Inverse finite element deformation reconstruction method based on element information matrix assembly

Abstract

The present invention discloses an inverse finite element deformation reconstruction method based on unit information matrix assembly, including: constructing a sub-unit information matrix; assembling and solving a global reconstruction equation group. The present invention constructs a sub-unit information matrix of each inverse unit, and uses a sparse matrix to realize the assembly and storage of the sub-information matrices of all inverse units to obtain a full unit information matrix, and then converts the full unit information matrix in the form of a sparse matrix into a full matrix method to obtain a global equivalent stiffness matrix and a global equivalent external force matrix, and finally establishes a global deformation reconstruction equation group, solves the equation group to obtain the global unit node displacement, and further uses a shape function interpolation method to obtain the deformation of any point in the structure to achieve deformation reconstruction, with a small data storage volume and a fast reconstruction speed, and can be used for real-time deformation monitoring, which greatly improves the reconstruction speed while meeting the requirements of aerospace structure deformation monitoring.

Description

Technical Field

[0001] The present invention belongs to the technical field of aerospace structural health monitoring, and in particular relates to an inverse finite element deformation reconstruction method based on unit information matrix assembly. Background Art

[0002] Large space array antennas will experience complex thermal deformation during on-orbit operation. This deformation is more difficult to predict than general aerospace structures because thermal excitation is caused by drastic high and low temperature changes and non-uniform temperature distribution. In addition, complex thermal coupling with antenna frames or other satellite structures must be considered. To ensure that large space antennas can work properly, it is necessary to correct the array amplitude and phase errors caused by structural deformation and perform feedback control on the element beams. Real-time and reliable deformation monitoring is the basis for correction and compensation.

[0003] Among the structural deformation monitoring methods based on strain information, fiber Bragg grating sensors have the characteristics of light weight, small size, strong anti-interference ability, and good electromagnetic insulation performance. They are suitable for deformation monitoring of on-orbit antenna structures densely covered with electromagnetic sensitive elements. The sensor obtains real-time strain and combines it with advanced signal processing and deformation reconstruction algorithms to achieve online real-time perception of aerospace structures. Among the deformation reconstruction algorithms that have been developed based on strain information, the inverse finite element algorithm has the advantages of not requiring structural materials and load characteristics and having high reconstruction accuracy, which has attracted widespread attention.

[0004] However, the method commonly used in the current inverse finite element method for unit assembly is to first determine the rows and columns of all elements of the equivalent stiffness matrix and equivalent external force matrix of each unit in the global equivalent stiffness matrix and equivalent external force matrix, and then transmit them to the corresponding positions. When the number of units is large and there is a requirement for assembly time, this method is time-consuming and occupies a large amount of data memory, affecting the real-time performance of deformation reconstruction. Summary of the invention

[0005] In view of the above-mentioned deficiencies of the prior art, the purpose of the present invention is to provide an inverse finite element deformation reconstruction method based on unit information matrix assembly, so as to solve the problem of poor real-time performance of the inverse finite element method reconstruction based on unit-by-unit assembly in the prior art.

[0006] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] The inverse finite element deformation reconstruction method based on unit information matrix assembly of the present invention comprises the following steps:

[0008] 1) Construct subunit information matrix: discretize the monitored structure into several inverse units with unit nodes, and number all inverse units and all unit nodes respectively; construct the least square error function of the analytical strain field and the measured strain of each inverse unit and optimize it to obtain the equivalent stiffness matrix and equivalent external force matrix of each inverse unit, and then organize the coefficients in the obtained equivalent stiffness matrix into a subunit equivalent stiffness information matrix according to the node number of each inverse unit. Similarly, organize the coefficients in the equivalent external force matrix into a subunit equivalent external force information matrix, forming N subunit equivalent stiffness information matrices and N subunit equivalent external force information matrices;

[0009] 2) Overall assembly and solving the global reconstruction equation group: Arrange the N sub-unit equivalent stiffness information matrices and the N sub-unit equivalent external force information matrices in sequence according to the inverse unit numbers; if there are common unit nodes in adjacent inverse units, merge the information with the same unit node numbers to obtain the full-unit equivalent stiffness information matrix and the full-unit equivalent external force information matrix stored in the form of sparse matrices, and then convert the sparse matrix into a full matrix to construct the global equivalent stiffness matrix and the global equivalent external force matrix to achieve overall assembly. Finally, establish the global reconstruction equation group to obtain the global unit node displacement to achieve deformation reconstruction.

[0010] Furthermore, the process of constructing the subunit information matrix in step 1) specifically includes:

[0011] 11) The monitored structure is discretized into N reverse units, and M unit nodes are formed according to the reverse unit type. The reverse units and unit nodes are numbered respectively, and the least square error function of the analytical strain field and the measured strain of each reverse unit is constructed and optimized to obtain the node displacement array a of each reverse unit when the error between the analytical strain and the measured strain is the smallest. e The set of equations satisfied is:

[0012] ke n ·a e =fe n ,n=1,2...N

[0013] Among them, n andfe n are the equivalent stiffness matrix and equivalent external force matrix of the nth inverse element respectively;

[0014] 12) Generate a global number of degrees of freedom of each reverse unit node according to the unit node number; each reverse unit has multiple unit nodes, assuming that the number of the jth unit node of the nth (n=1……N) reverse unit is m (m=1……M), and the degree of freedom of each unit node is dof, then all the degrees of freedom of the unit node are numbered as dof·(m-1)+1, dof·(m-1)+2……dof·m, and traverse all the unit nodes of the reverse unit to obtain the degree of freedom numbers of all the unit nodes of the nth (n=1……N) reverse unit as the equivalent stiffness matrix ke of the nth reverse unit n and the equivalent external force matrix fe n The row and column position indices of the elements;

[0015] 13) The equivalent stiffness matrix ke n The subscripts corresponding to all elements are stored in the row index vector I ke-n and column index vector J ke-n , and the equivalent stiffness matrix ke n Data vector V ke-n , [I ke-n J ke-n V ke-n ] is the sub-unit equivalent stiffness information matrix; similarly, the equivalent external force matrix fe n The subscripts corresponding to all elements are stored in the row index vector I fe-n and column index vector J fe-n , and the equivalent external force matrix fe n Data vector V fe-n , [I fe-n J fe-n V fe-n ] is the sub-unit equivalent external force information matrix; traverse all inverse units to obtain N sub-unit equivalent stiffness information matrices and N sub-unit equivalent external force information matrices.

[0016] Furthermore, the overall assembly and solving of the global reconstruction equations in step 2) specifically includes:

[0017] 21) Arrange all sub-unit equivalent stiffness information matrices in sequence according to the reverse unit number to obtain the full unit equivalent stiffness information concatenation matrix [I ke J ke V ke ], similarly, all sub-unit equivalent external force information matrices are arranged in sequence according to the reverse unit number to obtain the full-unit equivalent external force information splicing matrix [I fe J fe V fe ]:

[0018]

[0019]

[0020] 22) The equivalent stiffness information of all units is concatenated into a matrix [I ke J ke V ke ] have the same row index I ke and column index J ke Position V ke The elements on are accumulated to construct the full-unit equivalent stiffness information matrix S in the form of a sparse matrix ke Similarly, the equivalent external force information of the whole unit is spliced ​​into the matrix [I fe J fe V fe ] have the same row index I fe and column index J fe Position V fe The elements on the sparse matrix are accumulated to construct the full-unit equivalent external force information matrix S fe ;

[0021] 23) Full unit equivalent stiffness information matrix S ke and the equivalent external force information matrix S of the whole unit fe It is a sparse matrix, which realizes the compressed storage of the equivalent stiffness information of the whole unit and the equivalent external force information of the whole unit. By adding zero elements, the sparse matrix is ​​transformed into a full matrix, and the global equivalent stiffness matrix K and the global equivalent external force matrix F are obtained, and the global reconstruction equation group is established:

[0022] K·U=F

[0023] The structural constraint conditions are supplemented to solve the equation to obtain the global unit node displacement U, and the shape function interpolation method is further used to obtain the deformation of any point in the monitored structure, thereby realizing deformation reconstruction.

[0024] Beneficial effects of the present invention:

[0025] The present invention constructs a sub-unit information matrix of each inverse unit, and uses a sparse matrix to assemble and store the sub-information matrices of all inverse units to obtain a full-unit information matrix, and then converts the full-unit information matrix in the form of a sparse matrix into a full matrix method to obtain a global equivalent stiffness matrix and a global equivalent external force matrix, and finally establishes a global deformation reconstruction equation group, and solves the equation group to obtain the global unit node displacement. Further, the shape function interpolation method can be used to obtain the deformation of any point in the structure to achieve deformation reconstruction. The data storage volume is small and the reconstruction speed is fast. It can be used for real-time deformation monitoring, and the reconstruction speed is greatly improved while meeting the deformation monitoring needs of aerospace structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1Flow chart of the method of the present invention.

[0027] Figure 2 Schematic diagram of the composite honeycomb sandwich panel structure and fixture positions used in the embodiment.

[0028] Figure 3 Schematic diagram of reverse unit division.

[0029] Figure 4a Schematic diagram of the simulated strain field of the positive strain on the upper surface of the honeycomb sandwich panel along the X-axis direction as the algorithm input of the inverse finite element method.

[0030] Figure 4b Schematic diagram of the simulated strain field of the positive strain on the upper surface of the honeycomb sandwich panel along the Y-axis direction as the algorithm input of the inverse finite element method.

[0031] Figure 4c Schematic diagram of the simulated strain field of the shear strain on the upper surface of the honeycomb sandwich panel in the XY plane as the algorithm input of the inverse finite element method.

[0032] Figure 5a Schematic diagram of the simulated strain field of the positive strain on the lower surface of the honeycomb sandwich panel along the X-axis direction as the algorithm input of the inverse finite element method.

[0033] Figure 5b Schematic diagram of the simulated strain field of the positive strain on the lower surface of the honeycomb sandwich panel along the Y-axis direction as the algorithm input of the inverse finite element method.

[0034] Figure 5c Schematic diagram of the simulated strain field of the shear strain on the lower surface of the honeycomb sandwich panel in the XY plane as the algorithm input of the inverse finite element method.

[0035] Figure 6a Schematic diagram of the simulated displacement field of the honeycomb sandwich panel in the direction perpendicular to the panel surface.

[0036] Figure 6b Schematic diagram of the reconstructed displacement field perpendicular to the panel surface obtained by using the inverse finite element method to reconstruct the deformation of the honeycomb sandwich panel.

[0037] Figure 6c Schematic diagram of the error distribution between the reconstructed displacement field and the simulated displacement field. DETAILED DESCRIPTION

[0038] In order to facilitate the understanding of those skilled in the art, the present invention is further described below in conjunction with embodiments and drawings. The contents mentioned in the implementation modes are not intended to limit the present invention.

[0039] Reference Figure 1 As shown, the inverse finite element deformation reconstruction method based on unit information matrix assembly of the present invention comprises the following steps:

[0040] 1) Construct subunit information matrix: discretize the monitored structure into several inverse units with unit nodes, and number all inverse units and all unit nodes respectively; construct the least square error function of the analytical strain field and the measured strain of each inverse unit and optimize it to obtain the equivalent stiffness matrix and equivalent external force matrix of each inverse unit, and then organize the coefficients in the obtained equivalent stiffness matrix into a subunit equivalent stiffness information matrix according to the node number of each inverse unit. Similarly, organize the coefficients in the equivalent external force matrix into a subunit equivalent external force information matrix, forming N subunit equivalent stiffness information matrices and N subunit equivalent external force information matrices;

[0041] Specifically, the process of constructing the subunit information matrix in step 1) specifically includes:

[0042] 11) The monitored structure is discretized into N reverse units, and M unit nodes are formed according to the reverse unit type. The reverse units and unit nodes are numbered respectively, and the least square error function of the analytical strain field and the measured strain of each reverse unit is constructed and optimized to obtain the node displacement array a of each reverse unit when the error between the analytical strain and the measured strain is the smallest. e The set of equations satisfied is:

[0043] ke n ·a e =fe n ,n=1,2...N

[0044] Among them, n andfe n are the equivalent stiffness matrix and equivalent external force matrix of the nth inverse element respectively;

[0045] 12) Generate a global number of degrees of freedom of each reverse unit node according to the unit node number; each reverse unit has multiple unit nodes, assuming that the number of the jth unit node of the nth (n=1……N) reverse unit is m (m=1……M), and the degree of freedom of each unit node is dof, then all the degrees of freedom of the unit node are numbered as dof·(m-1)+1, dof·(m-1)+2……dof·m, and traverse all the unit nodes of the reverse unit to obtain the degree of freedom numbers of all the unit nodes of the nth (n=1……N) reverse unit as the equivalent stiffness matrix ke of the nth reverse unit n and the equivalent external force matrix fe n The row and column position indices of the elements;

[0046] 13) The equivalent stiffness matrix ke n The subscripts corresponding to all elements are stored in the row index vector I ke-n and column index vector J ke-n, and the equivalent stiffness matrix ke n Data vector V ke-n , [I ke-n J ke-n V ke-n ] is the sub-unit equivalent stiffness information matrix; similarly, the equivalent external force matrix fe n The subscripts corresponding to all elements are stored in the row index vector I fe-n and column index vector J fe-n , and the equivalent external force matrix fe n Data vector V fe-n , [I fe-n J fe-n V fe-n ] is the sub-unit equivalent external force information matrix; traverse all inverse units to obtain N sub-unit equivalent stiffness information matrices and N sub-unit equivalent external force information matrices.

[0047] 2) Overall assembly and solving of the global reconstruction equation group: Arrange the N sub-unit equivalent stiffness information matrices and the N sub-unit equivalent external force information matrices in sequence according to the inverse unit numbers; if there are common unit nodes in adjacent inverse units, merge the information with the same unit node number to obtain the full-unit equivalent stiffness information matrix and the full-unit equivalent external force information matrix stored in the form of sparse matrices to achieve overall assembly, and then convert the sparse matrix into a full matrix to construct the global equivalent stiffness matrix and the global equivalent external force matrix, and finally establish the global reconstruction equation group to obtain the global unit node displacement to achieve deformation reconstruction;

[0048] Specifically, the overall assembly and solving of the global reconstruction equations in step 2) specifically includes:

[0049] 21) Arrange all sub-unit equivalent stiffness information matrices in sequence according to the reverse unit number to obtain the full unit equivalent stiffness information concatenation matrix [I ke J ke V ke ], similarly, all sub-unit equivalent external force information matrices are arranged in sequence according to the reverse unit number to obtain the full-unit equivalent external force information splicing matrix [I fe J fe V fe ]:

[0050]

[0051]

[0052] 22) The equivalent stiffness information of all units is concatenated into a matrix [I ke J ke V ke ] have the same row index I ke and column index Jke Position V ke The elements on are accumulated to construct the full-unit equivalent stiffness information matrix S in the form of a sparse matrix ke Similarly, the equivalent external force information of the whole unit is spliced ​​into the matrix [I fe J fe V fe ] have the same row index I fe and column index J fe Position V fe The elements on the sparse matrix are accumulated to construct the full-unit equivalent external force information matrix S fe ;

[0053] 23) Full unit equivalent stiffness information matrix S ke and the equivalent external force information matrix S of the whole unit fe It is a sparse matrix, which realizes the compressed storage of the equivalent stiffness information of the whole unit and the equivalent external force information of the whole unit. By adding zero elements, the sparse matrix is ​​transformed into a full matrix, and the global equivalent stiffness matrix K and the global equivalent external force matrix F are obtained, and the global reconstruction equation group is established:

[0054] K·U=F

[0055] The structural constraint conditions are supplemented to solve the equation to obtain the global unit node displacement U, and the shape function interpolation method is further used to obtain the deformation of any point in the monitored structure, thereby realizing deformation reconstruction.

[0056] The object of deformation and reconstruction in this embodiment is a composite honeycomb sandwich panel structure, such as Figure 2 As shown in the figure, the dimensions are 3125mm×1500mm×20mm (length×width×thickness), and a heating element is set on the lower surface. During the heating process, there is a temperature gradient in the thickness direction, which causes the honeycomb sandwich panel to warp. The honeycomb sandwich panel is divided into 29×23 four-node reverse shell elements according to the gradient of the simulated displacement of 1mm, as shown in the figure. Figure 3 As shown in Figure 2, each unit has 4 nodes and each node has 6 degrees of freedom. The strain field distribution of the normal strain and shear strain on the upper and lower surfaces of the honeycomb sandwich panel is shown in Figure 2. Figure 4a-4c and Figure 5a-5c As shown, the simulation strain and analytical strain field of each inverse unit are extracted to construct the least squares error function, and the equivalent stiffness matrix ke and equivalent external force matrix fe of each inverse unit are obtained, where the matrix size of the equivalent stiffness matrix ke is 24×24, and the matrix size of the equivalent external force matrix fe is 24×1.

[0057] Generate a global number of the degrees of freedom of each reverse unit node according to the unit node number; assuming that the four unit nodes of the nth reverse unit are numbered i, j, k, s, then the global number of the degrees of freedom of the reverse unit node is (6×i-5…6×i, 6×j-5…6×j, 6×k-5…6×k, 6×s-5…6×s), and convert the equivalent stiffness matrix ke n The subscripts corresponding to all elements are stored in the row index vector I ke-n and column index vector J ke-n , and the data vector V ke-n , where the size of the row index vector, column index vector and data vector is 576×1, and the sub-unit equivalent stiffness information matrix [I ke-n J ke-n V ke-n ]; Similarly, the equivalent external force matrix fe n The subscripts corresponding to all elements are stored in the row index vector I fe-n and column index vector J fe-n , and the data vector V fe-n , where the size of the row index vector, column index vector and data vector are all 24×1, and the equivalent external force information matrix of the subunit is obtained [I fe-n J fe-n V fe-n ].

[0058] The equivalent stiffness information matrix of all 667 inverse units is arranged in sequence according to the inverse unit number to obtain the equivalent stiffness information concatenation matrix of the whole unit [I ke J ke V ke ], the size of the matrix is ​​384192×3; the equivalent external force information matrices of all 667 inverse units are arranged in sequence to obtain the equivalent external force information concatenation matrix of the whole unit [I fe J fe V fe ], the matrix size is 16008×3.

[0059] will have the same row index I ke and column index J ke The data vector V at position ke The elements of are accumulated to obtain the equivalent stiffness information matrix S of the whole unit ke , similarly will have the same row index I fe and column index J fe The data vector V at position fe The elements of are accumulated to obtain the equivalent external force information matrix S of the whole unit fe .

[0060] S ke and S feThe compressed storage of each element and its corresponding position in the equivalent stiffness matrix and equivalent external force matrix of all inverse units and the overall assembly of the units are realized in the form of sparse matrices.

[0061] In S ke and S fe The vacant positions are supplemented with zero elements to realize the conversion from sparse matrix to full matrix, and the global equivalent stiffness matrix K with a size of 4320×4320 and the global equivalent external force matrix F with a size of 4320×1 are obtained respectively, and the global reconstruction equation group is established:

[0062] K·U=F

[0063] By adding structural constraints and solving the equations, the global unit node displacement U can be obtained. The shape function interpolation method can be used to obtain the deformation of any point in the structure, thereby realizing deformation reconstruction. The reconstruction result is as follows: Figure 6a-6c shown.

[0064] Comparing the reconstruction accuracy and assembly speed based on the unit information matrix assembly and the unit stacking one by one, the results show that the reconstruction displacement calculated by the two unit assembly methods is equal in 0.01mm accuracy, and the maximum error position reconstruction error is 8.86%. The assembly time based on the unit information matrix is ​​28.43ms, while the assembly time of the unit stacking one by one is 52.94ms. Therefore, the overall assembly based on the unit information matrix can speed up by more than 40%, effectively improving the real-time performance of deformation reconstruction.

[0065] The present invention has many specific application paths. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principle of the present invention. These improvements should also be regarded as the protection scope of the present invention.

Claims

1. An inverse finite element deformation reconstruction method based on unit information matrix assembly, characterized in that: Here are the steps: 1) Construct subunit information matrix: discretize the monitored structure into several inverse units with unit nodes, and number all inverse units and all unit nodes respectively; construct the least square error function of the analytical strain field and the measured strain of each inverse unit and optimize it to obtain the equivalent stiffness matrix and equivalent external force matrix of each inverse unit, and then organize the coefficients in the obtained equivalent stiffness matrix into a subunit equivalent stiffness information matrix according to the node number of each inverse unit. Similarly, organize the coefficients in the equivalent external force matrix into a subunit equivalent external force information matrix, forming N subunit equivalent stiffness information matrices and N subunit equivalent external force information matrices; 2) Overall assembly and solving the global reconstruction equation group: Arrange the N sub-unit equivalent stiffness information matrices and the N sub-unit equivalent external force information matrices in sequence according to the inverse unit numbers; if there are common unit nodes in adjacent inverse units, merge the information with the same unit node numbers to obtain the full-unit equivalent stiffness information matrix and the full-unit equivalent external force information matrix stored in the form of sparse matrices, and then convert the sparse matrix into a full matrix to construct the global equivalent stiffness matrix and the global equivalent external force matrix to achieve overall assembly, establish the global reconstruction equation group to obtain the global unit node displacement to achieve deformation reconstruction; The process of constructing the subunit information matrix in step 1) specifically includes: 11) The monitored structure is discretized into N reverse units, and M unit nodes are formed according to the reverse unit type. The reverse units and unit nodes are numbered respectively, and the least square error function of the analytical strain field and the measured strain of each reverse unit is constructed and optimized to obtain the node displacement array a of each reverse unit when the error between the analytical strain and the measured strain is the smallest. e The set of equations satisfied is: the n a e =fe n ,n=1,2...N Among them, n andfe n are the equivalent stiffness matrix and equivalent external force matrix of the nth inverse element respectively; 12) Generate a global number of degrees of freedom of each reverse unit node according to the unit node number; each reverse unit has multiple unit nodes. Assuming that the number of the jth unit node of the nth reverse unit is m, m=1...M, and the degree of freedom of each unit node is dof, then all the degrees of freedom of the unit node are numbered as dof·(m-1)+1, dof·(m-1)+2...dof·m. Traverse all the unit nodes of the reverse unit to obtain the degree of freedom numbers of all the unit nodes of the nth reverse unit as the equivalent stiffness matrix ke of the nth reverse unit. n and the equivalent external force matrix fe n The row and column position indices of the elements; 13) The equivalent stiffness matrix ke n The subscripts corresponding to all elements are stored in the row index vector I ke-n and column index vector J ke-n , and the equivalent stiffness matrix ke n Data vector V ke-n , [I ke-n J ke-n V ke-n ] is the sub-unit equivalent stiffness information matrix; similarly, the equivalent external force matrix fe n The subscripts corresponding to all elements are stored in the row index vector I fe-n and column index vector J fe-n , and the equivalent external force matrix fe n Data vector V fe-n , [I fe-n J fe-n V fe-n ] is the subunit equivalent external force information matrix; traverse all inverse units to obtain N subunit equivalent stiffness information matrices and N subunit equivalent external force information matrices; The overall assembly and solving of the global reconstruction equations in step 2) specifically include: 21) Arrange all sub-unit equivalent stiffness information matrices in sequence according to the reverse unit number to obtain the full unit equivalent stiffness information concatenation matrix [I ke J ke V ke ], similarly, all sub-unit equivalent external force information matrices are arranged in sequence according to the reverse unit number to obtain the full-unit equivalent external force information splicing matrix [I fe J fe V fe ]: 22) The equivalent stiffness information of all units is concatenated into a matrix [I ke J ke V ke ] have the same row index I ke and column index J ke Position V ke The elements on are accumulated to construct the full-unit equivalent stiffness information matrix S in the form of a sparse matrix ke Similarly, the equivalent external force information of the whole unit is spliced ​​into the matrix [I fe J fe V fe ] have the same row index I fe and column index J fe Position V fe The elements on the sparse matrix are accumulated to construct the full-unit equivalent external force information matrix S fe ; 23) Full unit equivalent stiffness information matrix S ke and the equivalent external force information matrix S of the whole unit fe It is a sparse matrix, which realizes the compressed storage of the equivalent stiffness information of the whole unit and the equivalent external force information of the whole unit. By adding zero elements, the sparse matrix is ​​transformed into a full matrix, and the global equivalent stiffness matrix K and the global equivalent external force matrix F are obtained, and the global reconstruction equation group is established: K·U=F The structural constraint conditions are supplemented to solve the equation to obtain the global unit node displacement U, and the shape function interpolation method is further used to obtain the deformation of any point in the monitored structure, thereby realizing deformation reconstruction.

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