Nonlinear State Equations Research Articles

A pressure based solver for simulation of non-equilibrium wet steam flows

This paper describes a pressure-based solver based on the finite volume approach for the simulation of high-speed wet steam flows with non-equilibrium condensation. The governing mathematical model consists of the system of Navier–Stokes equations for common mixture density, velocity, and enthalpy equipped with an appropriate nonlinear equation of state based on IAPWS formulation and with additional equations for liquid mass fraction and droplet radius distribution. The proposed method has been implemented into open-source package OpenFOAM and validated using available experimental data for nozzle flows. Finally, it was used for simulations of flows in turbine cascades.

Internal gravity waves in a stratified layer atop a convecting liquid core in a non-rotating spherical shell

SUMMARY Seismic and magnetic observations have suggested the presence of a stably stratified layer atop Earth’s core. Such a layer could affect the morphology of the geomagnetic field and the evolution of the core, but the precise impact of this layer depends largely on its internal dynamics. Among other physical phenomena, stratified layers host internal gravity waves (IGW), which can be excited by adjacent convective motions. Internal waves are known to play an important role on the large-scale dynamics of the Earth’s climate and on the long-term evolution of stars. Yet, they have received relatively little attention in the Earth’s outer core so far and deserve detailed investigations in this context. Here, we make a first step in that direction by running numerical simulations of IGW in a non-rotating spherical shell in which a stratified layer lies on top of a convective region. We use a nonlinear equation of state to produce self-consistently such a two-layer system. Both propagating waves and global modes coexist in the stratified layer. We characterize the spectral properties of these waves and find that energy is distributed across a wide range of frequencies and length scales, that depends on the Prandtl number. For the control parameters considered and in the absence of rotational and magnetic effects, the mean kinetic energy in the layer is about 0.1 per cent that of the convective region. IGW produce perturbations in the gravity field that may fall within the sensitivity limit of present-day instruments and could potentially be detected in available data. We finally provide a road map for future, more geophysically realistic, studies towards a more thorough understanding of the dynamics and impact of internal waves in a stratified layer atop Earth’s core.

Asymmetries in gravity currents attributed to the nonlinear equation of state

Abstract

Scalar and vector perturbations in a universe with nonlinear perfect fluid

We study a three-component universe filled with dust-like matter in the form of discrete inhomogeneities (e.g., galaxies) and perfect fluids characterized by linear and nonlinear equations of state. Within the cosmic screening approach, we develop the theory of scalar and vector perturbations. None of the energy density contrasts associated with the distinct components is treated as small. Consequently, the derived equations are valid at both sub- and super-horizon scales and enable simulations for a variety of cosmological models.

Gravitational Collapse and Singularity Removal in Rastall Theory

In the present work, we study spherically symmetric gravitational collapse of a homogeneous fluid in the framework of Rastall gravity. Considering a nonlinear equation of state (EoS) for the fluid profiles, we search for a class of nonsingular collapse solutions and the possibility of singularity removal. We find that depending on the model parameters, the collapse scenario halts at a minimum value of the scale factor at which a bounce occurs. The collapse process then enters an expanding phase in the postbounce regime, and consequently the formation of a spacetime singularity is prevented. We also find that, in comparison to the singular case where the apparent horizon forms to cover the singularity, the formation of apparent horizon can be delayed allowing thus the bounce to be causally connected to the external universe. The nonsingular solutions we obtain satisfy the weak energy condition (WEC) which is crucial for physical validity of the model.

Chaotic behavior of turbine regulating system for hydropower station under effect of nonlinear turbine characteristics

This paper studies the chaotic behavior of turbine regulating system (TRS) for hydropower station under effect of nonlinear turbine characteristics (NTC). Firstly, the nonlinear fourth-order state equation that describes the dynamic behavior of TRS under effect of NTC is established and verified. Then, the chaotic behaviors of TRS, i.e. existence of chaotic motion and path to chaotic motion, are studied. The effects of sensitivity parameters on the chaotic behavior of TRS are analyzed. Finally, the effect mechanism of NTC on chaotic behavior of TRS is revealed. The guidance for practical applications is clarified. The results show that, with the change of governor parameters, the dynamic behavior of TRS transforms among damped oscillation, persistent oscillation, frequency division and chaos phenomenon. The chaotic motion only appears under the particular value ranges of governor parameters. The path to chaotic motion of TRS is period doubling bifurcation. The chaotic motion is more likely to appear when the stability of TRS is worse. The reasonable selection of sensitivity parameters can eliminate or weaken the chaotic motion. The chaos phenomenon is caused by the nonlinear speed characteristics (NSC), not the nonlinear head characteristics. The effect of NSC is related to the values of governor parameters.

Differential stability of discrete optimal control problems with mixed contraints

Differential stability of parametric discrete optimal control problems in Banach spaces with nonconvex cost functions, nonlinear state equations and mixed constraints is studied. By establishing an abstract result on the subdifferentials of the value function of a parametric mathematical programming problem with geometrical, equality and inequality constraints, we derive formulas for computing the Mordukhovich subdifferential and the singular subdifferential of the value function to a parametric discrete optimal control problem.

Transformation of nonlinear discrete-time state equations into the observer form: extension to non-reversible case

Transformation of nonlinear discrete-time state equations into the observer form: extension to non-reversible case

Research on Estimation Method of Fuel Cell Health State Based on Lumped Parameter Model

The increasingly serious environmental pollution and the shortage of social energy have promoted the rapid development of fuel cell vehicles. The major factor which limits the commercialization of fuel cell vehicles is durability. Accurately estimating the state and parameters of a fuel cell is critical to extending the life of the fuel cell. To address this challenge, we extended a proton exchange membrane fuel cell (PEMFC) lumped parameter model and incorporated new algorithms that are essential to estimate the health of the fuel cell in a range-extended fuel cell car. The unscented Kalman filter (UKF) algorithm has been used to estimate the ohmic internal resistance of the fuel cell in real time. By using the unscented transformation (UT) method, the linearization of the nonlinear state equation is avoided, and the filtering accuracy is improved without increasing the complexity of the system. By comparing simulation and experimental results, the feasibility and accuracy of the algorithm in this paper are further verified. This method has high estimation accuracy and is suitable for an embedded system. The research of this method is an important basis for improving the control strategy of fuel cell vehicles. Reasonable use of fuel cells can extend battery life, and this method is of great significance to the commercialization of fuel cell vehicles.

Parameterizing the Impact of Unresolved Temperature Variability on the Large‐Scale Density Field: Part 1. Theory.

AbstractUnresolved temperature and salinity fluctuations interact with a nonlinear seawater equation of state to produce significant errors in the ocean model evaluation of the large‐scale density field. It is shown that the impact of temperature fluctuations dominates the impact of salinity fluctuations and that the error in density is, to leading order, proportional to the product of a subgrid‐scale temperature variance and a second derivative of the equation of state. Two parameterizations are proposed to correct the large‐scale density field: one deterministic and one stochastic. Free parameters in both parameterizations are fit using fine‐resolution model data. Both parameterizations are computationally efficient as they require only one additional evaluation of a nonlinear equation at each grid cell. A companion paper will discuss the climate impacts of the parameterizations proposed here.

A Novel Alignment Method for SINS with Large Misalignment Angles Based on EKF2 and AFIS.

In order to achieve the fine alignment of strapdown inertial navigation (SINS) under large misalignment angles, a novel filtering alignment method is proposed based on the second-order extended Kalman filter (EKF2) and adaptive fuzzy inference system (AFIS). Firstly, the quaternion is employed to represent the attitude errors of SINS. A second-order nonlinear state equation is made based on the nonlinear velocity error model and attitude error model, and the linear measurement equation is based on the velocity outputs from SINS. Then, the filtering scheme is designed based on EKF2 and AFIS. The error estimation and fine alignment can be achieved by using the proposed filtering scheme. The results of Monte Carlo Simulation show that the errors of pitch, roll and yaw misalignment angles quickly decrease to about 14″, 15″ and 7.62′ respectively in 350 s under the condition of any misalignment angles with pitch error from −40° to 40°, roll error from −40° to 40°, and yaw error from −50° to 50°. Even when the initial misalignment angles are all very large such as (80°, 120°, 170°), the proposed nonlinear alignment method still can converge normally by utilizing the adaptive fuzzy inference system (AFIS) to adjust the covariance matrix Pk/k−1. Finally, the turntable experiment was performed, and the effectiveness and superiority of the proposed method were further verified by compared with other nonlinear methods.

Time-domain analysis and experimental investigation of electromagnetic wave coupling to RF/microwave nonlinear circuits

ABSTRACT This paper presents an efficient time-domain analysis of RF/microwave nonlinear circuits in the presence of the incident uniform plane wave. In this paper, a set of differential equations, including the linear equations due to incident field coupling and nonlinear state equations for the active part of the circuit, are derived and simultaneously solved using the finite difference time domain (FDTD) method. Moreover, a numerical approach with global convergence is used to guarantee the stability of the proposed algorithm. Furthermore, for validation of the proposed method, the theoretical and experimental studies using a measurement setup, including a transverse electromagnetic (TEM) cell, are carried out for performance analysis of a low noise amplifier illuminated by the high-power incident plane wave. By comparison with measurement results, satisfactory accuracy is obtained, which indicates the ability of the proposed method in dealing with the radiated susceptibility of RF/microwave circuits, involving the electromagnetic interference (EMI) problem.

Turbulent bubbly channel flows: Effects of soluble surfactant and viscoelasticity

Interface-resolved direct numerical simulations are performed to examine the combined effects of soluble surfactant and viscoelasticity on the structure of a bubbly turbulent channel flow. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant concentrations. The latter coupling is achieved through a non-linear equation of state which relates the surface tension to the surfactant concentration at the interface. The two-fluid Navier-Stokes equations are solved using a front-tracking method, augmented with a very efficient FFT-based pressure projection method that allows for massively parallel simulations of turbulent flows. It is found that, for the surfactant-free case, bubbles move toward the wall due to inertial lift force, resulting in formation of wall layers and a significant decrease in the flow rate. Conversely, a high-enough concentration of surfactant changes the direction of lateral migration of bubbles, i.e., the contaminated bubbles move toward the core region and spread out across the channel. When viscoelasticity is considered, viscoelastic stresses counteract the Marangoni stresses, promoting formation of bubbly wall-layers and consequently strong decrease in the flow rate. The formation of bubble wall-layers for combined case depends on the interplay of the inertial and elastic, and Marangoni forces.

Second‐order necessary optimality conditions for a discrete optimal control problem with nonlinear state equations

SummaryIn this article, we study second‐order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function, nonlinear state equations and mixed constraints. In order to achieve these conditions, we first establish an abstract result on the second‐order necessary optimality conditions for a mathematical programming problem and then we derive the second‐order necessary optimality conditions for a discrete optimal control problem. The main result of this article is illustrated by two examples.

Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach

We introduce a general multi-material topology optimization framework for large deformation problems that effectively handles an arbitrary number of candidate hyperelastic materials and addresses three major associated challenges: material interpolation, excessive distortion of low-density elements, and computational efficiency. To account for many nonlinear elastic materials, we propose a material interpolation scheme that, instead of interpolating multiple material parameters (such as Young’s modulus), interpolates multiple nonlinear stored-energy functions. To circumvent convergence difficulties caused by excessive distortions of low-density elements under large deformations, an energy interpolation scheme is revisited to account for multiple candidate hyperelastic materials. Computational efficiency is addressed from both structural analysis and optimization perspectives. To solve the nonlinear state equations efficiently, we employ the lower-order Virtual Element Method in conjunction with tailored adaptive mesh refinement and coarsening strategies. To efficiently update the design variables of the multi-material system, we exploit the separable nature and improve the ZPR (Zhang–Paulino–Ramos) update scheme to account for positive sensitivities and update the design variables associated with each volume constraint in parallel. Four design examples with three types of nonlinear material models demonstrate the efficiency and effectiveness of the proposed framework.

A Federated Derivative Cubature Kalman Filter for IMU-UWB Indoor Positioning.

The inertial measurement unit and ultra-wide band signal (IMU-UWB) combined indoor positioning system has a nonlinear state equation and a linear measurement equation. In order to improve the computational efficiency and the localization performance in terms of the estimation accuracy, the federated derivative cubature Kalman filtering (FDCKF) method is proposed by combining the traditional Kalman filtering and the cubature Kalman filtering. By implementing the proposed FDCKF method, the observations of the UWB and the IMU can be effectively fused; particularly, the IMU can be continuously calibrated by UWB so that it does not generate cumulative errors. Finally, the effectiveness of the proposed algorithm is demonstrated through numerical simulations, in which FDCKF was compared with the federated cubature Kalman filter (FCKF) and the federated unscented Kalman filter (FUKF), respectively.

Attitude dynamics aiding for three-dimensional passive target tracking of strap-down seeker based on instrumental variable Kalman filter

The accuracy of three-dimensional (3D) passive target tracking under strap-down system is usually affected by the measurement accuracy of attitude angular rate and attitude angle. In order to save the problem, a novel 3D passive target tracking method based on instrumental variable Kalman filter (IVKF) aided by the attitude dynamic is proposed. At first, the maneuvering target motion model is established based on the “current” statistical model and the filtering equation of MEMS inertial measurement unit (IMU) is also set up. Then, linearize the nonlinear state equations and replace the nonlinear measurement equations with pseudolinear equations. The 3D pseudolinear Kalman filter (PLKF) algorithm is derived according to the linear Kalman filter (KF). To counter the severe bias problems with PLKF, bias compensation and recursive instrumental variable (IV) methods are considered. In order to enhance observability of the system, a 3D motion tracking sliding-mode guidance law is deduced. Finally, some mathematical simulations were made to verify the effectiveness of the proposed method. The simulation results show the effect of the measurement accuracy and complexity of the algorithm are reduced, which proves the validity of the method.

Quark stars with isotropic matter in Ho\u0159ava gravity and Einstein\u2013\xe6ther theory

We study non-rotating and isotropic strange quark stars in Lorentz-violating theories of gravity, and in particular in Hořava gravity and Einstein-æther theory. For quark matter we adopt both linear and non-linear equations of state, corresponding to the MIT bag model and color flavor locked state, respectively. The new structure equations describing hydrostatic equilibrium generalize the usual Tolman–Oppenheimer–Volkoff (TOV) equations of Einstein’s general relativity. A dimensionless parameter nu measures the deviation from the standard TOV equations, which are recovered in the limit nu rightarrow 0. We compute the mass, the radius as well as the compactness of the stars, and we show graphically the impact of the parameter nu on the mass-to-radius profiles for different equations of state describing quark matter. The energy conditions and stability criteria are also considered, and they are all found to be fulfilled.

A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

A novel notion for constructing a well-balanced scheme – a gradient-robust scheme – is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient — which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces for low Mach numbers to a recent inf–sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM–FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible.

Effective Utilization of Renewable Energy Sources in Fog Computing Environment via Frequency and Modulation Level Scaling

Fog computing introduces a distributed processing capability close to end users. The proximity of computing to end users leads to lower service time and bandwidth requirements. Energy consumption is a matter of concern in such a system with a large number of computing nodes. Renewable energy sources can be utilized to lessen the burden on the main power grid and reduce the carbon footprint, but due to fluctuations, the effective utilization of renewable energy sources needs proper resource management. In this article, we deal with properly managing the resources in a fog environment where the fog nodes are equipped with onsite renewable energy. This article aims to design an efficient mechanism to dynamically dispatch requests among computing nodes and scale frequency and modulation level, based on the current workload and the availability of renewable energy sources, to minimize the service time while keeping the renewable energy utilization and stability at a satisfactory level. We state the problem as the design of a controller for a system with time-varying nonlinear state equations. Accordingly, we borrow the Lyapunov optimization technique from the control theory to design the request dispatching mechanism and prove its asymptotic optimality. We perform extensive simulations to evaluate the effectiveness of the proposed method. The simulation results demonstrate that our proposed method outperforms the naive time-aware baseline scheme up to 26% and 39%, respectively, in terms of service time and renewable energy utilization.