Phased array plane displacement field inversion method based on combination of modal extension and inverse finite element method

Abstract

The invention relates to the field of structural health monitoring, particularly discloses a phased array plane deformation high-precision inversion method combining modal extension and an inverse finite element method, and aims to solve the problem that the traditional inverse finite element method is insufficient in displacement field inversion precision under the condition of sparse strain sensor layout. The method comprises the following steps of: firstly, constructing an experimental model and a simulation model, and respectively extracting displacement and strain modal data of a plurality of orders before the experimental model and the simulation model; secondly, a typical deformation working condition is applied to the experimental model, and sparse strain data are collected; then, expanding the sparse strain field into a full strain field by using a modal expansion technology; then, inputting the expanded full-field strain field into an inverse finite element algorithm, and carrying out inversion to obtain full-field displacement distribution of the structure; and finally, verifying the effectiveness of the method by comparing the inversion displacement with the real displacement. According to the method, the problem of information loss of sparse strain data is effectively solved through the modal extension technology, and the displacement inversion precision of the inverse finite element method under the sparse measurement condition is remarkably improved. Experimental results show that the method can improve the array plane inversion precision of the phased array from 30%-40% of a traditional method to more than 90%, and can meet the requirements of the phased array antenna for array plane deformation high-precision perception and electrical performance compensation in actual engineering.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring, and in particular to a high-precision inversion method for phased array surface deformation that combines modal extension and inverse finite element method, so as to improve the accuracy of phased array surface displacement field inversion under sparse strain field. Background Technology

[0002] Phased array antennas are widely used in modern radar, satellite communications, astronomical observation, and other fields. Their array shape directly affects beam pointing accuracy, sidelobe level, and overall electrical performance. During long-term service, due to continuous exposure to complex environments such as wind loads, thermal loads, and vibration loads, the antenna structure is prone to deformation, causing the array shape to deviate from the ideal design shape. This leads to electrical performance degradation phenomena such as decreased antenna gain, beam pointing deviation, and increased sidelobe level. Therefore, achieving high-precision displacement field inversion of phased array antenna structures is of great significance for ensuring their stable and reliable electrical performance.

[0003] Currently, this technology, which requires only discrete strain data from the structural surface to achieve high-precision inversion of the displacement field, has become a popular technique in the field of phased array surface deformation inversion. It eliminates the need for prior knowledge such as material properties and load information. The principle is to calculate the full-field displacement distribution of the entire structure by measuring the strain data on the structural surface and utilizing the strain-displacement relationship. Specifically, it involves obtaining discrete strain data using strain sensors deployed on the structural surface and reconstructing the displacement field of the entire structure using an inverse finite element method.

[0004] However, in practical engineering, limitations such as sensor cost, wiring difficulty, weight constraints, number of data acquisition channels, and reliability requirements often necessitate the use of sparse strain sensor layouts. With sparse strain data, the displacement field obtained by the inverse finite element method will have significant errors compared to the actual displacement field, making it difficult to meet the high-precision requirements for electrical performance compensation.

[0005] Therefore, in order to ensure the performance of phased array antennas during service, a new technical solution is urgently needed that can meet the high-precision inversion requirements of phased array antennas for surface deformation in practical engineering applications under sparse strain data. Summary of the Invention

[0006] The purpose of this invention is to overcome the problem of insufficient accuracy when directly performing displacement field inversion of phased array antenna surfaces using the inverse finite element method under sparse strain field conditions. Specifically, this invention aims to provide a high-precision displacement field inversion method that combines mode extension and the inverse finite element method. By locally correcting the finite element modes through experimental modes and extending the sparse strain field to a full-field strain field, and then combining this with inverse finite element inversion, high-precision inversion of the full-field displacement field of phased array antenna surfaces can be achieved with a small number of sensors.

[0007] To achieve the above objectives, this invention provides a high-precision displacement field reconstruction method based on a combination of modal extension and inverse finite element method, the technical solution of which is as follows:

[0008] 1. Within the theoretical framework of the inverse finite element method, the phased array antenna surface is simplified into a 450mm × 250mm × 3mm plate shell structure made of aluminum alloy, with specific material properties of density 2.7g / cm³, elastic modulus 70GPa, and Poisson's ratio 0.3. Four-node inverse shell elements (iQS4) are selected to discretize it into a mesh of 18 × 10⁻¹⁰ elements, each element being 25mm × 25mm and containing 209 nodes. In Abaqus simulation analysis, a cantilever plate structure is formed by fixing one short side as an experimental model to analyze the displacement modes and their relationship with the three directions (…). , , The strain mode shapes of the first 10 modes were obtained, and the displacement and strain corresponding to 36 specified points were extracted as experimental modal data.

[0009] 2. A mass block made of aluminum alloy is added to the existing model as a simulation model to simulate deviations between the finite element model and the actual structure due to modeling errors or inaccurate material parameters in real-world engineering. Modal analysis is performed on this simulation model, extracting the displacement mode shapes of all 209 nodes under the first 10 modes, and the full-field strain mode shapes in three directions at the centroids of all 180 elements (including...). , , (Three directions) to establish a finite element modal database.

[0010] 3. Apply both bending and torsional deformation conditions to the experimental model and collect sparse strain field data from 15 selected elements. Using the collected sparse strain data, along with the first 10 displacement / strain modes of the experimental model and the first 10 mode shapes of the simulation model as input, modal extension calculations are applied to obtain the extended full-field strain field. .

[0011] 4. The full-field strain field obtained in step 3 through modal expansion As input, it is substituted into the inverse finite element algorithm to obtain the high-precision inverted displacement field of the array surface.

[0012] 5. Extract displacement data from 209 nodes of the experimental model under two sets of working conditions, compare them with the displacement fields inverted by the inverse finite element method under the original and sparse conditions, and calculate the inversion accuracy index. To verify the inversion performance of this method. Attached Figure Description

[0013] Figure 1 This is a flowchart of the displacement field inversion method of the present invention;

[0014] Figure 2 This is a schematic diagram of the experimental model of the present invention in simulation software;

[0015] Figure 3 This is a schematic diagram of the simulation model of the present invention in the simulation software;

[0016] Figure 4 This is a schematic diagram of the sensor layout in the experimental model of the present invention;

[0017] Figure 5 This is a comparison diagram of the inversion results of the present invention under bending deformation;

[0018] Figure 6 This is a comparison chart of the inversion results of the present invention under torsional deformation;

[0019] Specific implementation steps

[0020] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0021] The present invention provides a phased array antenna surface deformation sensing method based on a combination of mode extension and inverse finite element method, which is used for high-precision inversion of the displacement field of phased array antenna surfaces. The phased array antenna surface may include vehicle-mounted phased array antennas, spaceborne phased array antennas, shipborne phased array antennas, etc. This invention uses a spaceborne phased array antenna surface as an example to illustrate this high-precision displacement field inversion method.

[0022] Reference Figure 1 The implementation steps of this example include the following:

[0023] Step 1: Constructing experimental and simulation models

[0024] 1.1) Within the theoretical framework of the inverse finite element method, the phased array antenna surface is simplified as a rectangular plate structure of 450mm × 250mm × 3mm, made of aluminum alloy. Its material properties are: density ρ = 2.7 g / cm³, elastic modulus E = 70 GPa, and Poisson's ratio ν = 0.3.

[0025] 1.2) The shell structure was discretized using a four-node inverse shell element (iQS4), resulting in 180 elements (18×10), each element measuring 25mm×25mm, for a total of 209 nodes. In Abaqus finite element software, all degrees of freedom along one short side of the shell structure were constrained to simulate the cantilever plate boundary conditions. (Refer to...) Figure 2The structure was used as an experimental model for modal analysis, which analyzed the displacement mode shapes at the first 10 natural frequencies, as well as the corresponding strain mode shapes (including in-plane normal strain). , and shear strain Select 36 nodes and extract their displacement and strain data for the first 10 modes as experimental modal data for subsequent mode extension.

[0026] 1.3) Add a mass block of the same material, aluminum alloy, to the experimental model described in step 1.2) to simulate the difference between the finite element model and the actual structure caused by modeling errors or inaccurate material parameters in actual engineering. (Refer to...) Figure 3 The structure was used as a finite element model, and the same modal analysis was performed on this model. The displacement mode shapes of all 209 nodes under the first 10 modes, as well as the full-field strain mode shapes in three directions at the centroid of all 180 elements, were extracted as the finite element modal data required for subsequent modal expansion.

[0027] Step 2: Apply typical working conditions to the experimental model and collect sparse strain data.

[0028] 2.1) Static analysis was performed on the experimental model in Abaqus to simulate two typical deformation conditions that may occur during actual service. Condition 1 is bending deformation, simulating the bending response of the front surface under wind load. Condition 2 is torsional deformation, simulating the torsional response of the front surface under asymmetric load.

[0029] 2.2) From the above static analysis results, referring to... Figure 4 The strain values ​​at the centroids of 15 pre-selected elements are extracted. This strain dataset simulates a sparse strain field actually measured using sparsely arranged strain rosette sensors. .

[0030] Step 3: Extend the sparse strain field to the full-field strain field based on the modal method.

[0031] To address the issue that sparse strain data from only 15 measuring points cannot directly meet the requirements for high-precision inverse finite element analysis, the modal method is used to extend the sparse data to the entire field. 3.1) The displacement mode shapes at 36 specified points extracted from the experimental model in step 1 are used as experimental modal data. For each experimental mode... The degrees of freedom are divided into two groups: one group is the fitting degrees of freedom used for least squares fitting. The other set consists of the observational degrees of freedom used to assess the extended quality. That is, for the i-th mode, we have:

[0032]

[0033] In this embodiment, 20 of the 36 measurement points are divided into a fitting set, and the remaining 16 points are divided into an observation set.

[0034] 3.2) Using the modal data of the finite element model constructed in step 1, extract the full-field displacement mode shapes. First, Separate into a mode shapes with a degrees of freedom corresponding to the experimental modal measurement points. and the remaining unmeasured d degrees of freedom mode shapes Secondly, Further, following the rules in step 3.1, the model is divided into fitting degrees of freedom corresponding to the experimental modes. and the observation degrees of freedom ,now that:

[0035]

[0036] 3.3) For the i-th experimental mode, calculate the distance between its frequency and the frequencies of each mode in the finite element modal database. Reorder the finite element mode shapes according to the frequency distance from closest to furthest, establishing a priority candidate list for that experimental mode. Then, construct a series of mode shape clusters based on the sorted list. The first cluster contains only the first mode in the list, i.e., the dominant mode with the closest frequency; the second cluster contains the first two modes in the list; and so on, until all finite element modes in the list are included.

[0037] 3.4) For each mode family (assuming it contains m finite element modes), introduce a diagonal selection matrix. To label the modes used by the current cluster. The projection vector of the i-th experimental displacement mode onto the fitted degrees of freedom is calculated using the least squares method. :

[0038]

[0039] In the formula: This indicates a pseudo-inverse operation. .

[0040] 3.5) Using the calculated projection vector, the estimated experimental modes at the observed degrees of freedom are inferred. :

[0041]

[0042] 3.6) To avoid overfitting or underfitting, calculate the Modal Confidence Criterion (MAC) value between the estimated experimental modes of the observation set and the actual measured experimental modes of the observation set:

[0043]

[0044] Traverse all modal clusters and find the one that makes The number of modes that reach the maximum value is denoted as the optimal number of modes. This refers to the optimal number of mode shapes. It ensures that the extended algorithm selects the physically best-matching combination of finite element modes.

[0045] 3.7) Utilizing the determined optimal number of modes corresponding projection vector Combined with the strain mode matrix of the full-field finite element model Calculate the full-field smooth extended mode of the i-th experimental strain mode:

[0046]

[0047] At this time, what was obtained It contains strain information required for all 180 elements in the inverse finite element method.

[0048] 3.8) Repeat steps 3.2 to 3.7 for the first K experimental modes to finally assemble the extended full-field experimental strain mode matrix. .

[0049] 3.10) Utilizing the extended full-field experimental strain modes Combined with the sparse strain data of the 15 elements collected in step 2 The full-field strain was calculated. This provides high-precision input data for subsequent inverse finite element displacement inversion:

[0050]

[0051] Step 4: Invert the full-field displacement field based on the inverse finite element method

[0052] 4.1) Substitute the extended strain field from step 3. A least-squares error function is constructed based on the relationship between measured strain and theoretical strain:

[0053]

[0054] In the formula, , and These are dimensionless weighted coefficients related to membrane strain, bending strain, and shear strain.

[0055] 4.2) By calculating the error function By taking the partial derivative of the function and setting it to zero, we can find the minimum value of the error function, i.e.:

[0056]

[0057]

[0058] in The element stiffness matrix, For element load vectors, , , These are the membrane plane strain transformation matrix, bending strain matrix, and transverse shear strain matrix, respectively.

[0059] 4.3) Assemble the element matrix according to the degrees of freedom of the nodes to form a global matrix, i.e.:

[0060]

[0061]

[0062] As the transformation matrix, and combining the displacement boundary conditions defined in 1.2), additional constraints are added to the global reconstructed equation system. After inverting these constraints, the global displacement can be obtained. :

[0063]

[0064] Step 5, Evaluation of Inversion Results

[0065] To verify the effectiveness of this method, the inversion results are compared with the actual displacement field of the experimental model in step 2. 5.1) From the simulation results of the experimental model in step 2, the actual displacements of all 209 nodes under the bending and torsional conditions are extracted. 5.2) Combine the inverted displacement field obtained in step 4 with the displacement field obtained by direct solution under sparse strain. With the real displacement field A comparison was made, and an inversion accuracy index was introduced. Evaluate the results:

[0066]

[0067] Comparison results reference Figure 5 and Figure 6 In this example, the displacement inversion accuracy of the traditional iFEM method under sparse strain fields is approximately 30%-40%. However, by employing the modal extension and inverse finite element combined method proposed in this invention, the displacement field inversion accuracy of the spaceborne phased array surface is significantly improved to over 90%, effectively overcoming the problem of insufficient accuracy of the traditional inverse finite element method under sparse strain fields.

[0068] The above embodiments demonstrate the entire process of this invention, from model construction, modal extension, inverse finite element inversion to verification. This invention is not limited to the disclosed embodiments; adjustments can be made to different structural forms, sensor types and layouts, and modal orders according to actual circumstances, all of which fall within the protection scope of this invention.

Claims

1. A high-precision inversion method for phased array surface deformation based on a combination of modal extension and inverse finite element method, characterized in that, Includes the following steps: (1.1) Within the theoretical framework of the inverse finite element method, the phased array antenna surface is simplified into a 450mm×250mm×3mm plate shell structure made of aluminum alloy, with specific material properties of density 2.7g / cm³, elastic modulus 70GPa, and Poisson's ratio 0.

3. A four-node inverse shell element (iQS4) is selected to discretize it into a mesh of 18×10 elements, each element being 25mm×25mm in size and containing 209 nodes. In Abaqus simulation analysis, one short side is fixed to form a cantilever plate structure as an experimental model to analyze the displacement modes and their relationship with the three directions (…). , , The strain mode shapes of the first 10 modes were obtained, and the displacement and strain corresponding to 36 points in the first 10 modes were extracted as experimental modal data. (1.2) A mass block with the same material properties as aluminum alloy is added to the same model as the simulation model. This is intended to simulate the deviation between the finite element model and the actual structure in actual engineering due to modeling errors or inaccurate parameters. Modal analysis is performed on the simulation model to extract the displacement mode shapes of all 209 nodes under the first 10 modes, as well as the three directions at the centroid of all 180 elements. , , Full-field strain mode shapes were analyzed, and a finite element modal database was established. (1.3) Apply bending and torsional deformation conditions to the experimental model and collect sparse strain field data of 15 selected elements; use the collected sparse strain data, the first 10 mode shapes of the experimental model and the first 10 mode shapes of the simulation model as input, and apply modal extension calculation to obtain the extended full-field strain field. ; (1.4) The full-field strain field obtained by modal expansion in step (1.3) As input, it is substituted into the inverse finite element algorithm to obtain the high-precision inverted displacement field of the array surface; (1.5) Extract displacement data of 209 nodes from the experimental model under two sets of working conditions, compare them with the displacement fields inverted by this method and the inverse finite element method under sparse conditions, and calculate the inversion accuracy index. To verify the inversion performance of this method.

2. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The extraction of the experimental modal data specifically includes: dividing 20 of the 36 measurement points into a fitting set and the remaining 16 points into an observation set; for the i-th order experimental mode... Its degrees of freedom are divided into fitting degrees of freedom. and observation degrees of freedom ,Right now 3. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The construction of the finite element modal database specifically includes: finite element full-field displacement mode shapes. Separate into a mode shapes with a degrees of freedom corresponding to the experimental modal measurement points. and the remaining d degrees of freedom mode shape ; further Divided into fitting degrees of freedom and the observation degrees of freedom ,Right now 4. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The modality expansion calculation specifically includes: (4.1) For the i-th experimental mode, calculate the distance between its frequency and the frequencies of each mode in the finite element mode database. Rearrange the finite element mode shapes from near to far according to the frequency distance, establish a priority candidate list, and construct a mode shape cluster. (4.2) For each mode cluster constructed in step (4.1), a diagonal selection matrix is ​​introduced. The projection vector is calculated using the least squares method, where † denotes the pseudo-inverse operation: (4.3) Using the projection vector obtained in step (4.2) By inversely estimating the experimental modes at the observed degrees of freedom: (4.4) Calculate the Modal Confidence Criterion (MAC) value. Iterate through all modal clusters and find the value that maximizes it. As the optimal number of modes: (4.5) Utilization The corresponding projection vector, combined with the strain mode matrix of the full-field finite element model. Calculate the full-field smooth extended mode of the i-th experimental strain mode: (4.6) Repeat the above process for the first K experimental modes to assemble the extended full-field experimental strain mode matrix. ; (4.7) Solving using step (4.6) Combining sparse strain data Calculate the strain in the entire field :

5. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The inverse finite element algorithm specifically includes: (5.1) Constructing a least squares error function based on measured strain and theoretical strain in , and represents the dimensionless weighting coefficients related to membrane strain, bending strain, and shear strain; (5.2) By calculating the error function By taking the partial derivative of the function and setting it to zero, we can find the minimum value of the error function: in The element stiffness matrix, For the element load vector: , , These are the membrane plane strain transformation matrix, bending strain matrix, and transverse shear strain matrix, respectively. (5.3) Assemble the element matrix into a global matrix according to the degrees of freedom of the nodes. Using the transformation matrix and displacement boundary conditions, solve for the total field displacement.

6. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The inversion accuracy index The calculation formula is: in For the true displacement field, To invert the displacement field, The number of nodes; under sparse strain field conditions, the inversion accuracy of the traditional inverse finite element method is about 30%-40%, while this method improves it to over 90%.

7. The phased array surface displacement field inversion method based on the combination of modal extension and inverse finite element method according to claim 1, characterized in that, The phased array antenna array includes vehicle-mounted phased array antennas, space-borne phased array antennas, or shipborne phased array antennas.

Images