Orthogonal Expansion Method Research Articles

Design optimization of continuous fiber composites with thermo-mechanical coupling and load uncertainties

This paper introduces a theoretical framework for the design optimization of continuous fiber composites reinforced with continuous fiber trajectories subject to thermo-mechanical coupling and load uncertainties. Different uniform temperature variations are applied in the structure to investigate the influence of ambient temperature change on the structural performance. To consider the external load uncertainties, a robust design optimization model is proposed where the loads are modeled as hybrid variables, namely magnitudes as random variables and directions as interval variables, with the robust objective determined through a hybrid orthogonal polynomial expansion method. Furthermore, we use a level-set function to represent the structural boundary, with its evolution driven by shape derivatives calculated based on uncertainty analysis. The continuous fiber paths are subsequently determined by the level-set isoline extracted from the structural boundary, which in turn influences the structural mechanical performance due to the material anisotropy of composites. The continuity of continuous fiber and the equal space between adjacent trajectories largely ensure the additive manufacturability of the composites. Three numerical examples are presented to demonstrate the effectiveness of the developed framework. The results show that the ambient temperature variations and load uncertainties largely impact the optimized topology and fiber infill patterns of composites, thus are important to be considered in the design stage. Moreover, the optimized structure can have a 5-fold stiffness per unit mass compared with the initial design thus largely increasing the material efficiency in carrying external uncertain loads.

Dynamic strain field estimation of fixed offshore support structures using limited acceleration measurements

The strain history affecting the fatigue life of fixed offshore support structures is one of the most important topics in structural monitoring. However, some critical fatigue locations are located below the seabed, and the installation of strain gauges at these locations entails significant maintenance costs. In contrast, acceleration which reflects the vibration characteristics of the structure is an easily accessible and more stable physical parameter. Therefore, this study presents a strain estimation scheme that uses limited acceleration measurements to estimate the dynamic strain field of the fixed offshore support structures. An adaptive displacement reconstruction method is used to obtain the displacements at the master degrees of freedom, then the global displacements are obtained by the evolutionary orthogonal response expansion method, and finally, these displacements are implemented into the numerical software by redefining the boundary conditions to obtain the dynamic strain field. The advantage of the proposed method over the traditional strain-solving forward problem is that it relies solely on measured acceleration and structural geometry information, without requiring displacement, strain, or modal information. The correctness of the proposed method is verified by a numerical jacket platform model with multi-harmonic and non-stationary random excitations. Furthermore, physical experiments under eccentric rotation and hammering excitations are performed to assess the applicability of the proposed method on a wind turbine model equipped with laser displacement and fibre bragg grating sensors as reference measurements. The results show that the proposed method can accurately predict the dynamic strains of jacket platform and wind turbine support structures, while the appropriate agreement between the estimated strains and the numerical results, and the optical measured response is observed, with maximum estimation errors of 1.80% and 4.63%, respectively. Summarily, the application of the approach enables a new type of strain monitoring applicable to fixed offshore support structures.

Wind turbine foundation ring fatigue reliability analysis under wind-load using SCADA system

Under the action of wind load, the foundation ring of the fan will generate stress concentration and alternating stress, leading to fatigue failure. Based on the average wind speed obtained from the supervisory control and data acquisition (SCADA) system, the orthogonal expansion method of random pulsating wind field and the number theory point selection method are used to calculate and simulate the corresponding pulsating wind speed time series, and then calculate the wind speed time series and wind load time series. Based on the ABAQUS finite element software, a model of a 5 MW wind turbine is established. The modal analysis is used to obtain the modal vibration pattern and the intrinsic frequency to verify the reasonableness of the structural modeling. Subsequently, the wind load time series is applied to the wind turbine model to obtain the stress time series using instantaneous modal dynamic analysis (Modal dynamics). The stress time series at the location of maximum stress concentration in the foundation ring is extracted, and the stress time series with the same stress amplitude is obtained through conversion, which has Wiener characteristics. After obtaining the stress amplitude using the rainflow counting method, the fatigue reliability is calculated based on the structural fatigue reliability analysis method considering the threshold crossing duration of the random process. This research holds reference significance for the calculation of fatigue reliability under approximate working conditions.

Analytical formulas for option prices under time-changed CARMA process

We consider the option pricing problem when the underlying asset price is driven by a continuous time autoregressive moving average (CARMA) process, time changed a Lévy subordinator or/and an absolutely continuous time change process. We derive the analytical formulas for the option prices by employing the orthogonal polynomial expansion method. Our method is based on the observation that the CARMA process belongs to the class of polynomial diffusion and the time variable and underlying state variables enter the polynomial expansion separately. We demonstrate the accuracy of the method through a number of numerical experiments. We also investigate the price sensitivities with respect to the key parameters that govern the dynamics of the underlying state and time change variables.

Neural-network-based wavefront solution algorithm for a wide field survey telescope.

The wide field survey telescope (WFST) is a 2.5m optical survey telescope currently under construction in China. The telescope employs a primary-focus optical design to achieve a wide field of view of 3deg, and its focal plane is equipped with four pairs of curvature sensors to perform wavefront sensing and active optics. Currently, there are several wavefront solution algorithms available for curvature sensors, including the iterative fast Fourier transform method, orthogonal series expansion method, Green's function method, and sensitivity matrix method. However, each of these methods has limitations in practical use. This study proposes a solution method based on a convolutional neural network model with a U-Net structure for the curvature wavefront sensing of the WFST. Numerical simulations show that the model, when properly trained, has a high accuracy and performs a curvature wavefront solution effectively. Upon a comparison with the sensitivity matrix method, this new method demonstrates its superiority. Finally, the study is summarized, and the drawbacks of the proposed method are discussed, which leads to direction for future optimizations.

Strain fields of Ms >6.0 earthquakes in Menyuan, Qinghai, China

In predicting earthquakes, it is a major challenge to capture the time factor and spatial isoline anomalies, and understand their physical processes, of the seismic strain field before a strong earthquake. In this study, the seismic strain field was used as representative of seismic activity. The natural orthogonal function expansion method was used to calculate the seismic strain field before the Menyuan Ms 6.4 earthquakes in 1986 and 2016, and the Ms 6.9 earthquake in 2022. Time factor and spatial isoline anomaly of the strain field before each earthquake was extracted. We also compared the evolution of the strain field with numerical simulation results under the tectonic stress system at the source. The results showed that the time factor before the earthquakes had high or low value anomalies, exceeding the mean square error of the stable background. The anomalies were concentrated in the first four typical fields of the strain field, which has multiple components. The abnormal contribution rate of the first typical field is the largest (accounting for 42%–49% of the total field). The long- and medium-term anomalies appear 3-4, and 1-2 years before the earthquake, respectively. There were no short or immediate-term anomalies within 3 months of the earthquake. In addition, during the evolution of the strain field, the abnormal area of the spatial isoline changed with the change in time. Usually, the intersection area of the two isoseismic lines of strain accumulation and strain release becomes a potential location for strong earthquakes. Finally, we found that the high strain field values of the 1986 and 2016 Ms 6.4 earthquakes were equivalent to the numerical simulation results, while the high strain field values of the 2022 Menyuan Ms 6.9 earthquakes were slightly different, but within the accepted error range. These results indicate that the two methods are consistent. We have shown that the natural orgthagonal method can be used to obtain the spatiotemporal anomaly information of strain field preceding strong earthquakes.

Inverse of initial stress in carbon fiber reinforced polymer laminates using lamb waves and deep neural network

The prediction of the initial stress in composites is essential for the non-destructive testing (NDT) and structural health monitoring (SHM) of carbon fibre reinforced polymer (CFRP). This paper examines the potential of Lamb waves in the inverse of initial stress by calculating the influence of initial stress on the dispersion characteristics of Lamb waves propagating in multilayered CFRP laminates. By introducing the mechanics of incremental deformation into the linear three-dimensional elasticity theory, the Legendre orthogonal polynomial expansion (LOPE) method is used to mathematically model the Lamb wave propagating in multilayered CFRP laminates subjected to horizontal and vertical homogeneous initial stresses. Then, a three-hidden-layers Feed Forward Deep Neural Network (DNN) with Back Propagation (BP) algorithm is constructed to invert the magnitude and direction of the initial stresses. The input features are the phase velocities of fundamental Lamb wave A0 mode at five different frequencies. Both training and testing samples are obtained by LOPE forward calculation. An ablation experiment is presented to compare the two different activation functions. Finally, the accuracy of the inverse is verified by comparing with the available outcomes of LOPE forward calculation.

Contactless Measurement of Current and Mutual Inductance in Wireless Power Transfer System Based on Sandwich Structure

The mutual inductance between the transmitter and receiver sides is critical for the efficiency of a wireless power transfer (WPT) system. This article proposes a novel online mutual inductance measurement method based on a local detection coil with a sandwich structure. The mutual inductance and the receiver current are obtained by measuring the induced voltage of the sandwich coil and the output voltage of the inverter, with contactless measurement from the receiver side. Using the ferrite substrate as the skeleton of the sandwich detection coil not only increases the differential mutual inductance between the receiver coil and the sandwich coil but also maintains the differential mutual inductance almost constant when horizontal misalignment occurs. Moreover, the differential mutual inductance is used to calculate the mutual inductance between the transmitter and receiver coils and the orthogonal eigenvalue expansion method is exploited to analyze the sandwich detection coil in a finite region. Then, the factors affecting the differential mutual inductance are analyzed and optimized, and a stable differential mutual inductance is obtained by formulating a standard coil design procedure. Finally, a prototype is built to verify the proposed method. The experimental results show that the differential mutual inductance remains almost constant when the horizontal misalignment reaches 50% of the outer radius of the transmitter/receiver coil, and the maximum calculated error of the detected mutual inductance is 2.18%.

Concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties

AbstractA concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties is discussed in this work. A robust topology optimization method is proposed for structures composed of periodic microstructures under thermal and mechanical coupled loads. The robust objective function is defined as a linear combination of the mean and standard variance under the worst case for the robust optimization model. An efficient hybrid orthogonal polynomial expansion (HOPE) method is developed to evaluate the robust objective function. The sensitivities for the robust topology optimization are then calculated based on the uncertainty analysis. Three numerical examples are provided to verify the effectiveness of the proposed method, and the Monte Carlo scanning test is used to validate the numerical accuracy of our proposed method. For comparison purpose, the topology optimizations under deterministic assumptions are also provided for these examples to show the importance of considering hybrid uncertainties.

An Efficient Polynomial Chaos Method for Stiffness Analysis of Air Spring Considering Uncertainties

Traditional methods for stiffness analysis of the air spring are based on deterministic assumption that the parameters are fixed. However, uncertainties have widely existed, and the mechanic property of the air spring is very sensitive to these uncertainties. To model the uncertainties in the air spring, the interval/random variables models are introduced. For response analysis of the interval/random variables models of the air spring system, a new unified orthogonal polynomial expansion method, named as sparse quadrature-based interval and random moment arbitrary polynomial chaos method (SQ-IRMAPC), is proposed. In SQ-IRMAPC, the response of the acoustic system related to both interval and random variables is approximated by the moment-based arbitrary orthogonal polynomial expansion. To efficiently calculate the coefficient of the interval and random orthogonal polynomial expansion, the sparse quadrature is introduced. The proposed SQ-IRMAPC was employed to analyze the mechanic performance of an air spring with interval and/or random variables, and its effectiveness has been demonstrated by fully comparing it with the most recently proposed orthogonal polynomial-based interval and random analysis method.

Reliability analysis of horizontally loaded pile considering spatial variability of soil parameters

In the design and analysis of pile foundations, the horizontal load is an important factor that needs to be considered. The reliability analysis can take the uncertainty of various parameters into consideration and thus reduce the design deviations, especially the uncertainty of soil parameters. This paper proposes a reliability analysis method considering the spatial variability of soil for the horizontally loaded pile. Firstly, a parameter sensitivity analysis was performed by the Sobol sensitivity method, and the critical variables, including the undrained shear strength Cu, the soil strain ε50, which represents the soil strain when principal stress difference reaches half (50%) of the maximum value in triaxial test, and the horizontal load and applied moment, are identified and taken as the random variables. Moreover, the spatial variability of soil parameters (Cu and ε50) are simulated as one-dimensional random field through the orthogonal series expansion method. The failure probabilities of pile foundation under the failure modes of deflection and strength are evaluated by Monte Carlo simulation. Finally, the proposed method is verified through case study in which the influences of the spatial variability of soil parameters, the uncertainty of the horizontal load on the failure probability were investigated. The results show that the uncertainty of horizontal loads and Cu make significant effects on the reliability of pile foundations. The increase of correlation length of Cu and ε50 will lead to the increase of the mean value of pile response, which has different effects on the failure probability under different failure modes.

A time-variant uncertainty propagation analysis method based on a new technique for simulating non-Gaussian stochastic processes

In this paper, the time-variant uncertainty propagation analysis is defined to solve the output stochastic process of a time-variant function with uncertainty. And a time-variant uncertainty propagation analysis method is constructed with the combination of an extended orthogonal series expansion method (extended OSE) and sparse grid numerical integration (SGNI). The SGNI serving as a classical uncertainty propagation method is utilized here to solve the moments and autocorrelation function at discrete time points of the time-variant performance function. And the extended OSE is proposed to simulate the output stochastic process based on the results from SGNI. By extended OSE, a non-Gaussian stochastic process is represented as the sumof orthogonal time functions with random coefficients, and these coefficients can be directly obtained by discretization of the target process. For these coefficients are correlated and non-Gaussian, the correlated polynomial chaos expansion method (c-PCE) is presented to represent them in terms of correlated standard Gaussian variables, and then the principal component analysis (PCA) is adopted to transform them into independent ones with dimension reduction. Finally we can obtain an explicit expression to represent the non-Gaussian process whatever it is stationary or non-stationary. Three illustrative examples are used to verify the performance of the extended OSE. In addition, two engineering problems are investigated to demonstrate the effectiveness of the time-variant uncertainty propagation method.

Fatigue reliability analysis of wind turbine tower under random wind load

It is important to understand the vibration response and fatigue characteristics of the wind turbine structure. The fatigue reliability of the tower flange and bolt is studied in this paper. Firstly, by using the orthogonal expansion method and the Numerical-Theoretical method of random dynamic action, the random fluctuating wind field is expanded into a set of scattered points to calculate the wind load. The numerical calculation of the dynamic response of the wind turbine is verified with the measured data. Then, a fatigue probability calculation method based on the probability density evolution method is proposed. It is an effective method to substitute the fatigue damage D obtained by a rain flow counting method into the probability density evolution equation to solve the probability density of fatigue damage. In the end, a 1.5 MW wind turbine tower was taken as an example for analyzing the fatigue damage of its flange and bolt which considers the wind speed and wind direction distribution. By using the probability density evolution method, the evolution probability density curve clusters of the structural responses can be given accurately and quantitatively; thus, the reliability of the structure under random load can be conveniently obtained.

Fatigue life estimation of continuous girder bridges based on the sequence of loading

A need exists to determine the fatigue life of multi-span continuous bridges in the design stage. It is difficult to conduct a full-scale fatigue test of a bridge, hence the simulation of bridge-vehicle interaction dynamics is important to estimate the fatigue life of bridges. The vehicles entering the bridge are random in nature. In the present study, the arrival time of the vehicles is considered a random variable following Poisson process to simulate the actual traffic. The bridge-vehicle coupled system equations have been derived wherein the vehicle arrival rate appears as random variable casting difficulty in obtaining a direct solution. The solution approach has been devised using Orthogonal Polynomial Expansion method. Response statistics so obtained for a multi-span continuous bridge has been used in fatigue life estimation involving crack initiation phase and crack propagation phase modifying conventional linear damage rule. The sequence of loading is also considered in the modified fatigue formulation. The effect of the vehicle speed, road surface irregularity and the arrival rate of the vehicle on fatigue life estimation of the bridge have been studied.

Coupled Lamb waves propagation along the direction of non-principal symmetry axes in pre-stressed anisotropic composite lamina

Based on the linear three-dimensional elastic theory and the mechanics theory of incremental deformation, using the Legendre orthogonal polynomial expansion (LOPE) method, the wave equations of Lamb wave propagation in the non-principal symmetry axes direction of anisotropic composite lamina was firstly derived when the initial stresses were applied in the horizontal and vertical directions. Subsequently, the coupled equations of Lamb wave propagation were numerically solved. In order to verify the accuracy of the process, the dispersion curves of phase velocity in an isotropic aluminum plate and anisotropic composite lamina without loaded initial stresses were beforehand calculated by LOPE method to compare with those obtained from the commercial software of DISPERSE® and available data. Then, a series of cases referring to unidirectional fiber reinforced composite lamina were further implemented to study the effects of horizontal initial stress and vertical initial stress on the fundamental Lamb wave modes. The results show that the phase velocity of the A0 mode is more sensitive to the implementation of initial stresses compared with the S0 mode and SH0 mode. Finally, the effect of fiber orientation was illustrated in details.

Extending the Theory of Liptser-Shiryaev Filters

Two types of approximate suboptimal Liptser-Shiryaev filters (LSFs) are developed. The first type is based on the generalized Kalman-Bucy filter; the second type, on the parameterization of the a posteriori distribution using the orthogonal expansion method and the quasi-moment method. In principle, the design methods of the LSFs (see below) can be used for obtaining an almost optimal filter with any degree of accuracy. The higher the maximum order of the orthogonal expansion parameters under consideration is, the higher the accuracy of approximation to the optimal estimate will be.

Dynamic failure processes and failure mechanism of soil slope under random earthquake ground motions

Understanding the failure processes and failure mechanism of slopes is essential in the seismic design of slopes. However, the effect of ground motion uncertainty on slope failure is unclear (for example, whether the slope failure processes differ under different ground motions). This study investigated the failure processes and failure mechanism of a soil slope subjected to random earthquake ground motions. Non-stationary random earthquake ground motions were generated based on the improved orthogonal expansion method and dynamic problems were solved using the FLAC3D software. We attempted to determine the slope sliding surface by analyzing the dynamic strain time history of monitoring points. Statistical results obtained using the probability density evolution method revealed that the slip surface extended from the toe to the crest of the slope during the earthquakes and that only one slip surface could be determined under an earthquake scenario. Additionally, the slip surface, time of failure, and failure duration differed under different earthquakes. The strain development patterns also differed and four typical failure modes were identified based on these patterns. The slope risk was assessed based on the sliding mass uncertainty. The proposed framework can be useful in evaluating slope failure processes and for seismic design in the field of slope engineering.

Using Refined Theory to Studied Elastic Wave Scattering and Dynamic Stress Concentrations in Plates with Two Cutouts

In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.

Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

A unique technique for analytical solution of 2-D dual phase lag bio-heat transfer problem with generalized time-dependent boundary conditions

The eigenfunction-based solutions for various heat conduction models (Fourier and non-Fourier) exhibit mismatch at the boundaries due to the discrepancy between the boundary conditions (BCs) and the corresponding eigenfunction profiles. This mismatch leads to spurious oscillations and erroneous solutions near the boundaries. Recently developed finite integral transform (FIT) based analytical solution of the dual phase lag (DPL) bio-heat transfer problem with generalized time-dependent BCs in a 2-D planar skin tissue phantom also encounters the similar mismatch, especially for Dirichlet BCs. The present study aims at removing such mismatch from eigenfunction-based solutions, by the homogenization of generalized time-dependent BCs which is essentially carried out by subtracting an auxiliary function from the temperature of the domain of interest. This novel treatment yields a modified problem in terms of the modified temperature with homogeneous BCs and contains a modified source. For this, the above-mentioned auxiliary functions need to satisfy a set of conditions at the boundaries. However, the functions satisfying such conditions may not be unique. Therefore, an additional condition (pseudo-steady state condition) is imposed with a view to obtaining relatively simple, unique auxiliary functions. The current study also explores the homogenization process for 2-D bio-heat transfer problem for the tissue phantom in conjunction with the use of orthogonal eigenfunction expansion method (OEEM). The present developed solution approach is demonstrated using two test problems comprising of (a) constant surface temperature and (b) sinusoidal heat flux on the surface, respectively. In case of constant surface temperature, the present approach gets rid of large spurious oscillations near the boundary. In case of surface heat flux, temperature prediction by the present approach shows good agreement in the domain compare to that predicted by the FIT approach. Besides that, the flux distribution predicted on the boundary by present approach is a realistic non-zero value, whereas, the FIT approach produces zero heat flux for the same applied non-zero heat flux at the boundary. Moreover, for both the test problems, the developed novel approach facilitates significant reduction in number of terms in the series summation as compared to the corresponding FIT solutions, for the same level of accuracy.