Time-dependent Coordinate Transformation Research Articles

Novel Criteria of Stochastic Stability for Discrete-Time Markovian Jump Singular Systems via Supermartingale Approach

This article deals with the problem of stochastic stability for a class of discrete-time Markovian jump singular systems. A time-dependent coordinate transformation is provided, under which an equivalent form of the original Markovian jump singular systems can be obtained. This equivalent form not only shows the inherent state jump behavior at the switching instants, but also plays an important role in the stochastic stability analysis. Constructing a supermartingale over some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -algebra together with the property of conditional expectation, a necessary and sufficient condition is established to ensure that the equilibrium point of the underlying system is exponentially stable in mean square. Two numerical examples are given to illustrate the effectiveness of the developed results.

Acoustic black hole analogy to analyze nonlinear acoustic wave dynamics in accelerating flow fields

We present a physical model and a numerical method based on a space- and time-dependent Galilean-type coordinate transformation to simulate acoustic waves in the presence of an accelerating background flow field with sonic transition. Kinematically, the coordinate transformation is designed so as to maintain the well-posedness of the transformed wave equation, which is solved in a fixed computational domain using standard finite differences. Considering an acoustic black hole analogy, we analyze the nonlinear dynamics of acoustic waves in a stationary but non-uniformly accelerating flow field under the assumption of spherical symmetry. The choice of the acoustic black hole analogy is motivated by the fact that the steady-state spherical sonic horizon allows us to parameterize the wave-flow configuration in terms of a Helmholtz number He=c2/(λagh), which is expressed as a function of the speed of sound c, the emitted wavelength λa, and the flow acceleration at the sonic horizon, that is, the acoustic surface gravity gh. The results of the numerical simulations show that He describes geometrically similar sets of wave characteristics for different combinations of gh and λa. However, we also observe nonlinear variations of the wave amplitude along the wave characteristics, which are attributed to nonlinear Doppler modulations. It appears that these amplitude modulations depend on the acceleration of the flow field and can, therefore, differ for geometrically similar characteristics.

Ковариантный формализм гидродинамики гамильтоновых систем

The theory of hydrodynamic reduction of non-autonomous Hamiltonian mechanics (V. Kozlov, 1983) is presented in the geometric formalism of bundles over the time axis R. In this formalism, time is one of the coordinates, not a parameter; the connections describe reference frames and velocity fields of mechanical systems. The equations of the theory are presented in a form that is invariant with respect to time-dependent coordinate transformations and the choice of reference frames.

High-order finite difference method for the Schrödinger equation on deforming domains

A high-order finite difference discretisation of the Schrödinger equation on domains that deform in time is presented. For the domain deformation, a time-dependent coordinate transformation is used. The utilisation of summation-by-parts finite difference operators, combined with boundary conditions imposed weakly by a penalty technique, leads to a provably stable scheme. The convergence properties of the scheme are verified by numerical computations. The capabilities of the numerical scheme are demonstrated by simulations of Berry phases in deforming quantum billiards.

Fast simulations of cosmic large-scale structure with massive neutrinos

Accurate cosmological simulations that include the effect of non-linear matter clustering as well as of massive neutrinos are essential for measuring the neutrino mass scale from upcoming galaxy surveys. Typically, Newtonian simulations are employed and the neutrino distribution is sampled with a large number of particles in order to beat down the shot noise. Here we perform very efficient simulations with light massive neutrinos which require virtually no extra cost over matter-only simulations, and furthermore do not require tracer particles for the neutrinos. Instead, we use a weak-field dictionary based on the recently developed Newtonian motion approach, where Newtonian simulations for matter are paired with a linear relativistic Boltzmann code to allow for an absorption of the neutrino evolution into a time-dependent coordinate transformation. For this, only minimal modifications in existing N-body codes are required, which we have explicitly implemented in gevolution and gadget-2. Our fast method determines the non-linear matter power spectrum to permille-level precision when compared against state-of-the-art simulations that have been performed for 0.1 eV ⩽ ∑ mν ⩽ 0.3 eV.

Axisymmetric and Non-Axisymmetric Vibration of Thin Growing Viscoelastic Disc

Axisymmetric and non-axisymmetric in-plane vibrations of thin radially growing disc are considered in the frame of the Kelvin–Voigt model of linear viscoelasticity. The main focus of the research is on asymptotic behavior of solutions to the model. Mixed boundary value problems are formulated and converted into standard form by means of time dependent coordinate transformation. The boundary value problems obtained are transformed into infinite systems of ordinary differential equations using the Galerkin-Kantorovich method. It is shown that both axisymmetric and non-axisymmetric problems can be considered in terms of the same mathematical model. The simplified single mode model of the growing disc vibration is formulated and analysed. In the frame of this model the exact solutions are obtained in some interesting special cases. In the general case, solutions are derived using the Wentzel–Kramers–Brillouin (WKB) method. Depending on the exponent of radial growth, the vibration amplitude of the disc demonstrates qualitatively different behavior: if the exponent is less than 0.5 the vibration amplitude decays, if it is more than 0.5 the amplitude increases without bound no matter what size of damping coefficient is assumed. If the exponent equals 0.5 the vibration behavior depends on damping, namely there exist the critical damping coefficient and if the damping coefficient of a particular mode is less than this coefficient, the vibration amplitude grows without bound. Otherwise, if the damping coefficient of the mode is larger than the critical damping coefficient the vibration amplitude decays to zero. In the case where damping coefficient equals to the critical damping one, the amplitude of vibration remains constant. All of the qualitative results obtained for the single mode model are numerically tested and verified using a truncated multimodal system.

Adaptive Model-Free Coupling Controller Design for Multi-Axis Motion Systems

In this study, we introduce an adaptive model-free coupling controller while using recurrent fuzzy neural network (RFNN) for multi-axis system to minimize the contour error. The proposed method can be applied to linear or nonlinear multi-axis motion control systems following desired paths. By the concept of cross-coupling control (CCC), multi-axis system is transferred into a nonlinear time-varying system due to the time-dependent coordinate transformation; tangential, normal, and bi-normal components of desired contour. Herein, we propose a model-free adaptive coupling controller design approach for multi-axis linear motor system with uncertainty and nonlinear phenomena. RFNN establishes the corresponding adaptive coupling controller to treat the uncertain system with nonlinear phenomenon. The stability of closed-loop system is guaranteed by the Lyapunov method and the adaptation of RFNN is also obtained. Simulation results are introduced in order to illustrate the effectiveness.

Coordinate transformation methodology for simulating quasistatic elastoplastic solids

Molecular dynamics simulations frequently employ periodic boundary conditions where the positions of the periodic images are manipulated in order to apply deformation to the material sample. For example, Lees-Edwards conditions use moving periodic images to apply simple shear. Here, we examine the problem of precisely comparing this type of simulation to continuum solid mechanics. We employ a hypoelastoplastic mechanical model, and develop a projection method to enforce quasistatic equilibrium. We introduce a simulation framework that uses a fixed Cartesian computational grid on a reference domain, and which imposes deformation via a time-dependent coordinate transformation to the physical domain. As a test case for our method, we consider the evolution of shear bands in a bulk metallic glass using the shear transformation zone theory of amorphous plasticity. We examine the growth of shear bands in simple shear and pure shear conditions as a function of the initial preparation of the bulk metallic glass.

The possible states of a population of eusocial insects

A controlled dynamical system describing the growth of a population of eusocial insects is discussed. The approach differs in two ways from previous work. First, the full reachable set is determined (rather than just the optimal control maximizing the number of reproducibles). Second, a time-dependent coordinate transformation is used which renders the system non-autonomous, but eases the analytical derivation of the result. Finally, we comment on educational aspects which arose from the classroom use of the example discussed and report on the positive experiences made.

An explicit adaptive grid approach for the numerical solution of the population balance equation

Many engineering applications, such as the formation of soot in hydrocarbon combustion or the precipitation of nanoparticles from aqueous solutions, encompass a polydispersed particulate phase that is immersed in a reacting carrier flow. From a Eulerian perspective, the evolution of the particulate phase both in physical and in particle property space can be described by the population balance equation (PBE). In this article, we present an explicit solution-adaptive numerical scheme for discretizing the spatially inhomogeneous and unsteady PBE along a one-dimensional particle property space. This scheme is based on a space and time dependent coordinate transformation which redistributes resolution in particle property space according to the shapes of recent solutions for the particle property distribution. In particular, the coordinate transformation is marched in time explicitly. In comparison to many existing moving or adaptive grid approaches, this has the advantage that the semi-discrete PBE does not need to be solved in conjunction with an additional system of equations governing the movement of nodes in particle property space.By design, our adaptive grid technique is able to accurately capture sharp features such as peaks or near-discontinuities, while maintaining the semi-discrete system size and adhering to a uniform fixed grid discretization in transformed particle property space. This is particularly advantageous if the PBE is combined with a spatially and temporally fully resolved flow model and a standard Eulerian solution scheme is applied in physical space. In order to accommodate localized source terms and to control the grid stretching, we develop a robust scheme for modifying the coordinate transformation such that constraints on the resolution in physical particle property space are obeyed.As an example, we consider the precipitation of BaSO4 particles from an aqueous solution in a plug flow reactor. Our findings demonstrate that for a given accuracy of the numerical solution the explicit adaptive grid technique requires over an order of magnitude fewer grid points than a comparable fixed grid discretization scheme.

Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains

A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme.We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.

Sudden transition from a stable to an unstable harmonic trap as the adiabatic potential parameter is varied in a time-periodic harmonic trap

It has been shown already that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$ [Uzdin, Dalla Torre, Kosloff, and Moiseyev, Phys. Rev. A 88, 022505 (2013)]. Here we show that by using time-dependent linear coordinate transformation the time-dependent Schr\odinger equation with Hermitian time-dependent Hamiltonian is transformed into a time-dependent Schr\odinger equation with a time-independent harmonic oscillator with the dimensionless frequency of $\sqrt{1\ensuremath{-}{(\ensuremath{\mu}/2)}^{2}}$. At $\ensuremath{\mu}=2$ a transition to a non-Hermitian Hamiltonian is obtained as the potential well is transformed to a parabolic potential barrier. While in a harmonic trap noninteracting particles have classical periodic motions, they are pushed apart exponentially in time as the potential well is suddenly transformed into a parabolic potential barrier in the new variable representation.

Kubo formulas for viscosity: Hall viscosity, Ward identities, and the relation with conductivity

We derive from first principles the Kubo formulas for the stress-stress response function at zero wavevector that can be used to define the full complex frequency-dependent viscosity tensor, both with and without a uniform magnetic field. The formulas in the existing literature are frequently incomplete, incorrect, or lack a derivation; in particular, Hall viscosity is overlooked. Our approach begins from the response to a uniform external strain field, which is an active time-dependent coordinate transformation in d space dimensions. These transformations form the group GL(d,R) of invertible matrices, and the infinitesimal generators are called strain generators. These enable us to express the Kubo formula in different ways, related by Ward identities; some of these make contact with the adiabatic transport approach. For Galilean-invariant systems, we derive a relation between the stress response tensor and the conductivity tensor that is valid at all frequencies and in both the presence and absence of a magnetic field. In the presence of a magnetic field and at low frequency, this yields a relation between the Hall viscosity, the q^2 part of the Hall conductivity, the inverse compressibility (suitably defined), and the diverging part of the shear viscosity (if any); this relation generalizes a result found recently. We show that the correct value of the Hall viscosity at zero frequency can be obtained (at least in the absence of low-frequency bulk and shear viscosity) by assuming that there is an orbital spin per particle that couples to a perturbing electromagnetic field as a magnetization per particle. We study several examples as checks on our formulation.

Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type

We extend the results obtained in Sarlet and Prince (2010 J. Phys. A: Math. Theor. 43 445204) about a class of Lagrangian systems which admit alternative kinetic energy metrics to second-order mechanical systems with explicit time dependence. The main results are that a time-dependent alternative metric will have constant eigenvalues, and will give rise to a time-dependent coordinate transformation which partially decouples the system.

Dissipation‐induced subcritical flutter in the acoustics of friction

AbstractWe consider a gyroscopic system under the action of small dissipative and non–conservative positional forces, which has its origin in the models of rotating elastic bodies of revolution in frictional contact such as the singing wine glass or the squealing disc/drum brakes. The spectrum of the unperturbed gyroscopic system forms a spectral mesh in the plane ‘frequency versus gyroscopic parameter’ with double semi–simple purely imaginary eigenvalues at the nodes. In the subcritical range of the gyroscopic parameter the eigenvalues involved into the crossings have the same Krein signature and thus their splitting due to changes in the stiffness matrix, which break the rotational symmetry of the body, cannot produce complex eigenvalues and, therefore, flutter. We establish that perturbation of the gyroscopic system by the dissipative forces with the indefinite matrix can lead to the subcritical flutter instability even if the rotational symmetry is destroyed. With the use of the perturbation theory of multiple eigenvalues we explicitly find the linear approximation to the domain of the subcritical flutter, which turns out to have a conical shape. The orientation of the cone in the three dimensional space of the parameters, corresponding to gyroscopic, damping, and potential forces, is determined by the sign of an explicit expression involving the entries of both the damping and potential matrices. With the use of a time–dependent coordinate transformation we demonstrate that the conical zones of flutter for the original autonomous system coincide with the zones of the subcritical parametric resonance of the rotationally symmetric flexible body with the load moving in the circumferential direction. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Influence of the Different Wall Oscillation Modes on the Heat Transfer Enhancement

We studied the influence of the different wall oscillation modes on the heat transfer characteristics of a two-dimensional channel numerically. In this study, the following two channels with different wall oscillation modes were used. One is a two-dimensional channel bounded by the fixed wall and the transversely oscillating wall (channel A) and the other is a two-dimensional channel whose upper and lower wall oscillate transversely in the same manner (channel B). The fully implicit finite difference method was used for the analysis of the conservation equations and the time-dependent coordinate transformations were applied to solve the moving boundary problem. The calculated results are summarized as follows. (1) The wall oscillation takes a remarkable effect on the heat transfer enhancement in a low Reynolds number region for each channel. (2) If we must avoid the increase in the pressure loss, the channel B is suitable for this requirement compared to the channel A.

Stability analysis of uniform and non-uniform annular passages conducting incompressible laminar flows for small and large amplitude oscillatory motions of the outer cylinder

A computational method is developed involving the simultaneous integration of the Navier–Stokes and structural equations for the purpose of studying the stability of concentric annular passages conducting incompressible laminar flows. It is assumed that one side of the annulus, i.e. the centre-body, is fixed and the outer cylindrical duct is flexibly supported. The outer cylinder is displaced from its equilibrium position and is then released. In this situation, the fluid part of the problem is solved by an accurate method using a three-point backward implicit scheme, followed by a pseudo-time iteration using an artificial compressibility factor. The fluid equations are discretized in space based on a finite-difference formulation and primitive variables, for which stretched staggered grids are used. The resulting equations are cast in delta form and are solved using an ADI scheme. The fluid forces acting on the vibrating cylinder are calculated from the integration of the unsteady pressure and shear stresses resulting from the unsteady primitive variables calculated. The equations of motion of the structure, subjected to the calculated fluid forces are solved using the Runge–Kutta scheme to obtain the displacement of the moving cylinder. The problem is solved: (a) for small-amplitude motions, by means of the so-called mean position (MP) analysis and (b) for large amplitude oscillations of the outer cylinder for which a time-dependent coordinate transformation (TDCT) is used to fix the computational domain. Both these approaches (MP and TDCT) are applied to uniform and non-uniform (backstep-shaped) annuli for translational motion of the cylinder. The problem is also solved for (i) rocking motion and uniform annuli and (ii) translational motion for diffuser-shaped annuli, with only MP analysis. When the MP analysis is considered, it is shown that, for translational motion of the outer cylinder, the most stable configuration is that of a uniform geometry and the least stable one is the backward step geometry. For rocking motion in uniform annular geometry, for some system parameters and high enough flow velocity, the outer cylinder can develop flutter (limit-cycle oscillation). The comparison between the results of MP and TDCT analyses for uniform and backstep geometries for translational motion of the cylinder indicates that the outer cylinder is less stable when TDCT is used and the coupled frequency of oscillation is also changed considerably.

Theoretical Modeling and Optimization of the Detection Performance: a New Concept for Electrochemical Detection of Proteins in Microfluidic Channels

In this work, we present a complete theoretical analysis of a new concept of electrochemical detector for application in proteomics upon considering two band microelectrodes performing in generator-collector mode. This concept of an original electrochemical detector is aimed at the detection of proteins following their separation in microfluidic biochips. The theoretical analysis is based on the use of the time-dependent coordinate transformation which allows performing precise modeling for a wide range of the key parameters governing the electrochemical detector performance. This allows defining a precise optimization procedure for its best efficiency upon considering the qualitative and quantitative effects of each of the main operational parameters.

Modeling of Carrier-based Aircraft Ski Jump Take-off Based on Tensor

A general mathematical model of carrier based aircraft ski jump take-off is derived based on tensor. The carrier, the aircraft body and the movable parts of the landing gears are treated as independent entities. These entities are assembled into a multi-rigid-body system with flexible links. Dynamical equations of each entity are derived on the basis of the Newton law and the Euler transformation. Using the invariance property of the tensor, the dynamical and kinematical equations are converted to tensor forms which are invariant under time dependent coordinate transformations. Then the tensor formed equations are expressed by the matrix operation. Differential equation group of the matrix form is formur lated for the programming. The closure of the model is discussed, and the simulation results are given.

Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems

We present a method for the parallel numerical simulation of transient three-dimensional fluid-structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved altematingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier-Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MAC-method to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables