Additional Higher Order Terms Research Articles

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.

Direct statistical simulation of low-order dynamosystems

In this paper, we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third order and has cubic nonlinearities while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics: timestepping, which is able to locate only stable solutions of the equations for the statistics, and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases, timestepping is a more robust protocol for finding meaningful solutions to DSS.

A strain-gradient formulation for fiber reinforced polymers: hybrid phase-field model for porous-ductile fracture

A novel numerical approach to analyze the mechanical behavior within composite materials including the inelastic regime up to final failure is presented. Therefore, a second-gradient theory is combined with phase-field methods to fracture. In particular, we assume that the polymeric matrix material undergoes ductile fracture, whereas continuously embedded fibers undergo brittle fracture as it is typical e.g. for roving glass reinforced thermoplastics. A hybrid phase-field approach is developed and applied along with a modified Gurson–Tvergaard–Needelman GTN-type plasticity model accounting for a temperature-dependent growth of voids on microscale. The mechanical response of the arising microstructure of the woven fabric gives rise to additional higher-order terms, representing homogenized bending contributions of the fibers. Eventually, a series of tests is conducted for this physically comprehensive multifield formulation to investigate different kinds and sequences of failure within long fiber reinforced polymers.

An ostentatious model of cosmological scalar–tensor theory

We consider a novel model of gravity with a scalar field described by the Lagrangian with higher-order derivative terms in a cosmological context. The model has the same solution for the homogeneous and isotropic universe as in the model with the Einstein gravity, notwithstanding the additional higher-order terms. Finally, a possible modification scenario is briefly discussed.

Gradient Elasto-plasticity with the Generalised Interpolation Material Point Method

The modelling of geomechanics problems can require a method that allows large deformations and non-linear material behaviour, in this respect the Generalised Material Point Method (GIMPM) is ideal. A fully implicit version of GIMPM has recently been developed for geomechanics problems and some aspects of its implementation are described here. An area that has received less attention in material point methods is that conventional analysis techniques constructed in terms of stress and strain are unable to resolve structural instabilities such as shear banding. This is because they do not contain any measure of the length of the microstructure of the material analysed, such as molecule size or grain structure. Gradient theories provide extensions of the classical equations with additional higher-order terms. The use of length scales makes it possible to model a finite thickness shear band which is not possible with traditional methods. Much work has been done on using gradient theories to include the effect of microstructure in the finite element method (and other numerical analysis techniques) however this yet to be combined with material point methods. In this paper the key equations that are required to extend the implicit GIMPM method to include gradient elasto-plasticity are detailed.

Nonlinear circuit analysis technique for microelectromechanical systems with a time‐variant capacitor and an AC power source

Microelectromechanical systems (MEMS) utilise time-variant capacitors as transducers in many applications. However, this kind of component can introduce harmonics and disturbances into the circuit with an AC power source, which is difficult to evaluate through closed-form solutions. This Letter proposes an iterative solution to analyse the dynamics of MEMS devices which contain a time-variant capacitor and an AC source. First, the expressions of the time-variant capacitor, AC source and their derivatives with respect to time are determined. Then, an initial solution that is sufficiently close to the actual solution is determined using linear circuit analysis. On the basis of the previous steps and the principles of the iterative method, an approximated solution combining the initial solution and its iteratively-derived higher-order terms is reached. Adding additional higher-order terms can improve the accuracy of the solution. A case study considering a MEMS device which has an AC power source and sinusoidal motion was performed using MATLAB Simulink. The simulation study further demonstrated that: (i) this iterative solution can effectively analyse the dynamics of MEMS devices with a time-variant capacitor and an AC power source; and (ii) computing additional higher-order terms derived from the initial solution can further improve the solution's accuracy.

Choptuik’s critical phenomenon in Einstein-Gauss-Bonnet gravity

We investigate the effects of higher order curvature corrections to Einstein's Gravity on the critical phenomenon near the black hole threshold, namely the Choptuik phenomenon. We simulate numerically a five dimensional spherically symmetric gravitational collapse of massless scalar field in Einstein-Gauss-Bonnet gravity towards a black hole formation threshold. When the curvature is sufficiently large the additional higher order terms, affect the evolution of the whole system. Since high curvature characterizes the region when the critical behavior takes place this critical behavior is destroyed. Both the self similarity and the mass scaling relation disappear. Instead we find a different behavior near the black hole threshold, which depends on the coupling constant of the higher order terms. The new features include a change of the sign of the Ricci scalar on the origin which indicates changes in the local geometry of space-time, and never occurs in classical general relativity collapse, and oscillations with a constant rather than with a diminishing length scale.

Analytic expansions of luni-solar gravity perturbations along rotating axes for trajectory optimization: Part 1: The dynamic system

An analytic form of the accelerations due to the luni-solar perturbations resolved along the rotating Euler-Hill frame is devised by using the expansion method. The addition of higher order terms to the main gravity gradient term linear in the spacecraft radial distance, carried out to the third order, provides a very high level of accuracy in accounting for the gravity perturbations experienced by a vehicle in orbit due to the sun and the moon. The nodal precession as well as the perigee advance of the lunar orbit is taken into account analytically by using the analytic lunar theory of de Pontécoulant. The analytic description of the apparent solar orbit and the motion of the moon remove the need to call an epherneris generator at each integration step during the numerical integration of the spacecraft trajectory, leading to the self-contained software for rapid and efficient optimal trajectory generation through iterations. Equinoctial elements are used to describe the spacecraft state and the luni-solar accelerations are given in terms of the apparent solar and lunar longitudes as well as Eulerian angles of the spacecraft orbit with respect to the inertial ecliptic system. The analysis is useful in optimal low-thrust orbit transfers complementing previous analyses carried out by this author, in which thrust and Earth zonal perturbations such as J 2, J 3 and J 4 in terms of the nonsingular equinoctial elements are included.

Saturn's internal planetary magnetic field

A model of Saturn's internal planetary magnetic field based on data from the Cassini prime mission has been derived. In the absence of a determination of the rotation rate, the model is constrained to be axisymmetric. Non‐axisymmetric models for a range of plausible planetary rotation periods have also been derived and we evaluate upper limits on the asymmetry of the internal magnetic field based on those models. We investigate whether a maximum in the non‐axisymmetric magnetic field can be identified at a particular rotation rate thus providing insight into the rotation rate of the planet's interior. No such peak can unambiguously be identified. An axisymmetric octupole model is adequate to fit the data and addition of higher order terms does not improve the goodness of fit. The largest value of the dipole tilt obtained from non‐axisymmetric models (<0.1°) confirm the high degree of symmetry of Saturn's magnetic field.

GRAVITATIONAL PHASE TRANSITIONS IN THE COSMOLOGICAL MANY-BODY SYSTEM

Gravitational many-body clustering of particles (e.g., galaxies) in an expanding universe may be regarded as a form of phase transition. We calculate its properties here and find that it differs in several ways from usual laboratory phase transitions. The cosmological case is never complete since it takes longer to evolve dynamically on larger spatial scales. To examine this, we calculate the effects of higher order corrections on the thermodynamic properties and distribution functions (which are known to agree with observations). The additional higher order terms are subdominant and decrease as the number of particles in the system increases. We also propose an order parameter for this hierarchical phase transition and discuss its relation to the Yang-Lee theory of phase transitions. These results also help to quantify earlier ideas of continuous clustering..

Bremsstrahlung cross-section with screening and Coulomb corrections at high energies

Using analytical formulae which are exact in Born approximation, the doubly differential bremsstrahlung cross-section with form-factor screening is calculated. For the atomic form factor parameters are applied which approximate self-consistent Dirac–Hartree–Fock–Slater calculations. The evaluation of the bremsstrahlung spectrum requires a single numerical integration. The results are superior to the customary Bethe approximation, in particular at the high-energy part of the spectrum. At high energies the screening correction can be added to any Coulomb-corrected cross-section without screening. In the present work, we are using a cross-section calculated by means of Sommerfeld–Maue functions with additional higher-order terms.

Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography

Evolution equations are derived for weakly nonlinear, multi-frequency and directional surface gravity waves propagating from deep to shallow water over weakly two-dimensional bottom topography. A uniform transition from cubic resonances in deep-intermediate water (Stokes regime) to quadratic near resonances in shallow water (Boussinesq regime) is obtained by extending the ordered solution to include additional higher-order terms for the bound wave components. The model assumes a leading-order, alongshore-uniform bottom with a two-dimensional depth perturbation that is incorporated through a Taylor series expansion of the bottom boundary condition. Numerical implementations of the model and comparisons to experimental data are presented that demonstrate the model's ability to describe: (i) cubic wave-wave interactions in deep-intermediate water depth; (ii) harmonic generation over a one-dimensional submerged obstacle; (iii) harmonic generation over two-dimensional topography.

Multidimensional Finite Differencing (MDFD) with Hypersphere-Close-Pack Grids

ABSTRACT This article concerns the use of hypersphere-close-pack (HCP) grids and quadratic multivariate Taylor polynomial interpolation functions with several additional higher-order terms for multidimensional finite differencing (MDFD). MDFD is a general methodology for the numerical solution of partial differential Equations (PDE) defined on irregular domains. MDFD uses domain embedding, which means that expensive body-fitted gridding is not required, but the methodology is flexible enough to allow any grid. The HCP grid allows efficient, stable, and robust implementation of MDFD. In this article, we discuss MDFD and related methods, summarize the HCP grids in two and three dimensions, outline the algorithms used, and present the results of a number of diagnostic examples. The examples concern numerical solution of the unsteady parabolic heat equation defined for a number of domains of different shapes. The examples suggest, among other things, that the algorithms work with domains that have no similarity to the grid geometry and that the solutions appear to have global O(h 2) convergence.

Weak Singularities of Planar Vector Fields Under Weak Perturbation

We characterize the elements of the set Hn of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in Hn, both on the plane and on the Poincare sphere. We use this information to characterize elements of the set Wn of smooth vector fields on ĝ2 beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in Wn. We also determine the set of elements of Hn that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.

Brownian motion in dynamically disordered media.

The motion of Brownian test particles in a model random potential with time dependent correlations is investigated using four methods: renormalized perturbation, perturbation using Martin, Siggia, and Rose functional formalism (MSR), the Edwards variational method on the MSR functional, and renormalization group with the MSR function. The disorder averaged one-particle propagators determined by the renormalized perturbation expansion and MSR perturbation expansion are identical to the second and possibly higher order, and the two-particle propagators determined by these perturbation methods are identical at the first and possibly higher order. The one-particle propagator determined by the Edwards method is identical to the perturbation expansions at the first order, but the second-order analogue of the Edwards method has a more complex expression, which reduces to the second-order perturbation expression with additional higher-order terms. The diffusion constant and two-particle correlations are calculated from these propagators and are used to determine the effects of the random potential on the Brownian particles. Generally, the diffusion rate decreases with the disorder strength and increases with the temporal decay rate. The two competing mechanisms result in an enhancement of the diffusion constant for weak potentials with fast temporal fluctuations. The system exhibits two-particle correlations that are inherently non-Gaussian and indicate clustering behavior. The diffusion constant is also determined from a simple one-loop renormalization group calculation. In the static limit, the diffusion constant calculated by the renormalization group recovers the results of Deem and Chandler [M.W. Deem and D. Chandler, J. Stat. Phys. 76, 911 (1994)].

Multisoliton formation from a single narrow pulse in ferrite films

The formation of solitons in ferrite films is modeled by a nonlinear Schroedinger equation, and numerical solution will indicate how multisoliton pulses form from a single pulse. Modeling is done using a narrow initial pulse of duration 13 ns with additional higher order terms in the equation. It is shown that the higher order terms will result in an asymmetric two-peak structure which is similar to experimentally observed soliton structures in yttrium iron garnet thin films.

Role of interfaces in higher order angular terms of magnetic anisotropies: ultrathin film structures

The role of interface inhomogeneities on macroscopic magnetic anisotropy terms was investigated using simple mathematical modeling. It is shown that the intrinsic anisotropies caused by the spin–orbit interaction in the presence of lateral inhomogeneities can create additional magnetic anisotropies with higher order angular terms. The lateral inhomogeneous distribution of the uniaxial anisotropy can result in the presence of a fourth order angular term. These anisotropies decrease the total energy, and their strength is affected by the exchange averaging effect. Large atomic terraces allow the local magnetic moment to deviate from the mean orientation resulting in negative higher order angular terms in magnetic anisotropies satisfying the sample symmetry. The negative sign results in the magnetic easy axes being oriented 45° with respect to the intrinsic uniaxial axis. It is shown that the additional higher order terms do not always have to satisfy the time inversion symmetry. Odd powers in directional cosines can arise when the exchange restoring torque becomes comparable to or weaker than torques from the internal fields. The additional higher order angular terms in magnetic anisotropies decrease usually more rapidly with the sample thickness than the intrinsic interface anisotropies. Their rapid decrease in strength with an increasing sample thickness allows one to distinguish them from the intrinsic interface anisotropies. It will be shown that the magnetic anisotropies originating in interface inhomogeneities can play an important role in the magnetic phase transitions of ultrathin film structures.

A novel higher order extending method in a MESFET channel current model for Volterra series analysis

We propose a new method to find the higher order Taylor series coefficients of a MESFET channel current model for Volterra series analysis from the measured lower order Taylor coefficients at several bias points. Generally, the third-order Taylor expansion is not sufficient to accurately characterize the linear amplifier distortion properties. The channel current model is improved with these additional higher order terms. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 20: 292–295, 1999.

ON WILSON SURFACES IN TOPOLOGICAL CHERN-SIMONS INTERACTION IN 3+1 DIMENSIONS

We comment on the regularization of the expectation values of Wilson surfaces for the bosonic string in 3+1 dimensions. We analyze the singular behaviors of propagators for the Chern-Simons action with the additional higher order terms.

Regge gravity on general triangulations

We investigate quantum gravity in four dimensions using the Regge approach on triangulations of the four-torus with general, non-regular incidence matrices. We observe that the simplicial lattice with originally 31 regular vertices tends to develop spikes at irregularly inserted vertices with low coordination numbers even for vanishing gravitational coupling. Different to the regular, hypercubic lattices almost exclusively used in previous studies, we find now that the observables depend on the measure. Computations with nonvanishing gravitational coupling still reveal the existence of a region with well-defined expectation values. However, the phase structure depends on the triangulation with the regular and additional vertices undergoing two separate transitions. Even with additional higher-order terms in the action the critical behavior of the system changes with varying (local) coordination numbers.