Nonlinear Problem Research Articles

Optimal control in models of virus propagation

Based on the SEIRD model, we consider that when multiple viruses of different virulence coexist, the more virulent virus will reinfect nodes already infected by the less virulent virus, which we call here Superexposed. Based on the state transitions, the corresponding differential equations and cost functions are established, then building the corresponding optimal control problem, where the vaccine efficiency and drug efficiency are controlled variables. This nonlinear optimal control problem is solved by Pontryagin’s maximum principle to finding the structure of the optimal control strategies. Based on the definition of the basic regeneration number, yielding the R0 value for the model, then discussed the final epidemic size. In the numerical analysis section, we validate the accuracy of the structure, fitting the behavior of each state and the effect of different parameter values.

Stability of Beam Bridges Under Bridge-Vehicle Interaction

In this paper, we provide an accurate and reliable formulation for simulating the interactions of both train/bridge subsystems and suitable for high-speed railway lines as well as for existing lines worldwide that are being renewed or modernized. We model the train as a series of suspended masses, taking into account the energy dissipation and the suspension system for each train vehicle. On the other hand, the bridge supporting the rails with irregular elevations will be modeled as an Euler-Bernoulli beam. The mathematical formulation of the interaction problem between the two subsystems requires the writing of two sets of equations, which interact with each other through contact forces. Using a one-dimensional finite element formulation, a series of equations are constructed by Modeling the beam structure. In addition, the suspended mass equations are first discretized using Newmark's finite difference formulas, which then reduce the degrees of freedom (DOF) of the vehicle to those of the bridge element. This solves the coupling problem between the two subsystems. The derived component is known as the vehicle/bridge interaction (VBI) element. On the other hand, an iterative procedure will be used subsequently to solve the non-linearity problem of the resulting system of differential equations. MATLAB programs provide results that identify the critical parameters influencing the bridge's dynamic stability.

Economic modelling and optimisation of SPRT-based quality control schemes for individual observations

In many industrial processes, it may not be practical to obtain sample sizes larger than one due to various reasons. As a result, practitioners must rely on control charts based on individual observations for statistical process monitoring and control. The sequential probability ratio test (SPRT) chart is an appropriate solution for such situations. Existing literature predominantly delves into the statistical aspects of designing or optimising an SPRT chart, neglecting the economic and associated design aspects. However, past experience suggests that statistical design may not always be optimal from an economic perspective. This paper aims to bridge this research gap by proposing a two-stage Markov chain approach to develop an economic model for the optimal design of the SPRT chart with individual observations. Following the principles of economic-statistical designs, optimal parameters are determined by solving a mixed-integer non-linear programming problem that minimises the long-run cost per item while ensuring compliance with practical constraints. Through an extensive study, we find that the economic performance of the SPRT chart surpasses that of competing individual control charts in many cases. This indicates that the economic-statistical design of the SPRT chart can yield significant economic benefits while maintaining desirable statistical properties. Moreover, enhancements have been introduced to refine the optimal design of the SPRT chart, retaining excellent overall performance across a wide range of shifts. Finally, we provide two industrial examples to illustrate the effectiveness of the proposed SPRT chart.

Clearance Nonlinear Control Method of Electro-Hydraulic Servo System Based on Hopfield Neural Network

The electro-hydraulic servo system has advantages such as high pressure, large flow, and high power, etc., which can also realize stepless regulation, so it is widely used in many engineering machineries. A linear model is sometimes only a simple approximation of an idealized model, but in an actual system, there may be nonlinear and transient variation characteristics in the systems. Coupling is reflected in the fact that the components or functional structures implemented by each system used for the design of hydraulic systems are not completely or independently related to each other, but affect each other. The nonlinear clearance between the actuator and the load reduces the control accuracy of the system and increases the impact, thus losing stable working conditions. In the paper, based on the nonlinear clearance problem of the electro-hydraulic servo system, a mathematical transfer model with clearance is established, and on this basis, a clearance compensation method based on the Hopfield neural network is proposed. In this way, clearance compensation can be realized by adjusting the parameters of neural network nodes, through simple and convenient operation. Finally, by setting different clearance values, the results of the step response and sine response curve before and after clearance compensation of the hydraulic system are compared, and the effectiveness of Hopfield neural network compensation clearance control is verified based on the comparison simulation results.

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

Geographic coordinate validation and assignment using an edge-constrained layout

Electric grids with buses that are mapped to geographic latitude and longitude are useful for a growing number of applications, such as data visualization, geomagnetically induced current calculations, and multi-energy coupled infrastructure simulations. This paper presents a methodology for validating the quality of geographic coordinates for a power system model, and to assign coordinates to buses with missing or low-quality coordinates. This method takes advantage of geographic indicators already intrinsic to a grid model, such as branch length as implied by impedance and susceptance parameters. The coordinate assignment process uses an approach inspired by graph drawing, that lays out the vertices (buses) and edges (transmission lines), formulated as a nonlinear programming problem with soft edge length constraints. The layout method is very computationally fast and scalable to large power system cases. The method is demonstrated in this paper using a 37-bus test case and a 6717-bus test case, both publicly available, along with a large actual grid model. The results show that, for cases with only a few errors in the coordinates, cases with no coordinates known beforehand, and others in between, this method is able to assign reasonable geographic coordinates to best match known data about the grid.

A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.

Optimization of the on/off status configuration of control units in an active vibration isolation system - experiment

Abstract Active vibration isolation (AVI) is an important measure to attenuate vibration transmission in underwater vehicles and multiple active control units are usually used to attenuate low frequency vibrations induced by power equipment. Under certain operation conditions, however, the pre-installed control units in an AVI system may be redundant. Hence, it is necessary to optimize the on/off status configuration of all the control units. An optimization approach is proposed for configuration optimization of control units and validated by experiments. The criterion of the proposed approach is derived from the vibration responses of the monitoring points. The involved optimization, formulated as a constrained 0-1 nonlinear programming problem, utilizes the on/off status of control units as the input variable and the number of active units as the constraint. Optimized configurations were obtained by solving the nonlinear programming problem and deployed to the experimental system. The test results have demonstrated that vibration of the monitoring points can be effectively suppressed by activating only a portion of control units, and the configurations corresponding to maximum vibration reduction vary with different excitations.

Optimal Unmanned Combat System-of-Systems Reconstruction Strategy with Heterogeneous Cost via Deep Reinforcement Learning

The unmanned combat system-of-systems (UCSoS) in modern warfare is comprised of various interconnected entities that work together to support mission accomplishment. The soaring number of entities makes the UCSoS fragile and susceptible to triggering cascading effects when exposed to uncertain disturbances such as attacks or failures. Reconfiguring the UCSoS to restore its effectiveness in a self-coordinated and adaptive manner based on the battlefield situation and operational requirements has attracted increasing attention. In this paper, we focus on the UCSoS reconstruction with heterogeneous costs, where the collaboration nodes may have different reconstruction costs. Specifically, we adopt the heterogeneous network to capture the interdependencies among combat entities and propose a more representative metric to evaluate the UCSoS reconstruction effectiveness. Next, we model the combat network reconstruction problem with heterogeneous costs as a nonlinear optimization problem and prove its NP-hardness. Then, we propose an approach called SoS-Restorer, which is based on deep reinforcement learning (DRL), to address the UCSoS reconstruction problem. The results show that SoS-Restorer can quickly generate reconstruction strategies and improve the operational capabilities of the UCSoS by about 20∼60% compared to the baseline algorithm. Furthermore, even when the size of the UCSoS exceeds that of the training data, SoS-Restorer exhibits robust generalization capability and can efficiently produce satisfactory results in real time.

Zonal free element method for free and forced vibration analysis of two- and three-dimensional structures

This paper presents a new numerical method called the zonal free element method (ZFREM) for the free and forced vibration analysis of elastodynamic problems. In this approach, a complex computational domain is divided into some simple zones and generates a series of regularly arranged nodes in each zone, which can improve the accuracy during the analysis of complex models. The distinguishing feature of the ZFREM is that an independent isoparametric element is formed by only one freely chosen surrounding node at each configuration node. In this method, the mass term exists only in the internal nodes, which can accelerate the assembly of the final system equations. Building upon this foundation, the present study developed the Krylov reduced dimensional iterative method, which approximates the solution of the equation system by constructing a lower-dimensional subspace. This approach avoids the complex equation transformations involved in traditional algorithms for solving eigenvalue problems, thereby further enhancing computational efficiency. Moreover, to tackle damped vibration problems encountered in engineering applications, the proposed method is further extended to solve the non-linear forced vibration problems. The accuracy and effectiveness of the method are verified by numerical examples of free and forced vibration problems.

Numerical approximation of the typhoid disease model via Genocchi wavelet collocation method

AbstractIn this paper, we have considered the fractional typhoid disease model and obtained the numerical approximation of the model via the innovative wavelet scheme called the Genocchi wavelet collocation method (GWCM) with the help of Caputo fractional derivative for the fractional order. The approach under consideration is a powerful tool for obtaining numerical solutions to fractional-order nonlinear differential equations. The GWCM approach yields accurate solutions that are very close to exact solutions for highly nonlinear problems by avoiding data rounding and just computing a few terms. The Genocchi wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main benefit of this method lies in its capability to reduce the computational complexity associated with solving systems of ODEs, resulting in accurate and efficient solutions. The results of the developed technique, the RK4 method, and the ND solver have been compared. The numerical outcomes demonstrate that the implemented technique is incredibly effective and precise for solving the Typhoid model of fractional order. This paper contributes to numerical analysis by introducing the Genocchi wavelet method as a robust tool for solving biological models.

Modeling, Analysis and Evaluation of a Novel Compact 6-DoF 3-RRRS Needle Biopsy Robot

Robot-assisted surgical systems have been widely applied for minimally invasive needle biopsies thanks to their excellent accuracy and superior stability compared to manual surgical operations, which lead to possible fatigue and misoperation due to long procedures. Current needle biopsy robots are normally customed designed for specific application scenarios, and only position-level kinematics are derived, preventing advanced speed control or singularity analysis. As a step forward, this paper aims to design a universal needle biopsy robot platform which features 6 DoF 3-RRRS (Revolute–Revolute–Revolute–Spherical) parallel structure. The analytical solutions to its nonlinear kinematic problems, including forward kinematics, inverse kinematics, and differential kinematics are derived, allowing fast and accurate feedback control calculations. A multibody simulation platform and a first-generation prototype are established next to provide comprehensive verifications for the derived robotic model. Finally, simulated puncture experiments are carried out to illustrate the effectiveness of the proposed method.

Monarch Butterfly Optimization-Based Genetic Algorithm Operators for Nonlinear Constrained Optimization and Design of Engineering Problems

Abstract This paper aims to present a hybrid method to solve nonlinear constrained optimization problems and engineering design problems (EDPs). The hybrid method is a combination of MBO with the crossover and mutation operators of the genetic algorithm (GA). It is called a hybrid monarch butterfly optimization with genetic algorithm operators (MBO-GAO). Combining MBO and GA operators is meant to overcome the drawbacks of both algorithms while merging their advantages. The self-adaptive crossover (SAC) and the real-valued mutation are the GA operators that are used in MBO-GAO. These operators are merged in a distinctive way within MBO processes to improve the variety of solutions in the later stages of the search process, speed up the convergence process, keep the search from getting stuck in local optima, and achieve a balance between the tendencies of exploration and exploitation. In addition, the greedy approach is presented in both the migration operator and the butterfly adjusting operator, which can only accept offspring of the monarch butterfly groups who are fitter than their parents. Finally, popular test problems, including a set of 19 benchmark problems, are used to test the proposed hybrid algorithm, MBO-GAO. The findings obtained provide evidence supporting the higher performance of MBO-GAO compared to other search techniques. Additionally, the performance of the MBO-GAO is examined for several engineering design problems. The computational results show that the MBO-GAO method exhibits competitiveness and superiority over other optimization algorithms employed for the resolution of engineering design problems.

Optimal coordination of directional overcurrent protection relays using hybrid optimizer

In order to minimize or maximize an objective function, electrical engineering problems might benefit from optimization methodologies inspired by nature. This study employs a hybrid particle swarm optimizer (HPSO) to get rid of local optimum solutions and find the best global one for directional overcurrent protection relays (DOPR) protection. DOPR coordination is a highly constrained, mixed-integral nonlinear optimization problem. An adaptive defense method has been created to counteract the intentionally challenging design. Therefore, design variables such as time multiplier setting (TMS) and plug setting (PS) are utilized for each relay in the circuit. The goal function is to reduce the amount of time spent to trace the fault by all the relays. Reducing the time required for each relay to trace the fault is the aim of function. After analyzing and comparing two test studies of the IEEE benchmark using various optimization strategies, we found that the results for the IEEE 15-bus system has minimum time of 8.9 s, which is improved by 22.92% to 82.89%, and those for the IEEE 30-bus system minimum time of 22.45 s that is improved by 4.41% to 39.6%. In terms of overall DOPR operation and computational time needed to get the global optimum solution, the acquired results are demonstrably superior to those obtained using conventional and state-of-the-art methods.

Dynamic optimization of nonlinear differential game problems using orthogonal collocation with analytical sensitivities

AbstractThis article presents an effective computational method based on the orthogonal collocation on finite element for nonlinear pursuit‐evasion differential game problems. The original problems are transformed into two dynamic optimization problems at first, so that the difficulty of obtaining the solution is reduced. To improve the convergence rate and the efficiency, the sensitivities describing the influence of control and interval parameters on state are derived through the discretized dynamic equations for the resulting nonlinear programming problem. The convergence speed is introduced to measure the performance in the upper level iteration. The main structure and the algorithm of the method are also given. Two demonstrative differential game problems with different scenarios from practice are studied. Compared with the approach without sensitivity information, the proposed method needs less function evaluations and saves at least 68.4% of the computational time. The research results show the effectiveness of proposed approach.

TDOA-based UWB indoor 1D localization via weighted sliding window filtering

Accurate localization of a person or object is a significant aspect in the field of industrial internet of things (IoT). Ultra-wideband (UWB) localization system, a promising technology, has been widely studied due to its high accuracy in 2D and 3D scenarios. Moreover, in narrow corridors and deep-well environments, the 1D localization is also an important application with the advantage of resource saving, however which attracts less attention. In this paper, we focus on the problem of 1D localization and propose a practical localization model with time difference of arrivals (TDOA). To solve the nonlinear 1D localization problem, we design an algorithm that integrates the dichotomous method and Newton's iterative method, which realizes an efficient solution. Meanwhile, in order to cope with the random errors in the real scenes, we introduce a sliding window algorithm to filter out the undesirable solutions. By observing the localization error map, we design a weighted sliding window algorithm to achieve further improvement of the localization accuracy. In addition, we derive the Cramér-Rao lower bound (CRLB) for 1D localization of TDOA. Simulations and practical experiments show that our proposed algorithm greatly benefits the localization accuracy of 1D TDOA and provides strong support for its practical application.

A Nonlinear Mixed Finite Element Method for the Analysis of Flexoelectric Semiconductors

Abstract In this paper, we develop a nonlinear mixed finite element method for flexoelectric semiconductors and analyze the mechanically tuned redistributions of free carriers and electric currents through flexoelectric polarization in typical structures. We first present a macroscopic theory for flexoelectric semiconductors by combining flexoelectricity and nonlinear drift-diffusion theory. To use C0 continuous elements, we derive an incremental constrained weak form by introducing Langrage multipliers, in which the kinematic constraints between the displacement and its gradient are guaranteed. Based on the weak form, we established a mixed C0 continuous nine-node quadrilateral finite element as well as an iterative process for solving nonlinear boundary-value problems. The accuracy and convergence of the proposed element are validated by comparing linear finite element method results against analytical solutions for the bending of a beam. Finally, the nonlinear element method is applied to more complex problems, such as a circular ring, a plate with a hole, and an isosceles trapezoid. Results indicate that mechanical loads and doping levels have distinct influences on electric properties.

A Nonlinear Programming Approach to Solving Interval-Valued Intuitionistic Hesitant Noncooperative Fuzzy Matrix Games

Initially, fuzzy sets and intuitionistic fuzzy sets were used to address real-world problems with imprecise data. Eventually, the notion of the hesitant fuzzy set was formulated to handle decision makers’ reluctance to accept asymmetric information. However, in certain scenarios, asymmetric information is gathered in terms of a possible range of acceptance and nonacceptance by players rather than specific values. Furthermore, decision makers exhibit some hesitancy regarding this information. In such a situation, all the aforementioned expansions of fuzzy sets are unable to accurately represent the scenario. The purpose of this article is to present asymmetric information situations in which the range of choices takes into account the hesitancy of players in accepting or not accepting information. To illustrate these problems, we develop matrix games that consider the payoffs of interval-valued intuitionistic hesitant fuzzy elements (IIHFEs). Dealing with these types of fuzzy programming problems requires a significant amount of effort. To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs. Here, we construct a novel aggregation operator of IIHFEs, viz., min-max operators of IIHFEs. This operator is suitable for applying the developed methodology. The cogency and applicability of the proposed methodology are verified through a numerical example based on the situation of conflict between hackers and defenders to prevent damage to cybersecurity. To validate the superiority of the proposed model along with the computed results, we provide comparisons with the existing models.

Collaborative Computation Offloading and Resource Management in Space–Air–Ground Integrated Networking: A Deep Reinforcement Learning Approach

With the increasing dissemination of the Internet of Things and 5G, mobile edge computing has become a novel scheme to assist terminal devices in executing computation tasks. To elevate the coverage and computation capability of edge computing, a collaborative computation offloading and resource management architecture was proposed in space–air–ground integrated networking (SAGIN). In this manuscript, we established a novel model considering the computation offloading cost constraints of the communication, computing and cache model in the SAGIN. To be specific, the joint optimization problem of collaborative computation offloading and resource management was modeled as a mixed integer nonlinear programming problem. To address this issue, this paper proposed a computation offloading and resource allocation strategy based on deep reinforcement learning (DRL). Differing from traditional methods, DRL does not need a well-established formulation or previous information, and it is capable of revising the strategy adaptively according to the environment. The simulation results demonstrate the proposed approach can achieve the optimal reward values in the case of different terminal device numbers. Furthermore, this manuscript provided the analysis with variant parameters of the proposed approach.

Numerical study of binary mixture and thermophoretic analysis near a solar radiative heat transfer

The main focus of the current article is to investigate the two-dimensional boundary layer flow of non-Newtonian fluid induced by a linear stretching surface in the streamwise direction. The flow features of non-Newtonian fluid (i.e. shear-thinning and thickening nature of the fluids) are explored with Carreau fluid model. In addition, thermophoresis, Brownian motion, thermal radiation (with Rosseland approximation), binary mixture and activation enthalpy are also incorporated in fluid model for better analysis of thermal and solutal properties. The mathematical modelling of under consider physical problem yields nonlinear partial differential system. The governing mathematical system is first simplified with scaling transformations for the high Reynolds number and then transformed into non-dimensional ordinary differential system by using the similarity variables. Since the governing mathematical system comprised on nonlinear boundary value problem, thus shooting method is utilized for the numerical solution of the fluid flow governing equations. The obtained results followed the qualitative manners of the previously reported data. The computed results for important physical quantities (i.e. velocity, temperature, and concentration) are presented through graphs and then interpreted physically. It is observed that the boundary layer thickness increased for shear thickening fluids (n>1) while it is reduced for shear thinning fluids (n<1). Also, the temperature of shear thinning fluid is higher than shear thickening fluid. In addition, heat generation and thermal radiations enhance the thermal boundary layer. Thermophoretic viscosity effect reduces the concentration profile significantly in the flow domain.