Search results for: nonlinear PDEs

Authors: Marlene Perez Villalpando, Kelly Joel Gurubel Tun

Abstract:

In this paper, we propose an optimal control strategy for a hydroelectric power plant subject to input disturbances like meteorological phenomena. The engineering characteristics of the system are described by a nonlinear model. The random availability of renewable sources is predicted by a high-order neural network trained with an extended Kalman filter, whereas the power generation is regulated by the optimal control law. The main advantage of the system is the stabilization of the amount of power generated in the plant. A control supervisor maintains stability and availability in hydropower reservoirs water levels for power generation. The proposed approach demonstrated a good performance to stabilize the reservoir level and the power generation along their desired trajectories in the presence of disturbances.

Keywords: hydropower, high order neural network, Kalman filter, optimal control

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Authors: A. Boulanoir, B. Ould Bouamama, A. Debiane, N. Achour

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The paper deals with robust Fault Detection and isolation with respect to parameter uncertainties based on linear fractional transformation form (LFT) Bond graph. The innovative interest of the proposed methodology is the use only one representation for systematic generation of robust analytical redundancy relations and adaptive residual thresholds for sensibility analysis. Furthermore, the parameter uncertainties are introduced graphically in the bond graph model. The methodology applied to the nonlinear industrial Electro-Mechanical Actuators (EMA) used in avionic systems, has determined first the structural monitorability analysis (which component can be monitored) with given instrumentation architecture with any need of complex calculation and secondly robust fault indicators for online supervision.

Keywords: bond graph (BG), electro mechanical actuators (EMA), fault detection and isolation (FDI), linear fractional transformation (LFT), mechatronic systems, parameter uncertainties, avionic system

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Authors: Mohammed Daher Albalwi

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The transcritical flow of a stratified fluid over an obstacle for negative forcing amplitude (hole) that generation upstream and downstream, connected by an unsteady solution, is examined. In the weakly nonlinear, weakly dispersive regime, the problem is formulated in the forced Korteweg-de Vries–Burgers framework. This is done by including the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. The results show that the influence of viscosity is crucial in determining various wave properties, including the amplitudes of solitary waves in the upstream and downstream directions, as well as the widths of the bores. We focused here on weak damping, and the results are presented for transcritical, supercritical, and subcritical flows. In general, the outcomes are not qualitatively similar to those from the forced Korteweg-de–Vries equation when the value of the viscous is small, interesting differences emerge as the magnitude of the value of viscous increases.

Keywords: Korteweg–de Vries–Burgers equation, soliton, transcritical flow, viscous flow

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Authors: Yingxin Hui

Abstract:

Understanding seismic response feature of the bridges crossing fault is the basis of the seismic fortification. Taking a multi-span bridge crossing active fault under construction as an example, the seismic ground motions at bridge site were generated following hybrid simulation methodology. Multi-support excitations displacement input models and nonlinear time history analysis was used to calculate seismic response of structures, and the results were compared with bridge in the near-fault region. The results showed that the seismic response features of bridges crossing fault were different from the bridges in the near-fault region. The design according to the bridge in near-fault region would cause the calculation results with insecurity and non-reasonable if the effect of cross the fault was ignored. The design of seismic fortification should be based on seismic response feature, which could reduce the adverse effect caused by the structure damage.

Keywords: bridge engineering, seismic response feature, across faults, rupture directivity effect, fling step

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Authors: William Anderson, Kyle Kobold, Oleg Yakimenko

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Green microgrids using mostly renewable energy (RE) for generation, are complex systems with inherent nonlinear dynamics. Among a variety of different optimization tools there are only a few ones that adequately consider this complexity. This paper evaluates applicability of two somewhat similar optimization tools tailored for standalone RE microgrids and also assesses a machine learning tool for performance prediction that can enhance the reliability of any chosen optimization tool. It shows that one of these microgrid optimization tools has certain advantages over another and presents a detailed routine of preparing input data to simulate RE microgrid behavior. The paper also shows how neural-network-based predictive modeling can be used to validate and forecast solar power generation based on weather time series data, which improves the overall quality of standalone RE microgrid analysis.

Keywords: microgrid, renewable energy, complex systems, optimization, predictive modeling, neural networks

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Authors: V. D. Thi, M. Li, M. Khelifa, M. El Ganaoui, Y. Rogaume

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This paper presents a two-dimensional model to study the heat and moisture transfer through porous building materials. Dynamic and static coupled models of heat and moisture transfer in porous material under low temperature are presented and the coupled models together with variable initial and boundary conditions have been considered in an analytical way and using the finite element method. The resulting coupled model is converted to two nonlinear partial differential equations, which is then numerically solved by an implicit iterative scheme. The numerical results of temperature and moisture potential changes are compared with the experimental measurements available in the literature. Predicted results demonstrate validation of the theoretical model and effectiveness of the developed numerical algorithms. It is expected to provide useful information for the porous building material design based on heat and moisture transfer model.

Keywords: finite element method, heat transfer, moisture transfer, porous materials, wood

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Authors: Donatello Cardone

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Bilinear hysteresis models are usually used to describe the cyclic behavior of high damping rubber bearings. However, they neglect a number of phenomena (such as the interaction between axial load and shear force, buckling and post-buckling behavior, cavitation, scragging effects, etc.) that can significantly influence the dynamic behavior of such isolation devices. In this work, an advanced hysteresis model is examined and properly calibrated using consolidated procedures. Results of preliminary numerical analyses, performed in OpenSees, are shown and compared with the results of experimental tests on high damping rubber bearings and simulation analyses using alternative nonlinear models. The findings of this study can provide an useful tool for the accurate evaluation of the seismic response of structures with rubber-based isolation systems.

Keywords: seismic isolation, high damping rubber bearings, numerical modeling, axial-shear force interaction

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Authors: Yul Y. Nazaruddin, Anas Y. Widiaribowo, Satriyo Nugroho

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Boiler is one of the critical unit in a petrochemical plant. Steam produced by the boiler is used for various processes in the plant such as urea and ammonia plant. An alternative method to optimize the boiler combustion system is presented in this paper. Adaptive Neuro-Fuzzy Inference System (ANFIS) approach is applied to model the boiler using real-time operational data collected from a boiler unit of the petrochemical plant. Nonlinear equation obtained is then used to optimize the air to fuel ratio using Genetic Algorithm, resulting an optimal ratio of 15.85. This optimal ratio is then maintained constant by ratio controller designed using inverse dynamics based on ANFIS. As a result, constant value of oxygen content in the flue gas is obtained which indicates more efficient combustion process.

Keywords: ANFIS, boiler, combustion process, genetic algorithm, optimization.

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Authors: B. X. Tchomeni, A. A. Alugongo, T. B. Tengen

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In this paper, de Laval rotor system has been characterized by a hinge model and its transient response numerically treated for a dynamic solution. The effect of the ensuing non-linear disturbances namely rub and breathing crack is numerically simulated. Subsequently, three analysis methods: Orbit Analysis, Fast Fourier Transform (FFT) and Wavelet Transform (WT) are employed to extract features of the vibration signal of the faulty system. An analysis of the system response orbits clearly indicates the perturbations due to the rotor-to-stator contact. The sensitivities of WT to the variation in system speed have been investigated by Continuous Wavelet Transform (CWT). The analysis reveals that features of crack, rubs and unbalance in vibration response can be useful for condition monitoring. WT reveals its ability to detect non-linear signal, and obtained results provide a useful tool method for detecting machinery faults.

Keywords: Continuous wavelet, crack, discrete wavelet, high acceleration, low acceleration, nonlinear, rotor-stator, rub

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Authors: Aymen Laadhari, Gábor Székely

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This paper is concerned with the development of a fully implicit and purely Eulerian fluid-structure interaction method tailored for the modeling of the large deformations of elastic membranes in a surrounding Newtonian fluid. We consider a simplified model for the mechanical properties of the membrane, in which the surface strain energy depends on the membrane stretching. The fully Eulerian description is based on the advection of a modified surface tension tensor, and the deformations of the membrane are tracked using a level set strategy. The resulting nonlinear problem is solved by a Newton-Raphson method, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the presented method. We show that stability is maintained for significantly larger time steps.

Keywords: finite element method, implicit, level set, membrane, Newton method

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Authors: Drissi Mokhtaria, Chouaih Abdelkader, Fodil Hamzaoui

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Our work is about a comparison of experimental and theoretical results of the electron charge density distribution and the electrostatic potential around the M-Nitrophenol Molecule (m-NPH) kwon for its interesting physical characteristics. The molecular experimental results have been obtained from a high-resolution X-ray diffraction study. Theoretical investigations were performed under the Gaussian program using the Density Functional Theory at B3LYP level of theory at 6-31G*. The multipolar model of Hansen and Coppens was used for the experimental electron charge density distribution around the molecule, while we used the DFT methods for the theoretical calculations. The electron charge density obtained in both methods allowed us to find out the different molecular properties such us the electrostatic potential and the dipole moment which were finally subject to a comparison leading to an outcome of a good matching results obtained in both methods.

Keywords: electron charge density, m-nitrophenol, nonlinear optical compound, electrostatic potential, optimized geometric

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Authors: Ozden Saygili, Eser Cakti

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The masonry building stock in Istanbul and in other cities of Turkey are exposed to significant earthquake hazard. Determination of the safety of masonry structures against earthquakes is a complex challenge. This study deals with experimental tests and non-linear dynamic analysis of masonry structures modeled through discrete element method. The 1:10 scale model of Mustafa Paşa Mosque was constructed and the data were obtained from the sensors on it during its testing on the shake table. The results were used in the calibration/validation of the numerical model created on the basis of the 1:10 scale model built for shake table testing. 3D distinct element model was developed that represents the linear and nonlinear behavior of the shake table model as closely as possible during experimental tests. Results of numerical analyses with those from the experimental program were compared and discussed.

Keywords: dynamic analysis, non-linear modeling, shake table tests, masonry

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Authors: Guang-Ho Cha

Abstract:

This paper presents a context-sensitive media similarity search algorithm. One of the central problems regarding media search is the semantic gap between the low-level features computed automatically from media data and the human interpretation of them. This is because the notion of similarity is usually based on high-level abstraction but the low-level features do not sometimes reflect the human perception. Many media search algorithms have used the Minkowski metric to measure similarity between image pairs. However those functions cannot adequately capture the aspects of the characteristics of the human visual system as well as the nonlinear relationships in contextual information given by images in a collection. Our search algorithm tackles this problem by employing a similarity measure and a ranking strategy that reflect the nonlinearity of human perception and contextual information in a dataset. Similarity search in an image database based on this contextual information shows encouraging experimental results.

Keywords: context-sensitive search, image search, similarity ranking, similarity search

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Authors: Shuenn-Yih Chang, Chiu-Li Huang

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Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.

Keywords: dynamic analysis, high frequency, integration method, overshoot, weak instability

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Authors: Paolo Castaldo, Bruno Palazzo, Laura Lodato

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This study deals with the seizure effects of friction pendulum (FP) bearings on the seismic reliability of a 3D base-isolated nonlinear structural system, designed according to Italian seismic code (NTC08). The isolated system consists in a 3D reinforced concrete superstructure, a r.c. substructure and the FP devices, described by employing a velocity dependent model. The seismic input uncertainty is considered as a random variable relevant to the problem, by employing a set of natural seismic records selected in compliance with L’Aquila (Italy) seismic hazard as provided from NTC08. Several non-linear dynamic analyses considering the three components of each ground motion have been performed with the aim to evaluate the seismic reliability of the superstructure, substructure, and isolation level, also taking into account the seizure event of the isolation devices. Finally, a design solution aimed at increasing the seismic robustness of the base-isolated systems with FPS is analyzed.

Keywords: FP devices, seismic reliability, seismic robustness, seizure

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Authors: Gholamhosein Khosravi, Mohammad Azadi, Hamidreza Ghezavati

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In this paper, vibration of a nonlinear composite beam is analyzed and then an active controller is used to control the vibrations of the system. The beam is resting on a Winkler-Pasternak elastic foundation. The composite beam is reinforced by single walled carbon nanotubes. Using the rule of mixture, the material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are determined. The beam is cantilever and the free end of the beam is under follower force. Piezoelectric layers are attached to the both sides of the beam to control vibrations as sensors and actuators. The governing equations of the FG-CNTRC beam are derived based on Euler-Bernoulli beam theory Lagrange- Rayleigh-Ritz method. The simulation results are presented and the effects of some parameters on stability of the beam are analyzed.

Keywords: carbon nanotubes, vibration control, piezoelectric layers, elastic foundation

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Authors: Jin-Wei Liang, Hung-Yi Chen, Lung Lin

Abstract:

In this paper feedforward controller is designed to eliminate nonlinear hysteresis behaviors of a piezoelectric stack actuator (PSA) driven system. The control design is based on inverse Prandtl-Ishlinskii (P-I) hysteresis model identified using particle swarm optimization (PSO) technique. Based on the identified P-I model, both the inverse P-I hysteresis model and feedforward controller can be determined. Experimental results obtained using the inverse P-I feedforward control are compared with their counterparts using hysteresis estimates obtained from the identified Bouc-Wen model. Effectiveness of the proposed feedforward control scheme is demonstrated. To improve control performance feedback compensation using traditional PID scheme is adopted to integrate with the feedforward controller.

Keywords: the Bouc-Wen hysteresis model, particle swarm optimization, Prandtl-Ishlinskii model, automation engineering

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Authors: Paras Ram, Vikas Kumar

Abstract:

The transmission of Malaria with seasonal were studied through the use of mathematical models. The data from the annual number of Malaria cases reported to the Division of Epidemiology, Ministry of Public Health, Thailand during the period 1997-2011 were analyzed. The transmission of Malaria with seasonal was studied by formulating a mathematical model which had been modified to describe different situations encountered in the transmission of Malaria. In our model, the population was separated into two groups: the human and vector groups, and then constructed a system of nonlinear differential equations. Each human group was divided into susceptible, infectious in hot season, infectious in rainy season, infectious in cool season and recovered classes. The vector population was separated into two classes only: susceptible and infectious vectors. The analysis of the models was given by the standard dynamical modeling.

Keywords: ferrofluid, magnetic field, porous medium, rotating disk, Neuringer-Rosensweig Model

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Authors: Chaouch Souad, Abdou Latifa, Larbi Chrifi Alaoui

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This paper deals with sensorless backstepping control of induction motor using MRAS technique associated to sliding mode approach. A high order genetic algorithm structure is used to approximate a control law designed by the Backstepping technique, and to find the best parameters globally optimized. However, the Backstepping control approach is unsuitable for high performance applications because the need of a speed sensor for increased accuracy and the absence of any error decay mechanism. In this paper a nonlinear observer, obtained by combining sliding mode structure and model reference adaptive system (MRAS), is designed for the rotor flux and rotor speed estimations. To validate the proposed method, the results are presented for showing the improved drive characteristics and performances.

Keywords: Backstepping Control, Induction Motor, Genetic Algorithm, Sliding Mode observer

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Authors: Rouzbeh Ramezani, Renzo Di Felice

Abstract:

Nowadays, removal of CO₂ as one of the major contributors to global warming using alternative solvents with high CO₂ absorption efficiency, is an important industrial operation. In this study, three amines, including 2-methylpiperazine, potassium sarcosinate and potassium lysinate as potential additives, were added to the potassium carbonate solution as a base solvent for CO₂ capture. In order to study the absorption performance of CO₂ in terms of loading capacity of CO₂ and absorption rate, the absorption experiments in a blend of additives with potassium carbonate were carried out using the vapor-liquid equilibrium apparatus at a temperature of 313.15 K, CO₂ partial pressures ranging from 0 to 50 kPa and at mole fractions 0.2, 0.3, and 0.4. Furthermore, the performance of CO₂ absorption in these blend solutions was compared with pure monoethanolamine and with pure potassium carbonate. Finally, a correlation with good accuracy was developed using the nonlinear regression analysis in order to predict CO₂ loading capacity.

Keywords: absorption rate, carbon dioxide, CO2 capture, global warming, loading capacity

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Authors: Abbas Moradi, Mohsen Safajoy, Reza Yazdanparast

Abstract:

Analysis of blade-disc dovetail joints is one of the biggest challenges facing designers of aero-engines. To avoid comparatively expensive experimental full-scale tests, numerical methods can be used to simulate loaded disc-blades assembly. Mortar method provides a powerful and flexible tool for solving frictional contact problems. In this study, 2D frictional contact in dovetail has been analysed based on the mortar algorithm. In order to model the friction, the classical law of coulomb and moving friction cone algorithm is applied. The solution is then obtained by solving the resulting set of non-linear equations using an efficient numerical algorithm based on Newton–Raphson Method. The numerical results show that this approach has better convergence rate and accuracy than other proposed numerical methods.

Keywords: computational contact mechanics, dovetail joints, nonlinear FEM, mortar approach

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Authors: L. Bikulciene, E. Venskaityte, G. Jarusevicius

Abstract:

Non-stationary and nonlinear signals generated by living complex systems defy traditional mechanistic approaches, which are based on homeostasis. Previous our studies have shown that the evaluation of the interactions of physiological signals by using special analysis methods is suitable for observation of physiological processes. It is demonstrated the possibility of using deep physiological model, based interpretation of the changes of the human body’s functional states combined with an application of the analytical method based on matrix theory for the physiological signals analysis, which was applied on high risk cardiac patients. It is shown that evaluation of cardiac signals interactions show peculiar for each individual functional changes at the onset of hemodynamic restoration procedure. Therefore we suggest that the alterations of functional state of the body, after patients overcome surgery can be complemented by the data received from the suggested approach of the evaluation of functional variables interactions.

Keywords: cardiac diseases, complex systems theory, ECG analysis, matrix analysis

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Authors: Kaige Shi, Chao Jiang, Xin Li

Abstract:

The swirl gripper is an electrically activated noncontact handling device that uses swirling airflow to generate a lifting force. This force can be used to pick up a workpiece placed underneath the swirl gripper without any contact. It is applicable, for example, in the semiconductor wafer production line, where contact must be avoided during the handling and moving of a workpiece to minimize damage. When a workpiece levitates underneath a swirl gripper, the gap height between them is crucial for safe handling. Therefore, in this paper, we propose a method to estimate the levitation gap height by detecting pressure at two points. The method is based on theoretical model of the swirl gripper, and has been experimentally verified. Furthermore, the force between the gripper and the workpiece can also be estimated using the detected pressure. As a result, the nonlinear relationship between the force and gap height can be linearized by adjusting the rotating speed of the fan in the swirl gripper according to the estimated force and gap height. The linearized relationship is expected to enhance handling stability of the workpiece.

Keywords: swirl gripper, noncontact handling, levitation, gap height estimation

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Authors: M. J. Uddin, M. M. Rahman

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Transient natural convective flow dynamics of nanofluids in a horizontal homocentric annulus using nonhomogeneous dynamic model has been experimented numerically. The simulation is carried out for four different shapes of the inner wall, which is either cylindrical, elliptical, square or triangular. The outer surface of the annulus is maintained at constant low temperature while the inner wall is maintained at a uniform temperature; higher than the outer one. The enclosure is permeated by a uniform magnetic field having variable orientation. The Brownian motion and thermophoretic deposition phenomena of the nanoparticles are taken into account in model construction. The governing nonlinear momentum, energy, and concentration equations are solved numerically using Galerkin weighted residual finite element method. To find the best performer, the local Nusselt number is demonstrated for different shapes of the inner wall. The heat transfer enhancement for different nanofluids for four different shapes of the inner wall is exhibited.

Keywords: nanofluids, annulus, nonhomogeneous dynamic model, heat transfer

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Authors: Hyungoo Kang, Minsung Kim, Jinkoo Kim

Abstract:

This study investigates the performance of a 2D and 3D steel moment frame subjected to vehicle collision at a first story column using LS-DYNA. The finite element models of vehicles provided by the National Crash Analysis Center (NCAC) are used for numerical analysis. Nonlinear dynamic time history analysis of the 2D and 3D model structures are carried out based on the arbitrary column removal scenario, and the vertical displacement of the damaged structures are compared with that obtained from collision analysis. The analysis results show that the model structure remains stable when the speed of the vehicle is 40km/h. However, at the speed of 80 and 120km/h both the 2D and 3D structures collapse by progressive collapse. The vertical displacement of the damaged joint obtained from collision analysis is significantly larger than the displacement computed based on the arbitrary column removal scenario.

Keywords: vehicle collision, progressive collapse, FEM, LS-DYNA

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Authors: Aurore Cauquis, Philippe Heinrich, Mario Ricchiuto, Audrey Gailler

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In this talk, we look for an accurate time scheme to model the propagation of waves. Several numerical schemes have been developed to solve the extended weakly nonlinear weakly dispersive Boussinesq Equations. The temporal schemes used are two Lax-Wendroff schemes, second or third order accurate, two Runge-Kutta schemes of second and third order and a simplified third order accurate Lax-Wendroff scheme. Spatial derivatives are evaluated with fourth order accuracy. The numerical model is applied to two monodimensional benchmarks on a flat bottom. It is also applied to the simulation of the Algerian tsunami generated by a Mw=6 seism on the 18th March 2021. The tsunami propagation was highly dispersive and propagated across the Mediterranean Sea. We study here the effects of the order of temporal discretization on the accuracy of the results and on the time of computation.

Keywords: numerical analysis, tsunami propagation, water wave, boussinesq equations

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Authors: M. P. Nanda Kumar, K. Dheeraj

Abstract:

The design of a feedback controller, so as to minimize a given performance criterion, for a general non-linear dynamical system is difficult; if not impossible. But for a large class of non-linear dynamical systems, the open loop control that minimizes a performance criterion can be obtained using calculus of variations and Pontryagin’s minimum principle. In this paper, the open loop optimal trajectories, that minimizes a given performance measure, is used to train the neural network whose inputs are state variables of non-linear dynamical systems and the open loop optimal control as the desired output. This trained neural network is used as the feedback controller. In other words, attempts are made here to solve the “inverse optimal control problem” by using the state and control trajectories that are optimal in an open loop sense.

Keywords: inverse optimal control, radial basis function, neural network, controller design

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Authors: Rachid Marzoug, Noureddine Lakouari, Beatriz Castillo Téllez, Margarita Castillo Téllez, Gerardo Alberto Mejía Pérez

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This paper developed a two-lane cellular automata model to explain the relationship between car accidents at a signalized intersection and traffic-related parameters. It is found that the increase of the lane-changing probability P?ₕ? increases the risk of accidents, besides, the inflow α and the probability of accidents Pₐ? exhibit a nonlinear relationship. Furthermore, depending on the inflow, Pₐ? exhibits three different phases. The transition from phase I to phase II is of first (second) order when P?ₕ?=0 (P?ₕ?>0). However, the system exhibits a second (first) order transition from phase II to phase III when P?ₕ?=0 (P?ₕ?>0). In addition, when the inflow is not very high, the green light length of one road should be increased to improve road safety. Finally, simulation results show that the traffic at the intersection is safer adopting symmetric lane-changing rules than asymmetric ones.

Keywords: two-lane intersection, accidents, fatality risk, lane-changing, phase transition

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Authors: J. Bartus, J. Odrobinak

Abstract:

The paper presents a nonlinear analysis 3D model of composite steel and concrete beams with web openings using the Finite Element Method (FEM). The core of the study is the introduction of basic modeling techniques comprehending the description of material behavior, appropriate elements selection, and recommendations for overcoming problems with convergence. Results from various finite element models are compared in the study. The main objective is to observe the concrete failure mechanism and its influence on the structural performance of numerical models of the beams at particular load stages. The bearing capacity of beams, corresponding deformations, stresses, strains, and fracture patterns were determined. The results show how load-bearing elements consisting of concrete parts can be analyzed using FEM software with various options to create the most suitable numerical model. The paper demonstrates the versatility of Ansys software usage for structural simulations.

Keywords: Ansys, concrete, modeling, steel

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Authors: A. A. Behroozpoor, M. M. Mazarei

Abstract:

In this paper, a fuzzy recurrent neural network is proposed for solving the classical quadratic control problem subject to linear equality and bound constraints. The convergence of the state variables of the proposed neural network to achieve solution optimality is guaranteed.

Keywords: REFERENCES [1] Xia, Y, A new neural network for solving linear and quadratic programming problems. IEEE Transactions on Neural Networks, 7(6), 1996, pp.1544–1548. [2] Xia, Y., & Wang, J, A recurrent neural network for solving nonlinear convex programs subject to linear constraints. IEEE Transactions on Neural Networks, 16(2), 2005, pp. 379–386. [3] Xia, Y., H, Leung, & J, Wang, A projection neural network and its application to constrained optimization problems. IEEE Transactions Circuits and Systems-I, 49(4), 2002, pp.447–458.B. [4] Q. Liu, Z. Guo, J. Wang, A one-layer recurrent neural network for constrained seudoconvex optimization and its application for dynamic portfolio optimization. Neural Networks, 26, 2012, pp. 99-109.

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