Inverse Finite Element Method Research Articles

Damage Evolution Mechanism of Railway Wagon Bogie Adapter 1035 Steel and Damage Parameter Calibration Based on Gursone-Tvergaarde-Needleman Model.

As a critical component of a train, the railway wagon bogie adapter has higher quality requirements. During the forging process, external loads can induce voids, cracks, and other defects in the forging, thereby reducing its service life. Hence, studying the damage behavior of the forging material, specifically AISI 1035 steel, becomes crucial. This study involved obtaining stress-strain curves for AISI 1035 steel through uniaxial tensile tests at temperatures of 900 °C, 1000 °C, and 1100 °C, with strain rates of 0.1 s-1, 1 s-1, and 10 s-1. Subsequently, SEM was used to observe samples at various deformation stages. The damage parameters, q1, q2 and q3 in the GTN model "a computational model used to analyze and simulate material damage which can effectively capture the damage behavior of materials under different loading conditions" were then calibrated using the Ramberg-Osgood model and stress-strain curve fitting. Image Pro Plus software v11.1 quantified the sample porosity as f0, fn, fc and fF. A finite element model was established to simulate the tensile behavior of the AISI 1035 steel samples. By comparing the damage parameters of f0, fn, fc and fF obtained by the finite element method and experimental method, the validity of the damage parameters obtained by the finite element inverse method could be verified.

A distributed dynamic load identification approach for thin plates based on inverse Finite Element Method and radial basis function fitting via strain response

Dynamic load identification is crucial for structural design and health monitoring. Traditional methods for distributed dynamic loads (DDLs) typically involve establishing a transfer matrix, which often leads to ill-posed problems. In this paper, a novel method based on the inverse Finite Element Method (iFEM) and radial basis function (RBF) fitting was proposed to identify DDLs from measured strain responses. The method begins with the reconstruction of the displacement field from discrete strain data using iFEM, followed by applying RBF fitting to obtain a continuous displacement field. To enhance the performance of the RBF fitting, a constrained matrix for force boundary conditions is introduced, along with a selection rule for the RBF shape parameter. The identified loads are subsequently determined by incorporating the well-fitted RBFs into the motion equations of thin plates. This approach eliminates the need for a transfer matrix, thereby avoiding ill-posedness, and enables the simultaneous identification of the spatial distribution and time history of the loads. Three numerical examples for the identification of concentrated loads, uniformly distributed loads, and globally distributed loads demonstrate the method's effectiveness, accuracy, and noise resistance, with additional validation provided through a comparison with the classic time-domain method.

A quadrilateral inverse plate element for real-time shape-sensing and structural health monitoring of thin plate structures

The inverse finite element method (iFEM) emerged as a powerful tool in shape-sensing and structural health monitoring (SHM) applications with distinct advantages over existing methodologies. In this study, a quadrilateral inverse-plate element is formulated via a sub-parametric approach using bi-linear and non-conforming cubic Hermite basis functions for engineering structures, which can be modeled as thin plates. Numerical validation involves dense and assumed sparse sensor arrangements for in-plane, out-of-plane, and mixed general loading conditions. iFEM analysis reveals efficient monotonic convergence to analytical and high-fidelity finite element reference solutions. After successful numerical validation, defect detection analysis is performed considering minute geometric discontinuities and structural stiffness reduction because of latent subsurface defects under tensile and transverse loading conditions. The inverse formulation successfully resolves the presence of simulated defects under a sparse sensor arrangement. The proposed inverse-plate element is accurate in the full-field reconstruction of shape-sensing profiles and reliable in defect identification and quantification in thin plate structures.

A four-node inverse curved shell element coupling MITC method for deformation reconstruction of plate and shell structures

In the field of structural deformation monitoring, the inverse finite element method (iFEM) has significant engineering value as a structural health monitoring technique that provides timely and reliable warnings for shell structures. However, existing inverse finite elements are mainly based on first-order shear deformation theory and kirchhoff-love theory, which are not suitable for deformation reconstruction in plate and shell structures of arbitrary thickness. This study integrates iFEM with the Mixed Interpolation of Tensorial Components (MITC) method to develop a novel four-node quadrilateral inverse curved shell element, named iMICS(inverse Mixed Interpolation Curved Shell)4, aimed at enhancing the accuracy and efficiency of deformation reconstruction in complex plate and shell structures. The method uses the MITC4 shell element as the kinematic framework and applies the least squares variational principle to achieve deformation reconstruction, effectively alleviating shear and membrane locking issues across structures of varying thickness. Numerical examples validate the superior performance of the iMICS4 element, demonstrating its promising application prospects.

Deformation monitoring for fixed-wing UAS through Inverse Mesh-free Method

This study proposes the Inverse Local Radial Point Interpolation Method (iLRPIM) for reconstructing the deformation of irregular plate structures. The proposed methodology eliminates the drawbacks of the existing inverse finite-element method (iFEM), where the Jacobian matrix results in reduced computational accuracy for irregular plate-shape sensing. To this end, this paper released from the element and used the Mesh-Free method to describe the calculated strains. Subsequently, the system equation was formulated by minimizing the error between the measured and calculated strains. The deformation parameters were then computed by solving this system equation. Additionally, compared to iFEM, iLRPIM allows for easier adjustment of the interpolation function, making it more suitable for real-world engineering applications. The effectiveness of iLRPIM has been validated through simulations and practical experiments using four typical models. In simulations of the UAS wing box equivalent model, peak errors were below 0.8 mm for a deformation range of 25 mm. In practical scenarios, peak errors were below 3.1 mm for a deformation range of 85 mm. These results underscore the system’s capability to accurately sense the shapes of irregular plate structures using iLRPIM.

A High-Precision Inverse Finite Element Method for Shape Sensing and Structural Health Monitoring

In the contemporary era, the further exploitation of deep-sea resources has led to a significant expansion of the role of ships in numerous domains, such as in oil and gas extraction. However, the harsh marine environments to which ships are frequently subjected can result in structural failures. In order to ensure the safety of the crew and the ship, and to reduce the costs associated with such failures, it is imperative to utilise a structural health monitoring (SHM) system to monitor the ship in real time. Displacement reconstruction is one of the main objectives of SHM, and the inverse finite element method (iFEM) is a powerful SHM method for the full-field displacement reconstruction of plate and shell structures. However, existing inverse shell elements applied to curved shell structures with irregular geometry or large curvature may result in element distortion. This paper proposes a high-precision iFEM for curved shell structures that does not alter the displacement mode of the element or increase the mesh and node quantities. In reality, it just modifies the methods of calculation. This method is based on the establishment of a local coordinate system on the Gaussian integration point and the subsequent alteration of the stiffness integration. The results of numerical examples demonstrate that the high-precision iFEM is capable of effectively reducing the displacement difference resulting from inverse finite element method reconstruction. Furthermore, it performs well in practical engineering applications.

A novel shape sensing approach based on the coupling of Modal Virtual Sensor Expansion and iFEM: Numerical and experimental assessment on composite stiffened structures

Shape sensing, i.e. the reconstruction of the displacement field of a structure from discrete strain measurements, is becoming crucial for the development of a modern Structural Health Monitoring framework. Nevertheless, an obstacle to the affirmation of shape sensing as an efficient monitoring system for existing structures is represented by its requirement for a significant amount of sensors. Two shape sensing methods have proven to exhibit complementary characteristics in terms of accuracy and required sensors that make them suitable for different applications, the inverse Finite Element Method (iFEM) and the Modal Method (MM). In this work, the formulations of these two methods are coupled to obtain an accurate shape sensing approach that only requires a few strain sensors. In the proposed procedure, the MM is used to virtually expand the strains coming from a reduced number of strain measurement locations. The expanded set of strains is then used to perform the shape sensing with the iFEM. The proposed approach is numerically and experimentally tested on the displacement reconstruction of composite stiffened structures. The results of these analyses show that the formulation is able to strongly reduce the number of required sensors for the iFEM and achieve an extremely accurate displacement reconstruction.

An optimized single-side inverse finite element method for displacement field reconstruction of wind turbine tower

Wind turbines are subjected to complex dynamic loads during actual operation, which are difficult to predict accurately in advance, which may lead to inaccurate structural strength assessment in the structural design stage, and thus bring safety risks to wind power towers. Inverse Finite Element Method (iFEM) is a reconstruction method of structural displacement distribution, which can estimate the full-field displacement distribution of tower, and used for structural on-line health monitoring. However, conventional iFEM require sensors to be placed on both sides of the structure, which greatly limits its engineering practicality. For this reason, a single-side inverse finite element (iFEM_S) was developed in this study to enhance the engineering applicability of iFEM. Both iFEM and iFEM_S are based on the principle of least squares variation, and the solution equations can be established when the measured strain is known. The core idea of iFEM_S is to find the optimal node displacement s to minimize the sum of mean square errors between the theoretical and measured strains on one side of the structure, thereby generating a set of algebraic equations. Solving these linear equations can achieve the structural displacement field. Additionally, A cyclic optimization algorithm (COA) was developed in this study to improve the robustness of reconstruction results through optimizing strain gauge locations, in which the condition number of pseudo stiffness matrix was used as the objective function, and the condition number is minimized, named iFEM_S_O. Finally, the reconstruction accuracy and robustness and of iFEM, iFEM_S and iFEM_S_O were verified through several numerical examples. The simulated reconstruction results indicated that iFEM_S_O is more accurate and stable than iFEM_S, and iFEM_S is more accurate and stable than iFEM.

Dynamic analysis and shape sensing of pipes subjected to water hammer by the inverse finite element method

Pipe structures are commonly employed in modern industries. However, the transient pressure induced by water hammer leads to a high risk of structural damage, highlighting the importance of Structural Health Monitoring (SHM). The inverse Finite Element Method (iFEM) offers a way to reverse structural deformations based on a set of discrete strain data. In this paper, the water hammer phenomenon in a Reservoir-Pipe-Valve (RPV) system is investigated. The element iQS4 is adopted to reconstruct dynamic deformations for the pipe using iFEM. The water hammer resulting from an instantaneous closure of the valve can generate significant fluid pressure, leading to severe pipe vibrations. The vibration deformations can be monitored by iFEM. The maximum error of the pipe nodal displacement is below 3 % and the average error is below 2 % in the proposed several sensor location cases, even the mesh of iFEM is coarser than the FE model. Among the sparse sensor location cases, the X path which is suitable for Fiber Bragg Grating (FBG) sensors has the highest accuracy of the displacement reconstruction. The results reveal a promising prospect in the real-time monitoring based on iFEM for pipes.

Study on Strain Field Reconstruction Method of Long-Span Hull Box Girder Based on iFEM

The box girder’s condition significantly impacts the safety and overall performance of the entire ship because it is the primary stress component of the hull construction. This work used experimental research on the long-span hull box girder based on IFEM (Inverse Finite Element Method) technology to ensure the structural safety of the hull box girder. Due to the limitations of conventional experiments in this technical field, such as their reliance on finite element data and lack of input from physical tests, numerous research methods combining the strain sensing data from physical tests with the strain data from virtual sensors were conducted. The strain fields of the top plate, side plate, and bottom plate were each reconstructed in turn, and the verifier measuring points in the physical model test were used to assess the accuracy of the reconstruction results. The findings demonstrate that the top plate, side plate, and bottom plate reconstructions had relative errors of 0.24–7.86%, 0.75–8.13%, and 3.31–2.52%, respectively. This enables the reconstruction of the strain field of the long-span hull box girder using physical test data and promotes the use of iFEM technology in the field of structural health monitoring of large marine structures.

A preliminary investigation of the inverse Finite Element Method applied to bridge structures

Deflection monitoring plays a key role in bridge Structural Health Monitoring, as deflection data may offer valuable insights into potential structural issues, such as cracks and corrosion. However, monitoring bridge deflections under operative loads presents challenges, particularly in scenarios where establishing a fixed reference point for displacement sensors is impractical, such as when a bridge spans a waterway or the sensors may be vulnerable to vandalism. In recent years, the inverse Finite Element Method (iFEM) has emerged as a possible solution. This method provides a load- and material-independent means of computing structural displacements from strain measurements, in real time at a minimal computational cost. Additionally, the iFEM can serve as a virtual sensing approach to monitor strains at locations lacking sensors, enabling the assessment of the fatigue life consumption at virtually any point in the structure. Despite its potential, applications of the iFEM in the civil engineering field are very limited, with no validation on full scale bridges has been performed to date. This study addresses this gap by investingating the application of the iFEM for bridge deflection monitoring through distributed strain measurements. The proposed approach is tested on experimental case studies, involving cross-validatiion of reference-free deflections inferred by the iFEM with those obtained from displacement sensors. This work contributes to the advancements of structural health monitoring methodologies in civil engineering, demonstrating the practicality and reliability of iFEM in bridge applications.

Shape sensing with iFEM on a type IV pressure vessel based on experimental measurements

The service life of a ship depends on its structural integrity, which can be undermined through time via different loading conditions. The incorporation of a Structural Health Monitoring (SHM) system could enhance the safety of the vessel because information about its health condition could potentially lead to the avoidance of catastrophic scenarios by the detection of structural damages and their progression, especially when they may be unobservable via physical inspection. Within this context, a model-based SHM method called inverse Finite Element Method (iFEM) has been recently developed to assess the displacement and stress profile of vessels. iFEM offers several advantages, including the independence from material properties and loading conditions, the lack of training procedures, and the limited computational demand, which makes it suitable for real-time applications, enabling to reconstruct the complete displacement, and thus, the strain field starting from discrete strain measures. In this work, we explore the application of the iFEM in challenging scenarios, regarding repeated stress fluctuations (e.g. fatigue) or other type of mechanisms like corrosion, buckling or even externally imposed damages from collision impacts or combat type damages (explosions, ballistic, etc.), depending on the specific case of naval vessels. An anomaly index is used to highlight the health state of the structure by comparing the strain predicted by the iFEM at some test locations with the strain measured by a test sensor in the same position. The two values will match only for the healthy structure, while their departure from each other will indicate anomalous conditions. Though the formulation of the diagnostic problem is general for an arbitrary component geometry and damage type, the proposed study is accompanied by some numerical examples applied to a realistic ship case.

Shape sensing of the thin-walled beam members by coupling an inverse finite element method with a refined quasi-3D zigzag beam theory

In structural safety field, the inverse finite element method (iFEM) is an effective methodology to reconstruct full-field displacement on beam, plate and shell structures, independently of the loading conditions and of the material properties. However, the current iFEM in principle requires uniform shear distribution over the thickness of beam, which is practically hardly possible due to these thin-walled beam with the general cross-section shape, such as I-section beam, hat-section beam and box-section beam, and there is no effective method to realize deformation online monitoring at home and abroad. To relieve this issue, a novel iFEM strategy is proposed to establish the shape sensing model of the thin-walled beam, where the thin-walled beam is replaced with an equivalent layered composite one based on a generalized layered global-local beam (GLGB) theory, and the improved quasi-3D zigzag shear deformation theory is presented to describe deformation field of the equivalent layered composite beam. The proposed iFEM method accounts for not only thickness stretching but also interlaminar continuity of shear stresses and displacements. Besides, the proposed iFEM formulation does not need any shear correction factors. Accuracy and effectiveness of the established shape sensing model are demonstrated through several case studies. The numerical results show that the proposed iFEM can accurately reconstruct the deformation of the thin-walled structure and the reconstruction accuracy can be improved by 5 %.

Damage diagnosis of plates and shells through modal parameters reconstruction using inverse finite-element method

Damage diagnosis of plates and shells through modal parameters reconstruction using inverse finite-element method

The STAGE method for simultaneous design of the stress and geometry of flexure mechanisms

Current design methods for flexure (or compliant) mechanisms regard stress as a secondary, limiting factor. This is remarkable because stress is also known as a useful design parameter. In this paper we propose the Stress And Geometry (STAGE) method, to design the geometry of a flexure mechanism together with a desired stress field. From this design, the stress-free to-be-fabricated geometry is computed using the inverse finite element method. To demonstrate the potential of the method, the geometry of the well-known crossed-flexure pivot is taken as example. We first show how this mechanism can be redesigned for the same functional geometry with various internal stresses. This results for a specific choice of stress field in a design of a crossed-flexure pivot with 23% lower peak stresses during motion as compared to the known designs, for a ±45° rotation. We then present a second example, of a Folded Leaf Spring (FLS). With a parameter sweep the optimal stress field is calculated, showing a peak stress reduction of 28% during motion. This result was validated with an experiment, showing a normalized mean absolute error of 5.5% between experiment and theory. With a second experiment it was verified that the functional geometry of the FLS with internal stresses was equal to the one without internal stresses, with geometric deviations smaller than half the thickness of the flexures.

Discontinuous Deformation Monitoring of Smart Aerospace Structures Based on Hybrid Reconstruction Strategy and Fiber Bragg Grating.

A hybrid enhanced inverse finite element method (E-iFEM) is proposed for real-time intelligent sensing of discontinuous aerospace structures. The method can improve the flight performance of intelligent aircrafts by feeding back the structural shape information to the control system. Initially, the presented algorithm combines rigid kinematics with the classical iFEM to discretize the aerospace structures into elastic parts and rigid parts, which will effectively overcome structural complexity due to fluctuating bending stiffness and a special aerodynamic section. Subsequently, the rigid parts provide geometric constraints for the iFEM in the shape reconstruction method. Meanwhile, utilizing the Fiber Bragg grating (FBG) strain sensor to obtain real-time strain information ensures lightweight and anti-interference of the monitoring system. Next, the strain data and the geometric constraints are processed by the iFEM for monitoring the full-field elastic deformation of the aerospace structures. The whole procedure can be interpreted as a piecewise sensing technology. Overall, the effectiveness and reliability of the proposed method are validated by employing a comprehensive numerical simulation and experiment.

Real-time shape sensing of large-scale honeycomb antennas with a displacement-gradient-based variable-size inverse finite element method

Real-time thermal deformation monitoring plays a crucial role in calibrating phase signals and maintaining satellite performance for large spaceborne antennas. The inverse finite element method (iFEM) represents a promising shape-sensing methodology applicable to monitor the three-dimensional displacement of structures through surface-measured strain. However, the high-accuracy reconstruction of large-scale structures involves a large number of inverse elements, which leads to the low level of computational efficiency. To address this issue, a displacement-gradient-based variable-size iFEM is proposed in this paper. According to the characteristics of deformation, this method optimizes the size of each inverse element to reduce the number of inverse elements required, which improves real-time performance and maintains the high accuracy of reconstruction. The real-time performance of the proposed method is verified by conducting a simulation test on a large-scale honeycomb antenna. According to the simulation results, the variable-size iFEM is applicable to achieve the same accuracy of reconstruction using fewer inverse elements than conventional iFEM. An experimental test is performed and the optimal variable-size discretization that balances accuracy and efficiency is determined. The results show that the maximum error of reconstructed displacement is less than 7.3 % and the reconstruction time is in no excess of 0.1 s.

A Critical Comparison of Shape Sensing Algorithms: The Calibration Matrix Method versus iFEM.

Two shape-sensing algorithms, the calibration matrix (CM) method and the inverse Finite Element Method (iFEM), were compared on their ability to accurately reconstruct displacements, strains, and loads and on their computational efficiency. CM reconstructs deformation through a linear combination of known load cases using the sensor data measured for each of these known load cases and the sensor data measured for the actual load case. iFEM reconstructs deformation by minimizing a least-squares error functional based on the difference between the measured and numerical values for displacement and/or strain. In this study, CM is covered in detail to determine the applicability and practicality of the method. The CM results for several benchmark problems from the literature were compared to the iFEM results. In addition, a representative aerospace structure consisting of a twisted and tapered blade with a NACA 6412 cross-sectional profile was evaluated using quadratic hexahedral solid elements with reduced integration. Both methods assumed linear elastic material conditions and used discrete displacement sensors, strain sensors, or a combination of both to reconstruct the full displacement and strain fields. In our study, surface-mounted and distributed sensors throughout the volume of the structure were considered. This comparative study was performed to support the growing demand for load monitoring, specifically for applications where the sensor data is obtained from discrete and irregularly distributed points on the structure. In this study, the CM method was shown to achieve greater accuracy than iFEM. Averaged over all the load cases examined, the CM algorithm achieved average displacement and strain errors of less than 0.01%, whereas the iFEM algorithm had an average displacement error of 21% and an average strain error of 99%. In addition, CM also achieved equal or better computational efficiency than iFEM after initial set-up, with similar first solution times and faster repeat solution times by a factor of approximately 100, for hundreds to thousands of sensors.

Inverse finite element method and support vector regression for automated crack detection with OFDR-Distributed fiber optic sensors

Crack detection is a crucial but challenging task. In this paper, a new method is introduced for automatical detection of the substrate cracks. An anomaly index is defined based on optical frequency domain reflectometry (OFDR) measured strain and inverse finite element method (iFEM) recovered strain. The index is sensitive to crack-induced strain but not sensitive to substrate strain. Leveraging on the self-defined anomaly index, the crack location can be identified with millimetre accuracy. Then, the detection of the crack depth is regarded as a typical multi-regression analysis and solved by support vector regression (SVR) method. Particle swarm optimization (PSO) approach is applied to find optimal parameters of the SVR kernel function. Experimental cases of a simple-supported beams with different crack locations and depths are performed, demonstrating the excellent prediction accuracy of the method. The proposed method has a promising potential in identification and quantification of cracks.

Dynamic deformation reconstruction of propulsion system pipe based on inverse finite element method

Pipes are necessary components of the propulsion system. The water hammer effect is easy to provoke severe vibration of the pipe, which is harmful to the safety of the propulsion system. Therefore, it is necessary to monitor the health status of the propulsion system pipe in real time. In this paper, the FSI analysis is adopted to study the water hammer pressure and pipe vibration response caused by valve closure in RPV system. The iFEM algorithm is performed to invert the dynamic deformation of the pipe in real time. Four sensor layout schemes are considered, and the influence of sensor layout on the reconstruction accuracy based on the iFEM algorithm is discussed. It is concluded that the iFEM algorithm is very promising for the dynamic deformation reconstruction of the pipe.