Boltzmann Integral Research Articles

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

A new mixed Boltzmann-BGK model for mixtures solved with an IMEX finite volume scheme on unstructured meshes

In this work, we consider a novel model for a binary mixture of inert gases. The model, which preserves the structure of the original Boltzmann equations, combines integro-differential collision operators with BGK relaxation terms in each kinetic equation: the first involving only collisions among particles of the same species, while the second ones taking into account the inter–species interactions. We prove consistency of the model: conservation properties, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the shape of a global Maxwell distribution. We also derive hydrodynamic equations under different collisional regimes. In a second part, to numerically solve the governing equations, we introduce a class of time and space high order finite volume schemes that are able to capture the behaviors of the different hydrodynamic limit models: the classical Euler equations as well as the multi-velocities and temperatures Euler system. The methods work by integrating the distribution functions over arbitrarily shaped and closed control volumes in 2D using Central Weighted ENO (CWENO) techniques and make use of spectral methods for the approximation of the Boltzmann integrals with high order Implicit-Explicit (IMEX) Runge Kutta schemes. For these methods, we prove accuracy and preservation of the discrete asymptotic states. In the numerical section we first show that the methods indeed possess the theoretical order of accuracy for different regimes and second we analyse their capacity in solving different two dimensional problems arising in kinetic theory. To speed up the computational time, all simulations are run with MPI parallelization on 64 cores, thus showing the potentiality of the proposed methods to be used for HPC (High Performance Computing) on massively parallel architectures.

A Generalized Two–Scale Approximation for Ocean Wave Models

The two-scale approximation (hereafter, TSA) was previously presented as a new method to approximate and estimate transfer rates in wind – wave spectra. It was shown to perform well for a variety of idealized and observed sea state conditions and to compare well with respect to the total Boltzmann integral for nonlinear quadruplet wave-wave interactions. Here, we present a generalized formulation of TSA, allowing for multiple peaked spectra, sheared spectra, sea – swell combinations, etc. This formulation is implemented in a modern operational wave model, WAVEWATCHIII™, and shown to provide a significant improvement over the standard approximation used in wave models, the discrete interaction approximation (DIA). Additional tests involve the simulation of waves generated in Hurricane Teddy (2020).

An unified formulation of strong non-local elasticity with fractional order calculus

The research of a formulation to model non-local interactions in the mechanical behavior of matter is currently an open problem. In this context, a strong non-local formulation based on fractional calculus is provided in this paper. This formulation is derived from an analogy with long-memory viscoelastic models. Specifically, the same kind of power-law time-dependent kernel used in Boltzmann integral of viscoelastic stress-strain relation is used as kernel in the Fredholm non-local relation. This non-local formulation leads to stress-strain relation based on the space Riesz integral and derivative of fractional order. For unbounded domain, proposed model can be defined in stress- and in strain-driven formulation and in both cases the stress–strain relation represent a strong non-local model. Also, the proposed strain driven and stress driven formulations defined in terms of Riesz operators are proved to be fully consistent each another. Moreover, the proposed model posses a mechanical meaning and for unbounded non-local rod is described and discussed in detail.

A study of the vibration isolation performance of a limited phononic crystal vibration isolator based on local resonance theory

In this study, a limited phononic crystal vibration isolation (LPCVI) model is constructed based on a vibration isolator used in the field of rail transit, and analyses of the characteristics of the bandgap, the vibration isolation effect, and the vibrational energy transfer of the model are presented. In this paper, the Boltzmann integration theory and the Bloch theorem are used to establish a mathematical model that analyzes the band structure based on the viscoelastic damping of the system. Additionally, by comparing the practical finite periodic structure model and the conventional mass-spring-damping vibration isolation model, explicit forms of the vibration isolation coefficients of the models are derived. It is found that when the external excitation frequency is within the forbidden band range, the vibration isolation coefficient of the LPCVI system with a harmonic oscillator is much smaller than that of the vibration isolation system with a general mass-spring. Furthermore, the Newmark-β integration method is adopted to solve the vibration equation of the LPCVI model. The energy input, distribution, and output of the system are obtained when the energy is under excitation in the forbidden band and bandpass frequencies. It is found that the external excitation does both positive and negative works on the vibration isolation system within a certain period under the action of the central frequency excitation of the forbidden band; therefore, the energy cannot be input into the isolation system. This makes it possible to achieve effective vibration isolation at lower frequencies.

Analytical Results for the Three-Body Radiative Attachment Rate Coefficient, with Application to the Positive Antihydrogen Ion H+

To overcome the numerical difficulties inherent in the Maxwell–Boltzmann integral of the velocity-weighted cross section that gives the radiative attachment rate coefficient α R A for producing the negative hydrogen ion H − or its antimatter equivalent, the positive antihydrogen ion H ¯ + , we found the analytic form for this integral. This procedure is useful for temperatures below 700 K, the region for which the production of H ¯ + has potential use as an intermediate stage in the cooling of antihydrogen to ultra-cold (sub-mK) temperatures for spectroscopic studies and probing the gravitational interaction of the anti-atom. Our results, utilizing a 50-term explicitly correlated exponential wave function, confirm our prior numerical results.

A BGK model for charge transport in graphene

Abstract The classical Boltzmann equation describes well temporal behaviour of a rarefied perfect gas. Modified kinetic equations have been proposed for studying the dynamics of different type of gases. An important example is the transport equation, which describes the charged particles flow, in the semi-classical regime, in electronic devices. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the standard Boltzmann integral term. These new equations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook (BGK) model. In this work we propose and analyse a BGK model for charge transport in graphene.

Relaxation of a nonequilibrium phonon distribution induced by femtosecond laser irradiation

Ultrafast laser irradiation of solids leads to a thermodynamic nonequilibrium within and between the electron and phonon subsystems of the material. Due to electron-electron and phonon-phonon collisions, both subsystems relax into respective new thermodynamic equilibria within a characteristic thermalization time, which is different for each one of them. Moreover, they equilibrate their temperatures by electron-phonon coupling. The relaxation of the electronic nonequilibrium and its effect on the electron-phonon coupling was the subject of a number of studies, and it is comparably well understood, while the nonequilibrium within the phononic subsystem is usually neglected and the influence of the nonequilibrium phonons on other relaxation processes is unclear. Our calculations show significant differences in the energy transfer rate between the electrons and the phonons depending on whether a nonequilibrium distribution is assumed for the phonons or not. Here, we present a model to study the relaxation of the nonequilibrium phonon subsystem. Collisions between phonons are described within the frame of Boltzmann integrals. From this, an energy-dependent relaxation time can be extracted and inserted into a relaxation-time approach. Within the frame of this model, we study the thermalization of a phonon distribution induced by ultrafast laser irradiation. We show that the thermalization time of such a distribution is of the order of some hundreds of picoseconds. Moreover, we discuss the energy transfer between Fermi-distributed electrons and nonequilibrium phonons and compare it to the energy transfer for equilibrium distributions in both subsystems.

Numerical simulations of ocean surface waves under hurricane conditions: Assessment of existing model performance

Using the well-observed hurricane case Ivan (2004) as an example, we investigate and intercompare the performance of two wave models under hurricane conditions. One is the WAVEWATCH III model (WW3) and the other is the University of Miami Wave Model (UMWM). Within WW3, four different source term packages (ST2/3/4/6) of wind input, wave breaking dissipation and swell decay are chosen for comparison purposes. Based on the comparisons between model results and measurements from various platforms, we concluded that UMWM shows less accuracy than WW3 in specification of bulk wave parameters. This is possibly because (i) UMWM-estimated drag coefficient does not clearly show a saturation trend when wind speeds are beyond ∼ 35 m s−1 and (ii) the four-wave interaction term of UMWM disagrees evidently with the full solution of the Boltzmann integral in detail. Among the four WW3 source term packages, the older parameterization ST2 is basically the least accurate because of its systematic underestimation of high waves. The remaining three packages (ST3/4/6) are performed well under Ivan. However, we also find that they tend to overestimate energy of waves traveling in the oblique and opposing winds. It is shown that enhancing the strength of negative wind input properly can effectively improve model skills in such situations. Limited by the uncertainty in the wind forcing, we could not determine the most accurate package among ST3/4/6 unambiguously.

On the accuracy of the direct discrete simulation of the Landau collision integral by the Boltzmann integral

The Landau (Fokker–Planck) integral of Coulomb collisions is an integral component of the physical and mathematical models of both laboratory and space plasma, in which the intermediate collisionality regime is important. This paper discusses the method for the direct statistical simulation of the Monte Carlo type for a kinetic equation with a nonlinear operator of the Landau–Fokker–Planck (LFP) Coulomb collisions. This method is based on the approximation of the Landau collision integral by the Boltzmann collision integral. The paper has two main objectives: firstly, to obtain numerical estimates of the order of approximation of the Landau collision integral by the Boltzmann integral and, secondly, to explore the possibilities of optimizing the algorithm for multiply charged ions. The results are illustrated by the calculations of the problem on the relaxation of the initial distribution to equilibrium for one and two components.

Direct discrete simulation of the kinetic Landau-Fokker-Planck equation with an alternating external electromagnetic field

A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.

Physics-based potentials for the coupling between backbone- and side-chain-local conformational states in the UNited RESidue (UNRES) force field for protein simulations.

The UNited RESidue (UNRES) model of polypeptide chains is a coarse-grained model in which each amino-acid residue is reduced to two interaction sites, namely, a united peptide group (p) located halfway between the two neighboring α-carbon atoms (Cαs), which serve only as geometrical points, and a united side chain (SC) attached to the respective Cα. Owing to this simplification, millisecond molecular dynamics simulations of large systems can be performed. While UNRES predicts overall folds well, it reproduces the details of local chain conformation with lower accuracy. Recently, we implemented new knowledge-based torsional potentials (Krupa et al. J. Chem. Theory Comput. 2013, 9, 4620–4632) that depend on the virtual-bond dihedral angles involving side chains: Cα···Cα···Cα···SC (τ(1)), SC···Cα···Cα···Cα (τ(2)), and SC···Cα···Cα···SC (τ(3)) in the UNRES force field. These potentials resulted in significant improvement of the simulated structures, especially in the loop regions. In this work, we introduce the physics-based counterparts of these potentials, which we derived from the all-atom energy surfaces of terminally blocked amino-acid residues by Boltzmann integration over the angles λ(1) and λ(2) for rotation about the Cα···Cα virtual-bond angles and over the side-chain angles χ. The energy surfaces were, in turn, calculated by using the semiempirical AM1 method of molecular quantum mechanics. Entropy contribution was evaluated with use of the harmonic approximation from Hessian matrices. One-dimensional Fourier series in the respective virtual-bond-dihedral angles were fitted to the calculated potentials, and these expressions have been implemented in the UNRES force field. Basic calibration of the UNRES force field with the new potentials was carried out with eight training proteins, by selecting the optimal weight of the new energy terms and reducing the weight of the regular torsional terms. The force field was subsequently benchmarked with a set of 22 proteins not used in the calibration. The new potentials result in a decrease of the root-mean-square deviation of the average conformation from the respective experimental structure by 0.86 Å on average; however, improvement of up to 5 Å was observed for some proteins.

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier–Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar–Gross–Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler–Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures.

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.

Coupled viscoelastic–viscoplastic modeling of homogeneous and isotropic polymers: Numerical algorithm and analytical solutions

A coupled viscoelastic–viscoplastic (VE–VP) model is implemented and studied. The total strain is the sum of VE and VP parts, and the Cauchy stress is given by a linear VE model as a Boltzmann integral of the history of VE strains. The proposed computational algorithm features fully implicit integration, return mapping based on a two-step VE predictor/VP corrector strategy, and a consistent tangent operator. The algorithm is applied to J 2 VP coupled with VE. Very compact expressions are obtained which are form-identical to classical elasto-viscoplasticity (EVP) provided that the constant linear elastic shear and bulk moduli are replaced with incremental relaxation moduli which are appropriate functions of the time increment. Two different integration methods to obtain the incremental moduli are proposed and assessed. Closed-form solutions for uniaxial tension and simple shear are developed, based on an original solution method for integro-differential equations. The analytical results enable to illustrate the constitutive model and provide unambiguous benchmarks for numerical algorithms. Model predictions are compared with experimental data and reasonable correlation is obtained.

Spectral Analysis and Riesz Basis Property for Wave Equation with Boltzmann Damping

AbstractIn this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration frequency point of view. This suggests us to investigate the system with memory counted from the vibrating starting moment. In the latter case, it is shown that the spectrum of system determines completely the dynamic behavior of the vibration: There is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As the consequences, the spectrum-determined growth condition and exponential stability are concluded. The results of this paper expositorily demonstrate the proper modeling the elastic systems with Boltzmann damping.

Adiabatic initial conditions for perturbations in interacting dark energy models

We present a new systematic analysis of the early radiation era solution in an interacting dark energy model to find the adiabatic initial conditions for the Boltzmann integration. In a model where the interaction is proportional to the dark matter density, adiabatic initial conditions and viable cosmologies are possible if the early-time dark energy equation of state parameter is $w_e > -4/5$. We find that when adiabaticity between cold dark matter, baryons, neutrinos and photons is demanded, the dark energy component satisfies automatically the adiabaticity condition. As supernovae Ia or baryon acoustic oscillation data require the recent-time equation of state parameter to be more negative, we consider a time-varying equation of state in our model. In a companion paper [arXiv:0907.4987] we apply the initial conditions derived here, and perform a full Monte Carlo Markov Chain likelihood analysis of this model.

Observational constraints on an interacting dark energy model

We use observations of cosmic microwave background anisotropies, supernova luminosities and the baryon acoustic oscillation signal in the galaxy distribution to constrain the cosmological parameters in a simple interacting dark energy model with a time-varying equation of state. Using a Monte Carlo Markov Chain technique we determine the posterior likelihoods. Constraints from the individual data sets are weak, but the combination of the three data sets confines the interaction constant $\Gamma$ to be less than 23% of the expansion rate of the Universe $H_0$; at 95% CL $-0.23 < \Gamma/H_0 < +0.15$. The CMB acoustic peaks can be well fitted even if the interaction rate is much larger, but this requires a larger or smaller (depending on the sign of interaction) matter density today than in the non-interacting model. Due to this degeneracy between the matter density and the interaction rate, the only observable effect on the CMB is a larger or smaller integrated Sachs-Wolfe (ISW) effect. While SN or BAO data alone do not set any direct constraints on the interaction, they exclude the models with very large matter density, and hence indirectly constrain the interaction rate when jointly analysed with the CMB data. To enable the analysis described in this paper, we present in a companion paper [arXiv:0907.4981] a new systematic analysis of the early radiation era solution to find the adiabatic initial conditions for the Boltzmann integration.

A Two-Scale Approximation for Efficient Representation of Nonlinear Energy Transfers in a Wind Wave Spectrum. Part II: Application to Observed Wave Spectra

Abstract In Part I of this series, a new method for estimating nonlinear transfer rates in wind waves, based on a two-scale approximation (TSA) to the full Boltzmann integral (FBI) for quadruplet wave–wave interactions, was presented, and this new method was tested for idealized spectral data. Here, the focus is on comparisons of the TSA and the discrete interaction approximation (DIA) with the FBI for observed wave spectra from field measurements. Observed wave spectra are taken from a wave gauge array in Currituck Sound and a directional waverider off the coast near the Field Research Facility at Duck, North Carolina. Results show that the TSA compares much more favorably to the FBI than does the DIA, even for cases in which the parametric component of the formulation does not capture the spectral energy distribution very well. These results remain valid for the TSA estimates when the FBI results are significantly affected by the directional distribution of energy. It is also shown that although nonlinear transfers are substantially weaker in swell portions of the spectrum these interactions contribute significantly to the spectral evolution and net energy balance in long-distance swell propagation.

Assessing one-dimensional diffusion in nanoporous materials from transient concentration profiles

The use of interference microscopy has enabled the direct observation of transient concentration profiles generated by intracrystalline transport diffusion in nanoporous materials. The thus accessible intracrystalline concentration profiles contain a wealth of information which cannot be deduced by any macroscopic method. In this paper, we illustrate five different ways for determining the concentration-dependent diffusivity in one-dimensional systems and two for the surface permeability. These methods are discussed by application to concentration profiles evolving during the uptake of methanol by the zeolite ferrierite and of methanol by the metal organic framework (MOF) manganese(II) formate. We show that the diffusivity can be calculated most precisely by means of Fick's 1st law. As the circumstances permit, Boltzmann's integration method also yields very precise results. Furthermore, we present a simple procedure that enables the estimation of the influence of the surface barrier on the overall uptake process by plotting the boundary concentration versus the overall uptake.