Load Inversion Method for Jacket Platform Structures Based on Strain Measurement Data

Abstract

Due to the difficulty of directly measuring external loads on jacket platform structures and the challenges in accurately expressing them through analytical formulas, this study proposes a load inversion method based on local strain measurement data to obtain the time–history curves of structural loads. The method establishes a mapping relationship between unknown loads and measured strains based on the quasi-static superposition principle. An Improved Sine Cosine Algorithm, combined with an Opposition-Based Learning, is introduced to optimize the placement of strain sensors. The unknown loads are solved using a least squares approach integrated with Tikhonov regularization. The method was validated through indoor loading experiments under eight conditions, where the inverted load time–history curves accurately reflected the periodic characteristics of the applied loads, achieving a maximum Mean Absolute Relative Error (MARE) of 6.91%, demonstrating high stability and accuracy. The further application of the method to an in-service jacket platform in a marine environment yielded inverted wind and wave loads with a maximum MARE of 11.63% compared to loads calculated from measured wind and wave data, validating the method’s practical applicability and robustness. This approach offers a more accurate load basis for the safety assessment and residual life prediction of jacket platform structures.

1. Introduction

Jacket offshore platforms are among the most widely used types of offshore platforms worldwide due to their mature technology and broad applicability [1,2]. These structures operate in highly complex and variable marine environments over the long term, where external loads are diverse and often coupled. Such loads are difficult to describe accurately using fixed analytical expressions [3,4]. The majority of the jacket structure is submerged, posing significant challenges for the placement of force sensors, which in turn makes the direct measurement of structural loads impractical [5]. The precision of load characterization is critically linked to the reliability of fatigue life assessments and residual strength evaluations, making the safety assessment of in-service jacket platforms a challenging task.

Load identification from structural response represents one of the three classical inverse problems in structural dynamics and has garnered growing attention in recent decades [6]. Such techniques are increasingly needed in aerospace [7], automotive [8], and civil engineering [9,10], with considerable progress reported. Unlike forward analysis, load identification is an inverse problem and cannot be solved by directly integrating differential equations [11]. Instead, methods such as matrix inversion or deconvolution are typically employed [12,13]. However, the inversion process is susceptible to measurement noise, making robust sensing and numerically stable algorithms crucial to ensure accurate identification.

After the sensor type is determined, the robustness of test signals is primarily influenced by the sensor placement strategy, which directly affects the accuracy of load inversion. In load identification based on structural response, the key goal of sensor layout optimization is to reduce the number of sensors while maximizing the identifiability of structural response and parameters. Morris B.K. et al. [14] derived an analytical formulation for the influence coefficient method in beam structures and used the minimum condition number of the influence matrix as the optimization target. The optimized layout improved robustness against input noise. Zhu et al. [15] used a combined D-optimality and C-optimality design to reduce the dimensionality of a model used in wind load identification for jacket structures, while optimizing both the location and orientation of strain gauges. The reduced model improved precision, and the optimized sensor locations yielded more accurate results. Liangou T. et al. [16] optimized strain sensor layouts on wind turbine blades using a genetic algorithm combined with D-optimality, enhancing the stability of measured data and thus the reliability of load identification. Shen Y. et al. [17] proposed a strain-energy-based sensor placement method using genetic algorithms, which reduced redundancy while maintaining adequate sensitivity to structural damage. Li K. et al. [18] developed a generally applicable sensor layout method based on an extrapolated-strain Fisher Information Matrix (FIM). Under fixed noise levels and kernel functions, the objective function depends solely on sensor locations, meaning the layout is dictated entirely by the geometry of the structure. Numerical and experimental validation confirmed the framework’s effectiveness, robustness, and applicability to large and complex structures. Once the sensor layout is fixed, the choice of inversion algorithm becomes the dominant factor influencing identification accuracy.

Similar to other mathematical approaches to inverse problems, load identification methods can be broadly categorized into time-domain and frequency-domain techniques [19]. Frequency-domain load identification methods were developed earlier and are relatively more mature. A commonly used method for identifying loads in the frequency domain involves multiplying the system response by the pseudoinverse of the Frequency Response Function (FRF) [12,20]. Two typical frequency-domain methods for load identification are the direct matrix inversion method and the pseudoinverse excitation method [21]. The matrix inversion method is suitable for harmonic excitations with periodic characteristics, while the pseudoinverse excitation method is applicable to both stationary and non-stationary random dynamic loads. The key to time-domain analysis lies in the convolution relationship between the load and the structural response, which enables the identification of the load time history based on this relationship [22]. Due to their increasingly widespread application in engineering, time-domain methods have attracted growing attention [23,24]. Compared to frequency-domain methods, time-domain methods are more suitable for dynamic load identification. XUE et al. [25] constructed two-dimensional Hermitian interpolating wavelets using tensor products of modified Hermitian wavelets expanded at each coordinate, and incorporated them into finite element formulations to address wave propagation and load identification. Their method accurately determines the location, waveform, and amplitude of loads applied to the structure. WANG et al. [26] found that estimating strain in structural failure zones using a least squares fitting (LSF) approach significantly reduces load identification errors in those regions. They thus proposed a load identification scheme combining zonal partitioning with strain fitting. JENSEN et al. [27] proposed a simple yet effective method for identifying load processes by calibrating an ARMA model, and validated their approach using a single-pile model subjected to random wave excitation. Both time-domain and frequency-domain approaches can be used to identify various types of input loads. Although time-domain methods offer broader engineering applicability, they typically involve more complex computations, greater computational cost, and lower numerical stability [13]. With the rapid advancement of neural networks and machine learning algorithms, data-driven models have increasingly been applied to load identification research [28]. Selecting appropriate solution methods based on the geometry and physical characteristics of the structure, as well as the nature of the load, can significantly improve the accuracy, efficiency, and robustness of load identification.

In summary, recent advances in sensor optimization strategies and inverse load reconstruction techniques have provided a solid foundation for identifying environmental loads in marine structures. Offshore environmental loads—such as wind, waves, and currents—are inherently complex, uncertain, and strongly coupled, presenting substantial challenges for the structural safety assessment of offshore platforms. To address these challenges, this study proposes a load inversion method specifically designed for jacket platform structures. By utilizing local strain response data as input, the method reconstructs the time histories of external loads, thereby improving the accuracy and engineering applicability of environmental load estimation for offshore structural analysis.

This paper is organized as follows: Section 2 provides the load inversion theory based on strain test data in metal structures. In Section 3, an optimal placement method of strain sensors based on the Improved Sine Cosine Algorithm (ISCA) and Opposition-Based Learning (OBL) is proposed. In Section 4, the load identification calculation of the indoor jacket platform structure is carried out, and the accuracy of the identification is verified by loading experimental data, while Section 5 applies the load identification method mentioned in this paper to an in-service platform in engineering. Finally, Section 6 summarizes the main conclusions.

2. Load Inversion Theory Based on Strain Measurements

Under non-extreme operating conditions, the deformations of the jacket platform structure induced by external loads remain within the elastic range. Accordingly, the following assumptions are made for the load identification study:

  (i)

The strain induced in the structure under loading is assumed to be entirely elastic.

 (ii)

All structural components are made of isotropic materials.

(iii)

All structural components are assumed to be composed of homogeneous materials.

Based on the above assumptions, the uniaxial strain at a local structural point can be considered as the superposition of strains induced by multiple loads at that location [29]. Except in regions of stress concentration and geometric discontinuities, the relationship between the uniaxial strain and the applied loads at other locations on the structure is expressed as shown in Equation (1).

$$\mathit{\epsilon }=\mathit{A}\mathit{F}$$

(1)

where $\mathit{\epsilon }$ denotes the m × 1 strain column vector of the local structure, $\mathit{A}$ represents the m × k sensitivity coefficient matrix, and $\mathit{F}$ is the n × 1 load vector. The $\mathit{\epsilon }$ can be obtained through experimental measurements. Assuming the sensitivity matrix $\mathit{A}$ is known, the unknown load vector can be identified using Equation (2).

$$\widehat{\mathit{F}}={\left({\mathit{A}}^T\mathit{A}\right)}^{-1}{\mathit{A}}^T\mathit{\epsilon }$$

(2)

in which $\widehat{\mathit{F}}$ denotes the approximate value of the unknown load obtained through numerical analysis. If the strain measurement errors are mutually independent and each has a standard deviation of $\sigma$, the covariance matrix of the identified load $\widehat{\mathit{F}}$ can be expressed as

$$\text{var} \widehat{\mathit{F}}={\sigma }^2({\mathit{A}}^T\mathit{A}{)}^{-1}$$

(3)

As shown in Equation (3), the magnitude of the covariance values of the identified loads depends on both the variance of the measured strain data and the sensitivity coefficient matrix. Smaller covariance values indicate weaker linear correlations among the identified loads, implying a lower degree of coupling between them.

The equation of motion for undamped vibrations induced by the unknown loads can be expressed as Equation (4).

$$\left[\mathit{M}\right]\left\{\ddot{x}\right\}+\left[\mathit{K}\right]\left\{x\right\}=\left\{\mathit{F}\right(t\left)\right\}$$

(4)

The equation of motion for the system’s free vibration can be expressed as Equation (5).

$$\left[\mathit{M}\right]\left\{\ddot{x}\right\}+\left[\mathit{K}\right]\left\{x\right\}=0$$

(5)

Representing the displacement function as an m × 1 vector, the free vibration response can be expressed by Equation (6). The system’s free vibration response is represented as a linear superposition of its modal contributions, where $q\left(t\right)$ denotes the participation factor of each mode.

$$\left\{x\left(t\right)\right\}={\sum }_{i=1}^n q_i\left(t\right){\varphi }_i=\left[\mathbf{\Phi }\right]\left\{q\left(t\right)\right\}$$

(6)

The modal matrix, composed of all modes ${\varphi }_i$ is defined in the form of an n × n matrix, as shown in Equation (7). By decoupling the equation of motion for the ith mode, Equation (8) is obtained.

$$\left[\mathbf{\Phi }\right]=\left[{\varphi }_1,\dots ,{\varphi }_n\right]$$

(7)

$${\ddot{q}}_i+{\omega }_i^2{q}_i=({\mathbf{\Phi }}^T\mathit{F}{)}_i$$

(8)

For damped forced vibrations, the system’s equation of motion can be expressed in the form of Equation (9).

$$\left[\mathit{M}\right]\left\{\ddot{x}\right\}+\left[\mathit{C}\right]\left\{\dot{x}\right\}+\left[\mathit{K}\right]\left\{x\right\}=\left\{\mathit{F}\left(t\right)\right\}=\left\{\mathit{F}{e}^{i\overrightarrow{\omega }t}\right\}$$

(9)

Substituting the displacement response $x\left(t\right)=\mathit{X}{e}^{i\overline{\omega }t}$ into Equation (9) yields:

$$\left[\mathit{X}\right]={\left[\mathit{K}-\mathit{M}{\overline{\omega }}^2+i\mathit{C}\overline{\omega }\right]}^{-1}\left[\mathit{F}\right]=\left[\alpha \left(\overline{\omega }\right)\right]\left[\mathit{F}\right]$$

(10)

By substituting Equation (6) into Equation (9), the system’s equation of motion can be reformulated in the form of Equation (11).

$$\left[\mathit{M}\right][\mathbf{\Phi }]\left[\ddot{q}\right(t\left)\right]+\left[\mathit{C}\right][\mathbf{\Phi }]\left[\dot{q}\right(t\left)\right]+\left[\mathit{K}\right][\mathbf{\Phi }]\left[q\right(t\left)\right]=\mathit{F}\left(t\right)$$

(11)

Based on the assumption of structural continuity, the differential relationship between linear strain and displacement at any location on the structure can be expressed by Equation (12).

$$\left\{\epsilon \right\}=\mathit{D}\left\{x\right\}$$

(12)

Substituting Equation (6) into Equation (12) yields Equation (13), which can be further rearranged to obtain Equation (14).

$$\mathit{D}\left(\right\{x\left(t\right)\left\}\right)=\mathit{D}\left(\right[\Phi \left]\right)\left\{q\right(t\left)\right\}$$

(13)

$$\left\{\epsilon \right(t\left)\right\}=\left[{\mathbf{\Psi }}^{\epsilon }\right]\left\{q\right(t\left)\right\}$$

(14)

where $\left[{\mathbf{\Psi }}^{\epsilon }\right]$ denotes the modal strain matrix of the structure, which contains the strain modes corresponding to all vibration modes of the entire structure.

As indicated by Equation (14), the strain response at time t can be represented as a linear superposition of modal strains. Similar to displacement modes, strain modes characterize the dynamic properties of the structure. However, strain modes more directly reflect the internal dynamic behavior of the structure.

Assuming the modal strain matrix $\left[{\mathbf{\Psi }}^{\epsilon }\right]$ is known and $\left\{\epsilon \left(t\right)\right\}$ can be obtained through experimental testing, the modal participation factors can be calculated using Equation (15).

$$\left\{q\right(t\left)\right\}={\left(\left[{\mathbf{\Psi }}^{\epsilon }{]}^T\right[{\mathbf{\Psi }}^{\epsilon }]\right)}^{-1}\left[{\mathbf{\Psi }}^{\epsilon }{]}^T\right\{\epsilon \left(t\right)\}$$

(15)

Once the modal participation factors have been determined, the unknown loads in the undamped forced vibration equation can be computed using Equation (4), while those in the damped forced vibration equation can be obtained using Equation (9).

3. Load Inversion for Jacket Platform Structures Based on Optimized Strain Sensor Placement

According to load inversion theory, the sensor placement scheme and the solution method for the inversion equations are the key factors affecting computational accuracy. To address this, a load inversion method tailored for jacket platform structures is proposed, and its technical workflow is illustrated in Figure 1.

As illustrated in Figure 1, the load inversion process for jacket structures consists of four main steps: finite element analysis (FEA), optimal sensor placement (OSP), strain data acquisition, and load inversion solution. The primary objective of the FEA is to obtain the structural modal strain matrix, which serves as the basis for OSP. The placement of sensors significantly affects both the robustness and informational completeness of the measurement data. An optimized placement strategy based on an ISCA-OBL is better suited for load inversion in jacket structures. The experimentally acquired strain data are preprocessed to remove trends and high-frequency noise from the original signals, resulting in clean strain time histories. The load time–history curves are obtained by substituting the strain time–history data into the inversion equations and solving them via the least squares (LS) method combined with Tikhonov regularization.

3.1. Formulation of Objective Functions for OSP

The fundamental principles for optimal sensor placement in engineering structures vary depending on the focus of the study and generally follow four categories [30]:

  (i)

Maximizing the information content of measurement data.

 (ii)

Ensuring concentrated distribution of modal energy.

(iii)

Identification of sensitive regions based on influence coefficients.

(iv)

Effective independence of measurement information.

In load identification studies of jacket platform structures based on strain measurement data, sensor placement should adhere to the principle of information maximization. If the form of the load is known in advance, sensors should be positioned in regions where strain is most sensitive, as determined by influence coefficients.

As shown in Equation (3), a larger ${\mathit{A}}^{\mathsf{T}}\mathit{A}$ determinant value corresponds to a smaller covariance of the identified loads, indicating that the sensor network provides a greater amount of information. Assuming that the sensor measurement errors are mutually independent, the FIM can be expressed as Equation (16).

$$FIM={\mathit{A}}^{\mathsf{T}}\mathit{A}$$

(16)

When selecting sensitive regions for sensor placement based on influence coefficients, it is essential to ensure that the column vectors of the sensitivity matrix are linearly independent and that the matrix condition number is minimized. Therefore, the objective function for optimizing strain sensor placement on jacket structures must simultaneously maximize the determinant of the information matrix and minimize the condition number of the sensitivity matrix. The objective function is defined in Equation (17).

$$f=w_1\cdot \text{log} det\left(\text{FIM}\right)-w_2\cdot \text{log}\left(\text{cond}\right(\text{FIM}\left)\right)$$

(17)

After normalization, Equation (18) is obtained:

$$f=w_1{\displaystyle \frac{\text{log} det\left(\text{FIM}\right)}{|\text{log} det(\text{FIM}){|}_{max}}}-w_2{\displaystyle \frac{\text{log}\left(cond\right(\text{FIM}\left)\right)}{\left|\text{log}\right(cond\left(\text{FIM}\right)){|}_{max}}}$$

(18)

Given that jacket platform structures are characterized by high redundancy, diverse load types, strong load coupling, and constraints on sensor placement, controlling the condition number is generally more critical than maximizing information content. Therefore, when assigning weighting coefficients, the following condition should typically be satisfied: $w_1$ ≤ $w_2$. The proposed theoretical framework is valid under non-extreme operating conditions. To balance information content and computational stability, the weighting coefficients were set to $w_1$ = 0.4 and $w_2$ = 0.6. If the optimization results yield $det\left(\mathrm{F}\mathrm{I}\mathrm{M}\right)$ < 1 × 10⁴ or $cond\left(\mathrm{F}\mathrm{I}\mathrm{M}\right)$ < 1 × 10³, the coefficients can be adjusted accordingly. This weighting strategy improves the robustness of sensor placement optimization by ensuring that the selected configuration captures sufficient structural response information while maintaining numerical stability in the inversion process.

3.2. Sensor Optimization Algorithm Based on ISCA-OBL

The most commonly used approach for maximizing the determinant of FIM is the D-optimality design [31]. The combination of the greedy algorithm with D-optimality design is frequently employed for OSP in load identification due to its simplicity and ease of implementation [32]. The core idea of the greedy algorithm is to make the optimal choice at each step based on the current state. It offers advantages such as ease of implementation, fast computation, and low spatial complexity. However, due to its lack of backtracking and correction mechanisms, it often fails to balance global and local optima, and the final solution may not be optimal. To address these limitations, we propose a sensor placement optimization algorithm based on ISCA-OBL.

(1)

Improved Sine Cosine Algorithm

The Sine Cosine Algorithm (SCA) searches for and iterates toward the optimal solution by combining sine and cosine functions, leveraging their periodicity and oscillatory behavior [33]. Compared with other evolutionary algorithms, SCA exhibits superior global search capabilities, enabling efficient exploration of the solution space for global optima. Moreover, its search process is relatively insensitive to initial solutions, demonstrating strong robustness.

The SCA divides the iterative process into two stages: global search and local search. During the global search phase, large random perturbations are applied to the current set of solutions to explore unknown regions of the solution space, thereby expanding the search range and increasing the likelihood of identifying potential global optima. In the local search phase, small random perturbations are introduced to the solution set to thoroughly explore the neighborhood of the current solutions, thus refining the solution space and improving both the accuracy and quality of local optima. The corresponding iteration equation is shown in Equation (19):

$$X_i^j\left(t+1\right)=\left\{\begin{array}{c}{X}_i^j\left(t\right)+r_1\text{sin}\left(r_2\right)\left|r_3{P}^j\left(t\right)-X_2^j\left(t\right)\right|,r_4>0.5\\ X_i^j\left(t\right)+r_1\text{cos}\left(r_2\right)\left|r_3{P}^j\left(t\right)-X_i^j\left(t\right)\right|,r_4<0.5\end{array}\right.$$

(19)

where t denotes the current iteration number, and $X_i^j\left(t\right)$ represents the j-th component of the position of the i-th individual at iteration t. $r_1\in (0, 1]$. It is used to regulate the transition between the global and local search phases of the algorithm. $r_2\in (0, 2\pi ]$ It defines the direction in which the current solution moves toward the current best solution, as well as the extreme values of the iteration step size. $r_3\in (0, 2]$ Its role is to randomly amplify the influence of the best solution on the movement distance of candidate solutions. $r_4\in \left[0, 1\right]$. It characterizes the randomness between the two update equations in the SCA. $P^j\left(t\right)$ represents the j-th component of the best candidate solution in the solution set at iteration t.

Although the SCA provides sufficient stochastic search capability through global and local search processes, parent individuals may experience oscillatory behavior in the later stages of the iteration due to the influence of inertia weights. To address this issue, an Improved Sine Cosine Algorithm (ISCA) is proposed by incorporating a weight update mechanism into the original SCA to adjust the fitness values of individuals. This mechanism dynamically updates the fitness of individuals based on their historical performance during the iteration process, thereby effectively balancing the weights between global and local search. By adaptively adjusting individual fitness, ISCA can more flexibly respond to changes in the search space, thus improving both the convergence speed and the search efficiency of the algorithm. The iteration equation of ISCA is given in Equation (20).

$$X_i^j\left(t+1\right)=\left\{\begin{array}{c}Av{g}_t^j+r_1\text{sin}\left(r_2\right)|r_3{P}^j\left(t\right)-X_i^j\left(t\right)|,r_4>0.5\\ Av{g}_t^j+r_1\text{cos}\left(r_2\right)|r_3{P}^j\left(t\right)-X_i^j\left(t\right)|,r_4<0.5\end{array}\right.$$

(20)

in which $Av{g}_t^j$ denotes the mean position of the candidate solution set in the j-th dimension, which is calculated as shown in Equation (21).

$$Av{g}_t^j={\displaystyle \frac{\text{sum}\left(W_{x_1\left(t\right)}{x}_1\left(t\right),W_{x_2\left(t\right)}{x}_2\left(t\right),\dots ,W_{x_c\left(t\right)}{x}_c\left(t\right)\right)}{c_t}}$$

(21)

where $c_t$ represents the size of the candidate solution $Av{g}_t^j$ set involved in the computation. $c_t=\left[m-{\displaystyle \frac{t}{m}}\right]$, where m denotes the size of the candidate solution set. $W_{x_i\left(t\right)}$ represents the weight $x_i\left(t\right)$ obtained at iteration t, which is calculated as shown in Equation (22).

$$W_{x_i\left(t\right)}=1-{\displaystyle \frac{i}{m}}$$

(22)

in which $x_i\left(t\right)$ refers to the candidate solution with the i-th highest fitness at iteration t.

$$r_1=a{\left(1-{\displaystyle \frac{t}{T}}\right)}^2$$

(23)

where $a$ is a control constant, typically set to 2, T denotes the maximum number of iterations.

(2)

Opposition-Based Learning Strategy

The core concept of the OBL strategy is to introduce opposite solutions within the solution space and eliminate those with lower fitness during the iteration process, thereby enhancing the diversity and global exploration capability of the search. Within an acceptable computational cost, this strategy improves both the effectiveness and convergence of the algorithm [34].

In the OBL strategy, the opposite number $\stackrel{\_}{x}$ of a real-valued variable $x$ is defined as shown in Equation (24).

$$\stackrel{\_}{x}=l+u-x$$

(24)

where $l$ denotes the lower bound of the real number, and $u$ denotes the upper bound.

Accordingly, $X=\left(x_1,x_2,\dots ,x_n\right)\in R^n$, the opposite solution of $x_i\in \left[l_i,u_i\right]$ a multi-dimensional variable $\stackrel{\_}{X}$ can be defined as

$$\stackrel{\_}{X}=\left({\stackrel{\_}{X}}_1,{\stackrel{\_}{X}}_2,\dots ,{\stackrel{\_}{X}}_n\right),{\stackrel{\_}{X}}_i=l_i+u_i-x_i,i=\mathrm{1,2},\dots ,n$$

(25)

Due to the locally redundant degrees of freedom in the geometry of jacket platform structures, local structural responses tend to be weakly sensitive to variations in global structural responses. As a result, conventional strain sensor optimization strategies often fail to achieve both global and local optimality. To address this issue, the ISCA and OBL strategies are introduced to optimize the strain sensor placement scheme. ISCA leverages the properties of sine and cosine functions to more efficiently locate the optimal solution within the search space, while avoiding oscillations in the later search stages caused by inertia weights. This enhances both the computational accuracy and convergence speed of the algorithm. The OBL strategy enhances the exploration of the search space and facilitates rapid convergence to either a global or local optimum. The combined ISCA-OBL approach enables more efficient search for the optimal solution across the solution space, offering improved adaptability and robustness when addressing complex and diverse hyperparameter optimization problems. The computational workflow is illustrated in Figure 2.

Step 1: Generate the matrix based on the strain mode matrix obtained from FEA, and use it as the input data.

Step 2: Initialize all parameters in the optimization algorithm. The number of sensors m and the number of loads to be identified n must satisfy the condition: $m\ge n$, $w_1$ ≤ $w_2$.

Step 3: Generate the initial population based on the predefined number of sensors using the tent map.

Step 4: Use the objective function value defined in Equation (18) as the fitness value. Based on fitness, divide the population $P_m$ by selecting individuals: those with higher fitness form a sub-population $P_{good}$ of size N, while those with lower fitness form a sub-population $P_{bad}$ of size N.

Step 5: Optimize the population using the ISCA method as defined in Equation (20), and output a sub-population $P_{good}$ of size N. Simultaneously, optimize the population $P_1$ using the OBL strategy as defined in Equation (25), and output a sub-population $P_2$ of size N.

Step 6: Combine the sub-populations $P_1$ and $P_2$ obtained through ISCA and OBL optimization to form a new population $P_M$ of size 2N.

Step 7: Sort the individuals in the population in ascending order of fitness values, and select the top $N$ individuals with the best fitness to form a new population $P_{new}$. If the termination condition is met, output the optimal solution—the best combination of parameters. Otherwise, return to Step 4.

Step 8: Determine and output the final sensitivity matrix ${\mathit{A}}_c$.

3.3. Load Inversion Method Based on LS and Tikhonov Regularization

The initial sensitivity coefficient matrix $\mathit{A}$ can be generated using a Finite Element (FE) model. It is essential to ensure that the element types used in the FE model provide surfaces suitable for strain sensor placement. Specifically, shell elements, plate elements, or solid elements that account for surface effects should be used in the predefined sensor layout regions. In addition, to avoid errors caused by approximated strain calculations within elements, the discretization of the geometric model must consider the compatibility between element size and the physical dimensions of the strain sensors. Once the FE model is established, Equation (1) can be extended into the form presented in Equation (26).

$$\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ ⋮\\ {\epsilon }_{\mathrm{p}}\end{array}\right]=\left[\begin{array}{cccc}{A}_{\mathrm{1,1}}& A_{\mathrm{1,2}}& \cdots & A_{1,n}\\ A_{\mathrm{2,1}}& A_{\mathrm{2,2}}& \cdots & A_{2,n}\\ ⋮& ⋮& & ⋮\\ A_{p,1}& A_{p,2}& \cdots & A_{p,n}\end{array}\right]\left[\begin{array}{c}{\mathrm{F}}_1\\ {\mathrm{F}}_2\\ ⋮\\ {\mathrm{F}}_{\mathrm{n}}\end{array}\right]$$

(26)

where p denotes the number of elements within the candidate region for strain sensor placement. This region is typically determined through finite element analysis by identifying areas where strain exhibits significant variation under applied loads, combined with practical engineering considerations. n represents the number of loads to be identified. The resulting sensitivity $\mathit{A}$ coefficient matrix is of size p × n. The strain in a specific direction of the m-th element under the influence of n external loads can be expressed in the form of Equation (27).

$${\epsilon }_m=A_{m,1}{F}_1+A_{m,2}{F}_2+\cdots +A_{m,n}{F}_n$$

(27)

Within the range of small elastic deformations, the unidirectional strain of the m-th element in Equation (27) can be expressed as the combined effect of n loads, where the sensitivity coefficient represents the contribution of each load. When the number of sensors to be deployed is m and only one load is to be identified, Equation (26) can be reformulated into the form shown in Equation (28).

$$\left[\begin{array}{c}{\epsilon }_{1,1}\\ {\epsilon }_{2,1}\\ ⋮\\ {\epsilon }_{m,1}\end{array}\right]=\left[\begin{array}{cccc}{A}_{1,1}& A_{1,2}& \cdots & A_{1,n}\\ A_{2,1}& A_{2,2}& \cdots & A_{2,n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{p,1}& A_{p,2}& \cdots & A_{p,n}\end{array}\right]\left[\begin{array}{c}{F}_1\\ 0\\ ⋮\\ 0\end{array}\right]$$

(28)

If the number of loads to be identified is n, Equation (28) can be extended as follows:

$$\left[\begin{array}{cccc}{\epsilon }_{1,1}& {\epsilon }_{1,2}& \cdots & {\epsilon }_{1,n}\\ {\epsilon }_{2,1}& {\epsilon }_{2,2}& \cdots & {\epsilon }_{2,n}\\ ⋮& ⋮& & ⋮\\ {\epsilon }_{m,1}& {\epsilon }_{m,2}& \cdots & {\epsilon }_{m,n}\end{array}\right]=\mathit{A}\left[\begin{array}{cccc}{F}_1& 0& \cdots & 0\\ 0& F_2& \cdots & 0\\ ⋮& ⋮& & ⋮\\ 0& 0& \cdots & F_n\end{array}\right]$$

(29)

Within the linear elastic and small deformation range, when n loads are applied, the strain in a specific direction of the m-th element can be represented as the superposition of the strains caused by each individual load:

$$\begin{array}{c}{\epsilon }_{m,1}=A_{m,1}{F}_1\\ \begin{array}{c}{\epsilon }_{m,2}=A_{m,1}{F}_2\\ ⋮\\ {\epsilon }_{m,n}=A_{m,n}{F}_n\end{array}\end{array}$$

(30)

$${\epsilon }_m={\epsilon }_{m,1}+{\epsilon }_{m,2}+\cdots +{\epsilon }_{m,n}$$

(31)

Assuming that the applied loads are n unit loads, Equation (29) can be rewritten in the form of Equation (32).

$$\left[\begin{array}{cccc}{\epsilon }_{c\mathrm{1,1}}& {\epsilon }_{c\mathrm{1,2}}& \cdots & {\epsilon }_{c1,n}\\ {\epsilon }_{c\mathrm{2,1}}& {\epsilon }_{c\mathrm{2,2}}& \cdots & {\epsilon }_{c2,n}\\ ⋮& ⋮& & ⋮\\ {\epsilon }_{cm,1}& {\epsilon }_{cm,2}& \cdots & {\epsilon }_{cm,n}\end{array}\right]={\mathit{A}}_I\mathit{I}={\mathit{A}}_I$$

(32)

where the sensitivity coefficient matrix ${\mathit{A}}_I$ can be represented by the calculated strain vector matrix ${\epsilon }_c$, which is obtained through finite element analysis. Assuming that the sensor placement region contains p elements, n unit loads are applied at the designated load identification locations. After applying the geometric boundary conditions, step-by-step static analysis is performed to obtain the calculated strain vector matrix ${\epsilon }_c$ for the p elements.

Once the final sensitivity coefficient matrix is ${\mathit{A}}_c$ determined, a load coefficient matrix $C$ is introduced, and Equation (1) can be reformulated as Equation (33).

$$\left[{\mathit{A}}_c\right]\left[\mathit{C}\right]\left[\mathit{I}\right]=\left[\epsilon \right]$$

(33)

The load coefficient matrix $\mathit{C}$ can be expressed as the product of the pseudoinverse of the sensitivity matrix ${\mathit{A}}_c$ and the measured strain matrix $\mathit{C}$, as shown in Equation (34). Once the load coefficient matrix is obtained, the magnitudes of the unknown loads can be determined.

$$\left[\mathit{F}\right]=\left[\mathit{C}\right]={\left\{\left[{{\mathit{A}}_c]}^T{[\mathit{A}}_c\right]\right\}}^{-1}{{\mathit{A}}_c}^T\left[\mathit{\epsilon }\right]$$

(34)

If Equation (34) is solved directly using the LS method, the solution may become unstable and highly sensitive to errors when the system ${\mathit{A}}_c$ is nearly singular or ill-conditioned. To address this issue, a regularization term in the form of Equation (35) is introduced, transforming the original problem into a regularized optimization problem. The optimal solution can then be expressed by Equation (36).

$$\underset{\mathbf{F}}{min} \parallel A_c\mathrm{F}-\epsilon {\parallel }^2+\lambda {\parallel \mathrm{F}\parallel }^2$$

(35)

$$\mathrm{F}=({{\mathit{A}}_c}^T{\mathit{A}}_c+\lambda \mathit{I}{)}^{-1}{{\mathit{A}}_c}^T\epsilon$$

(36)

In which ${{\mathit{A}}_c}^T{\mathit{A}}_c+\lambda \mathit{I}$ is typically positive definite, ensuring the existence and numerical stability of its inverse, which significantly enhances the robustness of the solution process.

4. Experimental Verification of Load Inversion on an Indoor Jacket Platform Structure

4.1. Finite Element Modeling and Analysis of the Structure

A simplified scaled model of a jacket platform structure from a specific offshore region was constructed in the laboratory, as shown in Figure 3. The total height of the model is 2.3 m, with a total mass of 117.5 kg. The geometric dimensions and material properties of each component are listed in

Table 1. Geometric size table of indoor jacket platform structure.

Part Name	Location	Size/mm	Material
Horizontal transverse brace	1st~4th floors	Φ20 × 2	Steel 20
Horizontal inclined brace	The first floor	Φ20 × 2	Steel 20
	The third floor	Φ14 × 2	Steel 20
Vertical inclined brace	1st~3rd floors	Φ16 × 2	Steel 20
	The 4th floor	Φ14 × 2	Steel 20
Main pile	1st~4th floors	Φ34 × 2	Steel 20
Deck		900 × 750 × 6	Q235 steel

.

Using the geometric parameters provided in Table 1, the global FE model of the indoor jacket platform structure was developed in the FE software ANSYS Workbench 2022R2. The applied loads and boundary conditions are shown in Figure 4.

In Figure 4, the top plate and excitation plate are meshed using Shell 181 elements, while the remaining structural components are meshed using Beam188 elements. Load application regions are determined based on the position of the exciters, and fixed boundary conditions are applied to the pile legs. Material properties are assigned according to the materials listed in Table 1, with a Young’s modulus of 2.06 × 10¹¹ Pa, a Poisson’s ratio of 0.3, and a density of 7850 kg/m³. The applied loads consist of two concentrated forces oriented perpendicular to the excitation plate. These two forces lie in the same plane along the Z-axis, while their components in the X and Y directions follow different sinusoidal cycles. By varying the phase and amplitude of these loading curves over time, in-plane forces of varying magnitudes and directions can be synthesized.

A transient dynamic analysis was performed by applying sinusoidal loads with a period of 5 s and a peak amplitude of 1000 N in both the X and Y directions of the FE model. The resulting axial strain distribution at the time of peak loading is shown in Figure 5.

As shown in Figure 5, regions with higher levels of axial strain are observed at the 4th floor horizontal bracing and the base of the jacket legs, indicating that these locations are highly sensitive to changes in loading. In both engineering practice and experimental setups, strain sensors should be avoided in areas with stress concentration or geometric discontinuities. Based on the actual geometric proportions of the structure and the finite element analysis results, the horizontal bracing of the fourth layer was selected as the candidate region for sensor placement. This region was re-meshed using Shell 181 elements to match the size of the strain sensors, and the updated local model is shown in Figure 4.

A modal analysis was conducted on the FE model of the laboratory-scale jacket platform structure. The first three natural frequencies are listed in

Table 2. Natural frequency of indoor jacket platform structure.

Modal Order	First-Order	Second-Order	Third-Order
Frequency/Hz	36.299	40.388	77.85

, and the corresponding mode shapes are shown in Figure 6.

As shown in the mode shapes presented in Figure 6, the first three modes correspond to translational motion along the X-axis, translational motion along the Y-axis, and rotational motion about the Z-axis, respectively, which collectively capture the global dynamic behavior of the structure. If the target load to be identified is a concentrated force applied within the horizontal plane, it can be decomposed into two orthogonal forces along the X and Y directions, as illustrated in Figure 4. Since these two components induce only translational motion in the X and Y directions, the first two modes are sufficient to characterize the global response of the laboratory-scale jacket platform structure under such loading conditions.

4.2. Optimized Placement of Structural Strain Sensors

A static analysis was performed by applying unit loads in the X and Y directions, respectively, to the load identification reference model shown in Figure 4. A total of 100 sections were selected along 4th floor horizontal bracing, with six measurement points uniformly distributed within each section. In the designated sensor placement area, 600 measurement points were selected as candidate locations for the strain sensors. Based on the strain results at each measurement point, the initial strain sensitivity matrix was constructed, where Parameter 1 and Parameter 2 represent the axial strain responses under unit loads in the X and Y directions, respectively. The heat map of the initial sensitivity matrix is presented in Figure 7.

As shown in Figure 7, the spatial distribution of strain sensitivity is non-uniform and generally sparse, with the overall sensitivity of Parameter 1 being lower than that of Parameter 2. Due to the influence of the structural geometry, significant differences are observed between the sensitivity distributions of Parameter 1 and Parameter 2: the peak sensitivity region for Parameter 1 is primarily located between measurement points 180~280, while that for Parameter 2 is concentrated in the range of 130~280. During the initial population generation stage, it is recommended to prioritize these regions. Placing sensors within these intervals is expected to significantly enhance the information content for load identification, thereby improving identification accuracy.

Sensor placement optimization design calculations were performed using the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and ISCA-OBL algorithm. The x-axis represents the number of iterations, and the y-axis shows the fitness values calculated using Equation (18). The convergence curves are presented in Figure 8.

Figure 8 shows the fitness value variation curves during the optimization process for the three algorithms. In terms of convergence speed, ISCA-OBL performs the best, with the curve rapidly decreasing in the first 50 iterations and reaching a stable fitness value by the 143rd iteration. The PSO curve decreases more gradually in the early stages but improves progressively with increasing iterations, stabilizing at the 176th iteration. The GA curve shows the slowest rate of decline, with an initial fitness value similar to that of PSO, but the improvement is notably slower, reaching a stable fitness value by the 183rd iteration. Comparing the final fitness values of the three algorithms, the result obtained by ISCA-OBL is superior. In conclusion, the ISCA-OBL algorithm is more suitable for the optimization of strain sensor placement on jacket platform structures.

Analysis of the values det(FIM) and cond(FIM) during the ISCA-OBL optimization process reveals that the maximum value of det(FIM) reached 1.47 × 10⁷, while the det(FIM) of the final optimized result was 1.31 × 10⁷. This indicates that the FIM derived from the sensitivity matrix in this optimization process had a relatively large determinant, implying a strong sensitivity of the system response to the parameters and a high efficiency in utilizing the strain information. The theoretical minimum of the condition number is 1. The optimized result yielded cond(FIM) = 1.036, indicating a high degree of linear independence among the matrix column vectors, good matrix invertibility, and strong robustness against input perturbations, thereby ensuring high stability in the inverse computation. These results demonstrate that the strain data acquired by the optimally placed sensors not only capture rich structural response information but also exhibit strong noise resistance. The optimized sensor locations correspond to measurement points 149 and 519, whose spatial positions are shown in Figure 9.

As shown in Figure 9, strain sensors A# and B# are placed on the outer sides of the fourth-level horizontal bracing, near the mid-span. Their cross-sectional angular positions are approximately 0° and 180°, respectively. Both sensors are positioned at a considerable distance from the nearby tubular joints, effectively minimizing the influence of stress concentration around the joints. Geometrically, the two sensors are symmetrically arranged with respect to the centerline in the Z-directional plane.

4.3. Structural Loading Experiment

The accuracy of load identification is closely related to both the characteristics of the applied loads and the geometric and physical properties of the structure. Considering the periodic nature of marine environmental loads, the applied load frequency range was set between 0.1 and 1 Hz. According to the principles of dynamic similarity and scale modeling, the corresponding peak load values were determined to lie between 800 N and 2000 N. Furthermore, three types of periodic loads—sinusoidal, triangular, and square waves—were applied. Based on a combination of varying influencing factors, eight experimental conditions were designed to validate the load identification performance of the indoor jacket platform model, as detailed in

Table 3. Indoor jacket platform structure load identification condition table.

Working Condition	Number of Sensors	Number of Loads	Types of Loads	Frequency of Load/Hz	Amplitude of Load/N
1#	2	2	Square wave	0.2	1500
2#	2	2	Sine wave	0.2	1500
3#	2	2	Triangular wave	0.2	1500
4#	2	2	Sine wave	0.1	1500
5#	2	2	Sine wave	0.3	1500
6#	2	2	Sine wave	0.3	1000
7#	2	2	Sine wave	0.3	800
8#	2	2	Sine wave	0.3	2000

.

In Table 3, conditions 1# to 3# are designed to compare different load types, conditions 4# and 5# examine the effects of varying load frequencies, while conditions 6# to 8# investigate the influence of different load magnitudes. The loading experiment testing and data acquisition system primarily consists of strain transducer, wireless nodes, wireless base station, and computer. The connections between these devices are shown in Figure 10.

As shown in Figure 10, the strain sensors are placed on the indoor jacket platform structure, with all sensors connected to the wireless node module via transmission cables. The wireless local area network established by the wireless base station serves as a data relay, transmitting the test data from the wireless node module to the computer. The sensor placement locations are referenced in Figure 9, and the specific parameters of the strain sensors used are provided in

Table 4. Strain transducer parameter specifications.

Model	Range	Sensitivity	Excitation Voltage	Operating Temperature	Accuracy
ST350	±4000 µε	500 µε/mV	1~10 V	−50 °C~+80 °C	<±1%

.

Eight loading experiments were conducted on the indoor jacket platform structure, with a sampling frequency of 50 Hz and a test duration of 600 s. Data acquisition and storage were performed using a wireless structural monitoring system. Taking condition 2# as an example, the strain time–history data before and after preprocessing for a 200 s interval are shown in Figure 11.

As shown in Figure 11, the strain measurements exhibit a more uniform distribution around the zero baseline after preprocessing, with a significantly reduced offset relative to the original baseline. This preprocessing step effectively minimizes the errors caused by incomplete zeroing of the strain sensors.

4.4. Analysis of Load Inversion Results

The axial strain data at the sensor locations shown in Figure 11 are extracted to construct the sensitivity coefficient matrix ${\mathit{A}}_c$. One set of preprocessed strain time history data $\epsilon$, together with the matrix ${\mathit{A}}_c$, is substituted into Equation (36) for load inversion. The regularization parameter λ is determined using the Generalized Cross Validation (GCV) method. The GCV calculation formula is given in Equation (37).

$$GCV(\lambda )={\displaystyle \frac{\parallel {\mathit{A}}_c\mathbf{f}\left(\lambda \right)-\epsilon {\parallel }_2^2}{{\left(\parallel I-H\left(\lambda \right){\parallel }_F\right)}^2}}$$

(37)

Since cond(FIM) = 1.036, indicating a well-conditioned matrix ${\mathit{A}}_c$, the regularization parameter is preset within the range of 1 × 10⁻⁶ to 1 × 10⁻³. By substituting the known ${\mathit{A}}_c$ and $\epsilon$ into Equation (37), the optimal value of $\mathsf{\lambda }$ can be obtained within the preset range. Substituting this $\mathsf{\lambda }$ into Equation (36) yields the time history curve of the identified load.

The strain time histories $\epsilon$ and ${\mathit{A}}_c$ corresponding sensitivity matrices from conditions 1# to 8# were substituted into Equation (36) and Equation (37), respectively, to perform load identification calculations. This yielded the reconstructed load time history curves for all eight scenarios. Given the variations in load type, magnitude, and frequency across these working conditions, the effects of these three factors on the accuracy of load identification are discussed separately below.

(1)

Load Identification Results for Different Load Types

The load time history curves obtained by inverse calculation based on the strain measurement data from conditions 1# to 3# are shown in Figure 12, Figure 13 and Figure 14.

As shown in Figure 12, Figure 13 and Figure 14, the identified load curves closely follow the experimental curves over most of the time domain. All three loading conditions exhibit a periodicity of 5 s, and the periodic patterns of the identified loads are in perfect agreement with the applied experimental loads, demonstrating the robustness and accuracy of the inverse load identification method described in Section 3.3. Among the three loading conditions, the square wave shows the smallest peak error, while the triangular wave exhibits the largest. This is primarily because the square wave has a simpler waveform and a longer peak duration, allowing more strain data points to be captured in the peak region. In contrast, the shorter peak duration of the triangular wave results in fewer strain measurements at the peak, reducing the accuracy of peak identification.

To further analyze the error distribution over the entire time history, the Absolute Relative Error (ARE) between the identified load curve and the applied load curve is calculated using Equation (38).

$${\text{ARE}}_i=\left|{\displaystyle \frac{F_i^{\text{iden}}-F_i^{\text{true}}}{F_i^{\text{true}}+\delta }}\right|\times 100\%$$

(38)

where $F_i^{\text{iden}}$ represents the identified value at the iiith time point, $F_i^{\text{true}}$ denotes the corresponding true applied load, and $\delta$ is a small constant introduced to prevent division by zero.

The ARE curves comparing the identified loads and experimentally applied loads for conditions 1# to 3# are shown in Figure 15, Figure 16 and Figure 17.

As shown in Figure 15, Figure 16 and Figure 17, the ARE curves for conditions 1# to 3# exhibit clear periodic fluctuations that closely match the periodic features of the experimental loading curves. Each cycle displays a consistent variation pattern, indicating that the load identification algorithm maintains good stability throughout the entire time history. Comparing the non-peak regions of the curves in Figure 15, Figure 16 and Figure 17, the ARE values in the X direction are consistently higher than those in the Y direction across all three conditions, suggesting that the overall identification accuracy in the Y direction is superior to that in the X direction. The maximum ARE values are 20% in Figure 15, 8% in Figure 16, and 12% in Figure 17. To further compare the global identification accuracy across the three cases, the Mean Absolute Relative Error (MARE) and Root Mean Square Error (RMSE) for each condition is calculated using Equations (39) and (40).

$$\text{MARE}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N {\text{ARE}}_i$$

(39)

$${\displaystyle \frac{\sqrt{\frac{1}{N}\sum _i (F_i^{\text{iden}}-F_i^{\text{true}}{)}^2}}{\text{range}\left(F^{\text{true}}\right)}}\times 100\%$$

(40)

For condition 1#, the MARE and RMSE% are 6.91% and 17.17% in the X direction, and 5.79% and 14.21% in the Y direction, respectively. For condition 2#, the MARE and RMSE% are 5.35% and 5.16% in the X direction, and 4.22% and 3.91% in the Y direction. For condition 3#, the MARE and RMSE% are 6.47% and 11.92% in the X direction, and 5.58% and 9.32% in the Y direction. A comprehensive comparison of the three cases indicates that the type of periodic load significantly influences the accuracy of the load identification results. In terms of peak load identification error, the square wave yields the smallest error, while the triangular wave results in the largest. This suggests that peak identification accuracy is affected by the density of strain measurement data points—higher point density leads to more accurate and stable peak recognition. Based on the MARE values, the square wave load exhibits the lowest overall identification accuracy, whereas the sinusoidal load achieves the highest overall accuracy.

(2)

Load Identification Results under Different Frequencies

The load time–history curves obtained by inverse analysis based on the strain measurement data from conditions 4# and 5# are shown in Figure 18 and Figure 19, respectively.

As shown in Figure 18 and Figure 19, the identified load curves closely match the experimental curves over most of the time. The identified load frequency is 0.1 Hz for condition 4# and 0.3 Hz for condition 5#, both of which are fully consistent with the experimental loading cycles. The peak error for condition 4# is noticeably smaller than that for condition 5#. Due to the lower frequency in condition 4#, more strain data points are available in the peak region, resulting in higher peak identification accuracy. The ARE curves between the identified and experimental loads for both cases are shown in Figure 20 and Figure 21, respectively.

As illustrated in Figure 20 and Figure 21, the ARE curves exhibit periodic variations consistent with the experimental loading cycles. The error curve for condition 5# shows more frequent and pronounced oscillations, indicating that the load frequency has a significant influence on the ARE trend. Comparing the peak values of the curves reveals that the maximum ARE for condition 4# is 8%, whereas that for condition 5# reaches 10%, suggesting that lower load frequencies result in more stable ARE values. The average values of the two curves in Figure 20 yield a MARE of 5.27% in the X direction and 4.11% in the Y direction. The RMSE% for condition 4# is calculated as 5.12% in the X direction and 3.39% in the Y direction. The average values of the two curves in Figure 21 result in a MARE of 6.18% in the X direction and 4.69% in the Y direction. The RMSE% for condition 5# is calculated as 5.75% in the X direction and 4.06% in the Y direction. These results indicate that lower load frequencies lead to reduced peak ARE values and improved overall load identification accuracy.

(3)

Identification Results under Different Load Sizes

The load time–history curves identified from the strain measurement data of conditions 6#, 7#, and 8# are shown in Figure 22, Figure 23 and Figure 24.

As shown in Figure 22, Figure 23 and Figure 24, the identified load curves closely follow the experimental curves for most of the time duration. All three conditions exhibit a load identification frequency of 0.3 Hz, which perfectly matches the applied experimental loading. A comparison of the relative peak errors in each case indicates that condition 8# has the smallest relative peak error, suggesting that larger load sizes lead to more accurate peak identification. The ARE values between the identified and experimental loading curves for conditions 6# to 8# are calculated and plotted in Figure 25, Figure 26 and Figure 27.

As illustrated in Figure 25, Figure 26 and Figure 27, the ARE curves exhibit periodic variations that are consistent with the experimental loading curves, with similar oscillation trends and frequencies observed across all three cases. A comparison of the peak values reveals that the maximum ARE is 8% for condition 8#, 11% for condition 7#, and 10% for condition 6#. The MARE values for the two directions in Figure 25 are calculated as 6.25% in the X direction and 4.78% in the Y direction. The RMSE% for condition 6# is calculated as 6.69% in the X direction and 5.41% in the Y direction. The MARE values for the two directions in Figure 26 are calculated as 6.35% in the X direction and 4.82% in the Y direction. The RMSE% for condition 7# is calculated as 7.51% in the X direction and 6.69% in the Y direction. The MARE values for the two directions in Figure 27 are calculated as 5.51% in the X direction and 4.36% in the Y direction. The RMSE% for condition 8# is calculated as 5.26% in the X direction and 3.96% in the Y direction. These results indicate that larger load magnitudes lead to lower maximum ARE values, more uniform and stable error distributions, and an overall reduction in error level, thereby improving the accuracy of load identification.

The MARE values corresponding to the eight conditions are summarized and presented in the form of a bar chart, as shown in Figure 28.

As shown in Figure 28, the MARE values in the X direction are consistently higher than those in the Y direction across all eight loading cases, indicating that the identification accuracy is higher for loads in the Y direction. This can be attributed to the fact that loads in the Y direction induce deformation along the shorter span of the jacket structure, resulting in larger structural displacements and greater axial strains in the 4th-floor horizontal braces under the same loading conditions. Consequently, the global strain sensitivity is higher, which is consistent with the strain sensitivity distribution observed in the heatmap in Figure 7. A comparative analysis of the MARE values across the eight conditions reveals that loading type, frequency, and peak magnitude all influence identification accuracy to varying degrees. Among these factors, loading type has the most significant impact, as indicated by the comparison of conditions 1# to 3#. In contrast, loading magnitude has the least impact, as observed from the comparison of conditions 5# to 8#.

5. Engineering Applications

The proposed load identification method is applied to a jacket platform structure located in a specific offshore region, providing a basis for subsequent residual life assessment. Based on geometric drawings, a FE model of the four-leg platform is established. The deck structure is meshed using Shell 181 elements, while the remaining structural components are meshed using Beam 188 elements. The regions of applied loads and boundary constraints are illustrated in Figure 29.

As shown in Figure 29, the jacket structure is primarily subjected to current-induced loads below the splash zone, wave and wind loads within the splash zone, and wind loads above the splash zone. For platform structures located in shallow-water regions, the influence of current loads on fatigue life and strength analysis is considerably less significant than that of wind and wave loads. As a result, engineering efforts focus more on the variations in the magnitude and direction of wind and wave loads [35].

5.1. Field Monitoring Experiment

An in-service jacket platform located in a certain offshore region is equipped with an environmental monitoring system for real-time acquisition of marine environmental parameters [36]. The system includes an anemometer with a sampling frequency of 0.33 Hz, which provides time–history data of wind speed and direction for wind load estimation. Additionally, a radar wave gauge with a sampling frequency of 2 Hz is used to record variations in wave height and period, supplying wave elevation time histories for wave load calculations. To obtain localized structural response time–history data, strain sensors must be installed at appropriate positions. The sensor placement strategy can be determined using the method proposed in Section 3.

When a jacket platform structure is subjected to wind and wave loads, any plane along the height direction can be selected. Within this plane, both the wind and wave loads can be considered constant in magnitude. Furthermore, wind and wave loads in any direction within the plane can be represented as the resultant of load components in the X and Y directions. By leveraging the coefficient relationships of wind and wave loads across different height levels of the structure, the number of load components requiring identification can be significantly reduced. Based on the structural geometry and pressure coefficients, the variations of wind and wave load coefficients in the X and Y directions concerning height are plotted, as shown in Figure 30 and Figure 31.

Figure 31 illustrates the distribution of wind load coefficients along the height of the jacket structure. The X direction corresponds to the longitudinal (long-edge) axis, while the Y direction corresponds to the transverse (short-edge) axis. Due to geometric effects, the overall coefficient levels in the X direction are greater than those in the Y direction. The sudden increase at an elevation of 4 m is attributed to the geometric characteristics of the horizontal bracing members. Overall, the wind load coefficients exhibit an increasing trend with height. This variation is influenced by both the structural geometry and the height-dependent nature of wind pressure coefficients.

Figure 32 shows the distribution of wave load coefficients along the height of the jacket structure. The magnitude of the load coefficients is directly influenced by the effective area subjected to wave forces within the splash zone. Since the variation in these coefficients is governed solely by geometric characteristics, the wave load coefficient curve appears as a series of horizontal segments within the splash zone.

Taking the wind and wave load components in the X and Y directions at an elevation of 0 m as unit loads, the load values at other heights can be estimated based on the coefficient relationships shown in Figure 30 and Figure 31. After processing the wind and wave loads at various heights using this method, the number of loads to be identified is simplified to four components, requiring at least four strain sensors. The four load coefficients obtained through load inversion correspond to the wind and wave loads in the X and Y directions at an elevation of 0 m. By further applying the coefficient relationships, the wind and wave load values in the X and Y directions at other heights can be determined. This method significantly reduces the number of loads to be identified, thereby effectively lowering both the testing costs and the computational cost required for sensor placement optimization.

By applying the four load components to the finite element model of the jacket structure, a static analysis is performed to obtain the initial strain sensitivity matrix for sensor placement optimization calculations. Based on practical engineering considerations and the optimization results of the ISCA-OBL algorithm, the OSP is determined to be on the vertical diagonal braces located beneath the spider deck. The sensors are symmetrically distributed within the same horizontal plane, which enhances their ability to capture structural information. The specific locations and orientations are illustrated in Figure 32.

The test data from the eight strain sensors located at the positions shown in Figure 32 were first processed using a fourth-order polynomial regression method to remove trend components. A low-pass filter with a cutoff frequency of 12 Hz was then applied, followed by smoothing using a moving average method with a window width of 21. The comparison of strain time–history curves before and after preprocessing is shown in Figure 33.

As shown in the curves in Figure 33, the high-frequency noise components in the raw data have been effectively eliminated, and the baseline drift has been significantly mitigated, resulting in a marked improvement in the overall signal quality.

As shown in Figure 32, the wind speed time history recorded by the anemometer during a specific time is presented in Figure 34, while the wave height time history recorded by the wave radar during the same time is shown in Figure 35.

5.2. Wind and Wave Load Identification Analysis

The wind load on the jacket structure is generally calculated based on the mean wind speed, and the wind pressure can be expressed by the following equation [37]:

$$F_W={\displaystyle \frac{1}{2}}{\rho }_z{U}_{T,z}^2{K}_{w}A$$

(41)

where ${\rho }_z$ represents the air density at height Z, in units of kg/m³. $U_{T,z}^2$ denotes the average wind speed over some time T at height Z, in units of m/s. A is the projected area in the windward direction, measured in m². $K_w$ is the shape coefficient for wind loading.

Wave loads acting on the jacket platform structure within the splash zone can be calculated using Morison’s equation, as the structural geometry is considered small-scale. The mathematical expression of Morison’s equation is given in Equation (42) [35].

$$F_H={\displaystyle \frac{1}{2}}{C}_D\rho D{u}_x\left|u_x\right|+C_M\rho {\displaystyle \frac{\pi D^2}{4}}{\displaystyle \frac{\partial u_x}{\partial t}}$$

(42)

In which the first term on the right-hand side represents the horizontal drag force, where $C_D$ is the drag coefficient. The second term corresponds to the inertia force, with $C_M$ denoting the inertia coefficient. $\rho$ is the seawater density (kg/m³), $D$ is the diameter of the cylindrical member (m), and u_x and ${\displaystyle \frac{\partial u_x}{\partial t}}$ are the water particle velocity and acceleration, respectively, at the centroid height along the axis of the cylinder.

Given that the water depth at the analyzed engineering jacket platform is 112 m and no extreme weather conditions occurred during the monitoring period, the wave surface profile is described using the Airy linear wave theory. Under this framework, the terms $u_x$ and ${\displaystyle \frac{\partial u_x}{\partial t}}$ in Equation (42) can be expressed by Equation (43).

$$\left\{\begin{array}{l}{u}_x={\displaystyle \frac{\pi H}{T}}{\displaystyle \frac{\text{cosh} kz}{\text{sinh} kd}}\text{cos}\omega t\\ {\displaystyle \frac{\partial u_x}{\partial t}}={\displaystyle \frac{-2{\pi }^{2}H}{T^2}}{\displaystyle \frac{\text{cosh} kz}{\text{sinh} kd}}\text{sin}\omega t\\ k={\displaystyle \frac{2\pi }{L}},\omega ={\displaystyle \frac{2\pi }{T}}\end{array}\right.$$

(43)

By substituting the wind speed and wave height data from Figure 35 and Figure 36 into Equations (41), (42), and (43), respectively, the time history curves of wind and wave loads can be obtained. These curves serve as references for validating the results of wind and wave load identification.

The load inversion equations were solved using the LS method combined with Tikhonov regularization, and the results were synthesized to obtain the time history curves of wind direction and wind load size, as shown in Figure 36.

As shown in Figure 36a, the identified wind load direction generally agrees with the wind direction measured by the anemometer. However, larger deviations occur at moments when the wind direction changes significantly. As shown in Figure 36b, the identified wind load magnitude is generally smaller than the values calculated from wind speed data using Equation (38). This discrepancy arises because Equation (38) relies on empirical coefficients and does not accurately account for variations in the exposed area due to wind direction changes, leading to an overestimation of the effective windward area of the structure.

The ARE between the identified wind direction and the measured data is calculated, and the corresponding curve is shown in Figure 37.

As shown in Figure 37, the ARE of the identified wind direction ranges from 0% to 5%, with most data points fluctuating within the 1~5% interval. The calculated MARE of wind direction identification is 2.65%. This demonstrates that the proposed load inversion method achieves high accuracy and reliability in identifying load direction.

As shown in Figure 38, the ARE of the identified wind load magnitudes is mainly distributed between 5% and 17%, with a calculated MARE of 10.12%, indicating an average relative deviation of 10.12% between the identified results and the measured data. The RMSE is 4.4 kN, accounting for 11.04% of the average wind load size. These results meet the accuracy requirements for load identification in engineering practice and provide a reliable basis for subsequent fatigue damage assessment and residual life prediction.

The time–history curve of wave loads obtained through inverse load identification is compared with the wave load curve calculated using Morison’s equation, as shown in Figure 39.

In Figure 39, the black curve represents the time–history of wave loads obtained through load identification, while the red curve corresponds to the wave loads calculated using Morison’s equation based on wave gauge measurements. The two curves exhibit consistent trends over time; however, the identified load magnitudes are generally lower than those computed using Morison’s equation. This discrepancy is primarily attributed to the sensitivity of Morison’s equation to the hydrodynamic parameter values, where empirically determined coefficients can introduce significant errors. The ARE between the identified results and those calculated via Morison’s equation is shown in Figure 40.

As shown in Figure 40, the ARE between the identified wave load magnitudes and those calculated using the Morison equation primarily ranges from 1% to 23%, with a computed MARE of 11.63%, indicating a high overall average accuracy. The RMSE was calculated to be 25.88 kN, accounting for 34.29% of the average wave load magnitude. This error may be related to the Morison equation neglecting nonlinear interactions between waves and fluid–structure coupling effects, though the specific mechanisms require further investigation. Despite the relatively high RMSE%, the key accuracy indicator, MARE, still meets the precision requirements for engineering applications. These results validate the effectiveness and reliability of the proposed method in practical engineering applications, providing wave load inputs that comply with design standards for fatigue damage assessment and residual life prediction of jacket platform structures.

6. Conclusions

The marine environmental loads acting on jacket platform structures are diverse and complex, making them difficult to measure directly and challenging to characterize accurately using analytical expressions. This complexity poses significant obstacles to the accuracy of subsequent structural safety assessments. To address this issue, a load inversion method based on strain measurement data was proposed for identifying environmental loads acting on jacket platform structures. The proposed method is developed under the assumptions of isotropy, homogeneity, and elastic deformation, and applies to in-service conditions, excluding extreme events.

(1)

An optimized sensor placement algorithm is proposed based on an ISCA-OBL strategy. The determinant and condition number of the FIM are selected as objective functions for multi-objective optimization. The candidate sensor region is determined using FE dynamic analysis results, and the number and locations of strain sensors are identified within the ISCA-OBL framework. The optimized sensor configuration ensures that the collected strain data captures richer structural response information while exhibiting enhanced robustness against noise.

(2)

Eight loading experiments were conducted on a laboratory-scale jacket platform structure, during which strain measurement data were collected. Load identification was performed using a least squares method combined with Tikhonov regularization based on the experimental strain data. Comparison between the identified load curves and the experimental input reveals that load type, frequency, and peak magnitude all influence the accuracy of the inversion results. Among these factors, load type has the most significant impact, while peak magnitude has the least. The maximum MARE across the eight conditions is 6.91%, demonstrating that the proposed load identification method exhibits high stability and accuracy when applied to indoor jacket platform structures.

(3)

The proposed method was further applied to an in-service jacket platform in a specific marine area using field-measured strain monitoring data. The inverted environmental loads showed a MARE of 2.65% for wind direction, 10.12% for the size of wind load, and 11.63% for the size of wave load. These results confirm the effectiveness and reliability of the proposed method for real-world engineering applications and indicate that it can provide accurate load input data for structural safety assessment and residual life prediction of jacket platforms.