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We consider the approximation properties of primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. We show how the previous convergence results as well as the Aubin-Nitsche type duality arguments can be extended to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The arguments are based on the approximation theory of quadrilateral vector finite element spaces associated to the numerical flux variable of the DPG approximation. The theoretical results are validated by a numerical experiment that features also a comparison between the primal DPG method and a conventional least squares finite element method with the same number of degrees of freedom. [ABSTRACT FROM AUTHOR]

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly. In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path's history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified. [ABSTRACT FROM AUTHOR]

We investigate the coarse-graining of host-guest systems under the perspective of the local distribution of pore occupancies, along with the physical meaning and actual computability of the coarse-interaction terms. We show that the widely accepted approach, in which the contributions to the free energy given by the molecules located in two neighboring pores are estimated through Monte Carlo simulations where the two pores are kept separated from the rest of the system, leads to inaccurate results at high sorbate densities. In the coarse-graining strategy that we propose, which is based on the Bethe-Peierls approximation, density-independent interaction terms are instead computed according to local effective potentials that take into account the correlations between the pore pair and its surroundings by means of mean-field correction terms without the need for simulating the pore pair separately. Use of the interaction parameters obtained this way allows the coarse-grained system to reproduce more closely the equilibrium properties of the original one. Results are shown for lattice-gases where the local free energy can be computed exactly and for a system of Lennard-Jones particles under the effect of a static confining field. [ABSTRACT FROM AUTHOR]

A very simple equation, FAppij=[(∂2(Va i-Va j)/∂Q2)/(Va i-Va j)]1/2/2, giving a reliable magnitude of non-adiabatic coupling terms (NACTs, Fij's) based on adiabatic potential energies only (Va i and Va j) was discovered, and its reliability was tested for several prototypes of same-symmetry interstate crossings in LiF, C2, NH3Cl, and C6H5SH molecules. Our theoretical derivation starts from the analysis of the relationship between the Lorentzian dependence of NACTs along a diabatization coordinate and the well-established linear vibronic coupling scheme. This analysis results in a very simple equation, α=2κ/Δc, enabling the evaluation of the Lorentz function αα parameter in terms of the coupling constant κκ and the energy gap Δc (Δc= Va i-Va jQc) between adiabatic states at the crossing point Qc. Subsequently, it was shown that Qc corresponds to the point where FAppij exhibit maximum values if we set the coupling parameter as κ=[(Va i-Va j)⋅(∂2(Va i-Va j)/∂Q2)]1/2Qc/2. Finally, we conjectured that this relation could give reasonable values of NACTs not only at the crossing point but also at other geometries near Qc. In this final approximation, the pre-defined crossing point Qc is not required. The results of our test demonstrate that the approximation works much better than initially expected. The present new method does not depend on the selection of an ab initio method for adiabatic electronic states but is currently limited to local non-adiabatic regions where only two electronic states are dominantly involved within a nuclear degree of freedom. [ABSTRACT FROM AUTHOR]

We formulate mixed semiclassical-classical (SC-Cl) propagators by adding a further approximation to the phase-space SC propagators, which have been formulated in our previous paper [S. Koda, J. Chem. Phys. 143, 244110 (2015)]. We first show that the stationary phase approximation over the operation of the phase-space van Vleck propagator on initial distribution functions results in the classical mechanical time propagation. Then, after dividing the degrees of freedom (DOFs) of the total system into the semiclassical DOFs and the classical DOFs, the SC-Cl van Vleck propagator and the SC-Cl Herman-Kluk (HK) propagator are derived by performing the stationary phase approximation only with respect to the classical DOFs. These SC-Cl propagators are naturally decomposed to products of the phase-space SC propagators and the classical mechanical propagators when the system does not have any interaction between the semiclassical and the classical DOFs. In addition, we also numerically compare the original phase-space HK (full HK) propagator and the SC-Cl HK propagator in terms of accuracy and efficiency to find that the accuracy of the SC-Cl HK propagator can be comparable to that of the full HK propagator although the latter is more accurate than the former in general. On the other hand, we confirm that the convergence speed of the SC-Cl HK propagator is faster than that of the full HK propagator. The present numerical tests indicate that the SC-Cl HK propagator can be more accurate than the full HK propagator when they use a same and finite number of classical trajectories due to the balance of the accuracy and the efficiency. [ABSTRACT FROM AUTHOR]

In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates φ and ξ are obtained using Lagrange's equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter ε will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations. [ABSTRACT FROM AUTHOR]

A mixed quantal-semiquantal theory is presented in which the semiquantal squeezed-state wave packet describes the heavy degrees of freedom. Starting from the mean-field equations of motion that are naturally derived from the time-dependent variational principle, we introduce the stochastic particle description for both the quantal and semiquantal parts in an aim to take into account the interparticle correlation, in particular the "quantum backreaction" beyond the mean-field approximation. A numerical application on a model of O2 scattering from a Pt surface demonstrates that the proposed scheme gives correct asymptotic behavior of the scattering probability, with improvement over the mixed quantum-classical scheme with Bohmian particles, which is comprehended by comparing the Bohmian and the stochastic trajectories. [ABSTRACT FROM AUTHOR]

Systems of interacting charges and fields are ubiquitous in physics. Recently, it has been shown that Hamiltonians derived using different gauges can yield different physical results when matter degrees of freedom are truncated to a few low-lying energy eigenstates. This effect is particularly prominent in the ultra-strong coupling regime. Such ambiguities arise because transformations reshuffle the partition between light and matter degrees of freedom and so level truncation is a gauge dependent approximation. To avoid this gauge ambiguity, we redefine the electromagnetic fields in terms of potentials for which the resulting canonical momenta and Hamiltonian are explicitly unchanged by the gauge choice of this theory. Instead the light/matter partition is assigned by the intuitive choice of separating an electric field between displacement and polarisation contributions. This approach is an attractive choice in typical cavity quantum electrodynamics situations. [ABSTRACT FROM AUTHOR]

Here, I present a practical recipe for obtaining contravariant vibration-rotation metric tensors, and thus the kinetic energy operators, when some degrees of freedom are constrained rigidly. An element of the contravariant metric tensor is obtained as a sum of dot products of contravariant measuring vectors, which are obtained from their unconstrained counterparts by adding a frozen mode correction. The present method applies in principle for any choice of shape coordinates and a body-frame for which the contravariant measuring vectors can be evaluated. In contrast to the existing methods, the present method does not involve evaluation of covariant metric tensors, matrix inversions, chain rules of derivation, or numerical differentiation. It is applied in the sequel paper [L. Partanen, J. Pesonen, E. Sjöholm, and L. Halonen, J. Chem. Phys. 139, 144311 (2013)] to study the effects of several different approximations to the kinetic energy operator, when the two large-amplitude OH-torsional motions in H2SO4 are of interest. [ABSTRACT FROM AUTHOR]

Expressions for reaction rate constants in multidimensional chemical systems are derived by applying semiclassical approximation to the quantum path integrals of the ImF formulation of reaction rate theory. First, the transverse degrees of freedom orthogonal to the reaction coordinate are treated within the steepest descent approximation, after which the semiclassical approximation is applied to the remaining reaction coordinate. Thus derived, the semiclassical expressions account for the multidimensional nature of quantum effects and accurately incorporate nuclear quantum effects such as multidimensional tunneling and zero point energies. The obtained expressions are applicable in the broad temperature range from the deep tunneling to high-temperature regimes. The present paper provides derivation of the semiclassical instanton expressions proposed by Kryvohuz [J. Chem. Phys. 134, 114103 (2011)]. [ABSTRACT FROM AUTHOR]

Mixed quantum/classical (MQC) simulations treat the majority of a system classically and reserve quantum mechanics only for a few degrees of freedom that actively participate in the chemical process(es) of interest. In MQC calculations, the quantum and classical degrees of freedom are coupled together using pseudopotentials. Although most pseudopotentials are developed empirically, there are methods for deriving pseudopotentials using the results of quantum chemistry calculations, which guarantee that the explicitly-treated valence electron wave functions remain orthogonal to the implicitly-treated core electron orbitals. Whether empirical or analytically derived in nature, to date all such pseudopotentials have been subject to the frozen core approximation (FCA) that ignores how changes in the nuclear coordinates alter the core orbitals, which in turn affects the wave function of the valence electrons. In this paper, we present a way to go beyond the FCA by developing pseudopotentials that respond to these changes. In other words, we show how to derive an analytic expression for a pseudopotential that is an explicit function of nuclear coordinates, thus accounting for the polarization effects experienced by atomic cores in different chemical environments. We then use this formalism to develop a coordinate-dependent pseudopotential for the bonding electron of the sodium dimer cation molecule and we show how the analytic representation of this potential can be used in one-electron MQC simulations that provide the accuracy of a fully quantum mechanical Hartree-Fock (HF) calculation at all internuclear separations. We also show that one-electron MQC simulations of Na2+ using our coordinate-dependent pseudopotential provide a significant advantage in accuracy compared to frozen core potentials with no additional computational expense. This is because use of a frozen core potential produces a charge density for the bonding electron of Na2+ that is too localized on the molecule, leading to significant overbinding of the valence electron. This means that FCA calculations are subject to inaccuracies of order ∼10% in the calculated bond length and vibrational frequency of the molecule relative to a full HF calculation; these errors are fully corrected by using our coordinate-dependent pseudopotential. Overall, our findings indicate that even for molecules like Na2+, which have a simple electronic structure that might be expected to be well-treated within the FCA, the importance of including the effects of the changing core molecular orbitals on the bonding electrons cannot be overlooked. [ABSTRACT FROM AUTHOR]

The microscopic structure of fluids of simple spheres is well known. However, the constituents of most real-life fluids are non-spherical, leading to a coupling between the rotational and translational degrees of freedom. The structure of simple dense fluids of spheroids - ellipsoids of revolution - was only recently determined by direct experimental techniques [A. P. Cohen, E. Janai, E. Mogilko, A. B. Schofield, and E. Sloutskin, Phys. Rev. Lett. 107, 238301 (2011)]. Using confocal microscopy, it was demonstrated that the structure of these simple fluids cannot be described by hard particle models based on the widely used Percus-Yevick approximation. In this paper, we describe a new protocol for determining the shape of the experimental spheroids, which allows us to expand our previous microscopy measurements of these fluids. To avoid the approximations in the theoretical approach, we have also used molecular dynamics simulations to reproduce the experimental radial distribution functions g(r) and estimate the contribution of charge effects to the interactions. Accounting for these charge effects within the Percus-Yevick framework leads to similar agreement with the experiment. [ABSTRACT FROM AUTHOR]

We have investigated the thermodynamic properties of titanium carbide (TiC) nanowire within the framework of density functional theory and quasi-harmonic approximation via calculating the temperature dependence of a number of thermodynamic quantities including entropy, number of microstates, total and free energies, and specific heat. The level of disorder of the nanowire has been found to be larger than that of the bulk mainly due to expansion in only one direction, which accordingly results in acquiring more spatial degrees of freedom. A linear function of temperature has been also found for the low-temperature specific heat of the nanowire being in a remarkable agreement with the general T n -law for Debye systems. Results firmly establish a direct correlation between the spatial expansion of a TiC compound and its low-temperature specific heat and entropy. [ABSTRACT FROM AUTHOR]

We establish an approximation theory for Pearson's chi-squared statistics in situations where the number of cells is large, by using a high-dimensional central limit theorem for quadratic forms of random vectors. Our high-dimensional central limit theorem is proved under Lyapunov-type conditions that involve a delicate interplay between the dimension, the sample size, and the moment conditions. We propose a modified chi-squared statistic and introduce an adjusted degrees of freedom. A simulation study shows that the modified statistic outperforms Pearson's chi-squared statistic in terms of both size accuracy and power. Our procedure is applied to the construction of a goodness-of-fit test for Rutherford's alpha-particle data. [ABSTRACT FROM AUTHOR]

The vibrational coupled cluster (VCC) equations are analyzed in terms of vibrational Mo\ller-Plesset perturbation theory aiming specifically at the importance of four-mode couplings. Based on this analysis, new VCC methods are derived for the calculation of anharmonic vibrational energies and vibrational spectra using vibrational coupled cluster response theory. It is shown how the effect of four-mode coupling and excitations can be efficiently and accurately described using approximations for their inclusion. Two closely related approaches are suggested. The computational scaling of the so-called VCC[3pt4F] method is not higher than the fifth power in the number of vibrational degrees of freedom when up to four-mode coupling terms are present in the Hamiltonian and only fourth order when only up to three-mode couplings are present. With a further approximation, one obtains the VCC[3pt4] model which is shown to scale with at most the fourth power in the number of vibrational degrees of freedom for Hamiltonians with both three- and four-mode coupling levels, while sharing the most important characteristics with VCC[3pt4F]. Sample calculations reported for selected tetra-atomic molecules as well as the larger dioxirane and ethylene oxide molecules support that the new models are accurate and useful. [ABSTRACT FROM AUTHOR]

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schrödinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI. [ABSTRACT FROM AUTHOR]

We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into 'core' and 'reservoir' modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems. [ABSTRACT FROM AUTHOR]

We derive a many-site version of the non-Markovian time-convolutionless polaron master equation [Jang et al., J. Chem Phys. 129, 101104 (2008)] to describe electronic excitation dynamics in multichromophoric systems. By treating electronic and vibrational degrees of freedom in a combined frame (polaron frame), this theory is capable of interpolating between weak and strong exciton-phonon coupling and is able to account for initial non-equilibrium bath states and spatially correlated environments. Besides outlining a general expression for the expected value of any electronic system observable in the original frame, we also discuss implications of the Markovian and Secular approximations highlighting that they need not hold in the untransformed frame despite being strictly satisfied in the polaron frame. The key features of the theory are illustrated using as an example a four-site subsystem of the Fenna-Mathews-Olson light-harvesting complex. For a spectral density including a localised mode, we show that oscillations of site populations may only be observed when non-equilibrium bath effects are taken into account. Furthermore, we illustrate how this formalism allows us to identify the electronic and vibrational components of the oscillatory dynamics. [ABSTRACT FROM AUTHOR]

The solution of the time-dependent Schrödinger equation for systems of interacting electrons is generally a prohibitive task, for which approximate methods are necessary. Popular approaches, such as the time-dependent Hartree-Fock (TDHF) approximation and time-dependent density functional theory (TDDFT), are essentially single-configurational schemes. TDHF is by construction incapable of fully accounting for the excited character of the electronic states involved in many physical processes of interest; TDDFT, although exact in principle, is limited by the currently available exchange-correlation functionals. On the other hand, multiconfigurational methods, such as the multiconfigurational time-dependent Hartree-Fock (MCTDHF) approach, provide an accurate description of the excited states and can be systematically improved. However, the computational cost becomes prohibitive as the number of degrees of freedom increases, and thus, at present, the MCTDHF method is only practical for few-electron systems. In this work, we propose an alternative approach which effectively establishes a compromise between efficiency and accuracy, by retaining the smallest possible number of configurations that catches the essential features of the electronic wavefunction. Based on a time-dependent variational principle, we derive the MCTDHF working equation for a multiconfigurational expansion with fixed coefficients and specialise to the case of general open-shell states, which are relevant for many physical processes of interest. [ABSTRACT FROM AUTHOR]

Internal conversion is an inherently quantum mechanical process. To date, 'on the fly' computation of internal conversion rates is limited to harmonic approximations, which would seem to be especially unsuitable, given that the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the semiclassial initial value representation (SCIVR) approach which is in principle amenable to on the fly studies even with 'many' degrees of freedom. In this paper we study the applicability of the Herman-Kluk (HK) SCIVR to a model system with two coupled and anharmonic degrees of freedom. We find that (a) the HK SCIVR is a good approximation to the exact quantum dynamics; (b) computation of the first order correction to the HK-SCIVR approximation corroborates the accuracy; (c) by studying a large parameter range, we find that the harmonic approximation is mostly unsatisfactory; and (d) for the specific model used, the coupling between the modes was found to be relatively unimportant. These results imply that the HK-SCIVR methodology is a good candidate for on the fly studies of internal conversion processes of 'large' molecules. [ABSTRACT FROM AUTHOR]

We investigate the structure formation of amphiphilic molecules at planar walls using density functional theory. The molecules are modeled as (hard) spheres composed of a hydrophilic and hydrophobic part. The orientation of the resulting Janus particles is described as a vector representing an internal degree of freedom. Our density functional approach involves fundamental measure theory combined with a mean-field approximation for the anisotropic interaction. Considering neutral, hydrophilic, and hydrophobic walls, we study the adsorption of the particles, focusing on the competition between the surface field and the interaction-induced ordering phenomena. Finally, we consider systems confined between two planar walls. It is shown that the anisotropic Janus interaction yields pronounced frustration effects at low temperatures. [ABSTRACT FROM AUTHOR]

The frozen Gaussian approximation to the quantum propagator may be a viable method for obtaining 'on the fly' quantum dynamical information on systems with many degrees of freedom. However, it has two severe limitations, it rapidly loses normalization and one needs to know the Gaussian averaged potential, hence it is not a purely local theory in the force field. These limitations are in principle remedied by using the Herman-Kluk (HK) form for the semiclassical propagator. The HK propagator approximately conserves unitarity for relatively long times and depends only locally on the bare potential and its second derivatives. However, the HK propagator involves a much more expensive computation due to the need for evaluating the monodromy matrix elements. In this paper, we (a) derive a new formula for the normalization integral based on a prefactor free HK propagator which is amenable to 'on the fly' computations; (b) show that a frozen Gaussian version of the normalization integral is not readily computable 'on the fly'; (c) provide a new insight into how the HK prefactor leads to approximate unitarity; and (d) how one may construct a prefactor free approximation which combines the advantages of the frozen Gaussian and the HK propagators. The theoretical developments are backed by numerical examples on a Morse oscillator and a quartic double well potential. [ABSTRACT FROM AUTHOR]

We introduce a new formula for the acceleration weight factor in the hyperdynamics simulation method, the use of which correctly provides an exact simulation of the true dynamics of a system. This new form of hyperdynamics is valid and applicable where the transition state theory (TST) is applicable and also where the TST is not applicable. To illustrate this new formulation, we perform hyperdynamics simulations for four systems ranging from one degree of freedom to 591 degrees of freedom: (1) We first analyze free diffusion having one degree of freedom. This system does not have a transition state. The TST and the original form of hyperdynamics are not applicable. Using the new form of hyperdynamics, we compute mean square displacement for a range of time. The results obtained agree perfectly with the analytical formula. (2) Then we examine the classical Kramers escape rate problem. The rate computed is in perfect agreement with the Kramers formula over a broad range of temperature. (3) We also study another classical problem: Computing the rate of effusion out of a cubic box through a tiny hole. This problem does not involve an energy barrier. Thus, the original form of hyperdynamics excludes the possibility of using a nonzero bias and is inappropriate. However, with the new weight factor formula, our new form of hyperdynamics can be easily implemented and it produces the exact results. (4) To illustrate applicability to systems of many degrees of freedom, we analyze diffusion of an atom adsorbed on the (001) surface of an fcc crystal. The system is modeled by an atom on top of a slab of six atomic layers. Each layer has 49 atoms. With the bottom two layers of atoms fixed, this system has 591 degrees of freedom. With very modest computing effort, we are able to characterize its diffusion pathways in the exchange-with-the-substrate and hop-over-the-bridge mechanisms. [ABSTRACT FROM AUTHOR]

The forward-backward (FB) approximation as applied to semiclassical initial value representations (SCIVR’s) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FB-SCIVR has proven elusive. In this paper, we provide a new derivation of a forward-backward phase space SCIVR expression (FBPS-SCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPS-SCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPS-SCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPS-SCIVR term already provides an excellent approximation. [ABSTRACT FROM AUTHOR]

A semiclassical (SC) approximation to the quantum mechanical propagator for nonadiabatic systems is derived. Our derivation starts with an exact path integral expression that uses canonical coherent states for the nuclear degrees of freedom and spin coherent states for the electronic degrees of freedom. A stationary path approximation (SPA) is then applied to the path integral to obtain the SC approximation. The SPA results in complex classical trajectories of both nuclear and electronic degrees of freedom and a double ended boundary condition. The root search problem is solved using the previously proposed “real trajectory local search” algorithm. The SC approximation is tested on three simple one dimensional two-state systems proposed by Tully [J. Chem. Phys. 93, 1061 (1990)], and the SC results are compared to Ehrenfest and surface hopping predictions. Excellent agreement with quantum results is reached when the SC trajectory is far away from caustics. We discuss the origin of caustics in this SC formalism and the strengths and weaknesses of this approach. [ABSTRACT FROM AUTHOR]

Abstract A quadrilateral bi-cubic G 1 -conforming finite element for the analysis of Kirchhoff plates is presented. The rational version of the Gregory patch proposed by Loop et al. (2009) is the starting point of our formulation. This version of the Gregory patch consists in rational enhancement of the bi-cubic Bézier interpolation representing a suitable tool for designing G 1 -conforming quadrilateral element on C 0 -conforming un-structured meshes. The element includes as additional degrees of freedom the edge rotations like in the Loof-formulations but is only displacement based. Because of the presence of the rational functions, the second derivatives of the interpolation present a finite discontinuity at the corners of the element, that prevent the element from passing the bending patch test. The element so formulated does not present optimal rate of convergence under h -refinement operation. The formulation is enhanced enforcing these discontinuities to be zero by means of Lagrange multipliers. It is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh. In this way the rational conforming approximation collapses into a conforming re-arrangement of the complete bi-cubic Bézier interpolation. Some examples and benchmarks are presented in order to test the performance of the element for the Kirchhoff plate model. [ABSTRACT FROM AUTHOR]

Highlights • New controller based on resonant non-integer orders. ○ The experimental test involves balanced and non-balanced situations. • The non-integer order generates a new degree of freedom in the controller with high dynamic skills. • This controller emerges as a good option to be implemented in systems in which computational time is critical and memory dependence is important. Abstract This paper proposes a novel fractional proportional-resonant controller, which applies fractional order calculus to the well-known proportional-resonant controllers. The focus of the study is the current control loop of voltage source converters. The main merit of the proposed fractional controller formulation lies into the use of fractional exponents in the integro-derivative parts obtaining a controller with an extra degree of freedom. This degree of freedom allows the phase delay to be improved for a wide frequency range in comparison with the conventional proportional-resonant controllers. Furthermore, the obtained controller results in a lower order transfer function that reduces the computational burden when multiple current frequencies have to be tracked. As fractional integro-derivative exponents are not directly implementable, five mathematical approaches are explored, selecting the Chareff's approximation for the fractional controller operator's implementation. A tuning procedure for such a controller is also addressed. The new controller formulation is validated in a 20 kVA laboratory set-up based on a silicon-carbide converter, and it is implemented in a DSP. Two AC output converter's configurations are considered to demonstrate the controllers' tracking capability; short-circuited (balanced fault) output, and grid-connected operation. This last case is evaluated operating as active filter and delivering fundamental component to a non-ideal grid. [ABSTRACT FROM AUTHOR]

Recently, it has been shown that implicit rectangular descriptions can be successfully used for modelling and controlling broad classes of linear systems, including systems with internal switches (i.e. variable structure linear systems where the variation is driven by switching signals). This technique consists in finding first the degree-of-freedom, characterizing the internal variable structure, and then making it unobservable by means of an ideal proportional and derivative descriptor variable feedback. When the proportional and derivative feedback is approximated by a suitable proper controller, then the degree-of-freedom is only attenuated in an epsilon order (i.e. the degree of approximation). In this article, we propose two different ways for observing the descriptor variable for implicit rectangular systems, in the presence of column minimal indices blocks. The first one concerns a descriptor variable observer based on fault detection; an apparent failure signal characterizes the variation of structure, which observation is required to support the synthesis of a standard state observer (this approach is constrained to minimum phase systems, with respect to the output—degree-of-freedom transfer). The second method concerns a descriptor variable observer based on precise finite-time adaptive structure detection. [ABSTRACT FROM AUTHOR]

A bandwidth method based on power ratio is proposed to evaluate the system damping by using frequency response functions. For single-degree freedom systems, exact formula for calculating damping ratio from displacement frequency response function is established. Additionally, an approximate formula to estimate the damping ratio from acceleration frequency response function is also derived. Both are represented in terms of the power ratio and the bandwidths relative to the corresponding peak frequencies. In contrast to the well-known half-power method, the proposed method can be used for relatively large damping ratios by selecting a corresponding high power ratio. The accuracy of the proposed formulas in damping estimation is investigated for a four-degree of freedom system by numerical experiments. The results show that by increasing the power ratio, the estimation errors in damping ratios for the four-degree of freedom system can be significantly reduced. [ABSTRACT FROM AUTHOR]

This paper is motivated by the stability problem of nonconstant periodic solutions of time‐periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way. [ABSTRACT FROM AUTHOR]

In this paper we develop an adaptive finite element method for elliptic problems. First, we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration, for some norm H s . Then an a-posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations. Based on this new error estimator, the element-wise refinement strategy in each subdomain is provided for a given tolerance. A discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family. Four 2D numerical benchmark examples of different domains and two 3D numerical benchmark examples are tested to demonstrate the robustness of our method. When possible, our numerical results are also compared to corresponding results from existing methods. All the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom. [ABSTRACT FROM AUTHOR]

As an exact method, the Woodbury formula is used to solve local material nonlinearity problems in structural analysis, in which the calculation and factorization of the global stiffness matrix are avoidable compared to the direct method used in the conventional finite-element method (FEM). This study uses the time-complexity theory to evaluate the efficiency of the Woodbury formula. The results show that the time complexity increases as the number of the inelastic degrees of freedom (IDOF) increases, and the Woodbury formula is more efficient than the LDLT factorization method only when nonlinearity appears within local, small regions. For example, for a structure with a total of 10,000 degrees of freedom (DOF), the efficiency of the Woodbury formula is higher than the LDLT method when the number of IDOF is less than 10% of the total DOF (1,000). Although the nonlinearity only occurring in small partial domains is common for most engineering structures, this low efficiency threshold still limits the Woodbury formula application for some structures with large part of nonlinear regions. To extend this efficiency threshold value, a Woodbury approximation method (WAM) is proposed that incorporates the idea of a combined approximations (CA) approach into the framework of the Woodbury formula, in which the reduced-basis method and binomial series expansion are used to solve the system of linear equations whose scale depends on the number of IDOF. The accuracy considerations for the approximate solution are discussed and the convergence criterion for error evaluation is presented. Moreover, the time-complexity analysis indicates that the limit on the efficiency is enhanced greatly by the proposed WAM under the comparable computational accuracy to the Woodbury formula, such as expanding above percentage of 10 to 70%. Finally, a numerical example is presented to prove that the WAM can provide accurate results with high efficiency, and thus, has greater potential for solving nonlinear problems. [ABSTRACT FROM AUTHOR]

We compare cosmologic and spherically symmetric solutions to metric and Palatini versions of vector Horndeski theory. It appears that Palatini formulation of the theory admits more degrees of freedom. Specifically, homogeneous isotropic configuration is effectively bimetric, and static spherically symmetric configuration contains nonmetric connection. In general, the exact solution in metric case coincides with the approximative solution in Palatini case. The Palatini version of the theory appears to be more complicated, but the resulting nonlinearity may be useful: we demonstrate that it allows the specific cosmological solution to pass through singularity, which is not possible in metric approach. [ABSTRACT FROM AUTHOR]

Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cell-centred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundary conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required. [ABSTRACT FROM AUTHOR]

In this work a reliability based optimization (RBO) strategy of Tuned Mass Damper (TMD) parameters is presented. The strategy is based on an energetic approach. The strategy consists to optimize the TMD parameters so that we minimize the failure probability (objective function) characterized by the exceedence of the power dissipated in the primary structure of a certain threshold value during some interval time. The evaluation of the objective function is carried out using the classical Rice’s formula. The strategy is, firstly, applied to linear single-degree of freedom (SDOF) system, subjected to seismic motion, and then extended to linear multi-degree of freedom (MDOF) system. The use of the Rice’s formula requires the knowledge of the joint probability density function (PDF) of the considered processes; to this end, exact expression of the joint PDF is presented for the SDOF system and an approximation is presented for the evaluation of the failure probabilities for the MDOF system. By making use of the obtained joint PDF, for the SDOF system, as the a priori joint PDF, the approximation of the joint PDF, for the MDOF system, has been performed using the Minimum cross-entropy method (MinxEnt).To highlight the good effectiveness of the proposed strategy, a ten-story shear building, subjected to different earthquakes, is considered. The obtained results are compared with other from literature, and it has been shown the superiority of the proposed strategy. [ABSTRACT FROM AUTHOR]

We consider the use of nodal and edge Virtual Element spaces for the discretization of magnetostatic problems in two dimensions, following the variational formulation of Kikuchi. In addition, we present a novel Serendipity variant of the same spaces that allow to save many internal degrees of freedom. These Virtual Element Spaces of different type can be useful in applications where an exact sequence is particularly convenient, together with commuting-diagram interpolation operators, as is definitely the case in electromagnetic problems. We prove stability and optimal error estimates, and we check the performance with some academic numerical experiments. [ABSTRACT FROM AUTHOR]

A new class of continuous-time low-pass filter using a set of Jacobi polynomials, with all transmission zeros at infinity, is described. The Jacobi polynomial has been adapted by using the parity relation for Jacobi polynomials in order to be used as a filter approximating function. The resulting class of polynomials is referred to as a pseudo Jacobi polynomials, because they are not orthogonal. The obtained magnitude response of these filters is more general than the magnitude response of the classical ultraspherical filter, because of one additional degree of freedom available in pseudo Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripples or sharper cutoff slope. Monotonic, critical monotonic, or nearly monotonic passband filter approximating functions can be also generated. It is shown that proposed pseudo Jacobi polynomial filter approximation also includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, and many transitional filter approximations, as its special cases. Several examples are presented, and detailed formulas including the practical suggestions for their efficient implementation are also provided. The proposed nearly monotonic filter is compared with the least-square-monotonic filters, designed as critical monotonic, in details. The advantages of the new filters are discussed. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Physical limits on the amount of information carried by bandlimited waveforms radiated in one and three dimensions are considered. It is shown that the entropy of radiation can achieve the Bekenstein bound using a 'burst' of energy, whose density vanishes as the radiating system expands. In comparison, black body radiation of infinite bandwidth achieves the same entropy scaling, that is proportional to the volume of the space, but requires an energy density that remains constant as the system expands. Rather than following the standard statistical physics approach of counting the number of eigenstates of the Hamiltonian of the quantum wave field, our derivation first considers an optimal subspace approximation, and then determines the number of bits that are required to represent any waveform in the space spanned by this representation with a minimum quantized energy error. This favors a geometric interpretation where the complexity of state counting is replaced by the one of determining the minimum cardinality covering of the signal space by high-dimensional balls, or boxes, whose size is lower bounded by quantum constraints. All derivations are given for both deterministic and stochastic settings. [ABSTRACT FROM AUTHOR]

Composite basis functions for pyramidal elements on the spaces H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) and L 2 ( Ω ) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart–Thomas–Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, etc.) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions. [ABSTRACT FROM AUTHOR]

Dynamical regular response of an oscillator with two serially connected springs with nonlinear characteristics of cubic type and governed by a set of differential-algebraic equations (DAEs) is studied. The classical approach of the multiple scales method (MSM) in time domain has been employed and appropriately modified to solve the governing DAEs of two systems, i.e. with one- and two degrees-of-freedom. The approximate analytical solutions have been verified by numerical simulations. [ABSTRACT FROM AUTHOR]

The interaction between particles in the mean-field approximation of the many-body theory is often taken into account with the use of the semiclassical description of the particle motion. However, quantization of a part of the degrees of freedom becomes essential in certain cases. In this work, two such cases where nonlinear wave equations appear have been considered: electrons in a quantum well and excitons in a trap. In the case of indirect excitons in an annular trap, the one-dimensional Gross-Pitaevskii equation permits an analytical solution and it turns out that there can be no bound state in a one-dimensional symmetric potential well. This makes the problem qualitatively different from a similar one-body problem. In the case of electrons in a quantum well, the nonlinear integro-differential equation does not have an exact solution and the allowed energy levels have been found by the direct variational method. [ABSTRACT FROM AUTHOR]

The nonlinear two-point boundary value problem (TPBVP for short) u x x + u 3 = 0 , u ( 0 ) = u ( 1 ) = 0 , offers several insights into spectral methods. First, it has been proved a priori that ∫ 0 1 u ( x ) d x = π ∕ 2 . By building this constraint into the spectral approximation, the accuracy of N + 1 degrees of freedom is achieved from the work of solving a system with only N degrees of freedom. When N is small, generic polynomial system solvers, such as those in the computer algebra system Maple, can find all roots of the polynomial system, such as a spectral discretization of the TPBVP. Our second point is that floating point arithmetic in lieu of exact arithmetic can double the largest practical value of N . (Rational numbers with a huge number of digits are avoided, and eliminating M symbols like 2 and π reduces N + M -variate polynomials to polynomials in just the N unknowns.) Third, a disadvantage of an “all roots” approach is that the polynomial solver generates many roots – ( 3 N − 1 ) for our example – which are genuine solutions to the N -term discretization but spurious in the sense that they are not close to the spectral coefficients of a true solution to the TPBVP. We show here that a good tool for “root-exclusion” is calculating ρ ≡ ∑ n = 1 N b n 2 ; spurious roots have ρ larger than that for the physical solution by at least an order of magnitude. The ρ -criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings. [ABSTRACT FROM AUTHOR]

We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ≥ 1 and globally C k -continuous, with k = 0 or p − 1 , to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom. We show that B-spline basis functions of degree p ≥ 1 , which are C p − 1 -continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting. [ABSTRACT FROM AUTHOR]

We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C 0 -separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method “refined Isogeometric Analysis (rIGA)”. To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 , with p being the polynomial order of the discretization. Numerical results indicate that our proposed refined isogeometric analysis delivers a speed-up factor proportional to p 2 . In a 2 D mesh with four million elements and p = 5 , the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3 D mesh with one million elements and p = 3 , the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach. [ABSTRACT FROM PUBLISHER]