Nonlinear Solution Method Research Articles

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Comparison of nonlinear solution methods for magnetic equivalent circuits of saturated induction motors

This work compares three nonlinear solution methods for the performance of an induction motor’s magnetic equivalent circuit model with magnetic saturation. The interrelation between magnetic flux density and permeability introduces nonlinearities in the differential system of equations. Three popular nonlinear solution methods are selected for comparison, namely (i) the Gauss–Seidel method, (ii) the Newton–Raphson method and (iii) the inverse Broyden’s method. While all three methods have been applied in this context before, no comparison study has been published to the authors’ best knowledge. The study finds that the inverse Broyden’s method is most performant in terms of the number of required iterations, the computation time per iteration and the resulting total computation time. However, for substantial saturation levels, the authors recommend a hybrid implementation of multiple solution methods to obtain robust and reliable convergence.

Investigation on the vibration localization of mistuned bladed disk with frictional contact

The mistuning phenomenon is commonly seen in the bladed assemblies of turbomachinery, leading to an unreasonably localized energy distribution that may increase the risk of high cycle fatigue and potentially cause blade failure. The friction coupling interaction is often introduced to attenuate the effect of excessive amplification on the structural vibration. To investigate the vibration localization of damped and mistuned bladed-disk system, a typical mistuning characterization is applied to an improved four-degree-of-freedom lumped parameter model to explicit the vibratory properties with coupled frictional contact. The statistical findings of natural frequency and modal localization factor are obtained by employing the Monte Carlo simulation to explore the influence of mistuned variance and coupled stiffness on the modal localization of free and damped bladed-disk system, respectively. The forced response sensitivity featured in the mistuned variance is quantitatively analyzed from the perspective of vibration localization and energy distribution by the nonlinear solution method. The influence of excitation order and normal load are further, respectively, discussed on the response localization level of the mistuned system. The results can provide fundamental knowledge of the impact of the mistuning pattern on vibration localization for the early design of the aeroengine.

Numerical analysis of the performance of a composite marine propeller blade subject to structural blade oscillations

High-fidelity Large Eddy Simulations (LES) are conducted over the oscillating marine propeller blades to investigate the effects of blade oscillation on the unsteady flow behaviour as well as to analyse the performance of the composite marine propeller blades. The accuracy of the results is first verified against the reference experimental results. A nonlinear frequency-domain solution or harmonic solution method is also applied to this study, based on a high-fidelity LES model, in addition to the traditional time-domain solution method to investigate the capabilities of the frequency-domain method in predicting the unsteady flow parameters, vortex generation, and hydrodynamic damping to determine the stability of the propeller blades. Results obtained reveal that the oscillation of the blade has a great impact on the vortex generation process which could impose implications on the performance and structural integrity of the composite marine propeller blades.

Nonlinear dynamic optimization of marine riser system anti-recoil under a flexible multibody framework

Anti-recoil safety of marine risers is critical for deep water offshore drilling. The challenges of riser anti-recoil optimization stem primarily from two factors: (a) nonlinearities of the violent transient response with various system stages and types of design variables and (b) discontinuities of the objective and constraint functions expressed by max–min over the time domain. In this paper, a nonlinear dynamic optimization model and solution method of marine riser system anti-recoil under a flexible multibody framework is proposed. The objective and constraint functions are elaborately designed to protect the system from dynamic buckling and wellhead and riser collision. Corresponding treatments to system state changes and time step sampling are proposed for numerical sensitivity smoothing. In addition, an extra acceleration constraint is imposed to trim and regularize the feasible domain defined by the discontinuous clearance constraint. Following that, the global convergent method of moving asymptotes (GCMMA) is applied. The presented continuous optimization formulation is numerically tested on a real marine riser platform from the X oil field in the South China Sea with a water depth of 1547 m. The numerical results are compared with those computed using a genetic algorithm (GA), surrogate method and interior point method, and the effectiveness and efficiency of the proposed method is demonstrated.

VILO SLAM: Tightly Coupled Binocular Vision–Inertia SLAM Combined with LiDAR

For the existing visual–inertial SLAM algorithm, when the robot is moving at a constant speed or purely rotating and encounters scenes with insufficient visual features, problems of low accuracy and poor robustness arise. Aiming to solve the problems of low accuracy and robustness of the visual inertial SLAM algorithm, a tightly coupled vision-IMU-2D lidar odometry (VILO) algorithm is proposed. Firstly, low-cost 2D lidar observations and visual–inertial observations are fused in a tightly coupled manner. Secondly, the low-cost 2D lidar odometry model is used to derive the Jacobian matrix of the lidar residual with respect to the state variable to be estimated, and the residual constraint equation of the vision-IMU-2D lidar is constructed. Thirdly, the nonlinear solution method is used to obtain the optimal robot pose, which solves the problem of how to fuse 2D lidar observations with visual–inertial information in a tightly coupled manner. The results show that the algorithm still has reliable pose-estimation accuracy and robustness in many special environments, and the position error and yaw angle error are greatly reduced. Our research improves the accuracy and robustness of the multi-sensor fusion SLAM algorithm.

Effects of different distractor positions on the formation of expansion, stress and displacement patterns in surgically assisted rapid maxillary expansion without pterygomaxillary disjunction: a finite element analysis study

Skeletal orthopedic expansion of maxilla is accepted as a reliable method for the treatment of transverse maxillar deficiency in growing patients. The aim of the study was to evaluate the effects of different palatal distractor positions on the expansion, stress and displacement patterns of the structures of craniofacial complex in surgical assisted rapid maxillary expansion without pterygomaxillary disjunction (PTMD) with the help of finite element analysis(FEA). Four facial skeleton models with different distractor positions (first premolar, second premolar, first molar, second molar regions) were created. In all finite element models median and lateral osteotomies were performed, without PTMD. Stress distribution was evaluated after 5 mm activation of the transpalatal distraction in all models using the nonlinear solution method in FEA. Unilateral displacement(mm) and stress distribution(MPa) were measured in three directions (x, y, and z axes) of craniofacial and maxillofacial structures in the symmetrical finite element models. In all models, the unilateral transverse displacements of the anterior teeth were greater than those of the posterior teeth, and the greatest displacement was at the central incisor level. The greatest displacement values at the central incisor level, at the anterior nasal spine(ANS) and at the posterior nasal spine(PNS) levels was measured in Model-IV, III, II and I, respectively. Mean elemental stress(von Mises stress) in the medial pterygoid plate, screw and lateral pterygoid plate regions from highest to lowest was measured in Model-IV, III, II and I, respectively. The maxilla performed outward rotation and tipping movement in all models during the expansion period. Among the distractor positions, the second molar region was found to be the most advantageous one in terms of expansion pattern. Considering the patient’s anatomy and clinical conditions, placing the palatal distractor as posteriorly as possible will result in more effective maxillary expansion.

An efficient, robust and reliable initialization of phase fractions in Rachford-Rice system of equations for three-phase split calculation

Multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three phases may co-exist. Solution of the Richford-Rice system of equations is an important operation in the multiphase flash. Within the literature, solution methods have been proposed, based on optimization methods and on nonlinear equation solution methods. Many methods in these two approaches may fail if the initial estimate is too inaccurate, and may require a substantial amount of computation only to arrive at a trivial solution in cases where the assumed number of phases is too high. The present work deals with an initialization method for the solution of the Rachford-Rice system of equations based on approximating each of the functions by means of tangent lines. It is proposed to solve the system of equations by applying a two-step Newton-type method. Through extensive numerical experimentation, the efficiency, robustness and reliability of the proposed initialization method for the solution of the Rachford-Rice system of equations by means of a Newton-type method is shown.

Estimation of the Water Table Position in Unconfined Aquifers with MODFLOW 6.

The numerical estimation of the position of the water table in unconfined aquifers is important for many practical applications. Its determination through observations or analytical methods is restricted to a few cases. Therefore, it is often estimated through numerical simulations, which may be affected by numerical artifacts and/or poor stability. We use MODFLOW to estimate the position of the water table for a seemingly simple example problem and demonstrate difficulties that can be faced when performing this kind of numerical simulation. We explain the causes for the numerical challenges that originate from the properties of the mathematical equations that must be solved. Based on the results of more than 600 steady-state simulations, we show how the stability of the numerical solution can be affected by the values of physical parameters that define the problem (e.g., recharge rate, anisotropy ratio, and other parameters that control the numerical algorithm such as settings of the linear and nonlinear solution methods). Finally, we comment on some best practices to apply numerical simulations to estimate the water table position.

An Efficient Localized Nonlinear Solver for Simulating Natural Depletion of Unconventional Reservoirs with Complex Fracture Networks

Summary Simulating unconventional reservoirs efficiently and accurately poses a big challenge. Transient flow can last for a long period and sharp solution gradients appear because of the severe permeability contrast between fracture and matrix. Although high-resolution well and fracture models are required to achieve sufficient accuracy, they are computationally too demanding for field models with many hydraulic-fracture stages. This paper aims to develop a nonlinear solution method, which is applicable to discrete fracture models (DFMs) for unconventional reservoirs. The localization algorithm takes advantage of solution locality on timestep and Newton iteration levels. We study the new method through multiple natural-depletion cases containing discrete fractures and compositional models. It is found that a large extent of solution locality displays over iterations as well as timesteps. The developed nonlinear solver exhibits outstanding simulation efficiency compared with a standard solver. A significant speedup is achieved by focusing computations to the locales that undergo considerable solution changes. Moreover, nonlinear convergence performance is maintained, with no degradation of the solution results.

Load Bearing Capacity of Light Timber Frame Walls Under Fire

Light Timer Frame Walls are made of solid timber elements and are usually fire protected by other materials. This investigation determines the residual load bearing capacity of Light Timber Frame Walls for fire rating periods of 30, 60, 90 and 120 min. The timber frame is made with 3 studs and 2 tracks, using two different fire protection levels (one and two gypsum layers). The computational model includes the thermal analysis under standard fire, including all types of materials and a sequential mechanical analysis with incremental load applied for each fire rating periods, applied just for the load bearing material. Nonlinear solution methods are used with small tolerance values for solution convergence, to determine the correct temperature and displacement fields. The thermal and the mechanical properties are considered temperature dependent. The mechanical analysis considers large displacement behaviour, and the charring effect of wood is included by the reduction of the stiffness and strength of the timber. The results show that the load bearing capacity decreases with the fire exposure time. A new proposal is presented between the load bearing capacity and the fire rating and finally the char layer is compared with the current and future version of Eurocode 5, part 1.2.

Integrated modeling of regional and park-level multi-heterogeneous energy systems

Multi-heterogeneous energy (MHE) system includes both energy systems: regional-level and park-level. Energy hub station (EHS) is the energy interaction hub between regional-level energy system and park-level energy users. In order to achieve the multi-energy optimal dispatching, it is very essential to properly coordinate the lean modeling and fast solution of MHE system. In this paper, the component-level modeling and linearization method of MHE system is studied. Furthermore, an improved unified modeling and linearization method for energy hub station is proposed to obtain energy optimal dispatching. In the case study, 39-node power system and 20-node natural gas system are utilized as regional energy networks, and two energy hub stations are constructed as park-level energy hubs. Based on the hierarchical optimization structure, the simulation analysis is carried out. Compared with the nonlinear solution methods, the effectiveness of the proposed method is verified in terms of solution accuracy and calculation time.

An averaging principle for nonlinear parabolic PDEs via FBSDEs driven by G-Brownian motion

In this paper, we study the averaging problem for a class of forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) with rapidly oscillating coefficients, which corresponds to the singular perturbation problem of a kind of fully nonlinear partial differential equations (PDEs). With the help of the nonlinear stochastic analysis techniques and viscosity solution methods, we prove that the limit distribution of the solution is the unique viscosity solution to a fully nonlinear PDE.

POST-PEAK RESPONSE AUTOMATIC SOLUTIONS IN STRUCTURAL ENGINEERING PROBLEMS – A REVIEW

The ill-condition of stiffness matrix at the unstable region for example at the strain-softening region, the load control method will not be valid to give the solution therefore the displacement control method is essential to use. The stiffness matrix is changed at different steps by used load control and solution is found using any nonlinear solution method of Newton’s family. Therefore, Newton’s method will not give a solution to the unstable region. The iteration method and solution process may also stop when the diagonal element of the stiffness matrix becomes negative or zero if Newton's method is adopted. These difficulties need to be overcome and a non-positive definite stiffness matrix is retained and used for iteration in the arc-length method. A scientific review of literature has been made to pinpoint the difficulties faced in the prediction of complete load-displacement response of structures and techniques available to overcome the same. Displacement control methods have been reviewed with emphases on the arc length methods which can be integrated with the traditional finite element method.

Numerical Nonlinear Static Analysis of Cutout-Borne Multilayered Structures and Experimental Validation

The influence of two different types of nonlinear strain-displacement kinematics (Green–Lagrange and von Kármán) and their importance are investigated in this analysis by computing the static deflection and stress (normal and shear) values of cutout abided composite structures. The structural deformation behavior under the external loading conditions is derived with a higher-order polynomial without hampering the strain and stress (in-plane and out-of-plane) continuity. The mathematical formulation of the curved structure is derived, and the governing equation order is reduced using nonlinear finite element steps. A generic computer code is prepared using the derived mathematical formulation in conjunction with the isoparametric finite element technique in a MATLAB environment. The final output (that is, the nonlinear deflection and stress data) is predicted using a robust nonlinear solution method (Picard’s iteration). The suitability of full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of a higher-order displacement field model for the laminated structure is established by comparing the experimental deflection data and published analytical solutions. Furthermore, a series of numerical examples is solved to show the influences of different individual and/or the combined parametric influences on the structural deflections and the normal/shear stress (in-plane and out-of-plane) values, including the variable geometrical shapes.

Saddle cycles: Solving rational expectations models featuring limit cycles (or chaos) using perturbation methods

Unlike linear ones, nonlinear business cycle models can generate sustained fluctuations even in the absence of shocks (e.g., via limit cycles/chaos). A popular approach to solving nonlinear models is perturbation methods. I show that, as typically implemented, these methods are incapable of finding solutions featuring limit cycles or chaos. Fundamentally, solutions are only required not to explode, while standard perturbation algorithms seek solutions that meet the stronger requirement of convergence to the steady state. I propose a modification to standard algorithms that does not impose this overly strong requirement. Dynamic equilibrium economies computational methods nonlinear solution methods limit cycles chaos C63 C68 E37

An Experimental and Numerical Study on the Aerodynamic Performance of Vibrating Wind Turbine Blade with Frequency-Domain Method

A highly efficient nonlinear frequency-domain solution method is proposed and employed to investigate the aerodynamic and aeromechanical performances of an oscillating wind turbine blade aerofoil in this study. Extensive validations of a frequency-domain method against an experiment as well as a typical time-domain solution method are provided in this paper. An experiment is also designed and conducted to measure pressure distributions over an aerofoil as well as to validate the numerical model. Unsteady pressure distributions and aeroelasticity parameters of the oscillating NACA0012 aerofoil are computed at various angles of attack and Reynolds numbers. Results indicate that the difference of unsteady pressure distributions between the two surfaces of the aerofoil becomes larger as the angle of attack is increased, whereas the flow separation on the suction surface is reduced by raising the Reynolds number. The turbulent flow develops in the downstream region due to the laminar vortex shedding at lower Reynolds numbers. It is also revealed that the Reynolds number has an impact on the aeroelasticity, and the aerodynamic damping value is larger at higher Reynolds numbers. The comparison between the frequency-domain method and the time-domain method shows that the frequency-domain method is not only accurate but also computationally very efficient as the computation time is reduced by 90%.

Optimal vehicle trajectory planning scheme considering passengers’ comfort for advanced driving assistance

Speed guidance is a significant application in the driver assistance system or driverless automated system. Considering passengers’ comfort under different road conditions, this paper proposes an optimal vehicle speed trajectory planning strategy using the nonlinear programming solution method. In this method, the Pearl curve is taken as the basic model to describe the time-interval characteristics so that the speed and acceleration variation can be further obtained. Taking the weighted dimensionless average accelerated speed and vehicle travelling time as optimal objective, and the speed limit of intersection, speed limit of section, the position of the intersection, the maximum acceleration and deceleration rates, and the arrival time at the downstream intersection as constraints, a multi-variable and single-objective optimization model is built and calculated using Genetic Algorithm after discretization process of the original continuous acceleration variation curve. Results from case study shows that the proposed method presents substantial performance in planning a vehicle trajectory with optimal comfort on the basis that the vehicle arrives the downstream intersection as early as possible.

High-fidelity CFD simulations of two wind turbines in arrays using nonlinear frequency domain solution method

Aerodynamics of a wind turbine within windfarms is strongly influenced by the wake of neighbouring turbines. In particular, the performance of a wind turbine can be dramatically reduced depending on its location in the wake region of an upstream turbine. A detailed investigation of the effect of the upstream turbine on the downstream turbine with respect to their distances is essential for the design and optimisation of wind farm layouts. Conventional time domain solution methods, such as Unsteady Reynolds Averaged Navier Stokes (URANS) based Computational Fluid Dynamics (CFD) model of wind turbines in arrays, can provide a detailed analysis of this interaction effect. These methods are, however, impractical due to a high computational cost required for modelling turbines in array configurations. In this paper, a novel modelling and computational method is proposed to simulate two wind turbines in arrays by considering them as a multi-stage turbine. A nonlinear frequency domain solution method is then employed to model flow nonlinearities due to their interactions. The distances between the turbines are varied, and the effects of the upstream wind turbine on the downstream one are thoroughly investigated. Extensive validations of the nonlinear frequency domain solution method against the conventional time domain solution method reveal that the proposed frequency domain solution method provides accurate results while reducing the computational cost by one to two orders of magnitude.

Large deflection stability of axially functionally graded tapered cantilever column

This article studies the large deflection stability of axially functionally graded (AFG) cantilever column for analyzing post-buckling behavior. Consideration is given to a nonlinearly tapered column with a regular polygon cross section, whose volume is constant. The governing differential equations of the problem are derived based on the large deflection beam theory. To calculate the elastica and buckling load of the AFG column, the differential equations are solved with the direct integration method together with the nonlinear solution method. The computed results are compared with those reported in literature and obtained from finite element method. Numerical examples are provided to ascertain the effects of the geometric and material parameters on the elastica and buckling load. The elastica shape is shown to depend on applied load and column properties, and the optimal shape such that the AFG column with fixed volume enables to bear the maximum buckling load is discussed.