C. de Mooij
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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. ...
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. ...
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.
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.
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.
...
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.
Structural health monitoring (SHM) is a growing field of research, as it has the potential to simultaneously improve the reliability of structures and reduce their maintenance cost. SHM requires accurate stress and strain information, preferably for the entire structure. Unfortunately, it is often infeasible to instrument every part of the structure, making it necessary to estimate the stress and strain fields based on data from a limited number of sensors. One promising technique for making this estimate is the inverse finite element method (iFEM), which can be applied to any combination of geometry and loading conditions. In addition, it can also process several different types of sensor data. In this study, benchmark problems based on the MacNeal and Harder linear elastic problem set for FEM algorithms were extended to test the accuracy of iFEM algorithms. As the benchmarks use linear elastic materials, small displacements and strains, the iFEM implementation was also limited to these conditions. Accurate iFEM estimates can be obtained for the benchmark problems for which accurate FEM solutions can be obtained with solid elements, specifically 3-dimensional 20 node hexahedral elements with reduced integration (C3D20R), based on either displacement sensors, strain sensors, or both combined, and provided that a sufficient number of sensors is used. The iFEM algorithms generally produce more accurate estimates of displacements than of strains. The addition of Tikhonov regularization does not result in a significant increase in accuracy for either the displacement or strain distribution estimates and can even deteriorate the results in certain cases.
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Structural health monitoring (SHM) is a growing field of research, as it has the potential to simultaneously improve the reliability of structures and reduce their maintenance cost. SHM requires accurate stress and strain information, preferably for the entire structure. Unfortunately, it is often infeasible to instrument every part of the structure, making it necessary to estimate the stress and strain fields based on data from a limited number of sensors. One promising technique for making this estimate is the inverse finite element method (iFEM), which can be applied to any combination of geometry and loading conditions. In addition, it can also process several different types of sensor data. In this study, benchmark problems based on the MacNeal and Harder linear elastic problem set for FEM algorithms were extended to test the accuracy of iFEM algorithms. As the benchmarks use linear elastic materials, small displacements and strains, the iFEM implementation was also limited to these conditions. Accurate iFEM estimates can be obtained for the benchmark problems for which accurate FEM solutions can be obtained with solid elements, specifically 3-dimensional 20 node hexahedral elements with reduced integration (C3D20R), based on either displacement sensors, strain sensors, or both combined, and provided that a sufficient number of sensors is used. The iFEM algorithms generally produce more accurate estimates of displacements than of strains. The addition of Tikhonov regularization does not result in a significant increase in accuracy for either the displacement or strain distribution estimates and can even deteriorate the results in certain cases.
Conference paper
(2017)
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Maria Skandali, Sotiris Koussios, Jos Sinke, Rinze Benedictus, Cornelis de Mooij, Kaspar Jansen
Heating the thermoset tape to a higher temperature than is used in the conventional automated tape laying process can affect the cure cycle, void content, interlaminar shear strength and residual stresses of the final product. In this study, the effects of different heating strategies on the degree of cure in thermoset tapes were modelled. This was done for two tape laying speeds and by heating different combinations of the incoming tape, the previously laid tape and the mould. The model predicts the thermochemical profile of two different materials and is validated by experimental data from thermocouples and a pyrometer.
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Heating the thermoset tape to a higher temperature than is used in the conventional automated tape laying process can affect the cure cycle, void content, interlaminar shear strength and residual stresses of the final product. In this study, the effects of different heating strategies on the degree of cure in thermoset tapes were modelled. This was done for two tape laying speeds and by heating different combinations of the incoming tape, the previously laid tape and the mould. The model predicts the thermochemical profile of two different materials and is validated by experimental data from thermocouples and a pyrometer.
Sensor Fusion for Shape Sensing
Theory and Numerical Simulation
Computing global strain distributions in complex aerospace and wind energy structures in quasi-real time is an important challenge for the aerospace and wind energy industry. Shape sensing is being considered as a potential means for obtaining global strain fields of complex structures. This is achieved through the use of inverse finite element methods (iFEM). Traditional shape sensing techniques use a variety of sensors in isolation. These isolated approaches lead to a number of drawbacks, including significant errors in the strain and/or displacement distributions that they determine. This study focuses on reducing these errors by combining data from various sensor types.
In order to achieve this objective, a new iFEM approach was developed to improve how shape sensing determines a structure’s deformation from distributed sensors. The method utilizes a procedure that minimizes a quadratic error functional, based on the difference between the theoretical and the measured strains and displacements. The analytical iFEM equations are discretized for use in a numerical model, which is used to analyze simulated sensor data. The simulated data is obtained from a FEM analysis of a structure loaded in bending, torsion and shear, in addition to experiencing combined loads. As with previous inverse methods, the error functional weights were employed to manage missing measurements and Tikhonov regularization was applied to guarantee smoothness of the numerical solution.
In this study, it has been shown that the new iFEM methodology can determine the structural deformations to within 1% of the FEM results for each of the load cases. In addition, the methodology is capable of achieving the same accuracy as single sensor type iFEM methods with a smaller total number of sensors. The single sensor type iFEM methods that were considered made use of only strain data or only displacement data. Finally, an experimental setup consisting of a simple cantilever plate structure was constructed in order to experimentally verify the newly developed iFEM algorithm. The experimental strain and displacement data were obtained utilizing a Rayleigh backscattering fiber optic distributed sensing system and MEMS sensors, respectively. Both of these sensor types were mounted on the structure and subjected to the load cases that were simulated in the FEM analysis.
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Computing global strain distributions in complex aerospace and wind energy structures in quasi-real time is an important challenge for the aerospace and wind energy industry. Shape sensing is being considered as a potential means for obtaining global strain fields of complex structures. This is achieved through the use of inverse finite element methods (iFEM). Traditional shape sensing techniques use a variety of sensors in isolation. These isolated approaches lead to a number of drawbacks, including significant errors in the strain and/or displacement distributions that they determine. This study focuses on reducing these errors by combining data from various sensor types.
In order to achieve this objective, a new iFEM approach was developed to improve how shape sensing determines a structure’s deformation from distributed sensors. The method utilizes a procedure that minimizes a quadratic error functional, based on the difference between the theoretical and the measured strains and displacements. The analytical iFEM equations are discretized for use in a numerical model, which is used to analyze simulated sensor data. The simulated data is obtained from a FEM analysis of a structure loaded in bending, torsion and shear, in addition to experiencing combined loads. As with previous inverse methods, the error functional weights were employed to manage missing measurements and Tikhonov regularization was applied to guarantee smoothness of the numerical solution.
In this study, it has been shown that the new iFEM methodology can determine the structural deformations to within 1% of the FEM results for each of the load cases. In addition, the methodology is capable of achieving the same accuracy as single sensor type iFEM methods with a smaller total number of sensors. The single sensor type iFEM methods that were considered made use of only strain data or only displacement data. Finally, an experimental setup consisting of a simple cantilever plate structure was constructed in order to experimentally verify the newly developed iFEM algorithm. The experimental strain and displacement data were obtained utilizing a Rayleigh backscattering fiber optic distributed sensing system and MEMS sensors, respectively. Both of these sensor types were mounted on the structure and subjected to the load cases that were simulated in the FEM analysis.
Sensor Fusion for Shape Sensing
Theory and Numerical Simulation
Computing global strain distributions in complex aerospace and wind energy structures in quasi-real time is an important challenge for the aerospace and wind energy industry. Shape sensing is being considered as a potential means for obtaining global strain fields of complex structures. This is achieved through the use of inverse finite element methods (iFEM). Traditional shape sensing techniques use a variety of sensors in isolation. These isolated approaches lead to a number of drawbacks, including significant errors in the strain and/or displacement distributions that they determine. This study focuses on reducing these errors by combining data from various sensor types.
In order to achieve this objective, a new iFEM approach was developed to improve how shape sensing determines a structure’s deformation from distributed sensors. The method utilizes a procedure that minimizes a quadratic error functional, based on the difference between the theoretical and the measured strains and displacements. The analytical iFEM equations are discretized for use in a numerical model, which is used to analyze simulated sensor data. The simulated data is obtained from a FEM analysis of a structure loaded in bending, torsion and shear, in addition to experiencing combined loads. As with previous inverse methods, the error functional weights were employed to manage missing measurements and Tikhonov regularization was applied to guarantee smoothness of the numerical solution.
In this study, it has been shown that the new iFEM methodology can determine the structural deformations to within 1% of the FEM results for each of the load cases. In addition, the methodology is capable of achieving the same accuracy as single sensor type iFEM methods with a smaller total number of sensors. The single sensor type iFEM methods that were considered made use of only strain data or only displacement data. Finally, an experimental setup consisting of a simple cantilever plate structure was constructed in order to experimentally verify the newly developed iFEM algorithm. The experimental strain and displacement data were obtained utilizing a Rayleigh backscattering fiber optic distributed sensing system and MEMS sensors, respectively. Both of these sensor types were mounted on the structure and subjected to the load cases that were simulated in the FEM analysis.
...
Computing global strain distributions in complex aerospace and wind energy structures in quasi-real time is an important challenge for the aerospace and wind energy industry. Shape sensing is being considered as a potential means for obtaining global strain fields of complex structures. This is achieved through the use of inverse finite element methods (iFEM). Traditional shape sensing techniques use a variety of sensors in isolation. These isolated approaches lead to a number of drawbacks, including significant errors in the strain and/or displacement distributions that they determine. This study focuses on reducing these errors by combining data from various sensor types.
In order to achieve this objective, a new iFEM approach was developed to improve how shape sensing determines a structure’s deformation from distributed sensors. The method utilizes a procedure that minimizes a quadratic error functional, based on the difference between the theoretical and the measured strains and displacements. The analytical iFEM equations are discretized for use in a numerical model, which is used to analyze simulated sensor data. The simulated data is obtained from a FEM analysis of a structure loaded in bending, torsion and shear, in addition to experiencing combined loads. As with previous inverse methods, the error functional weights were employed to manage missing measurements and Tikhonov regularization was applied to guarantee smoothness of the numerical solution.
In this study, it has been shown that the new iFEM methodology can determine the structural deformations to within 1% of the FEM results for each of the load cases. In addition, the methodology is capable of achieving the same accuracy as single sensor type iFEM methods with a smaller total number of sensors. The single sensor type iFEM methods that were considered made use of only strain data or only displacement data. Finally, an experimental setup consisting of a simple cantilever plate structure was constructed in order to experimentally verify the newly developed iFEM algorithm. The experimental strain and displacement data were obtained utilizing a Rayleigh backscattering fiber optic distributed sensing system and MEMS sensors, respectively. Both of these sensor types were mounted on the structure and subjected to the load cases that were simulated in the FEM analysis.
Sensor Fusion for Shape Sensing
Theory and Numerical Results
Shape sensing utilizing an inverse finite element method (iFEM) was considered for obtaining the displacement fields of two cantilever beams. Traditional iFEM techniques make use of only a single type of sensor, which can lead to errors in the strain and/or displacement distributions. This study has reduced these errors by combining data from multiple sensor types. A new iFEM algorithm was developed in order to minimize the error on computed strain profiles. The new iFEM algorithm incorporates Tikhonov smoothing in addition to compensating for the lack of sensors in areas of the structure where no sensors are present. The new iFEM algorithm was demonstrated for cantilever plates under various load cases. The preliminary results showed that the new iFEM algorithm is able to determine the structural deformations of two cantilever plates with greater accuracy than a traditional iFEM implementation that was found in the literature, using fewer sensors.
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Shape sensing utilizing an inverse finite element method (iFEM) was considered for obtaining the displacement fields of two cantilever beams. Traditional iFEM techniques make use of only a single type of sensor, which can lead to errors in the strain and/or displacement distributions. This study has reduced these errors by combining data from multiple sensor types. A new iFEM algorithm was developed in order to minimize the error on computed strain profiles. The new iFEM algorithm incorporates Tikhonov smoothing in addition to compensating for the lack of sensors in areas of the structure where no sensors are present. The new iFEM algorithm was demonstrated for cantilever plates under various load cases. The preliminary results showed that the new iFEM algorithm is able to determine the structural deformations of two cantilever plates with greater accuracy than a traditional iFEM implementation that was found in the literature, using fewer sensors.