Deformation Reconstruction Using the Inverse Finite Element Method and the Calibration Matrix Method

Abstract

Over the past century and a half, the aerospace industry has strived to create ever lighter and safer structures, driven by both economic demand and the insights derived from tragic accidents. In order to create increasingly optimized designs, scientists and engineers have developed not just more advanced technology, but also multiple design philosophies for structural integrity. From the building block approach, to safe-life, fail-safe, and damage tolerance, the aerospace industry is now gearing up to progress to condition-based maintenance (CBM) with Structural Health Monitoring (SHM) and the Holistic Structural Integrity Process (HolSIP). These approaches aim to detect or even predict the growth of damage throughout structural aircraft components to predict their remaining life.

SHM and HolSIP both depend on input from other systems, including Flight Load Monitoring (FLM). FLM takes the measurement data obtained from aircraft to detect damage, measure loads, or measure properties like strains and displacements that can be used to find the loads. Obtaining and analyzing this data is challenging; sensors are difficult to install and easily damaged. The resulting data is often noisy and incomplete. Existing algorithms for analyzing the data often need to be manually tuned and most only determine displacements or strains, requiring further processing to reconstruct the applied loads. Such algorithms, also known as shape-sensing techniques, attempt to translate the incomplete sensor data into complete strain and displacement fields or a complete set of applied loads. This is not straightforward, as such translations pose inverse problems, which are generally ill-posed with no guarantee that solutions exist or are unique.

Still, various attempts have been made and presented in the literature to solve this inverse problem using various approximate solutions. These include various visual methods, which observe and analyze patterns on a structure’s surface using one or more cameras, fitting and smoothing methods, which fit curves with an enforced smoothness to the measured values, various other methods that include dynamic programming, modal transformations and direct integration of strains and curvatures, the inverse finite element method (iFEM), which could be considered a variant of the fitting and smoothing methods that appears to be the state-of-the-art method for shape sensing, and the calibration matrix method (CM), which appears to be a promising alternative that relies on a superposition of a number of calibration load cases to reconstruct actual load cases.

This thesis focuses on the iFEM and CM techniques: iFEM appears to be the state-of-the-art method for shape sensing, with many variants demonstrated both numerically and experimentally in the literature for various structures for linear and, recently, geometrically nonlinear (GNL) deformation. It relies on the minimization of the value of an error functional which compares measured values, typically strains, curvatures or displacements, to their numerical counterparts, with the option of applying regularization to create smoother solutions. The method can reconstruct displacements and strains. The CM method was initially costly and time-consuming, as it relied on a physical calibration process involving the deformation of the real structure. In recent years, it has become feasible to perform the calibration numerically using a finite element model of the structure, which has been demonstrated in the literature. Using the principle of superposition, the deformation and applied loads of a real load case can be reconstructed based on the sensor measurements of the real load case and the deformations, applied loads and sensor measurements of a sufficient number of calibration load cases. This is achieved by assuming that the real load case can be represented (approximately) as a linear weighted sum of these calibration load cases and inverting this relationship.

The shape-sensing methods presented in the literature were found to have certain limitations, namely that there is a lack of an accurate and broadly applicable shape-sensing method, and that it is difficult to compare the existing methods, as each method is assessed differently. Visual methods produce strains with insufficient accuracy for shape-sensing applications. Fitting and smoothing techniques require careful manual tuning to each problem. Various other methods also need tuning or only determine the shape of the sensor itself. iFEM also requires tuning, but its parameters appear to be less problem-dependent than for other methods, which makes it more broadly applicable. However, it can only reconstruct displacements and strains, not applied loads, and the iFEM techniques from the literature could only be applied to thin-walled structures and trusses. The CM method also seems broadly applicable and has the benefit of reconstructing applied loads directly. However, it has been studied less extensively than iFEM and, prior to this work, had not been compared to it directly or demonstrated for GNL deformation or anisotropic materials. Additionally, the computational performance of these methods is unclear from the literature: are the methods capable of performing accurate reconstructions in real time with a modest amount of lightweight hardware?

The research presented in this thesis attempted to develop novel variants of broadly applicable shape-sensing methods and perform reliable evaluations of their accuracy and performance, in order to achieve highly accurate real-time reconstructions of displacement, strain and applied load distributions. To be broadly and easily applicable, these methods should work for both shell and solid elements, reconstruct both linear and GNL deformation, based on either displacement or strain measurements, or a combination of both types of data. The methods should require little or no manual tuning and work for various types of structures, boundary conditions and loads. To achieve a fair and reliable evaluation of the accuracy and performance of each method, a broad and reproducible set of linear and nonlinear benchmark problems from the literature was utilized.

For this research, multiple variants of shape-sensing methods were developed, namely several variants of the iFEM algorithm, which were the first to use solid elements, and both linear and nonlinear variants of the CM algorithm for both solid and shell elements. All these variants were demonstrated on a wide range of reproducible benchmark problems from the literature, including 23 linear benchmark problems and 11 variants of seven GNL benchmark problems. The linear methods were also demonstrated for a representative aerospace structure. Each problem was augmented with sets of displacement and strain sensors. The methods were also demonstrated using only displacement data, only strain data, and a combination of displacement and strain data, and the iFEM methods were demonstrated both with and without Tikhonov regularization.

The use of standard benchmark problems allowed for reliable and reproducible evaluations and comparisons of both the accuracy and computational performance of each method. The results for iFEM showed more accurate estimates for displacements than for strains and that Tikhonov regularization did not significantly improve the reconstruction accuracy and instead often reduced accuracy, particularly for reconstructed displacements based on strain sensor data. Displacement reconstructions based on displacement data reached errors of 9% or less, but strain reconstruction was unreliable.

The linear CM algorithm was evaluated using the same benchmark problems and compared to its iFEM counterpart, achieving greater accuracy with errors of 0.01% or less for both displacement and strain reconstructions. Additionally, CM also directly reconstructs the applied loads with an error of less than 1%. Both results were obtained using a reasonable number of calibration load cases and a realistic number of sensors, with a computational efficiency equivalent or superior to iFEM. To the author’s knowledge, this was the first direct comparison of these two shape-sensing methods to be published.

The GNL variant of CM that was developed also appears to be entirely novel. Under noiseless conditions, it was demonstrated to achieve force reconstruction errors of 1% or less for all GNL benchmark problems when using 96 or more strain sensors and of 2% or less for all but one problem when employing 96 displacement sensors or a combination of strain and displacement sensors. Displacement reconstructions achieved errors of 1% or less for all problems using just 32 sensors of either type. When subjected to noise, GNL CM reached displacement reconstruction errors around 1% and force reconstruction errors around 2.5%. These levels of accuracy were achieved for both isotropic and composite laminate materials, and for structures undergoing buckling deformation, and were generally greater than those achieved by the state-of-the-art GNL iFEM algorithm, which achieved displacement reconstruction errors of 6.32% or lower under noise-free conditions and 7.70% or lower with noise. This indicates that GNL CM is a reliable and more accurate method for finding the deformed shape of structures undergoing large deformations and rotations.

The aim of this research was achieved by developing variants of the iFEM and CM shape-sensing algorithms and demonstrating that these can perform real-time reconstructions of displacements, strains and applied load distributions. CM achieved acceptable or good accuracy for a broad range of well-defined problems.