Conference Agenda

The variational principle underlying the inverse Finite Element Method (iFEM) was first introduced in 2003 by Tessler and Spangler in a NASA Technical Memorandum, with the goal of enabling real-time structural health monitoring in next-generation aerospace vehicles. This foundational work was followed by a conference paper presented at the 2nd European Workshop on Structural Health Monitoring in Munich (2004), which elaborated the iFEM formulation for shear-deformable plate and shell structures. Since its inception, iFEM has been extensively validated and evaluated by numerous researchers. Its applications span a wide range of structural systems—including three-dimensional beams, plates, and shells—composed of metallic or laminated composite materials, and encompassing both linear and nonlinear deformation regimes under static and dynamic conditions.

The iFEM framework reconstructs full-field displacement, strain, and stress distributions across an entire structure using in-situ strain-gauge measurements taken at arbitrary surface locations. The implementation integrates seamlessly with the state-of-the-art commercial finite element software ABAQUS, providing an industrial-grade analysis capability ideally suited for Structural Health Monitoring (SHM) and repair management of aerospace, naval, and civil structures. A structural or stress engineer proficient in ABAQUS can readily employ the iFEM algorithm through a user-supplied routine that enables efficient coupling between iFEM and the ABAQUS environment.

In this presentation, we begin by reviewing two decades of progress in the development and application of the inverse Finite Element Method (iFEM). Key contributions are highlighted, with emphasis on their relevance to structural health monitoring (SHM) and related engineering applications. We then introduce a practical computational system built on a loosely coupled ABAQUS–iFEM framework, designed to support Digital Twin initiatives. The system’s architecture and computational components are described. Next, we present an updated iFEM formulation that employs dimensionally consistent kinematic variables for shear-deformable, C⁰-continuous shell elements. This refined formulation inherently provides the dimensionally consistent terms that underpin the weighted least-squares variational principle fundamental to iFEM. Finally, we discuss numerical results for a representative built-up naval structure instrumented with a distributed strain-sensor network.

The inverse finite element method (iFEM) has emerged as a powerful computational framework for real-time shape and strain sensing of structures using sparse experimental measurements. As a purely physics-based approach, iFEM reconstructs full-field displacement, rotation, and strain distributions by minimizing a weighted least-squares functional between analytically computed and experimentally measured section strains, without requiring any prior knowledge of the material properties or external loads. This capability makes iFEM particularly attractive for structural health monitoring (SHM) applications where only limited sensor information is available.

Despite its advantages, the accuracy and robustness of iFEM strongly depend on two critical parameters: the resolution of the inverse finite element mesh and the spatial density of available strain measurements. A coarse mesh may lead to insufficient representation of structural kinematics, while excessive mesh refinement without adequate sensor coverage can result in numerical instability and noise amplification. Similarly, sparse or non-optimally distributed sensors can significantly deteriorate reconstruction quality, especially for complex geometries. Although several weighting strategies have been proposed in the literature to mitigate these issues, a comprehensive evaluation of the combined effects of mesh resolution and data sparsity on iFEM performance remains largely unexplored.

In this study, a systematic parametric investigation is conducted to assess how mesh discretization density and sensor availability influence the deformation reconstruction and strain-sensing accuracy of the four-node quadrilateral inverse-shell element (iQS4) formulation within the iFEM framework. Three representative benchmark cases including a clamped flat plate, a curved cylindrical shell, and a T-beam structure are examined under various loading conditions. Each configuration is analyzed using different mesh resolutions and strain sensor distributions to capture a wide range of practical SHM scenarios. The reconstructed deformation and strain fields obtained from iFEM are quantitatively compared against high-fidelity forward finite element analysis (FEA) results in terms of displacement and strain error metrics.

The findings reveal that there exists an optimal balance between mesh refinement and sensor density that minimizes reconstruction error while maintaining computational efficiency. Furthermore, the study identifies critical zones where sensor placement has the greatest impact on the fidelity of the reconstructed fields. Based on these insights, practical guidelines and best practices for sensor layout design and mesh configuration in iFEM/iQS4-based SHM systems are proposed. The outcomes of this research contribute to a deeper understanding of iFEM performance limitations and provide valuable recommendations for implementing reliable and cost-effective shape and strain sensing in engineering structures.

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.