Method for eliminating ill-conditioning of mathematical model for frequency domain identification of large-area distributed random ice load

Abstract

The present application relates to the field of ship structure safety evaluation, and discloses a large-area distributed random ice load frequency domain identification mathematical model ill-posed elimination method, first, by establishing a finite element model containing the hull plate, the rib and the longitudinal structure, the ice load is converted into an equivalent concentrated force model, and multiple point strain response measuring points are arranged on the surface of the structure, then, frequency response data is obtained by applying a simple harmonic excitation to form a frequency response function matrix, and the high ill-posed frequency is optimized by a multiple extension search algorithm, the sub-matrix with the smallest condition number is screened out, the frequency domain identification model is reconstructed, finally, based on the optimized frequency response function sub-matrix, combined with the structure response spectrum and the inverse problem theory, the generalized inverse method is used to solve the ice load power spectral density function. By optimizing the frequency response function matrix through the multiple extension search algorithm and the generalized inverse method, the stability and identification accuracy of the model are significantly improved, and the ill-posed problem caused by the high ill-posed frequency in the existing method is solved.

Description

Technical Field

[0001] This invention relates to the field of ship structural safety assessment, specifically to a method for eliminating ill-conditioned mathematical models for frequency domain identification of large-area distributed random ice loads. Background Technology

[0002] With the development of shipping in ice-covered waters, the dynamic ice loads generated by ship-ice collisions pose a severe challenge to the structural safety of ships. To effectively monitor ice loads, the indirect identification method of determining loads by inverse structural response has become the mainstream technology due to its advantages such as safety and convenience.

[0003] However, during high-speed icebreaking, ships encounter non-stationary random ice loads with complex time-frequency characteristics, which places higher demands on the accuracy of load identification, especially in the frequency domain. The core technical problem facing current frequency domain load identification technology is the severe ill-conditioned nature of the mathematical models it relies on during the solution process. This problem arises from the ill-conditioned characteristics of the frequency response function matrix in the model, leading to low accuracy and poor stability in load identification results.

[0004] To address this issue, existing technologies primarily employ two approaches: one is algorithms such as Tikhonov regularization, but these are ineffective when handling complex random ice loads; the other is optimizing sensor placement to reduce the condition number of the frequency response function matrix, such as C-optimal or D-optimal design methods. However, these optimization methods only significantly improve the ill-conditioned nature of the matrix at a single frequency point, and have limitations in controlling global ill-conditioned nature across the entire operating frequency band. They are unable to effectively suppress the ill-conditioned problem prevalent across a wide frequency domain and cannot meet the requirements for accurate and stable identification of large-area distributed random ice loads.

[0005] Therefore, existing technologies still have significant gaps in how to effectively eliminate the global ill-conditioning of frequency domain identification mathematical models for broadband, large-area random ice loads, and new solutions are urgently needed. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a method for eliminating ill-conditioned mathematical models for frequency domain identification of large-area distributed random ice loads, which solves the problems of ill-conditioned frequency response function matrix and ill-posed inversion caused by excessively large condition number in the identification of ice loads on ship structures.

[0007] To achieve the above objectives, the present invention provides the following technical solution: a method for eliminating ill-conditioning in a frequency domain identification mathematical model of large-area distributed random ice load, comprising the following steps:

[0008] S1. Establish a finite element model including the hull plate, ribs and longitudinal girder structure, construct an equivalent concentrated force model under ice load, and set up multiple strain response measurement points on the structural surface as the initial array of sensors.

[0009] S2. Set multiple nodes in the ice load area, apply simple harmonic excitation of unit amplitude to each node one by one, obtain strain response frequency data at the structural measuring points, and establish the frequency response function matrix to form a frequency domain structure identification model.

[0010] S3. In the frequency domain identification model, based on the ill-conditioned analysis of the frequency response function matrix, the frequency point corresponding to the maximum condition number is identified, and the high ill-conditioned frequency response matrix is ​​extracted.

[0011] S4. For the highly ill-conditioned matrix, a multi-extension search algorithm is used to select the sub-matrix with the smallest condition number from multiple measurement point combinations, forming the optimized frequency response sub-matrix, and determining the corresponding optimized measurement point layout scheme.

[0012] S5. Reconstruct the structural frequency domain identification model using the optimized frequency response function submatrix, collect the measured strain response of the optimized measurement points, and construct the response power spectral density function.

[0013] S6. Based on the inverse problem theory and the generalized inverse matrix solution method, combined with the reinforcement identification model and structural response spectrum, the power spectral density function of ice load is identified, which is used to reflect the random excitation intensity distribution of each load node in the frequency domain.

[0014] Preferably, the equivalent concentrated force model in step S1 is constructed by discretizing the ice extrusion pressure into multiple concentrated normal nodal forces acting on the surface of the hull. The nodes are evenly distributed along the bow outer plate area to form a two-dimensional grid arrangement, which is used to simulate the spatial distribution characteristics of the actual icebreaking load.

[0015] Preferably, the sensor deployment method in step S1 includes the following:

[0016] The strain gauges of the sensor in the outer plate area are arranged along the axial and transverse directions of the structure, respectively, at 0 degrees and 90 degrees.

[0017] The strain gauges for the sensors in the rib and longitudinal girder areas are arranged at an angle of ±45 degrees.

[0018] The sensor array is arranged in a rectangular, equally spaced pattern to form a multi-channel structural response observation network.

[0019] Preferably, in step S2, the frequency response function matrix is ​​a complex domain matrix with row dimension representing the number of sensors and column dimension representing the number of ice load nodes. The matrix elements are the complex strain response at a certain sensor location generated by a unit harmonic excitation acting on a certain load node at a given frequency.

[0020] Preferably, the identification method based on ill-conditioned analysis of the frequency response function matrix in step S3 is as follows: within a preset frequency scanning range, the condition number of the frequency response function matrix at each frequency is calculated. The condition number is the product of the norm of the matrix and the norm of its generalized inverse matrix. The frequency with the largest condition number is taken as the high ill-conditioned frequency.

[0021] Preferably, the multiple expansion retrieval algorithm in step S3 includes the following steps:

[0022] Select an initial set of row vectors from the high ill-conditioned frequency response matrix as the initial measurement point set;

[0023] The initial combination is replaced row by row to generate several candidate submatrices. The condition number is recalculated after each replacement.

[0024] A full combinatorial traversal strategy is adopted to traverse all possible combinations of test point permutations;

[0025] The submatrix with the smallest condition number is selected as the optimized frequency response submatrix, and the measurement point positions corresponding to its row indices are recorded to form the final optimized layout scheme.

[0026] Preferably, the optimized frequency response sub-matrix in step S5 is used to reconstruct the structural frequency domain identification model, wherein the model input is the ice load power spectral density matrix at each frequency, and the output is the strain response spectrum of the optimized measurement point, which is constructed through the structural power transfer relationship.

[0027] Preferably, the method for constructing the response power spectral density function in step S5 includes:

[0028] Collect the strain time series of the optimized measuring points under actual ice load excitation;

[0029] The frequency domain strain response is obtained by performing a discrete Fourier transform on the strain sequence;

[0030] Calculate the autocorrelation and cross-correlation spectral densities between each response channel to construct the structural response spectral matrix.

[0031] Preferably, the step of solving the ice load power spectral density function in step S6 includes:

[0032] A frequency domain inverse problem model is constructed based on the structural response spectrum matrix and the optimized frequency response function submatrix.

[0033] The input spectral matrix is ​​solved using the Moore-Penrose generalized inverse method;

[0034] The identification result is the diagonal elements of the spectral matrix, which represent the self-power spectral density at each ice load node and are used as the final output.

[0035] This invention provides a method for eliminating ill-conditioned mathematical models of large-area distributed random ice load frequency domain identification. It has the following beneficial effects:

[0036] 1. This invention provides better optimization results for highly ill-conditioned matrices than the C-optimal design method and the D-optimal design method. By optimizing the highly ill-conditioned frequencies in the frequency response function matrix through multiple extended retrieval algorithms, it achieves a significant reduction in the condition number at that frequency. Compared with the C-optimal design method and the D-optimal design method, which only optimize a single frequency and lead to an increase in the condition number at other frequencies, this invention can perform more comprehensive optimization of highly ill-conditioned matrices, effectively mitigating the negative impact of highly ill-conditioned frequencies on model stability and recognition accuracy.

[0037] 2. Compared with the C-optimal design method and the D-optimal design method, the present invention can provide an optimized frequency response function matrix with lower global conditions. In the optimization process, it not only focuses on the condition number of individual frequencies, but also reduces the condition number of the entire frequency response function matrix through a global optimization scheme, thereby improving the numerical stability of the matrix. Compared with the traditional C-optimal and D-optimal design methods that only optimize the condition number of specific frequencies, the present invention can achieve a lower global condition number in the entire frequency domain, thereby improving the stability of the inversion process and the accuracy of the results.

[0038] 3. Compared with the C-optimal design method and the D-optimal design method, the present invention can achieve more flexible subjective control of computational cost. In the optimization process of multi-frequency points, the optimization method of the present invention can flexibly adjust the complexity of optimization calculation as needed. Compared with the fixed calculation strategy of the C-optimal design method and the D-optimal design method, the solution of the present invention provides a more flexible control method, which can adjust the degree of optimization according to the actual computing resources and recognition accuracy requirements, and reduce unnecessary computational burden.

[0039] 4. The recognition results of this invention show a more significant accuracy advantage compared to those optimized using the C-optimal design method and the D-optimal design method. Because this invention employs a combination of multiple extended searches and generalized inverse methods in the optimization process of the frequency response function matrix, the optimized matrix provides more stable and accurate recognition results during the inversion process. Compared to the C-optimal design method and the D-optimal design method, the optimization results of this invention show a significant accuracy advantage, especially when facing high ill-conditioned frequencies, effectively improving the accuracy of model inversion and thus providing a more reliable basis for ship structural safety assessment. Attached Figure Description

[0040] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0041] Figure 2 This is the mechanical model of the random ice load equivalent of the present invention;

[0042] Figure 3 This is a schematic diagram of the initial measurement point arrangement and sensor arrangement method of the present invention;

[0043] Figure 4 This is a schematic diagram of the ice load distribution of the node to be identified in this invention;

[0044] Figure 5 This is a flowchart of the optimization algorithm of the present invention;

[0045] Figure 6 The results of ill-condition elimination in the mathematical model of this invention;

[0046] Figure 7 This is a schematic diagram of the sensor optimization arrangement scheme of the present invention;

[0047] Figure 8 This is the result of ice load spectrum identification in this invention. Detailed Implementation

[0048] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] Please see the appendix Figure 1 -Appendix Figure 8 This invention provides a method for eliminating ill-conditioned mathematical models for frequency domain identification of large-area distributed random ice loads, comprising the following steps:

[0050] S1. Establish a finite element model including the hull plate, ribs and longitudinal girder structure, construct an equivalent concentrated force model under ice load, and set up multiple strain response measurement points on the structural surface as the initial array of sensors.

[0051] Before constructing a frequency domain mathematical model to identify the random dynamic ice loads experienced by a ship's structure in an ice-covered environment, it is necessary to first model the ship's structure and its loading environment. This embodiment takes the bow region of an icebreaker as the research object and establishes an initial mechanical system covering the observation network of ice load distribution and structural response.

[0052] Generally, to improve the spatial resolution and practical adaptability of the identification results, a finite element model capable of characterizing the ice-ship coupling behavior needs to be constructed, supplemented by a multi-point measurement array for subsequent calculation of the frequency response function matrix. In this embodiment, the model includes the main load-bearing structures, including the bow plating, ribs, and longitudinal girder.

[0053] In this embodiment, to characterize the actual distribution of ice extrusion loads on the structural surface, the ice loads are considered as concentrated normal forces at multiple points, and an equivalent mechanical model is constructed using discretization. The ice extrusion force is simplified as a concentrated force acting perpendicularly at the nodes of the outer plate of the structure, distributed along the normal direction of the outer plate, and constructed as follows:

[0054] In one specific implementation, let the total number of load nodes be... These nodes are arranged in a regular pattern on the outer bow plate area, covering the flat ice contact zone. The node spacing in the x direction (lateral) is 0.4 m, and the node spacing in the z direction (vertical) is 0.3 m, ultimately forming a node array of approximately 1.2 m × 10.4 m, with a total of 133 nodes.

[0055] Ice load nodes can be defined as:

[0056] ;

[0057] in, Indicates the first Concentrated ice load at each node, in N; =133 represents the total number of nodes. All nodes act along the normal direction, forming a two-dimensional discrete field, approximating the random loading of actual ice blocks.

[0058] As an alternative, to ensure the identifiability of the frequency response function, this embodiment employs a network of strain gauges strategically arranged within the structure to form a response observation network. The arrangement of the measuring points follows the principle of the principal stress directions of the structure, thereby enhancing the directional sensitivity of the strain response.

[0059] Specifically:

[0060] In the outer plate area, strain gauges are arranged in two directions: 0° (hull axis) and 90° (lateral).

[0061] In the rib and longitudinal girder areas, strain gauges are arranged at an angle of ±45° to capture shear strain.

[0062] The measuring point array adopts a rectangular uniform arrangement with a measuring point spacing of 0.1 m, covering an inner area of ​​11.2 m × 4.2 m, and precisely covering all areas with significant stress.

[0063] In one possible implementation, let the total number of measuring points be m, and the measuring point number be denoted as:

[0064] ;

[0065] in, Indicates the first The sensor locations are indicated by position coordinate vectors. The measurement points cover the ribs, longitudinal girder, and interior of the outer panels. Points in the outer panel area are positioned 0.1 m from the structural edge to ensure observational sensitivity.

[0066] To establish the frequency domain model, a unit amplitude harmonic excitation needs to be applied to the structural model to simulate a single-point ice load input, and the strain response at all measuring points needs to be recorded to form the frequency response function. This process will be carried out in the subsequent S2 step.

[0067] Generally, to ensure the stability of the inverse problem solution, it is necessary to pay attention to the ill-conditioned characteristics of the frequency response function matrix and ensure that the number and location of measurement points are sufficiently linearly independent of the composition of the frequency response matrix. This embodiment retains a certain degree of redundancy in the measurement point layout to provide a data foundation for subsequent dimensionality reduction optimization.

[0068] S2. Set multiple nodes in the ice load area, apply simple harmonic excitation of unit amplitude to each node one by one, obtain strain response frequency data at the structural measuring points, and establish the frequency response function matrix to form a frequency domain structure identification model.

[0069] After completing the structural model and observation network construction, in order to establish a frequency domain response model that can accurately reflect the structure's performance under ice loads, it is necessary to further identify the dynamic response behavior of the structure when excited at each ice load node. By applying a unit amplitude harmonic excitation signal to the preset ice load nodes and combining the strain measurement results at the observation points, the response mapping relationship of the structure in the frequency domain can be established, ultimately forming a frequency response function matrix to characterize the frequency domain dynamic characteristics of the structure.

[0070] Generally, ice loads exhibit random uncertainty in actual processes, making it difficult to directly obtain their time-domain distribution characteristics. Therefore, frequency-domain modeling methods are more convenient for extracting system response characteristics and using them for subsequent inversion and identification. This embodiment establishes a frequency-domain structure identification model by constructing the frequency response function matrix.

[0071] In this embodiment, the Frequency Response Function (FRF) matrix is ​​constructed in the following way:

[0072] First, multiple nodes are defined within the ice load region, and a simple harmonic excitation of unit amplitude is applied sequentially to each node. The form of this excitation signal is as follows:

[0073] ;

[0074] in, Indicates the application of the first Excitation functions at each ice load node This represents the excitation angular frequency, expressed in rad / s. This represents the total number of ice load nodes. Harmonic excitation is applied with unit amplitude, perpendicular to the hull plating, and in the same direction.

[0075] While applying excitation at each node, through the arrays deployed on the inner surface of the structure... Each strain measurement point records the response signal. For any given excitation point... and any measuring point The strain response at this frequency can be obtained. It is a complex number used to simultaneously characterize the amplitude and phase information of the response.

[0076] Therefore, the constructed frequency response function matrix is ​​represented as:

[0077] ;

[0078] in, The frequency response function matrix, The total number of strain measurement points. The total number of ice load nodes, matrix elements When the frequency is At that time, the unit harmonic excitation acts on the first The first load node caused by the Complex strain response at each measuring point.

[0079] In one possible implementation, the frequency response function is measured using a frequency sweep method or a multi-frequency excitation technique, sampling point-by-point within the frequency range of interest, for example:

[0080] Frequency range is rad / s, corresponding to 1Hz to 50Hz;

[0081] Sampling was performed every 0.25Hz to obtain 200 independent frequency points;

[0082] For each frequency point, repeat the above unit excitation and response recording process;

[0083] A network is constructed at each frequency point. matrix.

[0084] Generally, the structural response is significantly affected by the orientation of the measurement points. Therefore, in actual measurements, it should be ensured that the strain gauge orientation is consistent with the principal stress direction to improve the sensitivity and accuracy of response identification. For oblique arrangements in rib and longitudinal girder regions, the strain response usually includes a shear component and should be incorporated into the complex response calculation process.

[0085] As an alternative, to avoid severe ill-conditioned problems in the frequency response function matrix, singular value analysis can be introduced to evaluate the matrix. The rank of the frequency response function matrix affects the stability of the structural input-output mapping; a low-rank matrix may lead to non-uniqueness in subsequent load identification. Therefore, the measurement point layout should cover multiple structural directions to ensure linear independence between the matrix column vectors.

[0086] Specifically, in this embodiment, the number of measurement points is m=9100, the number of load nodes is n=133, and the response matrix is ​​a complex field matrix of 9100×133.

[0087] At certain frequencies, if the matrix rank is found to be insufficient, it is necessary to further increase the observation dimension, for example:

[0088] Densifying the number of measuring points in the outer panel area;

[0089] Increase the number of strain gauges in the vertical direction;

[0090] Extend the observation area to the adjacent module structure.

[0091] Furthermore, to balance computational efficiency and recognition accuracy, frequency domain window functions or noise filtering methods are introduced in some embodiments to enhance the robustness of FRF measurements. Commonly used methods include Gaussian windows, Hamming windows, or weighted averaging techniques.

[0092] The final constructed frequency response function matrix It will serve as the core input for subsequent ice load inversion analysis and structural characteristic identification, and will be used throughout the entire identification process.

[0093] S3. In the frequency domain identification model, based on the ill-conditioned analysis of the frequency response function matrix, the frequency point corresponding to the maximum condition number is identified, and the high ill-conditioned frequency response matrix is ​​extracted.

[0094] After obtaining the complete frequency response function matrix, it is necessary to quantitatively analyze the ill-conditioned nature of the matrix at different frequencies to further improve the stability and inversion accuracy of structure identification. By identifying the frequency points with the most severe ill-conditioned behavior and performing in-depth processing on the frequency response function matrix at those frequencies, a foundation can be provided for subsequent observation point optimization and data dimensionality reduction.

[0095] Generally, the frequency response function matrix may exhibit characteristics such as rank deficiency and enhanced linear correlation at certain frequency points, leading to instability in the structural input-output mapping. These frequencies often correspond to system resonance frequencies, regions of dense mode activity, or weak structural characteristic response regions. Therefore, identifying ill-conditioned frequencies in the frequency domain is a crucial step in constructing a robust inversion model.

[0096] In this embodiment, the ill-conditioned analysis method uses the matrix condition number as the evaluation metric. Let a certain frequency point... The corresponding frequency response function matrix is:

[0097] ;

[0098] Its condition number Defined as the product of the norm of the matrix and the norm of its generalized inverse matrix, the expression is:

[0099] ;

[0100] in, represents the matrix norm, taking the 2-norm (i.e., the largest singular value); Representation matrix Moore-Penrose generalized inverse; The larger the value, the closer the matrix is ​​to being ill-conditioned.

[0101] In one possible implementation, the selected frequency scanning interval is: The condition number is calculated in rad / s, with a step size of 0.25 Hz, for each frequency point. Finally, the frequency point corresponding to the maximum condition number is determined. :

[0102] ;

[0103] High pathological frequency Then, extract the corresponding frequency response function matrix. This serves as the basis for subsequent analysis of the combination and optimization of observation points.

[0104] To further improve the numerical stability of the structural response matrix, this embodiment introduces a combinatorial optimization algorithm based on a multiple expansion retrieval strategy to extract submatrices for highly ill-conditioned frequency response matrices.

[0105] Specifically, the algorithm includes the following steps:

[0106] In the initial stage, from Selected from Rows form a submatrix, denoted as ,in This initial submatrix can be determined through random selection, principal component selection, or mode coverage principles, serving as the initial observation subset.

[0107] Alternatively, the initial combination can be set as a row index sequence. Its corresponding submatrix is ​​denoted as:

[0108] ;

[0109] Where represents the first in the original matrix Row vectors.

[0110] Initial composite index sequence Based on this, the remaining row vectors are stored sequentially. Add rows to obtain the high-dimensional matrix with the minimum condition number, and continue until the target dimension matrix is ​​obtained, at which point the expansion part ends.

[0111] Subsequently, a row-by-row replacement method was used to generate candidate sub-combinations. While maintaining... Without changing any of the original rows, replace one row vector with a new one to generate a new submatrix, and then recalculate its condition number:

[0112] ;

[0113] This process iterates continuously until all possible row combinations have been traversed, forming a combination set. The total number of combinatorial spaces is Efficiency can be improved through pruning or parallel computing.

[0114] Finally, the submatrix with the smallest condition number is selected from all candidate combinations. Its corresponding combination The final optimized measurement point scheme is recorded. Its condition number is denoted as:

[0115] ;

[0116] in, Indicate combination The corresponding submatrix.

[0117] In general, this method can effectively avoid the ill-conditioned dimension of the frequency response function matrix and provide a stable and interference-resistant input-output mapping structure for subsequent load inversion.

[0118] Specifically, the optimized combination selected in this embodiment consists of 133 strain measurement points, corresponding to a submatrix. In high pathological frequency It has the lowest condition number and significantly enhanced numerical stability.

[0119] As an extension, mechanisms such as combinatorial sparsity constraints and structural topological information weighting can be introduced to impose prior restrictions on the combinatorial process, thereby improving physical interpretability and computational efficiency.

[0120] S4. For the highly ill-conditioned matrix, a multi-extension search algorithm is used to select the sub-matrix with the smallest condition number from multiple measurement point combinations, forming the optimized frequency response sub-matrix, and determining the corresponding optimized measurement point layout scheme.

[0121] After identifying the high ill-conditioned frequency points and extracting their corresponding frequency response function matrices, further filtering and reconstruction of the row vector composition of the observation points in the original response matrix are still required. Since the structural system exhibits response redundancy or linear dependence at some measurement points, selecting the optimal combination of measurement points through multiple extended search algorithms helps improve the numerical stability of the submatrix and avoids ill-posed inversion problems caused by excessively large condition numbers.

[0122] Generally, directly performing inversion or load identification on the full response matrix faces problems such as excessive dimensionality, high noise sensitivity, and poor convergence. Therefore, compressing the observation dimensionality through the optimized frequency response submatrix is ​​a crucial step in structural identification modeling. This step addresses the aforementioned highly ill-conditioned matrix. The row vectors of the observation points are combined and optimized to obtain the final optimized layout scheme of the observation points.

[0123] In this embodiment, the multiple expansion retrieval algorithm is based on the aforementioned constructed high ill-conditioned frequency response function matrix. ,in This represents the initial total number of observation points. For the number of load nodes, This is the frequency point with the strongest pathological characteristics.

[0124] First, set the number of observation points after target compression to be... ,satisfy ,from Selected from Row vectors form candidate submatrices:

[0125] ;

[0126] in, Represents the first element in the original matrix. The response row vector corresponding to each observation point, and the row index combination Select a solution for the current measurement points.

[0127] To select the optimal sub-combination, multiple candidate combinations are constructed based on an initial set of combinations using a process of successive substitution and recursive expansion. The condition number of the submatrix corresponding to each combination is then calculated.

[0128] ;

[0129] in, This is the condition number of the current candidate submatrix; the smaller the number, the higher the numerical stability of the submatrix. for Moore-Penrose generalized inverse; matrix norm We still use the 2-norm, which is the maximum singular value.

[0130] In one possible implementation, a full combinatorial traversal is used to iterate through all possible combinations. The algorithm performs an exhaustive search for all possible combinations of measurement points. To reduce computational cost, a layer-by-layer pruning mechanism is introduced, retaining only locally optimal combinations for the next round of expansion. For example, in each round of expansion, only the combinations with the smallest number of conditions in the first K groups (e.g., K=50) are retained, and replacement attempts are continued based on these combinations in subsequent rounds.

[0131] Specifically, for the submatrix obtained after each replacement The condition number is recalculated for all cases.

[0132] ;

[0133] Finally, select the combination with the smallest condition number from all feasible combinations:

[0134] ;

[0135] The corresponding set of row indices is the optimized observation point layout scheme, denoted as . .

[0136] Alternatively, in actual deployment, the observation point locations can be converted into two-dimensional coordinate mappings to correspond to the actual structural layout diagram. Combining the structural panel numbers with the rib and compartment divisions, the final observation point layout scheme can be output in structural coordinate form.

[0137] In this embodiment, the initial number of measurement points is m=9100, and the final selected number of observation points is r=133, corresponding to an optimized frequency response sub-matrix dimension of 133×133, where 133 represents the number of ice load nodes. The optimized combination of measurement points significantly reduces the ill-conditioned nature of the original matrix, and its corresponding condition number... The size is significantly smaller than the original matrix, and the numerical stability is stronger.

[0138] In some embodiments, to further improve search efficiency, a greedy strategy or a dynamic programming method based on the rate of change of eigenvalues ​​can be combined to heuristically prune the combinatorial space. Such methods can complement the full combinatorial strategy and improve the scalability of the algorithm in large-scale structure recognition tasks.

[0139] S5. Reconstruct the structural frequency domain identification model using the optimized frequency response function submatrix, collect the measured strain response of the optimized measurement points, and construct the response power spectral density function.

[0140] After optimizing and selecting the highly ill-conditioned frequency response matrix and obtaining the optimal measurement point layout, it is necessary to further reconstruct the frequency domain structure identification model based on the selected measurement points to ensure the accuracy and stability of the model's physical response mapping. Since the frequency response function submatrix has achieved measurement point compression and ill-conditioning mitigation, the reconstructed model can significantly improve inversion performance while retaining the main structural dynamic characteristics. The model construction process needs to combine the actually acquired strain response data and establish the response power spectral density function through frequency domain analysis as the structural output expression.

[0141] Generally, a structural frequency domain model can be established through the convolution relationship between the frequency response function and the excitation power spectrum, where the ice load power spectral density matrix serves as the input and the strain response power spectral density function is the output, with the two connected through a power spectrum transfer relationship. Because the optimized submatrix corresponds to measurement points with low redundancy and high recognizability, the output mapping of this model is more representative.

[0142] In this embodiment, the structure frequency domain identification model is reconstructed using the following power transfer method:

[0143] ;

[0144] in:

[0145] : Represents the strain response power spectral density matrix corresponding to the optimized measurement point;

[0146] : is the optimized frequency response function submatrix obtained in step S4;

[0147] : Represents the input excitation power spectral density matrix corresponding to each ice load node;

[0148] :express The conjugate transpose of ;

[0149] ω represents angular frequency, measured in rad / s.

[0150] In one possible implementation, It can be estimated through prior scene simulation or ice-excited statistical models, and can also be used as a parameter to be estimated in the inverse recognition process. Corresponding This originates from the optimized processing of measured response data from the measurement points.

[0151] To construct the response power spectral density matrix, it is necessary to first collect the strain time series of the optimized measuring points under actual ice loading conditions. Assume the strain time series of a certain measuring point is... Sampling time is The sampling frequency is A total of Point data.

[0152] Perform a Fourier transform on the strain time series of each channel to obtain the frequency domain response:

[0153] ;

[0154] in, ; , for the first One frequency component; , for the first Frequency domain strain response of each channel.

[0155] Subsequently, by performing statistical processing on the frequency domain response, the autocorrelation and cross-correlation power spectral density functions are constructed:

[0156] ;

[0157] in, For channel With channel In frequency The cross-correlation power spectrum under the following conditions; Indicates complex conjugation; if This expression degenerates into the power spectral density.

[0158] This method can be used to construct complete frequency points. The response spectrum matrix below:

[0159] ;

[0160] Specifically, what was obtained It possesses a Hermitian structure, meaning it satisfies the conjugate symmetry property:

[0161] ;

[0162] This structure ensures that the subsequent inversion and estimation processes have physical consistency and numerical reversibility.

[0163] In some embodiments, to avoid window function leakage and frequency aliasing, the Fourier transform process can introduce preprocessing methods such as Hanning window and Welch piecewise overlapping averaging technique to improve the stability of spectral density estimation.

[0164] As an alternative, the structural response spectrum matrix can be repeatedly acquired and averaged under multiple measurement point groups and multiple ice load excitation conditions to finally obtain the frequency domain structural response statistical expression under multiple scenarios, providing input basis for the next stage of ice load inversion or spectrum decoupling analysis.

[0165] S6. Based on the inverse problem theory and the generalized inverse matrix solution method, combined with the reinforcement identification model and structural response spectrum, the power spectral density function of ice load is identified, which is used to reflect the random excitation intensity distribution of each load node in the frequency domain.

[0166] After completing the optimization of the frequency response function submatrix, the construction of the response power spectrum, and the initial establishment of the frequency domain identification model, the next step is the inversion identification process of the ice load power spectral density function. This step utilizes inverse problem theory, constructing a frequency domain inverse problem model and combining it with the generalized inverse matrix solution method to identify the power spectral density function of each ice load node. This process reflects the excitation intensity distribution of each node in the frequency domain and provides effective input for subsequent structural response analysis.

[0167] In general, solving the inverse problem involves the mapping relationship between the known output response and the system frequency response function. However, by utilizing the generalized inverse matrix method, we can infer the input excitation power spectrum from the observed structural response spectrum. This process can effectively obtain the self-power spectral density of each load node, which can then be used to analyze and optimize the load's influence.

[0168] In this embodiment, the optimized frequency response function submatrix obtained in the preceding steps is first used as the basis. and response power spectral density matrix A frequency domain inverse problem model is constructed. Specifically, it is assumed that the relationship between the structural response and the ice load excitation satisfies the following power spectral transfer equation:

[0169] ;

[0170] in, Let be the power spectral density matrix of the ice load to be identified, containing the excitation intensity of each ice load node in the frequency domain. The goal of the inverse problem is to extract the power spectral density matrix from the known... and China countered .

[0171] Specifically, the inverse problem is solved using the Moore-Penrose generalized inverse matrix method. The basic definition of the Moore-Penrose generalized inverse is:

[0172] ;

[0173] in, To solve for the ice load power spectral density matrix, for The pseudo-inverse matrix can stably deduce the ice load excitation information.

[0174] In one possible implementation, the inverse problem is solved band-by-band with multiple frequency bands and multiple measurement point combinations as inputs, combined with spectral decoupling techniques to improve the solution accuracy. This method effectively avoids ill-posedness in the solution process, ensuring the stability of the ice load power spectrum.

[0175] Finally, the identified ice load power spectral density matrix By extracting its diagonal elements, the self-power spectral density at each ice load node is obtained. ,Right now:

[0176] ;

[0177] These diagonal elements represent the random excitation intensity distribution of each ice load node at different frequencies, and serve as the final output of the node's power spectral density.

[0178] As an alternative, if there are multiple load nodes or multiple excitation conditions, the obtained power spectral density matrix can be weighted and averaged to further improve the robustness and accuracy of the results.

[0179] The method described in this embodiment successfully reversed the relationship between the structural response and the excitation spectrum in the frequency domain, thereby obtaining the power spectral density function of each ice load node in the structure, providing an important basis for structural analysis and optimization.

[0180] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for eliminating the ill-conditioning of a mathematical model for identifying the frequency domain of large-area distributed random ice loads, characterized in that, The method comprises the following steps: S1, a finite element model containing the hull plate, the rib and the longitudinal structure is established, an equivalent concentrated force model under the action of ice load is constructed, and a plurality of strain response measurement points on the structure surface are arranged as an initial sensor array; S2, a plurality of nodes in the ice load action area are set, a unit amplitude harmonic excitation is applied to each node one by one, the strain response frequency data of the structure measurement point is obtained, and a frequency response function matrix is established to form a frequency domain structure identification model; S3, in the frequency domain structure identification model, based on the ill-conditioned analysis of the frequency response function matrix, the frequency point corresponding to the maximum condition number is identified, and a high ill-conditioned frequency response matrix is extracted; The identification method based on the ill-conditioned analysis of the frequency response function matrix in the S3 step is that, in a preset frequency scanning range, the condition number of the frequency response function matrix at each frequency is calculated, the condition number is the product of the matrix norm and the generalized inverse matrix norm, and the frequency with the maximum condition number is taken as the high ill-conditioned frequency; S4, the high ill-conditioned frequency response matrix is subjected to a multiple expansion search algorithm, a sub-matrix with the minimum condition number is selected from a plurality of measurement point combinations, an optimized frequency response function sub-matrix is formed, and a corresponding optimized measurement point arrangement scheme is determined; The multiple expansion search algorithm in the S4 step comprises the following steps: An initial row vector combination is selected from the high ill-conditioned frequency response matrix as a starting measurement point set; The starting combination is replaced one by one to generate a plurality of candidate sub-matrices, and the condition number is recalculated after each replacement; A full combination traversal strategy is adopted to traverse all possible combinations of measurement point arrangements; The sub-matrix with the minimum condition number is selected as the optimized frequency response function sub-matrix, and the measurement point positions corresponding to the row indexes are recorded to form a final optimized arrangement scheme; S5, the frequency domain structure identification model is reconstructed using the optimized frequency response function sub-matrix, the measured strain response of the optimized measurement point is collected, and a response power spectral density function is constructed; S6, based on the inverse problem theory and the generalized inverse matrix solving method, the reinforced identification model and the structure response spectrum are combined to identify the ice load power spectral density function, which is used to reflect the random excitation intensity distribution of each load node in the frequency domain.

2. The method according to claim 1, wherein, In the S1 step, the equivalent concentrated force model is constructed by discretizing the ice extrusion force into a plurality of concentrated normal node forces acting on the hull shell surface, the nodes are uniformly arranged along the bow plate area to form a two-dimensional grid arrangement, and the spatial distribution characteristics of the actual ice breaking load are simulated.

3. The method for eliminating the ill-condition of the mathematical model of large-area distributed random ice load frequency domain identification according to claim 1, characterized in that, The sensor arrangement in the S1 step comprises the following contents: The strain gauges in the plate area are arranged in the axial and transverse directions of the structure, respectively, and the directions are 0 degrees and 90 degrees; The strain gauges in the rib and longitudinal area are arranged at angles of plus and minus 45 degrees; The sensor array is arranged in a rectangular equidistant manner to form a multi-channel structure response observation network.

4. The method for eliminating the ill-condition of the mathematical model of large-area distributed random ice load frequency domain identification according to claim 1, characterized in that, The frequency response function matrix in the S2 step is a complex domain matrix, the row dimension is the number of sensors, the column dimension is the number of ice load nodes, and the matrix element is the complex strain response of a certain sensor position under the action of a unit harmonic excitation on a certain load node at a given frequency.

5. The method of claim 1, wherein, The optimized frequency response function sub-matrix in the S5 step is used to reconstruct a frequency domain structure identification model, wherein a model input is an ice load power spectral density matrix at each frequency, and an output is a strain response spectrum of the optimized measuring point, and the structure power transfer relationship is used for construction.

6. The method for eliminating the ill-condition of the mathematical model of large-area distributed random ice load frequency domain identification according to claim 1, characterized in that, The construction method of the response power spectral density function in the S5 step comprises: acquiring a strain time sequence of the optimized measuring point under actual ice load excitation; performing discrete Fourier transform on the strain time sequence to obtain a frequency domain strain response; calculating autocorrelation and cross-correlation spectral densities between each response channel to form a structure response spectral matrix.

7. The method for eliminating the ill-conditioning of the mathematical model of large-area distributed random ice load frequency domain identification according to claim 1, characterized in that, The solving step of the ice load power spectral density function in the S6 step comprises: constructing a frequency domain inverse problem model based on the structure response spectral matrix and the optimized frequency response function sub-matrix; solving the input spectral matrix by using a Moore-Penrose generalized inverse method; the identification result is a diagonal element of the spectral matrix, which represents an autocorrelation spectral density at each ice load node and is used as a final output.

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