Time-independent Problems Research Articles

Electronic structure and carrier transfer in B-DNA monomer polymers and dimer polymers: Stationary and time-dependent aspects of a wire model versus an extended ladder model.

We employ two tight-binding (TB) approaches to systematically study the electronic structure and hole or electron transfer in B-DNA monomer polymers and dimer polymers made up of N monomers (base pairs): (I) at the base-pair level, using the onsite energies of base pairs and the hopping integrals between successive base pairs, i.e., a wire model and (II) at the single-base level, using the onsite energies of the bases and the hopping integrals between neighboring bases, i.e., an extended ladder model since we also include diagonal hoppings. We solve a system of M (matrix dimension) coupled equations [(I) M=N, (II) M=2N] for the time-independent problem, and a system of M coupled first order differential equations for the time-dependent problem. We perform a comparative study of stationary and time-dependent aspects of the two TB variants, using realistic sets of parameters. The studied properties include HOMO and LUMO eigenspectra, occupation probabilities, density of states and HOMO-LUMO gaps as well as mean over time probabilities to find the carrier at each site [(I) base pair or (II) base], Fourier spectra, which reflect the frequency content of charge transfer, and pure mean transfer rates from a certain site to another. The two TB approaches give coherent, complementary aspects of electronic properties and charge transfer in B-DNA monomer polymers and dimer polymers.

Full-counting statistics of time-dependent conductors

We develop a scheme for the computation of the full-counting statistics of transport described by Markovian master equations with an arbitrary time dependence. It is based on a hierarchy of generalized density operators, where the trace of each operator yields one cumulant. This direct relation offers a better numerical efficiency than the equivalent number-resolved master equation. The proposed method is particularly useful for conductors with an elaborate time-dependence stemming, e.g., from pulses or combinations of slow and fast parameter switching. As a test bench for the evaluation of the numerical stability, we consider time-independent problems for which the full-counting statistics can be computed by other means. As applications, we study cumulants of higher order for two time-dependent transport problems of recent interest, namely steady-state coherent transfer by adiabatic passage and Landau-Zener-St\"uckelberg-Majorana interference in an open double quantum dot.

Nonlinear dynamics of a thin liquid film deposited on a laterally oscillating corrugated surface in the high-frequency limit

The nonlinear dynamics of thin liquid films deposited on various periodically corrugated substrates, both left-right symmetric and asymmetric, subjected to lateral vibration in the high-frequency limit is investigated. The method used to derive the governing evolution equation is based on the long-wave approximation, multiscale time expansion, and averaging over the fast time scale. The resulting evolution equation contains the effects of gravity, capillarity, vibration, and the substrate topology. The initial-boundary-value problem associated with this evolution equation is numerically solved and the system behavior is investigated for a variety of parameter sets. Typical patterns emerging as a result of the film evolution include hump formation within the troughs of the substrate and homogenized coatings whose configuration resembles that of the substrate, as well as the possibility of film rupture. We show that the choice of the vibration parameters and the topological features of the substrate may be used for controlling the shape of the film interface as well as its properties such as the amplitude, continuity, or rupture. Together with the film profiles stationary in terms of the averaged film interface with respect to the slow time scale, time evolution of the total (comprising of the averaged and pulsating components) interfacial profiles and streamline maps is presented to illustrate the film flow. We carry out Floquet stability analysis of periodically replicated steady states for the time-independent problem, linear stability analysis based on a reduced low-order projection approximation for the time-dependent problem, and stability analysis with respect to disturbances of a larger wavelength. We have found also that in the case of two-dimensional corrugated substrates, the vast majority of two-dimensional steady-state flows in terms of the averaged film interface exhibit stability in three dimensions with respect to small perturbations in the transverse spatial direction.

Dynamic analysis of linear viscoelastic cross-ply laminated shallow spherical shells

This paper aims to propose an alternative approach for response of viscoelastic cross-ply laminated shallow spherical shells. In the proposed approach, the governing differential equations of cross-ply laminated shallow spherical shell are derived using the dynamic version of the principle of virtual displacements. Following this Laplace transform is employed in the transient analysis of laminated shell problem. In the transformed domain damping can be incorporated very easily. The transformed time-independent problem in spatial coordinate is solved numerically by Gauss elimination. Inverse transformation of the results into the time domain is performed by the modified Durbin’s method. Verification of the presented method is carried out by comparing the results with those obtained by Newmark method and ANSYS finite element software. Materials of the shell laminates are assumed to be linear elastic or viscoelastic. The novelty of the present study lies in the fact that a combination of Laplace transform and Navier method is employed in the analysis of damped response of laminated shells. The numerical sample results have proved that the presented method constitutes a highly accurate and efficient solution and it can be easily applied to the laminated composite shell problems.

A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation

In the current work, numerical solution of a two-dimensional fractional evolution equation has been investigated by using two different aspects of strong form meshless methods. In the first method a time discretization approach and a numerical technique based on the convolution sum are employed to approximate the appearing time derivative and fractional integral operator, respectively. It has been proven analytically that the time discretization scheme is unconditionally stable. Then a meshfree collocation method based on the radial basis functions is used for solving resulting time-independent discretization problem. As the second approach, a fully Kansa׳s meshfree method based on the Gaussian radial basis function is formulated and well-used directly for solving the governing problem. In this technique an explicit formula to approximate the fractional integral operator is computed. The given techniques are used to solve two examples of problem. The computed approximate solutions are reported through the tables and figures, also these results are compared together and with the other available results. The presented results demonstrate the validity, efficiency and accuracy of the formulated techniques.

Elementary solution to the time-independent quantum navigation problem

A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing ‘background’ Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of time-independent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalizations to higher-dimensional systems are discussed.

Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

In this paper we present a localized polynomial chaos expansion for partial differential equations (PDE) with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and, more importantly, in a much lower dimensional random space. In a postprocesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. In this paper we present the general mathematical framework of the methodology and use numerical examples to demonstrate the properties of the method.

Numerical Simulation of Non-linear Water Waves over Variable Bathymetry

Abstract We present a numerical method for the computation of surface gravity waves in the presence of variable bathymetry. The evolution is described by an alternative form of the usual Hamiltonian system where the substrate boundary value problem for the potential gives its place to an infinite system of second order partial differential equations on horizontal fields with appropriate boundary conditions and a linear algebraic constraint. We employ the finite difference method to solve this time-independent problem and the classic Runge-Kutta method for the time-integration of the evolution equations. Special care is taken for the consistent wave generation and absorption at the lateral boundaries of a reasonably sized computational domain. Simulations of non-linear waves in the presence of undulated bottoms (Bragg reflection) are presented and validated against experimental measurements.

Elastic and viscoelastic response of heterogeneous annular structures under arbitrary transient pressure

A novel approach is employed in the forced-vibration analysis of functionally-graded annular structures under conditions of axisymmetry. The material is heterogeneous in the sense that it is functionally-graded in the radial direction. The grading function may be an arbitrary continuous function of the radial coordinate. Forcing functions applied on the inner boundary are dynamic pressures which may be harmonic, an arbitrary continuous function of time, or impulsive. These conditions result in governing differential equations with variable coefficients. Analytical solutions of such equations cannot be obtained except for certain simple grading functions and pressures. Numerical approaches must be adopted to solve the problem on hand. The novelty of the present study lies in the fact that a combination of Laplace transform and Complementary Functions Method (CFM) is employed in the analysis. Laplace transformation gives a time-independent boundary-value problem in spatial coordinate which is then solved by CFM. Inverse transformation of the results into the time domain is performed by modified Durbin׳s method. In the Laplace domain, viscoelasticity is modeled easily by a simple change of variable. Benchmark solutions available in the literature are used to validate the results and to observe the convergence of the numerical solutions. The solution procedure is well-structured, simple and efficient and it can be readily applied to cylinders, disks and spheres. It is also well suited for problems in which the graded mechanical properties and applied pressures are supplied point-by-point in a discretized manner.

Time-Dependent Mechanism Reliability Analysis With Envelope Functions and First-Order Approximation

This work develops an envelope approach to time-dependent mechanism reliability defined in a period of time where a certain motion output is required. Since the envelope function of the motion error is not explicitly related to time, the time-dependent problem can be converted into a time-independent problem. The envelope function is approximated by piecewise hyperplanes. To find the expansion points for the hyperplanes, the approach linearizes the motion error at the means of random dimension variables, and this approximation is accurate because the tolerances of the dimension variables are small. The expansion points are found with the maximum probability density at the failure threshold. The time-dependent mechanism reliability is then estimated by a multivariable normal distribution at the expansion points. As an example, analytical equations are derived for a four-bar function generating mechanism. The numerical example shows the significant accuracy improvement.

Discontinuous finite element formulations for neutron transport in spherical geometry

We have developed the linear and quadratic Galerkin discontinuous finite element methods for the solution of both time-independent and time-dependent spherical geometry neutron transport problems. Discrete ordinates method is used for the angular discretization while the implicit method is utilized for temporal discretization in time-dependent problems. In order to assess the relative performance of the newly developed linear and quadratic discontinuous finite element spatial differencing methods relative to the previously developed linear discontinuous finite element and diamond difference discretizations, a computer code is developed and numerical solutions of the neutron transport equation for some benchmark problems are obtained. These numerical applications reveal that the newly developed quadratic discontinuous finite element method produces the most accurate results while the newly developed linear discontinuous finite element method follows as the second best discontinuous finite element method.

Generalized energy and time-translation invariance in a driven dissipative system

Driven condensed matter systems consistently pose substantial challenges to theoretical understanding. Progress in the study of such systems has been achieved using the Floquet formalism, but certain aspects of this approach are not well understood. In this paper, we consider the exceptionally simple case of the rotating Kekul\'e mass in graphene through the lens of Floquet theory. We show that the fact that this problem is gauge-equivalent to a time-independent problem implies that the ``quasienergies'' of Floquet theory correspond to a continuous symmetry of the full time-dependent Lagrangian. We use the conserved Noether charge associated with this symmetry to recover notions of equilibrium statistical mechanics.

Product approximations for a class of quantum anharmonic oscillators

In this article, we investigate analytically and numerically a class of non-autonomous Schrödinger equations in one space dimension describing the dynamics of quantum anharmonic oscillators driven by time-dependent quartic interactions. We do so within a suitably constructed Faedo–Galerkin scheme by analyzing several product approximations for their solutions, which involve various exponential operator splittings. Our main objective is to study the convergence rates and the accuracy of such approximations, among which there are extensions of the Trotter–Kato product formula and several other variants. Crucial to our analysis is the knowledge of the lowest energy level and of the corresponding eigensolution to the associated time-independent problem, which we also compute within the very same framework.

Structure and calculation of field-induced free-free transition matrix elements in many-electron atoms

In earlier work [Phys. Rev. A 65, 043412 (2002)] we examined the description of atom-radiation coupling in terms of the multipolar Hamiltonian rather than the minimal coupling one, and reached the conclusion that the simple operator ${D}_{SF}(r)={\begin{array}{cc}r,a rl{r}_{0} {r}_{0},a r\ensuremath{\ge}{r}_{0}\end{array}$ constitutes a good approximation to the full electric interaction, ${H}_{\mathrm{el}}$. Unlike the standard length form of the electric dipole approximation (EDA), ${D}_{SF}(r)$ does not increase indefinitely. Instead, it stops at ${r}_{0}=3\ensuremath{\lambda}/8$, where $\ensuremath{\lambda}$ is the wavelength, and then remains constant. This theoretically founded simplification of ${H}_{\mathrm{el}}$ has obvious computational advantages over ${H}_{\mathrm{el}}$, while at the same time accounts for effects beyond the EDA. The present paper discusses the elucidation of the singularities of the free-free matrix elements using ${D}_{SF}(r)$ and state-specific, energy-normalized, numerical scattering functions in polyelectronic atoms, and explains how their computation can be done efficiently. Accurate knowledge of such matrix elements is critical to the quantitative solution of time-dependent or time-independent problems involving various multiphoton processes that are induced by strong fields and involve off-resonance coupling between extended wave functions representing high-$n$ Rydberg states of the discrete spectrum or asymptotically unbound states of the continuum. For completeness, the singularities of the EDA are also derived and discussed within the present approach.

The Helmholtz Operator on Higher Dimensional Möbius Strips Embedded in $${\mathbb{R}^{4}}$$

In this paper we present explicit formulas for the fundamental solution to the Helmholtz operator on a higher-dimensional analogue of the Mobius strip in three real variables (embedded in \({\mathbb{R}^{4}}\)) with values in distinct pinor bundles. Herefore we use an approach that uses classical harmonic analysis methods combined with some Clifford analysis tools and adapt it to this special geometry. The fundamental solution is described in terms of generalizations of the Weierstrass \({\wp}\) -function that are adapted to the context of these geometries. As our main result we present an analytic integral representation formula to express the solutions of the inhomogeneous time-independent Klein-Gordon problem on Mobius strips.

Plasma radiation losses in the electrostatic limit

Steady state ion drift configurations are usually treated as time-independent problems, hence radiation losses are usually ignored. This assumption is not strictly true when the number of charge carriers is small and their acceleration is big. According to Larmor formula , accelerating charges radiate. In spite of that, even if single charges accelerate, in the continuum DC limit, there is no radiation. This challenge is encountered in any plasma application based on a steady state ion drift, because ions accelerate under the influence of the electric field. Even if we assume the model of an average speed, defined by the average gas mobility multiplied by the electric field, the velocity changes because the electric field is not constant in magnitude or direction. One remarks that in any curvilinear DC electric current path, charge carriers are accelerated, raising the question of when are radiation losses significant. This issue is best analyzed in a canonical configuration of charges in circular motion. One single charge simply radiates power according to Larmor formula. When distributing the charge over the circle, the destructive interference between the field contributions of the different charges reduces the radiated power so that it goes to 0 in the continuum DC limit.

Optimization Design of Structures Subjected to Transient Loads Using First and Second Derivatives of Dynamic Displacement and Stress

This paper developed an effective optimization method, i.e., gradient-Hessian matrix-based method or second order method, of frame structures subjected to the transient loads. An algorithm of first and second derivatives of dynamic displacement and stress with respect to design variables is formulated based on the Newmark method. The inequality time-dependent constraint problem is converted into a sequence of appropriately formed time-independent unconstrained problems using the integral interior point penalty function method. The gradient and Hessian matrixes of the integral interior point penalty functions are also computed. Then the Marquardt's method is employed to solve unconstrained problems. The numerical results show that the optimal design method proposed in this paper can obtain the local optimum design of frame structures and sometimes is more efficient than the augmented Lagrange multiplier method.

Optimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress

This paper developed an effective optimization method, i.e., gradient-Hessian matrix-based method or second order method, of frame structures subjected to the transient loads. An algorithm of first and second derivatives of dynamic displacement and stress withrespect to design variables is formulated based on the Newmark method. The inequality time-dependent constraint problem is converted intoa sequence of appropriately formed time-independent unconstrained problems using the integral interior point penalty function method. The gradient and Hessian matrixes of the integral interior point penalty functions are also computed. Then the Marquardt's method is employed to solve unconstrained problems. The numerical results show that the optimal design method proposed in this paper can obtain the local optimum design of frame structures and sometimes is more efficient than the augmented Lagrange multiplier method.

Transient energy density distribution of a rod under high-frequency excitation

Energy finite element analysis (EFEA) is an efficient way to solve high-frequency structural dynamics response problems. Up to now, EFEA has been used to deal with time-independent vibration problems. However, it is still necessary to understand the time dependent details of energy density distribution of a structure. To study the transient response of a rod under high-frequency sinusoidal excitation, the transient energy density governing equation for a rod is presented. The governing equation is solved, and the solution is verified using an analytical method. Example application to a rod is presented to illustrate the feasibility.

Functional optimization technique for numerical solution of the radiative transfer equation

A numerical method for solving the time-independent radiative transfer problem in a flat layer with given properties and temperature distribution is proposed. This method avoids the numerical diffusion; rather, it is based on a gradient procedure for the functional minimization of the residual of the radiative transfer integral equation. Means for suppressing computational instabilities are proposed that reduce requirements for the approximation of the operators in the optimization problem but do not change the problem objective functional.