements of each super - node into an array of N - dimensional size, and all adjacent row - indices are grouped into one segment. As Figure 4 shown, in an embodiment of this application, an 8 - dimensional matrix is divided into 3 super - nodes, , and the row - index segments of each super - node are circled with a dotted - line box and saved using a two - dimensional array:

[0053]

[0054] Among them, FN the array saves the number of segments each super - node has, Start , End the array saves the start and end information of each segment.

[0055] In the method for super - node approximate symbolic decomposition of the structural finite - element stiffness matrix in this application, in step three, constructing the super - node adjacency matrix and the super - node elimination tree. In this embodiment, according to the obtained segment information, construct the super - node adjacency matrix, and its matrix size isSN Rows SN Columns. For each supernode J , if there is the same row index segment between its segment and the diagonal matrix of its subsequent supernode K , then there is an adjacency relationship between these two supernodes, or it can be said that the supernode K is the ancestor supernode of the supernode J . The set of ancestor supernodes of the supernode J can be expressed as:

[0056]

[0057] According to Figure 4 the row index segments in, the supernode adjacency matrix and the supernode elimination tree are as shown in Figure 5 . Among them, the diagonal of the supernode adjacency matrix is filled with 1, the non-diagonal is represented by 1 for the existence of an adjacency relationship, and 0 for no adjacency relationship. Thus, the original symbolic factorization problem of the 8th-order matrix is successfully transformed into the symbolic factorization problem of the 3rd-order matrix. Subsequently, by applying the elimination tree generation algorithm, the supernode elimination tree can be quickly obtained to guide the parallel symbolic factorization process. The supernode elimination tree represented by a tree diagram is shown in the right figure of Figure 5 .

[0058] In the method for supernode approximate symbolic factorization of the structural finite element stiffness matrix of the present application, in step four, after the above-mentioned preparation stage is completed, the symbolic factorization principle is formally applied, and the supernodes and row index segments are used as the operation objects for symbolic factorization. When performing symbolic factorization, a fused supernode segment task pool and an added supernode segment task pool are respectively constructed. Among them, the fused supernode segment task pool is used to store tasks of the fusion type, and the added supernode segment task pool is used to store tasks of the addition type. Each thread preferentially retrieves tasks of the fusion type in each loop. In the preferred embodiment of the present application, OpenMP is used for parallel symbolic factorization. Combining the proposed task pool parallel template, two task pools for fusing supernode segments and adding supernode segments are constructed. Among them, tasks of the fusion type have priority. Once a certain supernode completes a task of the fusion type, the task of adding the supernode segment of its parent supernode can be put into the task pool to wait for retrieval and execution. After all tasks are executed, the positions of all fill elements will be included in the obtained supernode segments.

[0059] In this embodiment, the parallel supernode approximate symbolic factorization algorithm using OpenMP is specifically:

[0060]

[0061] Line 1 creates two task pools to store two different types of tasks respectively. task_addThe task number for storing the contribution segments of added sub - super - nodes to their parent super - nodes into their parent super - nodes, while task_merge stores the task number for fusing a certain super - node segment. For example Figure 5 In the matrix of, if super - node 2 is the parent super - node of super - node 0 and super - node 1, then both super - node 0 and super - node 1 need to add their contribution segments to super - node 2; then after super - node 2 receives the contribution segments of super - node 0 and super - node 1, to ensure that there are no duplicate segments, it is necessary to fuse the segments of super - node 2. Among them, if I represents a segment of super - node J , then the set of contribution segments of super - node J to its parent super - node F can be expressed as:

[0062]

[0063] The second row first adds all independent sub - super - node numbers to task_add , and then enters the OpenMP parallel region. Each thread preferentially retrieves the fusion - type task in each loop. If there is no fusion - type task, it retrieves the addition - type task. When retrieving the addition - type task, it should be noted that multiple sub - super - nodes cannot simultaneously add segments to the same parent super - node. Therefore, an additional auxiliary array is needed to limit it when retrieving tasks. For example, a SN - dimensional array J_lock can be used for status marking. If J_lock[J] = 1, it means that another thread is adding segments to super - node J , and the current thread cannot retrieve the addition task for the sub - super - node number of super - node J . After the task retrieval is completed, the specific task to be executed next is determined according to the task_type variable. When judging whether all tasks are completed, it can be judged whether all root super - nodes have completed the fusion task. Usually, there is only one root super - node, but in special cases, there may be multiple root super - nodes.

[0064] The supernode approximate symbolic factorization method for the structural finite element stiffness matrix of the present application. A supernode consists of consecutive multiple columns with similar row indices. Different from the traditional symbolic factorization method that divides supernodes after symbolic factorization is completed, the present application directly divides supernodes on the matrix after METIS reordering, enabling dimensionality reduction in the matrix dimension for symbolic factorization; based on the divided supernodes, the row index segments of each supernode are obtained, further reducing the dimension in terms of matrix row indices; according to the supernode division situation and row index segments, a supernode adjacency matrix is obtained, and based on this, a supernode elimination tree is obtained to guide symbolic factorization; different from the traditional symbolic factorization method that performs symbolic factorization with single columns and single row indices as the operation objects, the present application performs symbolic factorization with supernodes and row index segments as the basic operation objects. Since the number of supernodes and row index segments is much smaller than the number of matrix columns and single row indices, the time complexity of symbolic factorization is greatly reduced. Since the positions of the predicted fill elements in this method contain logical zero elements, which is an approximation of the exact mode, this symbolic factorization is called approximate symbolic factorization.

[0065] For the supernode approximate symbolic factorization method for the structural finite element stiffness matrix of the present application, finally, in step five, the structural finite element stiffness matrix after symbolic factorization obtained according to the above steps can efficiently obtain the structural response and guide the aircraft strength design and optimization.

[0066] As described above, the above is only the specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present application should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claimed rights.

Claims

1. A method for super-node approximate symbolic decomposition of structural finite element stiffness matrix, characterized in that: include: Step 1: Divide super nodes on the structural finite element stiffness matrix; Step 2: Obtain the row index fragment of the supernode; Step 3: construct a supernode adjacency matrix according to the row index fragments, and construct a supernode elimination tree according to the supernode adjacency matrix; The diagonal of the supernode adjacency matrix is ​​filled with 1, and the off-diagonal is filled with 1 to indicate the existence of an adjacency relationship, and 0 to indicate the absence of an adjacency relationship; Step 4: perform symbol decomposition according to the supernode elimination tree, taking the supernode and the row index fragment as operation objects; When performing symbol decomposition, a fusion supernode fragment task pool and an addition supernode fragment task pool are respectively constructed, wherein the fusion supernode fragment task pool is used to store fusion type tasks, and the addition supernode fragment task pool is used to store addition type tasks, and each thread takes priority to receive the fusion type tasks in each cycle; Step 5: Perform aircraft structural strength design based on the structural finite element stiffness matrix after symbolic decomposition.

2. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 1, characterized in that: In step 1, super nodes are divided on the structural finite element stiffness matrix, including: Obtain the structural finite element stiffness matrix after METIS reordering; Super nodes are divided on the structural finite element stiffness matrix according to boundaries between matrix blocks.

3. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 2, characterized in that: The boundaries between the matrix blocks are determined according to the following characteristics: The row where the boundary is located has only one non-zero element on the diagonal, and the number of non-zero elements in the previous row is greater than 1, then the number of non-zero elements in this row minus the number of non-zero elements in the previous row is less than 0, that is, RN j =1, RN j - RN j-1 <0, RN j For the j The non-zero arity of the row, RN j-1 For the j -1 non-zero arity of the row; Determine the threshold TH ,like RN j - RN j-1 < TH , then take the j The column where the boundary lies.

4. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 3, characterized in that: In step 2, obtaining the row index fragment of the supernode includes: Obtaining a row index fragment of a non-zero element of each supernode; All the row index fragments of each supernode are mapped into an array for array representation.

5. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 4, characterized in that: In step three, a supernode adjacency matrix is ​​constructed according to the row index fragments, including: If the super node J Segments and supernodes K If the diagonal matrix of has the same row index fragment, then the supernode J With SuperNode K There is an adjacency relationship, and a supernode adjacency matrix is ​​constructed according to the adjacency relationship.

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