The inverse Finite Element Method (iFEM) is a well-established approach for reconstructing displacement fields from strain measurements, offering notable advantages in structural health monitoring (SHM) of complex systems. However, its application to real-world structures such as stiffened panels is often limited by geometric complexity and sparse sensing capabilities. In such cases, direct iFEM implementation becomes challenging due to the inability to instrument all structural regions, especially around reinforcing elements. This work introduces a novel iFEM-based framework tailored for stiffened panel structures, combining a simplified strain-to-displacement reconstruction with a dehomogenization step. The procedure first applies iFEM to a reduced geometry (e.g., flat plate without stiffeners) using available strain data. Then, a post-processing step transfers the reconstructed displacement field onto the full, detailed geometry of the stiffened structure. This two-step approach enables accurate shape sensing even in the absence of strain sensors on complex features, significantly reducing modeling effort and sensor deployment requirements. The framework is experimentally validated and shows strong potential for SHM applications in marine and aerospace structures.

Semi-submersible platforms endure long-term service in harsh marine environments, often subjected to complex and unpredictable dynamic environmental loads. These platforms float on the water surface for extended periods, with waves directly impacting the columns and pontoons, inducing structural responses and movements. Therefore, establishing a Structural Health Monitoring (SHM) system is crucial for real-time assessment of the health status of semi-submersible platforms. This study employs the inverse Finite Element Method (iFEM) to monitor the operational state of a semi-submersible platform under dynamic wave loads. By comparing the total displacements derived from iFEM analysis with benchmark data from fluid-structure interaction simulations, an optimization method for sensor placement based on the Multi-Island Genetic Algorithm is proposed, ultimately determining the optimal sensor configuration.

Accurate full-field reconstruction of structural stress and strain remains a central challenge in structural health monitoring, particularly when only sparse sensing data are available during operation. While the inverse finite element method (iFEM) is widely recognised for its robustness in displacement and stress estimation, its practical implementation is limited by the requirement for dense and well-distributed in-situ strain measurements. This can make the method infeasible in complex and large-scale structures where sensor layouts are limited by cost, accessibility, or weight constraints.

Recent advances in operator-learning methods offer promising avenues to overcome this limitation by inferring missing field information directly from sparse observations. Approaches such as Deep Operator Networks (DeepONets) can learn continuous strain operators and enrich discrete strain data, enabling full-field reconstruction from a reduced sensing configuration.

This work introduces a two-step reconstruction framework that combines an operator-learning–based strain augmentation module with iFEM to recover the full stress and strain fields from limited strain measurements. First, the measured strains are augmented using DeepONet to estimate the strain values at the iFEM elements that do not have direct strain measurements. These augmented strains are then used within iFEM to compute displacements and stresses, ensuring the reconstruction remains grounded in physical constraints. The framework is adaptable to diverse sensor layouts and explicitly incorporates model parameters to account for uncertainties and potential sensor loss. The reconstructed fields enable the subsequent extraction of stresses at critical locations, supporting traditional integrity assessments and remaining useful life analyses.

The methodology is demonstrated on a numerical cantilever composite plate benchmark. A dataset of 400 laminate configurations is generated by varying the number of plies and ply orientations while keeping the geometry, loading, sensor layout, ply thickness, and material properties fixed. The generated layups include symmetric, unsymmetric, balanced, and unbalanced configurations, enabling the model to learn across a mechanically diverse laminate family. Results show that conditioning the DeepONet on the laminate stiffness matrices enables strain reconstruction across unseen layups, providing a compact and physically meaningful route for incorporating laminate variability into data-driven SHM workflows. The study demonstrates the feasibility of combining sparse sensing, laminate mechanics, and operator learning as a preliminary step towards more generalizable iFEM-based reconstruction frameworks for composite structures.

Shape sensing based on strain data is a key technique in Structural Health Monitoring (SHM) for reconstructing the displacement field of a deformed structure, from which the strain distribution can be calculated. A popular approach to the reconstruction of the displacement field is the inverse Finite Element Method (iFEM), where the deformed shape is interpolated using the trial (approximation) functions of the finite element method. The accuracy of such reconstruction is strongly influenced by the number of strain measurements and by the quality of the trial basis. This study presents the development of a hybrid-Trefftz iFEM, aiming to maximize the reconstruction accuracy while minimizing the number of required strain readings. Hybrid-Trefftz elements are well-suited for iFEM applications due to their ability to achieve high accuracy with very sparse meshes, by using trial functions that satisfy exactly the differential equation governing the problem. In this way, Trefftz elements embed physical insight into the model. This choice of trial functions makes hybrid-Trefftz elements insensitive to mesh distortion and high solution gradients, allowing for the use of larger finite elements and thereby reducing the number of required strain measurements.

The proposed method is numerically validated shape sensing trough representative examples that include various boundary and loading conditions, demonstrating its applicability across a wide range of stress states.

The capability of hybrid-Trefftz elements to deliver highly accurate results in challenging scenarios involving stress concentrations or with irregular element geometries, is promising to offer a strong foundation for future SHM development.

“Shape sensing” refers to the reconstruction of the displacement field of a structure based on in-situ strain measurements. There are various deformation field reconstruction techniques, of which the inverse Finite Element Method (iFEM) is considered to be one of the most effective. Structural deformation reconstruction technique is researched. Based on the least squares variational principle the inverse finite element method is applied to reconstruct the displacement of the ship stiffener in various cases. The effect of the mesh of the inverse finite element model on error of the reconstruction is researched. Feasibility of the inverse finite element method and its applicability in complex structures are verified through error analysis. To make the inverse finite element method more practical, an improved genetic algorithm is applied, which provides a basis for selection of the number and location of discrete measuring points The results show that the number of measurement points in the inverse finite element model can be significantly reduced by optimization of their layout, while maintaining high accuracy of reconstruction results, strengthening the practical application of iFEM in shape sensing analysis of the structures.