Structural finite element stiffness matrix super-node approximate symbol decomposition method

Abstract

The invention belongs to the field of structural finite element stiffness analysis, and particularly relates to a structural finite element stiffness matrix super-node approximate symbol decomposition method. The method comprises the steps of 1, dividing super nodes on a structural finite element stiffness matrix; step 2, acquiring a row index fragment of the super node; 3, constructing a super-node adjacent matrix according to the row index fragment, and constructing a super-node elimination tree according to the super-node adjacent matrix; step 4, according to the super node elimination tree, performing symbol decomposition by taking the super nodes and the row index fragments as operation objects; and 5, according to the structural finite element stiffness matrix after symbol decomposition, aircraft structural strength design is carried out. According to the structural finite element stiffness matrix super-node approximate symbol decomposition method provided by the invention, the symbol decomposition time is greatly shortened under the guidance of a dimension reduction thought.

Description

Technical Field

[0001] The present application belongs to the field of structural finite element stiffness analysis, and in particular relates to a method for super-node approximate symbolic decomposition of a structural finite element stiffness matrix. Background Art

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

[0003] The purpose of symbolic decomposition of the structural finite element stiffness matrix is ​​to predict the location of the filler element, open up enough memory space for numerical decomposition, and provide a reference for planning the decomposition process. Traditional symbolic decomposition is guided by the elimination tree of the reordered matrix and operates on the matrix with single column and single row index. L The time complexity of the algorithm is O (| L |),| L | is a matrix L As the size of the matrix and the number of non-zero elements in the matrix continue to grow, the time consumed by symbolic decomposition will increase dramatically.

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

[0005] The purpose of the present application is to provide a method for super-node approximate symbolic decomposition of a structural finite element stiffness matrix to solve at least one problem existing in the prior art.

[0006] The technical solution of this application is: A method for super-node approximate symbolic decomposition of a structural finite element stiffness matrix, comprising: 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; Step 4: perform symbol decomposition according to the supernode elimination tree, taking the supernode and the row index fragment as operation objects; Step 5: Perform aircraft structural strength design based on the structural finite element stiffness matrix after symbolic decomposition.

[0007] In at least one embodiment of the present application, in step 1, dividing super nodes on the structural finite element stiffness matrix includes: 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.

[0008] In at least one embodiment of the present application, the boundaries between the matrix blocks are determined according to the following features: 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 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 , then take the j The column where the boundary lies.

[0009] In at least one embodiment of the present application, 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.

[0010] In at least one embodiment of the present application, in step three, constructing a supernode adjacency matrix according to the row index fragments includes: 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.

[0011] In at least one embodiment of the present application, the diagonal lines in the supernode adjacency matrix are filled with 1, and the non-diagonal lines are filled with 1 to indicate the existence of an adjacency relationship, and 0 to indicate the absence of an adjacency relationship.

[0012] 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 constructed respectively, 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 has priority to receive the fusion type task in each cycle.

[0013] The invention has at least the following beneficial technical effects: The structural finite element stiffness matrix super-node approximate symbolic decomposition method of the present invention is guided by the idea of ​​dimensionality reduction, greatly reduces the time of symbolic decomposition, and solves the problem that the time complexity of traditional symbolic decomposition methods increases dramatically with the increase of matrix size and non-zero number of elements. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a flow chart of a method for super-node approximate symbolic decomposition of a structural finite element stiffness matrix according to an embodiment of the present application; Figure 2 is a schematic diagram of the distribution of non-zero elements of the structural finite element stiffness matrix before and after METIS reordering according to an embodiment of the present application; Figure 3 is a schematic diagram of the boundaries between matrix blocks in one embodiment of the present application; Figure 4 is a schematic diagram of a row index fragment of a supernode according to an implementation mode of the present application; Figure 5 It is a schematic diagram of a supernode adjacency matrix and a supernode elimination tree according to one embodiment of the present application. DETAILED DESCRIPTION

[0015] In order to make the purpose, technical scheme and advantages of the implementation of this application clearer, the technical scheme in the embodiment of this application will be described in more detail below in conjunction with the drawings in the embodiment of this application. In the drawings, the same or similar reference numerals throughout represent the same or similar elements or elements with the same or similar functions. The described embodiments are part of the embodiments of this application, not all of them. The embodiments described below with reference to the drawings are exemplary and are intended to be used to explain this application, and should not be construed as limitations on this application. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of this application. The embodiments of this application are described in detail below in conjunction with the drawings.

[0016] In the description of the present application, it should be understood that the terms "center", "longitudinal", "lateral", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply 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 understood as limiting the scope of protection of the present application.

[0017] The following is combined with Figures 1 to 5 This application is described in further detail.

[0018] This application provides a method for super-node approximate symbolic decomposition of a structural finite element stiffness matrix, such as Figure 1 As shown, the following steps are included: Step 1: Divide super nodes on the structural finite element stiffness matrix; Step 2: Get 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; Step 4: Based on the supernode elimination tree, the supernode and the row index fragment are used as the operation objects to perform symbolic decomposition; Step 5: Design the aircraft structure strength based on the structural finite element stiffness matrix after symbolic decomposition.

[0019] The supernode approximate symbol decomposition method of the structural finite element stiffness matrix of the present application first divides the supernodes. In step 1, the supernodes are divided on the structural finite element stiffness matrix, specifically including: Obtain the structural finite element stiffness matrix after METIS reordering; Supernodes are divided on the structural finite element stiffness matrix according to the boundaries between matrix blocks.

[0020] Directly divide the supernodes on the structural finite element stiffness matrix after METIS reordering. The METIS reordering algorithm reorders the matrix based on a multi-level nested partitioning algorithm, which can effectively reduce the matrix filler elements and improve the parallelism of the matrix. Figure 2 As shown in Figure 2, the non-zero element distribution of the matrix after METIS reordering has obvious block characteristics. Based on the reordered matrix, super-node division is performed according to the block characteristics of the matrix. Figure 3 As shown, how to find the boundaries between matrix blocks is the key. These positions have the following characteristics, and the boundaries between matrix blocks are determined based on these characteristics: The row where the boundary is located has only one non-zero element on the diagonal. If 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 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 , then take the j The column to use as the boundary.

[0021] Since only the lower triangular part of the matrix is ​​saved during symbolic decomposition, the row where the boundary lies has only one non-zero element on the diagonal.

[0022] The structural finite element stiffness matrix supernode approximate symbol decomposition method of the present application, in step 2, obtaining the row index fragment of the supernode includes: Get the row index fragment of the non-zero elements of each supernode; Map all row index fragments of each supernode into an array for array representation.

[0023] After the supernodes are divided, all non-zero row index fragments of each supernode are mapped to an N-dimensional array, and all adjacent row indexes are grouped into one fragment. Figure 4 As shown, in one embodiment of the present application, the 8-dimensional matrix is ​​divided into 3 super nodes. , the row index fragment of each supernode is circled with a dotted box and saved in a two-dimensional array:

[0024] in, FN The array stores the number of segments each supernode has. Start , End The array holds the start and end information for each segment.

[0025] In the method for supernode approximate symbol decomposition of the structural finite element stiffness matrix of the present application, in step 3, a supernode adjacency matrix and a supernode elimination tree are constructed. In this embodiment, a supernode adjacency matrix is ​​constructed based on the obtained segmentation information, and its matrix size is SN OK, SN For each supernode J , if its segment and its subsequent supernode K If the diagonal matrix of has the same row index fragment, then the two supernodes have an adjacency relationship, or the supernodes K It is a super node J The ancestor supernode of J The set of ancestor supernodes can be expressed as:

[0026] according to Figure 4 The supernode adjacency matrix and supernode elimination tree obtained from the row index fragment in are as follows Figure 5As shown in the figure, the diagonal of the supernode adjacency matrix is ​​filled with 1, the non-diagonal is filled with 1 to indicate the existence of adjacency relationship, and 0 to indicate no adjacency relationship. At this point, the original 8-order matrix symbol decomposition problem is successfully transformed into a 3-order matrix symbol decomposition problem. Then, the elimination tree generation algorithm is applied to quickly obtain the supernode elimination tree, which is used to guide the parallel symbol decomposition process. The supernode elimination tree represented by the tree diagram is as follows Figure 5 Middle right picture.

[0027] The structural finite element stiffness matrix supernode approximate symbolic decomposition method of the present application, in step 4, after the aforementioned preparation stage is completed, the symbolic decomposition principle is formally applied, and the symbolic decomposition is performed with the supernode and the row index fragment as the operation object. When performing symbolic decomposition, a fusion supernode fragment task pool and an addition supernode fragment task pool are constructed respectively, 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 has priority to receive the fusion type task each time in a cycle. In the preferred implementation of the present application, OpenMP is used for parallel symbolic decomposition. Combined with the proposed task pool parallel template, two task pools of fusion supernode fragment and addition supernode fragment are constructed, wherein the fusion type task has priority, and once a supernode completes the fusion type task, the task of adding the supernode fragment of its parent supernode can be put into the task pool to wait for collection and execution. After all tasks are executed, the positions of all filler elements will be included in the obtained supernode fragments.

[0028] In this embodiment, the parallel supernode approximate symbol decomposition algorithm of OpenMP is used, specifically:

[0029] Line 1 creates two task pools, one for each type of task. task_add It is used to store the task number of adding the contribution fragment of the child supernode to its parent supernode, and task_merge The task number for fusing a certain supernode fragment is stored. Figure 5 In the matrix, supernode 2 is the parent supernode of supernode 0 and supernode 1, so supernode 0 and supernode 1 need to add their contribution fragments to supernode 2 to supernode 2; then after supernode 2 receives the contribution fragments of supernode 0 and supernode 1, in order to ensure that there are no duplicate fragments, it is necessary to merge the fragments of supernode 2. I Represents a supernode J A fragment of , then the supernode J To its parent supernode F The set of contributing fragments of can be expressed as:

[0030] Line 2 first goes to task_add Add all independent child supernode numbers in the , and then enter the OpenMP parallel region. Each thread will first receive the fusion type task in each loop. If there is no fusion type task, it will receive the add type task. When receiving the add type task, it should be noted that multiple child super nodes cannot add fragments to the same parent super node at the same time, so an additional auxiliary array is required to limit when receiving tasks. For example, you can use SN Dimensional array J_lock Mark the status if J_lock[J] =1, it means that other threads are executing on the supernode. J To add a fragment, the current thread cannot claim the supernode J The task of adding the sub-supernode number of . The task is completed, according to task_type The variable determines which specific task is to be performed next. 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 there may be multiple root super nodes in special cases.

[0031] The present invention relates to a method for approximate symbolic decomposition of supernodes of structural finite element stiffness matrix. A supernode is composed of multiple continuous columns with similar row indices. Unlike the traditional symbolic decomposition method, which requires the supernode to be divided after the symbolic decomposition is completed, the present invention directly divides the supernode on the matrix after the METIS reordering, so that the symbolic decomposition is reduced in the matrix dimension; based on the divided supernodes, the row index fragments of each supernode are obtained, and the matrix row index is further reduced in dimension; based on the supernode division and the row index fragments, the supernode adjacency matrix is ​​obtained, and the supernode elimination tree is obtained based on this to guide the symbolic decomposition; unlike the traditional symbolic decomposition method, which uses a single column and a single row index as the operation object for symbolic decomposition, the present invention uses supernodes and row index fragments as the basic operation objects for symbolic decomposition. Since the number of supernodes and row index fragments is much smaller than the number of matrix columns and the number of single row indices, the time complexity of symbolic decomposition is greatly reduced. Since the position of the filler element predicted by this method contains logical zero elements, it is an approximation of the exact pattern, so this symbolic decomposition is called approximate symbolic decomposition.

[0032] The structural finite element stiffness matrix super-node approximate symbolic decomposition method of the present application, finally, in step five, the structural finite element stiffness matrix after symbolic decomposition obtained according to the above steps can efficiently obtain the structural response and guide the aircraft strength design and optimization.

[0033] The above is only a specific implementation 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 a person skilled in the art within the technical scope disclosed in the present application should be included in the protection scope of the present application. Therefore, the protection scope of the present application shall be based on the protection scope of the claims.

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; Step 4: perform symbol decomposition according to the supernode elimination tree, taking the supernode and the row index fragment as operation objects; 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 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 , 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.

6. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 5, characterized in that: The diagonal of the supernode adjacency matrix is ​​filled with 1, and the non-diagonal is filled with 1 to indicate the existence of an adjacency relationship, and 0 to indicate the absence of an adjacency relationship.

7. The method for super-node approximate symbolic decomposition of structural finite element stiffness matrix according to claim 6, characterized in that: In step 4, when performing symbol decomposition, a fusion supernode fragment task pool and an addition supernode fragment task pool are constructed respectively, 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 has priority to receive the fusion type task in each cycle.

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