Closed-form Method Research Articles

Solution for time-dependent resilience in the presence of gradual deterioration of performance

In this paper, a closed-form method is developed for the evaluation of time-dependent resilience (so named as it is a function of the service time of interest) of an aging object (e.g., a structure or system). These structures and systems often suffer from the deterioration of performances in a harsh service environment, causing the decline of serviceability. They are thus expected to be sufficiently resilient during their service lives, i.e., to have the ability to withstand disruptions to their performances. The proposed method takes into account the uncertainty associated with the performance deterioration process, the availability of resources that support the performance recovery, and the impact of a changing environment. The accuracy and improved efficiency of the proposed method are demonstrated through three examples. It is also shown through sensitivity analysis that the impact of a changing environment, and the availability of recovery-supporting resources play an essential role in the time-dependent resilience. The proposed resilience method can also be used to efficiently guide the design of new structures that meet predefined resilience goals.

A Closed-Form Inverse Kinematic Analytical Method for Seven-DOF Space Manipulator with Aspheric Wrist Structure

The seven-degree-of-freedom space manipulator, characterized by its redundant and aspheric wrist structure, is extensively used in space missions due to its exceptional dexterity and multi-joint capabilities. However, the non-spherical wrist structure presents challenges in solving inverse kinematics, as it cannot decouple joints using the Pieper criterion, unlike spherical wrist structures. To address this issue, this paper presents a closed-form analytical method for solving the inverse kinematics of seven-degree-of-freedom aspheric wrist space manipulators. The method begins by identifying the redundant joint through comparing the volumes of the workspace with different joints fixed. The redundant joint angle is then treated as a parametric joint angle, enabling the derivation of closed-form expressions for the non-parametric joint angles using screw theory. The optimal solution branch is identified through a comparative analysis of various self-motion manifold branches. Additionally, a hybrid approach, combining analytical and numerical methods, is proposed to optimize the parametric joint angle for a trajectory tracking task. Simulation results confirm the effectiveness of the proposed method.

A semi-analytical solution for critical buckling loads of orthotropic stiffened rectangular thin plates

Stiffened plates are widely used in various engineering fields as main load-carrying components. Semi-analytical methods are effective for studying the eigenbuckling problems of stiffened rectangular plates, but there are no semi-analytical solutions available for stiffened plates with arbitrary homogeneous boundary conditions. This work aims to develop a semi-analytical method for solving the eigenbuckling problems of orthotropic stiffened thin plates. In this method, base functions for constructing the mode functions of stiffened plate are the mode functions of a simple plate (a plate without stiffener), and the base functions are obtained with the extended separation-of-variable method, which is a closed-form solution method for the eigenvalue problems of plates. The critical buckling load is achieved by substituting the mode functions into the Rayleigh's principle. The present method can deal with arbitrary homogeneous boundary conditions without anticipating the forms of the base functions for different boundary conditions. The accuracy can be improved by using more superposition terms. Besides, an exact closed-form solution of simply supported stiffened plates is achieved, serving as benchmark solutions. Numerical experiments validate the accuracy of the present solutions, and the study on the minimum stiffener stiffness is conducted for different boundary conditions.

Closed-Form Method for Unified Far-Field and Near-Field Localization Based on TDOA and FDOA Measurements

When the near-field and far-field information of a target is uncertain, it is necessary to choose a suitable localization method. The modified polar representation (MPR) method integrates the two scenarios and achieves a unified localization with direction of arrival (DOA) estimation in the far field and position estimation in the near field. Previous studies have only proposed solutions for stationary environments and have not considered the motion factor. Therefore, this paper proposes a new unified positioning algorithm using multi-sensor time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements without prior target source information. The method represents the position of the target source using MPR and describes the localization problem as a weighted least squares (WLS) problem with two constraints. We first obtain the initial estimates by WLS without considering the constraints and then investigate a two-step error correction method based on the constraints. The first step corrects the initial estimate using the Taylor series expansion technique, and the second step corrects the DOA estimate in the previous step using the direct error compensation technique based on the properties of the second constraint. Simulation experiments show that the method is effective for the unified positioning of moving targets and can achieve the Cramer–Rao lower bound (CRLB).

High-resolution, large field-of-view label-free imaging via aberration-corrected, closed-form complex field reconstruction

Computational imaging methods empower modern microscopes to produce high-resolution, large field-of-view, aberration-free images. Fourier ptychographic microscopy can increase the space-bandwidth product of conventional microscopy, but its iterative reconstruction methods are prone to parameter selection and tend to fail under excessive aberrations. Spatial Kramers–Kronig methods can analytically reconstruct complex fields, but is limited by aberration or providing extended resolution enhancement. Here, we present APIC, a closed-form method that weds the strengths of both methods while using only NA-matching and darkfield measurements. We establish an analytical phase retrieval framework which demonstrates the feasibility of analytically reconstructing the complex field associated with darkfield measurements. APIC can retrieve complex aberrations of an imaging system with no additional hardware and avoids iterative algorithms, requiring no human-designed convergence metrics while always obtaining a closed-form complex field solution. We experimentally demonstrate that APIC gives correct reconstruction results where Fourier ptychographic microscopy fails when constrained to the same number of measurements. APIC achieves 2.8 times faster computation using image tile size of 256 (length-wise), is robust against aberrations compared to Fourier ptychographic microscopy, and capable of addressing aberrations whose maximal phase difference exceeds 3.8π when using a NA 0.25 objective in experiment.

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

Free vibration analysis of tall buildings using a generalized continuous model: simplified approach to soil-structure interaction

Currently, due to the simplifications imposed on continuous models, there is an absence of a generalized approximate solution to investigate all possible dynamic behaviors of a tall building. In an attempt to address this deficiency, this paper proposes a new generalized continuous model and its solution methods to generalize all types of behavior that may be encountered in structural systems and tall buildings. The continuous model results from a composite beam that serially couples the classical sandwich beam and a pure shear beam. Unlike existing continuous models, the serial coupling, inclusion of new kinematic fields, and inclusion of rotational inertia allow for the resolution of complex interaction among different types of bending and shear behaviors. Specifically, it is shown how local shear behavior simultaneously interacts with global bending behavior, global shear, and local bending. Two solutions to the dynamic problem are presented: a closed-form analytical solution method for structures with uniformity along their height, and a numerical method that combines the advantages of the continuous method and the transfer matrix method to resolve structural and geometric variability in height. The transfer matrix is analytically solved by maintaining a fixed 6 × 6 matrix range that is independent of the number of levels of the structure, drastically reducing computational cost by eliminating the classical need to calculate the inverse of the zero matrix. Numerical applications investigate the effect of rotational inertia and local shear on the accuracy of the proposed results compared to results from the finite element method. To this end, a parametric analysis of 1560 different cases covering a wide range of typical behaviors in individual structural systems and tall buildings is conducted. The results exhibit promising accuracy, showing that the qualities of the proposed continuous model significantly improve precision compared to classical continuous models and therefore justify its practical application by academics and practicing engineers. Furthermore, the proposed mathematical approach is easily applicable in various domains of composite materials engineering.

Influence of Chemical and Radiation on an Unsteady MHD Oscillatory Flow using Artificial Neural Network (ANN)

This paper delves into the intricate interplay between chemical and thermal radiation in the context of an unstable magnetohydrodynamic(MHD) oscillatory flow through a porous medium. The fluid under investigation is presumed to be incompressible, electrically conductive, and radiating with the additional influence of a homogeneous magnetic field applied perpendicular to the channel’s plane. Analytical closedform solutions are derived for the momentum, energy, and concentration equations providing a comprehensive understanding of the system’s behavior. The investigation systematically explores the impact of various flow factors, presenting their effects through graphical representations. The governing partial differential equations (PDE) of the boundary layer are transformed into a set of coupled nonlinear ordinary differential equations (ODE) using a closed-form method. Subsequently, an artificial neural network (ANN) is applied to these ODEs, and the obtained results are validated against numerical simulations. The temperature profiles exhibit oscillatory behavior with changes in the radiation parameter (N), revealing insights into the system’s dynamic response. Furthermore, the paper uncovers that higher heat sources lead to increased temperature profiles. Additionally, concentration profiles demonstrate a decrease with escalating chemical reaction parameters, with a reversal observed as the Schmidt number (Sc) increases. This study highlights the efficacy of an ANN model in providing highly efficient estimates for heat transfer rates from an engineering standpoint. This innovative approach leverages the power of artificial intelligence to enhance our understanding of complex fluid magnetohydrodynamics and porous media flows.

Evaluate Geometry of Radiance Fields with Low-Frequency Color Prior

A radiance field is an effective representation of 3D scenes, which has been widely adopted in novel-view synthesis and 3D reconstruction. It is still an open and challenging problem to evaluate the geometry, i.e., the density field, as the ground-truth is almost impossible to obtain. One alternative indirect solution is to transform the density field into a point-cloud and compute its Chamfer Distance with the scanned ground-truth. However, many widely-used datasets have no point-cloud ground-truth since the scanning process along with the equipment is expensive and complicated. To this end, we propose a novel metric, named Inverse Mean Residual Color (IMRC), which can evaluate the geometry only with the observation images. Our key insight is that the better the geometry, the lower-frequency the computed color field. From this insight, given a reconstructed density field and observation images, we design a closed-form method to approximate the color field with low-frequency spherical harmonics, and compute the inverse mean residual color. Then the higher the IMRC, the better the geometry. Qualitative and quantitative experimental results verify the effectiveness of our proposed IMRC metric. We also benchmark several state-of-the-art methods using IMRC to promote future related research. Our code is available at https://github.com/qihangGH/IMRC.

A complete closed-form method for transformation from Cartesian to geodetic coordinates

A complete closed-form method for transformation from Cartesian to geodetic coordinates

Building Vibration Measurement and Prediction during Train Operations

Urban societies face the challenge of working and living in environments filled with vibration caused by transportation systems. This paper conducted field measurements to obtain the characteristics of vibration transmission from soil to building foundations and within building floors. Subsequently, a prediction method was developed to anticipate building vibrations by considering the soil and structure interaction. The rigid foundation model was simplified into a foundation–soil system connected via spring damping, and the building model is based on axial wave transmission within the columns and attached floors. Building vibrations were in response to measured input vibration levels at the ground and were validated through field measurements. The influence of different building heights on soil and structure vibration propagation was studied. The results showed that the predicted vibrations match well with the measured vibrations. The proposed prediction model can reasonably predict the building vibration caused by train operations. The closed-form method is an efficient tool for predicting floor vibrations prior to construction.

Variational Bayesian Student's-t Mixture Model With Closed-Form Missing Value Imputation for Robust Process Monitoring of Low-Quality Data.

Due to record errors, transmission interruptions, etc., low-quality process data, including outliers and missing data, commonly exist in real industrial processes, challenging the accurate modeling and reliable monitoring of the operating statuses. In this study, a novel variational Bayesian Student's-t mixture model (VBSMM) with a closed-form missing value imputation method is proposed to develop a robust process monitoring scheme for low-quality data. First, a new paradigm for the variational inference of Student's-t mixture model is proposed to develop a robust VBSMM model, which optimizes the variational posteriors in an extended feasible region. Second, conditioned on the complete and partially missing data information, a closed-form missing value imputation method is derived to address the challenges of outliers and multimodality in accurate data recovery. Then, a robust online monitoring scheme that can maintain its fault detection performance in the presence of poor data quality is developed, where a novel monitoring statistic called the expected variational distance (EVD) is first proposed to quantify the changes in operating conditions and can be easily extended to other variational mixture models. Case studies on a numerical simulation and a real-world three-phase flow facility illustrate the superiority of the proposed method in missing value imputation and fault detection of low-quality data.

A Closed-Form Method With Low Noise Sensitivity for Locating a Moving Source At a Known Altitude

A Closed-Form Method With Low Noise Sensitivity for Locating a Moving Source At a Known Altitude

An Algorithm Developed for Smallsats Accurately Retrieves Landsat Surface Reflectance Using Scene Statistics

Closed-form Method for Atmospheric Correction (CMAC) is software that overcomes radiative transfer method problems for smallsat surface reflectance retrieval: unknown sensor radiance responses because onboard monitors are omitted to conserve size/weight, and ancillary data availability that delays processing by days. CMAC requires neither and retrieves surface reflectance in near real time, first mapping the atmospheric effect across the image as an index (Atm-I) from scene statistics, then reversing these effects with a closed-form linear model that has precedence in the literature. Five consistent-reflectance area-of-interest targets on thirty-one low-to-moderate Atm-I images were processed by CMAC and LaSRC. CMAC retrievals accurately matched LaSRC with nearly identical error profiles. CMAC and LaSRC output for paired images of low and high Atm-I were then compared for three additional consistent-reflectance area-of-interest targets. Three indices were calculated from the extracted reflectance: NDVI calculated with red (standard) and substitutions with blue and green. A null hypothesis for competent retrieval would show no difference. The pooled error for the three indices (n = 9) was 0–3% for CMAC, 6–20% for LaSRC, and 13–38% for uncorrected top-of-atmosphere results, thus demonstrating both the value of atmospheric correction and, especially, the stability of CMAC for machine analysis and AI application under increasing Atm-I from climate change-driven wildfires.

Model resolution-based deconvolution for improved quantitative susceptibility mapping.

Quantitative susceptibility mapping (QSM) utilizes the relationship between the measured local field and the unknown susceptibility map to perform dipole deconvolution. The aim of this work is to introduce and systematically evaluate the model resolution-based deconvolution for improved estimation of the susceptibility map obtained using the thresholded k-space division (TKD). A two-step approach has been proposed, wherein the first step involves the TKD susceptibility map computation and the second step involves the correction of this susceptibility map using the model-resolution matrix. The TKD-estimated susceptibility map can be expressed as the weighted average of the true susceptibility map, where the weights are determined by the rows of the model-resolution matrix, and hence a deconvolution of the TKD susceptibility map using the model-resolution matrix yields a better approximation to the true susceptibility map. The model resolution-based deconvolution is realized using closed-form, iterative, and sparsity-regularized implementations. The proposed approach was compared with L2 regularization, TKD, rescaled TKD in superfast dipole inversion, the modulated closed-form method, and iterative dipole inversion, as well as sparsity-regularized dipole inversion. It was observed that the proposed approach showed a substantial reduction in the streaking artifacts across 94 test volumes considered in this study. The proposed approach also showed better error reduction and edge preservation compared with other approaches. The proposed model resolution-based deconvolution compensates for the truncation of zero coefficients in the dipole kernel at the magic angle and hence provides a closer approximation to the true susceptibility map compared with other direct methods.

Seismic vulnerability of shield tunnels in interbedded soil deposits: Case study of submarine tunnel in Shantou Bay

This paper presents a numerical approach for deriving seismic fragility curves for a submarine shield tunnel subjected to transverse ground shaking in interbedded soil deposits. To simulate the nonlinear behavior of soils, PM4Sand and HS-small constitutive models were adopted. In contrast, the closed-form elastic design method or elastic–plastic design method was used for modeling the shield tunnel lining considering the effect of joints on the seismic performance of structures. Fully coupled numerical analyses provided a satisfactory interpretation of the seismic responses of soil–structure systems considering the effects of groundwater level and soil–tunnel interaction. Subsequently, free-field peak ground acceleration was selected as the intensity measure. The ratio of maximum bending moment to capacity bending moment or permanent joint rotation of the tunnel lining was selected as the damage measure, combined with incremental dynamic analysis, a series of fragility curves were generated. In this study, one-dimensional equivalent-linear earthquake site response analysis was implemented to validate the nonlinear dynamic analysis under weak earthquakes. However, the analysis overestimates the amplitude periodic vibrations in interbedded soil deposits under strong earthquakes. Additionally, numerical results show that the examined tunnel is prone to slight or moderate damage under ground shaking, and its joints are more vulnerable to damage than its segments. The fragility curves constructed were compared with existing fragility curves, highlighting the significant effects of local soil profiles, groundwater conditions, soil–tunnel interaction, and segmental joints on the seismic vulnerability of shield tunnels in a submarine environment. The developed fragility functions contribute to the prediction of seismic damage to submarine shield tunnels of similar characteristics, enabling the rapid and efficient response of structures to disasters in ocean engineering.

An Efficient Hybrid Multi-Station TDOA and Single-Station AOA Localization Method

Multi-station passive localization algorithms based on hybrid Time Difference of Arrival (TDOA) and Angle of Arrival (AOA) have been thoroughly studied. But it usually requires each station to obtain the AOA of the source and relies on the precise station position. These conditions are generally not satisfied in practical applications. In addition, the geometry between the source and these stations also affects the localization accuracy. This paper studies the passive source localization problem based on multi-station TDOA and single-station AOA measurements. First, we propose a closed-form solution source localization method based on multi-station TDOA and single-station AOA measurements, which requires only the reference station used to calculate TDOA to be able to observe the AOA measurements of the source. It solves the problem of hybrid localization of different numbers of TDOA and AOA measurements. Theoretical analysis proves that the performance of the proposed method can reach the Cramer-Rao Lower Bound (CRLB) under small noise conditions. Then, we model the relationship between the position of each station relative to the source and the localization accuracy. Through the D-optimality criterion, we obtain the optimal geometry between multiple stations and the source Simulation results confirm the validity of these theories.

Online learning using deep random vector functional link network

Deep neural networks have shown their promise in recent years with their state-of-the-art results. Yet, backpropagation-based methods may suffer from time-consuming training process and catastrophic forgetting when performing online learning. In this work we attempt to curtail them by employing the ensemble deep Random Vector Functional Link (edRVFL). As opposed to backpropagation-based neural networks that adjust weights iteratively, RVFL uses a closed-form solution method without iterative parameter learning. In addition, our approach allows the model to grow incrementally as new data is made available so that it can more resemble real-life learning scenarios. Our proposed online learning models were able to perform better on 72% of the datasets in the classification scenario and 80% of the datasets in the regression scenario, when compared to other available randomization-based online learning models in the literature. This is further supported by statistical comparisons which also show the stability of our network.

S-VIO: Exploiting Structural Constraints for RGB-D Visual Inertial Odometry

Vision-based localization is an essential problem for autonomous systems while the performance of vision-based odometry degrades in the challenging scenario. This letter presents S-VIO, an RGB-D visual inertial odometry (VIO) which fully uses multi-sensor measurements (i.e. depth, RGB and IMU), heterogeneous landmarks (i.e. points, lines and planes) and structural regularity of the environment to obtain a robust and accurate localization result. In order to detect the underlying structural regularity of the environment, a two-step Atlanta world inference method is proposed. Leveraging the gravity direction estimated by the VIO system, the proposed algorithm first generates horizontal Atlanta axis hypotheses from a set of recently optimized plane landmarks. Then the following plane landmarks and line clusters are used to filter out the occasionally observed axes based on the persistence of the observation. The remaining axes will survive and be saved in the Atlanta map for future re-observations. Particularly, an efficient mined-and-stabbed (MnS) method is applied to classify the structural line and extract the vanishing point from each line cluster. In addition, a closed-form initialization method for the structural line feature is proposed, which leverages the known direction to obtain a better initial estimation. Integrated with the above novelties, S-VIO is tested on two public real-world RGB-D inertial datasets. Experiments demonstrate that S-VIO has better accuracy and robustness compared to state-of-the-art VIO and RGB-D VIO algorithms. The code and the supplementary material will be available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/oranfire/SVIO</uri> .

Closed-Form Method for Atmospheric Correction (CMAC) of Smallsat Data Using Scene Statistics

High-cadence Earth observation smallsat images offer potential for near real-time global reconnaissance of all sunlit cloud-free locations. However, these data must be corrected to remove light-transmission effects from variable atmospheric aerosol that degrade image interpretability. Although existing methods may work, they require ancillary data that delays image output, impacting their most valuable applications: intelligence, surveillance, and reconnaissance. Closed-form Method for Atmospheric Correction (CMAC) is based on observed atmospheric effects that brighten dark reflectance while darkening bright reflectance. Using only scene statistics in near real-time, CMAC first maps atmospheric effects across each image, then uses the resulting grayscale to reverse the effects to deliver spatially correct surface reflectance for each pixel. CMAC was developed using the European Space Agency’s Sentinel-2 imagery. After a rapid calibration that customizes the method for each imaging optical smallsat, CMAC can be applied to atmospherically correct visible through near-infrared bands. To assess CMAC functionality against user-applied state-of-the-art software, Sen2Cor, extensive tests were made of atmospheric correction performance across dark to bright reflectance under a wide range of atmospheric aerosol on multiple images in seven locations. CMAC corrected images faster, with greater accuracy and precision over a range of atmospheric effects more than twice that of Sen2Cor.