Closed-form Solution Research Articles

Comprehensive design strategy of the NSD–TMD controlled system with closed-form frequency-amplitude solutions

A negative stiffness device (NSD) utilizing pre-compressed helical springs has been integrated with diverse vibration control techniques. Recently, the incorporation of NSD has been proposed to enhance the efficacy of a tuned mass damper (TMD) controlled system. This study introduces two design strategies for the NSD–TMD system, which functions either as a TMD with an adjustable operating bandwidth or as a pure non-linear energy sink. First, the non-linear force of NSD was expressed in a Taylor series approximation form incorporating a negative linear stiffness and a positive cubic stiffness term. Second, the equations of motion were formulated, and closed-form solutions were derived using the complexification–averaging method. The accuracy of the Taylor series approximation was substantiated within a specified range through a comparison of force–displacement curves. Third, the accuracy of the dynamic closed-form solutions was affirmed by comparing the amplitude–frequency curves with the numerical simulations. For a case study, an actual timber pedestrian bridge with decaying natural frequency was selected, and the two design strategies were examined individually. The vibration control performance of the TMD and NSD–TMD was also comparatively analyzed. The assessment focused on the peak force transmissibility value and the effective operating frequency bandwidth. Results highlighted a notable enhancement in operational performance, particularly for primary structures exhibiting diminishing modal frequencies. In a TMD-controlled system, the efficacy of vibration control deteriorates and may even amplify the dynamic response when a modal frequency shift occurs in the primary structure. Conversely, both design strategies of NSD–TMD can achieve lower transmissibility and broaden the operating bandwidth.

Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.

For some communicable endemic diseases (e.g., influenza, COVID-19), vaccination is an effective means of preventing the spread of infection and reducing mortality, but must be augmented over time with vaccine booster doses. We consider the problem of optimally allocating a limited supply of vaccines over time between different subgroups of a population and between initial versus booster vaccine doses, allowing for multiple booster doses. We first consider an SIS model with interacting population groups and four different objectives: those of minimizing cumulative infections, deaths, life years lost, or quality-adjusted life years lost due to death. We solve the problem sequentially: for each time period, we approximate the system dynamics using Taylor series expansions, and reduce the problem to a piecewise linear convex optimization problem for which we derive intuitive closed-form solutions. We then extend the analysis to the case of an SEIS model. In both cases vaccines are allocated to groups based on their priority order until the vaccine supply is exhausted. Numerical simulations show that our analytical solutions achieve results that are close to optimal with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size. In addition to being accurate and interpretable, the solutions are easy to implement in practice. Interpretable models are particularly important in public health decision making.

Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model.

We study the behavior of the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of temporal networks. We consider the multilayer representation of temporal networks, i.e., a set of networks linked through ordinal interconnected layers. We analyze the Laplacian matrix, known as supra-Laplacian, constructed through the supraadjacency matrix associated with the multilayer formulation of temporal networks, using a constant block Jacobi model which has closed-form solution. To do this, we assume that the interlayer weights are perturbations of the Kronecker sum of the separate adjacency matrices forming the temporal network. Thus we investigate the properties of the eigenvectors associated with the smallest eigenvalues (close to zero) of the supra-Laplacian matrix. Using arguments of perturbation theory, we show that these eigenvectors can be approximated by linear combinations of the zero eigenvectors of the individual time layers. This finding is crucial in reconsidering and generalizing the role of the Fielder vector in supra-Laplacian matrices.

Adaptive Bayesian speckle reduction in the 2D DOST domain using 2D MC-GARCH-M model: Preserving image edges and textural information

Image denoising and image quality enhancement are major issues in image processing. Additive and multiplicative noise removal is one of the main image enhancement approaches. In this paper, we propose a novel 2D Merged Complex Generalized Autoregressive Conditional Heteroscedasticity Mixture (MC-GARCH-M) model for 2D complex stochastic processes. This model effectively captures non-Gaussian statistics and all three types of dependencies within the processes (RRD, IID, and RID) using the proposed linear mapping transformation T. Our statistical analysis shows that the 2D MC-GARCH-M model provides the best fit to the non-Gaussian statistics of the mapped matrix of 2D DOST coefficients compared to Gaussian, Generalized Gaussian, and Stable distributions. We confirm the presence of heteroscedasticity in the mapped matrix using the Engle hypothesis test, verifying ARCH/GARCH effects. The 2D MC-GARCH-M model, with its location-dependent conditional variances, provides a flexible approach for noise removal and modeling complex stochastic processes. Experimental results on artificially speckled aerial and actual SAR images demonstrate that our method outperforms state-of-the-art approaches in terms of speckle removal and preserving the edges and textural information of the image. Key innovations include: a new statistical analysis using linear mapping T showing non-Gaussian heavy-tailed distributions; an adaptive Bayesian estimator, MCMAP, based on the heteroscedastic model for speckle noise removal; and the method's applicability to images with varying mutual information (MI) between real and imaginary parts of 2D DOST coefficients. Our method is adaptive and robust to initial parameter settings, effectively utilizing magnitude and phase information, providing an optimal closed-form solution, reducing memory and computational requirements, and demonstrating robustness to speckle power levels. While requiring more computational resources compared to thresholding methods, our approach is less demanding than other Bayesian algorithms, making it suitable for offline processing.

Vertical vibration control of structures with tuned liquid column dampers

Conventional tuned liquid column dampers (TLCDs) can only be used to suppress structural vibrations in horizontal directions. However, in reality, the vibration-induced motion of civil engineering structures normally consists of both vertical and horizontal components. In this study, a TLCD-based vibration control system, which is composed of a pair of inclined U-shaped containers, springs, dashpots, as well as supporting slopes, is proposed to control the vertical vibration of an engineering structure. The proposed control system (denoted as vertical TLCD) successfully extends the application of TLCD-type devices to vertical vibration reduction. First, the general configuration and operating principle of the proposed vertical TLCD are illustrated in detail. Second, the equations of motion are derived based on the Lagrange dynamic equations and further verified by two-way fluid–structure coupling numerical simulations using finite element method and finite volume method. Subsequently, the closed-form solution is derived for the optimized design of the vertical TLCD in the frequency domain. The dynamic characteristics of an optimized vertical TLCD-structure system are revealed. The effects of mass ratio, excitation amplitude and structural damping ratio on the optimization results are elaborated, respectively. Under the same optimized parameters, the vertical TLCD is found to be superior, compared to the vertical tuned mass damper (TMD). Finally, the effectiveness of the proposed system is demonstrated by applying it to the control of a bridge experiencing vertical vibrations under seismic loads. The findings of this study provide a comprehensive understanding of TLCD’s capabilities and identify its potential for vertical vibration control, which contributes to the development of more effective and versatile vibration mitigation strategies for a wide range of structures under different engineering scenarios.

The structural strain method for fatigue evaluation of welded components: Analytical treatment of reversed plasticity

The traction structural stress method has been adopted by various industry sectors, in addition to ASME Boiler and Pressure Code (B&PV), for fatigue life evaluation and assessment of welded components, especially for high-cycle fatigue applications. Its effectiveness has been validated and demonstrated by showing good data correlation in the form of the master S–N curve scatter band. For low cycle fatigue with the involvement of elastic-plastic deformation, the structural strain method has been recently developed by simply extending the current elastic-deformation-based structural stress method to more general elastic-plastic deformation. Closed-form structural strain solutions for elastic-perfectly plastic materials and numerical solutions for nonlinear strain hardening materials have been obtained under simple load-controlled loading and unloading cycling conditions. In this paper, the equations of nonlinear cyclic structural stress-strain curves under fully-reversed fatigue loading are analytically derived for the first time with a 3-bar model, which consists of three elastic-perfectly plastic bars. The general trends and patterns of the loading paths of the constructed cyclic structural stress-strain curves with the 3-bar model are validated with finite element analysis (FEA). Based on the insights gained from the 3-bar model and the FEA results, a simple procedure for constructing cyclic structural stress-strain curves from their corresponding monotonic loading curves are developed for both elastic-perfectly plastic model and the modified Ramberg-Osgood nonlinear strain hardening model. The effectiveness of this procedure is demonstrated by correlating low-cycle fatigue data of welded structures under pulsating and fully-reversed loading conditions.

A geometric solution to microphone array position calibration and its confidence interval analysis

Wireless acoustic sensor networks (WASNs) comprised of spatially distributed acoustic nodes have been increasingly popular in many areas such as speech enhancement, blind source separation, and audio surveillance, where node position information is generally required to implement relevant processing algorithms. In this article, we focus on the node position calibration only using direction-of-arrival (DOA) information with each node consisting of a microphone array. State-of-the-art techniques translate the position calibration into optimization of DOA-based cost functions, and have shown notable success. However, a compromise between position calibration accuracy and computational load still exists. Furthermore, none of the existing studies can offer a quantitative evaluation of calibration performance before the optimization procedure is finished. To address these problems, a geometric solution to array node position calibration is proposed using DOA measurements. For a given array node, DOAs of three sources are first utilized to construct cylinder, plane, and cone equations. Next, the two-dimensional projection of intersections among these equations is calculated using Sylvesterresultantmatrix. Finally, a closed-form solution of node position in three-dimensional space is established. Moreover, explicit mathematical derivation of the confidence interval analysis on the calibration results under given node angle estimation error condition is presented, indicating that the proposed method can provide predictable performance with corresponding probabilities. The Cramér-Rao bound (CRB) is further investigated. Simulation results reveal that as compared with existing methods our approach achieves comparable calibration accuracy with much less computational burden. Real-world experiments also verify its effectiveness.

A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.

Solving constrained consumption–investment problems by decomposition algorithms

Consumption–investment problems with maximizing utility agents are usually considered from a theoretical viewpoint, aiming at closed-form solutions for the optimal policy. However, such an approach requires that the model be relatively simple: even the inclusion of nonnegativity constraints can prevent the derivation of explicit solutions. In such cases, it is necessary to solve the problem numerically, but standard dynamic programming algorithms can only solve small problems due to the curse of dimensionality. In this paper, we adapt the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve dynamic constrained consumption–investment problems with stochastic labor income numerically. Unlike classical dynamic programming approaches, SDDP allows us to analyze problems with multiple assets, and an internal sampling procedure allows the problems to have a very large, or even infinite, number of scenarios. We start with a simpler problem for which a closed-form solution is known and compare it to the optimal policy obtained by SDDP. We then illustrate the flexibility of our approach by solving a defined contribution pension fund problem with multiple assets, for which no closed-form solution is available.

Physically Motivated Model of a Painting Brush for Robotic Painting and Calligraphy

Robot artistic painting and robot calligraphy do require brush models for brushstroke simulation and painting robot control. One of the main features of the brush is its compliance, which describes the relationship between the brush footprint shape and the pressure applied to the brush. In addition, during motion, the brush footprint position lags from the brush handle position in a complicated manner. To date, the question of creating a physically correct model of these effects and choosing the best method for the model parameter calibration has not been presented in the literature. In the current paper, we derive equations of the brush contact patch motion, give their closed-form solutions, and investigate three methods for the brush model calibration: capturing brush footprints on a matte glass with a camera, painting calibration brushstrokes, and capturing a brush shape side projection with a camera. As we show, calibration brushstrokes give us primary information on brush contact patch displacement during painting, and capturing the brush side projection allows the accurate estimation of the gap from the brush tip to the center of the contact patch. Capturing brush footprints is useful for creating a brushstroke executable model. As an example, a model for a round artistic brush was created and verified in three tests, including measuring the coordinates of an angular brushstroke center line, simulating an angular brushstroke, and writing a signature using a robotic setup.

A closed-form solution of dowel action based on beam on elastic foundation theory and fracture mechanics

This paper proposes a new mechanical model to describe the dowel action with the aim of using the model to gain a deeper understanding of the unstable dowel splitting cracking observed in shear experiments of beams without shear reinforcement. The model was developed by combining beam on elastic foundation (BEF) theory and fracture mechanics. The proposed model is able to predict the whole evolution process of dowel action until the propagation of the dowel splitting crack becomes unstable. The model theoretically proves that the development of a dowel splitting crack can become unstable under certain conditions, therefore leading to the unstable shear failure of the whole member. In addition to the derivation of the analytical model, the paper also validates the model using data from the literature. Finally, an analytical solution of the critical shear displacement that triggers the unstable dowel splitting crack is derived. It can be used to improve the failure criterion initially proposed in the Critical Shear Displacement Theory (CSDT).

Supervised maximum variance unfolding

Maximum Variance Unfolding (MVU) is among the first methods in nonlinear dimensionality reduction for data visualization and classification. It aims to preserve local data structure and in the meantime push the variance among data as big as possible. However, MVU in general remains a computationally challenging problem and this may explain why it is less popular than other leading methods such as Isomap and t-SNE. In this paper, based on a key observation that the structure-preserving term in MVU is actually the squared stress in Multi-Dimensional Scaling (MDS), we replace the term with the stress function from MDS, resulting in a model that is usable. The property of the usability guarantees the “crowding phenomenon” will not happen in the dimension reduced results. The new model also allows us to combine label information and hence we call it the supervised MVU (SMVU). We then develop a fast algorithm that is based on Euclidean distance matrix optimization. By making use of the majorization-mininmization technique, the algorithm at each iteration solves a number of one-dimensional optimization problems, each having a closed-form solution. This strategy significantly speeds up the computation. We demonstrate the advantage of SMVU on some standard data sets against a few leading algorithms including Isomap and t-SNE.

Unified theory for identifying vertical and radial damping ratios of curved bridges by two connected scanning vehicles

A unified theory is presented for identifying the vertical and radial damping ratios of a horizontal curved bridge by the correlation between two connected scanning vehicles using the variational mode decomposition (VMD) and synchrosqueezed wavelet transform (SWT). Firstly, closed-form solutions are derived for the out-of-plane and in-plane responses of the curved beam caused by the gravitational and centrifugal forces, respectively, of the two moving vehicles. Then, generalized contact formulas are presented for calculating the vertical and radial responses of the vehicle-bridge contact points from the vehicle’s responses, to avoid vehicle frequencies’ masking effect. Subsequently, the instantaneous amplitudes of the component responses of the curved bridge are obtained by the VMD-SWT technique. Finally, generalized damping formulas for retrieving the vertical and radial damping ratios of the curved bridge are established by utilizing the spatial correlation between the two vehicles. This paper has demonstrated that: (1) the vertical and radial damping ratios of the curved bridge can be successfully and precisely identified; (2) the generalized damping formula for curved beams has been attested to be robust against various factors; and (3) the adverse effect of road roughness on damping ratio recovery can be mitigated by the aid of accompany traffic.

Analytical solution for axially loaded floating pipe piles in homogeneous elastic soil layer

An analytical solution for the analysis of open-ended floating pipe piles in a homogeneous elastic soil layer is developed in this paper, based on the Tajimi-type elastic continuum model. Solving the governing equations of the outer and inner soil results in closed-form expressions for the external and internal frictional resistances developing at the soil-pile interfaces. The one-dimensional rod theory is adopted to model the pipe pile and its underlying soil hollow column. The external and internal frictional resistances are substituted into the governing equations of the pile and soil column to derive closed-form solutions for the vertical displacement and head stiffness of the pipe pile. The accuracy of the proposed model is verified by comparisons against existing numerical studies. Finally, parametric analyses are presented to discuss the sensitivity of the behavior of axially loaded floating pipe piles to the main problem parameters.

Investigating (2+1)-dimensional dissipative long wave system in water waves using three innovative integration norms

This research article investigates the dissipative long wave system in (2+1) dimensions. To analyze this system, three separate integrating strategies were used: the single manifold approach, the unified method, and the generalized projective Ricatti equation method. Novel soliton solutions were effectively obtained by applying these strategies to the suggested model. The resulting solutions are various, including mixed soliton-like solutions, dark solitons, rational solutions, triangular periodic solutions, and combined Jacobi elliptic wave function solutions. Using the aforementioned approaches, an extensive variety of solution types has been discovered for the proposed model. The dynamic character of these closed-form solutions has been adequately demonstrated using both three dimensional and two dimensional graphical representations. This visualization helps to better understand the behavior of the acquired solutions.

RETRACTED: A robust multi-target localization algorithm for ISAC systems

Localization for multi-target systems (MTS) in anchor-free scenarios is receiving increasing attention, particularly for MTS, such as the Internet of Vehicles and swarms. This paper proposes a state-of-the-art fusion model based on the odometer and ranging measurements to address the anchorfree localization problem for MTS. Regarding varying measurements as a topological structure of the position, we formulate the optimization problem as a Procrustes problem that fully utilizes measurement information. Singular value decomposition is used to obtain the closed-form solution to the problem, which places minimal demands on computational resources. We also introduce an adaptive mechanism with a trade-off factor to enhance the model's flexibility. Additionally, the model applies to 2D and 3D cases and is generally suitable for most odometer and ranging technologies. Simulation experiments demonstrate that the positioning accuracy of the proposed scheme significantly improves over conventional optimizationbased methods and outperforms the state-of-the-art.

A geometrically nonlinear Hellinger–Reissner shell element for the postbuckling analysis of variable stiffness composite laminate structures

Variable stiffness (VS) composite laminates provide larger freedom to design thin-walled structures than constant stiffness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the postbuckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress field interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures.First, the element performance is investigated by analysing a carbon fibre VS composite laminate plate subjected to compressive stresses. Numerical results obtained with MISS-4C are compared with those obtained with the MISS-4 element, showing good accuracy and a high convergence rate. Subsequently, the structural response of a glass fibre VS composite laminate girder of a short-length bridge is optimised through a multi-objective optimisation that exploits the robustness of the MISS-4C element and the efficiency of the multi-modal Koiter algorithm.

Flexural performance of GFRP-sheathed cold-formed steel composite panel filled with lightweight phosphogypsum

To develop lightweight, high-strength and environmentally friendly prefabricated panels, this paper proposed a novel GFRP plate-sheathed cold-formed steel (CFS) composite panel (GFRP-CFS panel) filled with lightweight phosphogypsum (LPG) and investigated its flexural performance experimentally, numerically and theoretically. Firstly, nine full-scale specimens were subjected to four-point bending test. The effects of the number, web height, thickness and arrangement of the longitudinal CFS studs, the number of lateral CFS studs, and the support connection on the flexural performance were discussed. Results indicate that the failure mode of the GFRP-CFS panels was debonding of GFRP plate in the compression zone at midspan and local buckling of CFS below the loading point. The strain of GFRP plate always lagged behind that of the CFS stud at the corresponding position, and the built-up cross section of the panel no longer conformed to the plane-section assumption. In addition, a nonlinear finite element modelling (FEM) analysis based on DIANA software was conducted to validate the test results. The load-deflection response, strain response and failure mode from the FEM analyses were in good agreement with the test results. Furthermore, based on the theoretical analysis considering the bond-slip effect between GFRP and CFS, a closed-form solution for the GFRP utilization coefficient was derived. The effect of different variables on the GFRP utilization coefficient was analyzed through parametric analysis. Finally, the calculation method for the flexural strength and ultimate deflection of GFRP-CFS panels were established, and the predicted results were in good agreement with test results. Therefore, the developed GFRP-CFS panels have excellent potential for prefabricated floor and wall applications.

Steady-state flow under surface recharge through unconfined aquifer with depth-decaying hydraulic conductivity: Analytical results

This paper presents some analytical results related to groundwater flow through an unconfined horizontal aquifer induced by areal recharge on the top and a decaying hydraulic conductivity. Under steady-state conditions, and in the context of Boussinesq equation, exact analytical solutions describing drainage into one or two penetrating channels at the aquifer boundaries have been derived in closed forms. The analytical expressions have been used to investigate the existence of water divide inside the unconfined aquifer. Based on the derived analytical solutions, a closed-form analytical formula is developed for calculating groundwater transit time within Dupuit-type flow system. The analytical transit time expression is simply expressed in terms of common mathematical functions. Analytical results are verified against a numerical model and show excellent agreements. The analytical transit time is validated against a numerical one obtained from a two-dimensional model representing a rectangular unconfined aquifer. Based on the explicit closed-form solution, a fully analytical inverse problem to estimate the decay parameter is proposed using a least-square objective function. The solution of the inverse problem simply corresponds to the solution of a nonlinear algebraic equation obtained from the definition of the objective function and the form of the analytical solution.

Joint optimization of dynamic pricing, quality, and production quantity in the presence of online reviews: A distributionally robust optimization approach

This paper investigates the joint pricing, product quality, and production quantity decisions of a firm in a two-stage scenario, considering potential market demand uncertainty and the existence of online reviews. Two decision models are established based on whether the firm adjusts product quality and prices according to consumer feedback from reviews: a static model (as a baseline) and a dynamic model. In the static model, the firm decides product quality, prices, and production quantity in the first stage, while in the second stage, the firm does not enhance product quality, and prices remain unchanged. In the dynamic model, in the first stage, the firm decides initial product quality and pricing, then in the second stage, adjusts product quality based on consumer feedback and sets pricing accordingly. We propose a distributionally robust optimization (DRO) method to address this problem, obtaining a closed-form solution for the expected profit function under the worst distribution, making it easy to determine the optimal decision. Through numerical simulation analysis, we find that the firm implementing dynamic decisions always make higher profits compared to those using static decisions. This highlights the superiority of dynamic decisions in adapting to market changes and improving product quality. We also analyze the impact of expected and standard deviation of potential market demand, as well as consumer private assessment weight, on business decisions and profits. Our research provides effective support for firms with limited market information, enabling them to better formulate joint pricing, quality, and production decisions.