Uncoupled Equations Research Articles

Liquefied Natural Gas Tanks for Suppressing Pitching Motions of FLNG Facility

Floating liquefied natural gas (FLNG) facility using partially filled tanks for control of pitch motion response to wave-exciting forces is investigated in this paper. The governing equations for sloshing analysis of rectangular tanks under pitch motion excitation are first established, then the spatial (boundary- value) partial derivatives are approximated by finite differences. The uncoupled pitch equation of FLNG is derived by assuming that pitch is uncoupled from other modes of vibration. By using state-space model to represent fluid-memory effect, the pitch equation can be transformed to first- order ordinary differential equations which can be solved with sloshing equations simultaneously with the given initial conditions. By using the proposed coupling model for FLNG facility and the liquefied natural gas (LNG) tanks, the performance of partially filled tanks for suppressing pitching motions of FLNG facility is numerically assessed. The parametric studies on the example FLNG show that there is a beneficial filling level by which the pitch motion of FLNG can be considerably reduced.

Pt-Symmetric Nonlinear Directional Fiber Couplers with Gain and Loss for Ultrashort Optical Pulses

In this paper, we study the higher-order nonlinear Schrdinger (HNLS) equations for a directional fiber optic coupler with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering (SRS) effects and gain in one waveguide and loss in the other. We have reduced the system equations to an uncoupled HNLS equation which has bright and dark soliton solutions. It is shown that the Hamiltonian for this equation has a PT-symmetric form. Furthermore the equilibrium points are found and the discussion about their stability is presented.

On radiating solitary waves in bi-layers with delamination and coupled Ostrovsky equations.

We study the scattering of a long longitudinal radiating bulk strain solitary wave in the delaminated area of a two-layered elastic structure with soft ("imperfect") bonding between the layers within the scope of the coupled Boussinesq equations. The direct numerical modelling of this and similar problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and averaging with respect to the fast space variable, leading to the coupled Ostrovsky equations in bonded regions and uncoupled Korteweg-de Vries equations in the delaminated region. We show that the semi-analytical approach agrees well with direct numerical simulations and use it to study the nonlinear dynamics and scattering of the radiating solitary wave in a wide range of bi-layers with delamination. The results indicate that radiating solitary waves could help us to control the integrity of layered structures with imperfect interfaces.

A new direct time integration method for the semi-discrete parabolic equations

A direct time integration method is presented for the solution of systems of first order ordinary differential equations, which represent semi-discrete diffusion equations. The proposed method is based on the principle of the analog equation, which converts the N coupled equations into a set of N single term uncoupled first order ordinary differential equations under fictitious sources. The solution is obtained from the integral representation of the solution of the substitute single term equations. The stability and convergence of the numerical scheme is proved. The method is simple to implement. It is self-starting, unconditionally stable, accurate, while it does not exhibit numerical damping. The stability does not demand symmetrical and positive definite coefficient matrices. This is an important advantage, since the scheme can solve semi-discrete diffusion equations resulting from methods that do not produce symmetrical matrices, e.g. the boundary element method. The method applies also to equations with variable coefficients as well as to nonlinear ones. It performs well when long time durations are considered and it can be used as a practical method for integration of stiff parabolic equations in cases where widely used methods may fail. Numerical examples, including linear as well as non linear systems, are treated by the proposed method and its efficiency and accuracy are demonstrated.

Simultaneous controllability of damped wave equations

We study the exact controllability of q uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.

Comparison of Coupled and Uncoupled Consolidation Equations Using Finite Element Method in Plane-Strain Condition

In the current paper, the consolidation settlement of a strip footing over a finite layer of saturated soil has been studied using the finite element method. In Biot’s coupled consolidation equations, the soil deformation and excess pore pressure are determined simultaneously in every time step which refers to the hydro-mechanical coupling. By considering a constant total stress throughout the time and by assuming that volume strain is a function of isotropic effective stress, uncoupled consolidation equations can be obtained using coupled consolidation equations. In these uncoupled equations, excess pore pressure and deformation are determined separately. In this approach, the excess pore pressure can be identified in the first stage. Using the calculated excess pore pressure, the soil deformation is determined through effective stress-strain analyses. A computer code was developed based on coupled and uncoupled equations that are capable of performing consolidation analyses. To verify the accuracy of these analyses, the obtained results have been compared with the precise solution of Terzaghi’s one-dimensional consolidation theory. The capability of these two approaches in estimation of pore water pressure and settlement and to show Mandel-Crayer’s effect in soil consolidation is discussed. Then, the necessity of utilizing coupled analyses for evaluating soil consolidation analysis was investigated by comparing the coupled and uncoupled analyses results.

Free vibration study of sandwich plates using a family of novel shear deformable dynamic stiffness elements: limitations and comparison with the finite element solutions

In this paper, the dynamic stiffness matrix of a completely free rectangular multi-layer plate element based on Reddy's higher-order shear deformation theory is derived. The reduction of the proposed model to the first-order shear deformation theory-based formulation is presented. Three coupled Euler-Lagrange equations of motion have been transformed into two uncoupled equations introducing a boundary layer function. The proposed model enables free transverse vibration analysis of rectangular multi-layer plates with (transversely) isotropic layers having arbitrary combinations of boundary conditions.The influence of transverse shear deformation is discussed along with the applicability of two shear deformable dynamic stiffness elements. Moreover, the influence of the boundary conditions on the free vibration characteristics of sandwich panels has been discussed. The natural frequencies obtained using different dynamic stiffness multi-layer plate elements have been validated against the solutions from the commercial software Abaqus and the previously verified numerical solutions using layered finite elements. The limitations of the model regarding the differences between material properties of the face and core layers within a sandwich plate are highlighted. The influence of face-to-core thickness ratio on natural frequencies is illustrated, while a variety of new results is provided as a benchmark for future investigations.

Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section

A novel method is proposed to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. For a wide class of Timoshenko beams, this method enables us to reduce the coupled governing differential equations with variable coefficients to a pair of uncoupled second-order differential equations of Sturm–Liouville type with respect to the rotation angle due to bending. The reduced equations contain two important parameters, one describing the variations of translational inertia and bending rigidity along the beam axis, and the other reflecting the comprehensive effect of rotatory inertia and shear deformation. A series of exact analytical solutions are derived from the reduced equations for the first time, and several examples are also provided as benchmarks.

Nonlinear propagation of ion-acoustic waves in self-gravitating dusty plasma consisting of non-isothermal two-temperature electrons

Nonlinear propagation of ion-acoustic waves in self-gravitating multicomponent dusty plasma consisting of positive ions, non-isothermal two-temperature electrons and negatively charged dust particles with fluctuating charges and drifting ions has been studied using the reductive perturbation method. It has been shown that nonlinear propagation of ion-acoustic waves in gravitating dusty plasma is described by an uncoupled third order partial differential equation which is a modified form of Korteweg–deVries equation, in contraries to the coupled nonlinear equations obtained by earlier authors. Quasi-soliton solution for the ion-acoustic solitary wave has been obtained from this uncoupled nonlinear equation. Effects of non-isothermal two-temperature electrons, gravity, dust charge fluctuation and drift motion of ions on the ion-acoustic solitary waves have been discussed.

Characterization of the static behavior of electrically actuated micro-plates using extended Kantorovich method

This paper presents a new approach based on Extended Kantorovich Method to simulate the static response of micro-plates to electrostatic actuation. The presented model accounts for the electric force nonlinearity of the excitation as well as the applied in-plane loads. Using a one term Galerkin approximation and following the extended Kantorovich procedure, the nonlinear partial differential equation governing the micro-plate deflection reduces to two un-coupled nonlinear ordinary differential equations with constant coefficients which can be solved iteratively with rapid convergence to yield the desired solution. A parametric study has also been performed to characterize the effect of various design parameters such as applied in-plane loads and micro-plate aspect ratio to pull-in limits of micro-plates. Results in some specific cases have been validated, comparing them with other theoretical and experimental findings reported in the literature as well as the results of the finite element simulation of the problem. It is shown that rapid convergence, high precision and independency of initial guess function make extended Kantorovich method an effective and accurate design tool for design optimization of micro-plates under electrostatic actuation.

Time Analysis of Dissipative Systems Elastic Response. Part 1

The article presents the results for the theory of time analysis of discrete dissipative systems, the vibrations of which are described with the help of ordinary differential equations with constant coefficients. The theory is based on the algebraic approach centered around the research of matrix quadratic equation which is characteristic for the homogeneous system of ordinary differential equations. The basic concepts of the theory applied to the analysis of elastic systems: mathematical apparatus, solution structure of matrix quadratic equation and properties of matrix ratios, are presented. The conditions, under which the structure of solutions of the quadratic equation corresponds coupled and uncoupled differential equations of motion, are considered. The scheme of the solution of solving dissipative system elastic response equations, that allows determining the displacement and speeds of the nodes of the construction as the result of the effects of arbitrary dynamic load is given. These equations are written in a non-trivial matrix form of Duhamel's integral.

Simulation of a Full Fuel Cell Membrane Electrode Assembly Using Pore Network Modeling

A pore network model has been applied to a both sides of a fuel cell membrane electrode assembly. The model includes gas transport in the gas diffusion layers and catalyst layers, proton transport in the catalyst layers and membrane, and percolation of liquid water. This paper presents an iterative algorithm to simulate a steady state isothermal cell with a 3D pore network model for constant voltage boundary condition. The proposed algorithm provides a simple method to couple the results of the anode and the cathode sides by iteratively solving the uncoupled equations of the transport processes. It was found that local water blockages at the GDL/CL interface not only affect concentration polarization, but also might change ohmic polarization of the cell. Depending on the liquid water configuration in the porous electrodes, the protons generated in the anode need to travel longer paths to reach the active sites of the cathode; consequently, the IR loss will be increased in the presence of liquid water. This finding highlights the strength of pore network models which resolve discrete water blockages in the electrodes.

Free vibration analysis of plate assemblies using the dynamic stiffness method based on the higher order shear deformation theory

This paper deals with the free vibration analysis of isotropic plate assemblies using the dynamic stiffness method (DSM) based on the Reddy’s higher-order shear deformation theory (HSDT). Using the proposed method, the isotropic rectangular plate assemblies of non-uniform thickness and material properties can be analyzed. The proposed model does not have any restrictions regarding the boundary conditions or the frequency limitations. It enables free vibration analysis of both thin and thick plates, making it advantageous in comparison with the conventional finite element method (FEM) regarding the computational cost and the accuracy of the results.Three coupled Euler-Lagrange equations of motion based on the HSDT have been transformed into two uncoupled equations of motion introducing a boundary layer function. The dynamic stiffness matrix for a completely free rectangular plate element has been derived using the superposition and the projection method.The proposed numerical model has been applied in the free vibration analysis of rectangular plate assemblies. Along with the convergence study, the results for natural frequencies have been validated against the existing data from the literature, the previous results from the authors as well as the results obtained by using the finite element software Abaqus. Excellent agreement has been obtained. Finally, a variety of new results is provided as a benchmark for future investigations.

Gravitational, shear and matter waves in Kantowski-Sachs cosmologies

A general treatment of vorticity-free, perfect fluid perturbations of Kantowski-Sachs models with a positive cosmological constant are considered within the framework of the 1+1+2 covariant decomposition of spacetime. The dynamics is encompassed in six evolution equations for six harmonic coefficients, describing gravito-magnetic, kinematic and matter perturbations, while a set of algebraic expressions determine the rest of the variables. The six equations further decouple into a set of four equations sourced by the perfect fluid, representing forced oscillations and two uncoupled damped oscillator equations. The two gravitational degrees of freedom are represented by pairs of gravito-magnetic perturbations. In contrast with the Friedmann case one of them is coupled to the matter density perturbations, becoming decoupled only in the geometrical optics limit. In this approximation, the even and odd tensorial perturbations of the Weyl tensor evolve as gravitational waves on the anisotropic Kantowski-Sachs background, while the modes describing the shear and the matter density gradient are out of phase dephased by π /2 and share the same speed of sound.

Probability density evolution method: Background, significance and recent developments

Investigation of a stochastic system from a viewpoint of studying the randomness propagation process in a physical system starts to play an important role in understanding the complete performance or behaviour of engineering systems. Following this path, the probability density evolution method (PDEM) has been developed by Li and Chen at the beginning of this century on the basis of the principle of probability preservation and its random event description. A family of generalized probability density evolution equation (GPDEE) was then derived. The systematic analysis indicates that the new family of equation actually reveals the logical fundamentals of randomness propagation in a physical system: the transition of the probabilistic structure in a stochastic system definitely relies on the change of physical state of the system. This paper devotes to a summary on the background and basic theoretical developments of the PDEM. Considering the limitation of probability conservative systems for the GPDEE, a novel concept of probability dissipation is introduced and a completely uncoupled partial differential equation is derived as well with respect to the evolutionary probability density function, which holds for any physical quantity of a probability dissipative system. For illustrative purposes, several engineering applications, including the dynamic reliability assessment of controlled structures with fractional derivative viscoelastic dampers and the stability analysis of structures under dynamic loading, are investigated, respectively.

Exact Solutions of Coupled Multispecies Linear Reaction-Diffusion Equations on a Uniformly Growing Domain.

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies

The dynamic stiffness matrix of completely free rectangular Mindlin plate element is presented in this paper. The system of three coupled equations of motion is transformed into two uncoupled equations introducing a boundary layer function. The dynamic stiffness matrix is derived by use of the superposition method and the projection method. Using the proposed method natural frequencies of individual plates and plate assemblies with arbitrary boundary conditions are computed and validated against the results available in the literature and the finite element analysis. High efficiency and accuracy of the results are demonstrated.

Axisymmetric dynamic response of the multi-layered transversely isotropic medium

By virtue of the precise integration method (PIM) and the technique of mixed variable formulations, solutions for the dynamic response of the multi-layered transversely isotropic medium subjected to the axisymmetric time-harmonic forces are presented. The planes of cross anisotropy are assumed to be parallel to the horizontal surface of the stratified media. Four kinds of vertically acting axisymmetric loads are prescribed either at the external surface or in the interior of the soil system. Thicknesses and number of the medium strata are not limited. Employing the Hankel integral transform in cylindrical coordinate, the axisymmetric governing equations in terms of displacements of the multi-layered media are uncoupled. Applying mixed variable formulations, more concise first-order ordinary differential matrix equations from the uncoupled motion equations can be obtained. Solutions of the ordinary differential matrix equations in the transformed domain are acquired by utilizing the approach of PIM. Since PIM is highly accurate to solve the sets of first-order ordinary differential equations, any desired accuracy of the solutions can be achieved. All calculations are based on the corresponding algebraic operations and computational efforts can be reduced to a great extent. Comparisons with the existing numerical solutions are made to confirm the accuracy of the present solutions proposed by this procedure. Several examples are illustrated to explore the influences of the type and degree of material anisotropy, the frequency of excitation and loading positions on the dynamic response of the stratified medium.

Optimal State Estimation of Spinning Ping-Pong Ball Using Continuous Motion Model

Precise trajectory prediction is a fundamental issue for ping-pong robot systems. Due to the difficulty of spin estimation and the complexity of the motion model, most existing algorithms ignore the effect of spin, which will result in a significant deviation in trajectory prediction of spinning ping-pong ball. Some literatures proposed to estimate spin state based on trajectory bias, but due to the limitations of the discrete motion model they derived, only polynomial fitting method can be used for spin estimation, rather than model-based method, which will cause inaccurate spin estimation and trajectory prediction. In this paper, we derive a continuous motion model (CMM) of spinning ping-pong ball based on forces analysis. During the derivation, the Fourier series is used to fit the velocity changing over time, which transforms the model from unsolvable coupled variable-coefficient differential equations to solvable uncoupled equations. On the strength of the CMM, a model-based optimal algorithm for ball's motion state <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> estimation is proposed. Using the initial trajectory acquired by a stereo vision system, the proposed method first estimates the motion state approximately with polynomial fitting, and then uses gradient descent method to achieve a model-based optimal estimation by minimizing a cost function corresponding to the differences between trajectory predictions and observations. We also prove that this optimization problem can be plotted as a convex optimization problem; thus, the globally optimal solution can be obtained. The experimental results confirm the effectiveness and accuracy of the proposed method.

Line source in a poroelastic layer bounded by an elastic space

SummaryThe fundamental solution of a continuous line source, injecting fluid at a constant rate over the thickness of a poroelastic reservoir bounded by infinite impermeable elastic layers, is derived in this paper. This idealized problem has applications in hydrogeology and in petroleum engineering, as it can be used to assess the mechanical perturbations caused by injection or withdrawal of fluid in the subsurface through a vertical well. Construction of the solution takes advantage of the uncoupling of the pore pressure field, which, in this particular case, is given by the classical singular solution of the diffusion equation for an infinite line source. The mechanical fields then are determined by solving an elasticity‐like problem with a body force field proportional to the time‐dependent pore pressure gradient. On account of the problem symmetries, the Navier equations of elasticity reduce to two uncoupled partial differential equations for the radial and vertical (axial) displacement components, which are solved by a twofold application of Fourier and Hankel transforms. The solution exhibits different regimes at small, intermediate, and large times. When the diffusion radius, proportional to the square root of time, is smaller than or comparable to the thickness of the permeable layer, the induced deformation is confined to a region with a characteristic dimension of the same order as the diffusion radius. At large time, when the diffusion radius is large compared with the permeable layer thickness, the deformation rate in the reservoir is essentially oedometric (uniaxial). The different regimes of solutions are justified with a conceptual model based on identifying the evolving characteristics of complementary interior and exterior domain problems. The derived solution can serve as a valuable benchmark for coupled reservoir simulators. It also provides insights in to such problems as waterflooding, shearing at reservoir/cap rock interfaces, and stress redistribution around producing wells. Copyright © 2015 John Wiley &amp; Sons, Ltd.