Fourth-order Expansion Research Articles

The general low-frequency prediction for asymptotically nonlinear material functions in oscillatory shear

We use a fourth-order fluid expansion to make general predictions for asymptotically nonlinear material functions in oscillatory shear, a characterization protocol sometimes known as medium-amplitude oscillatory shear. The calculation applies to any viscoelastic fluid in the terminal regime defined by the limit of Deborah number much less than one. Two viscous nonlinearities appear at third order, with shear stress scaling as ω3, and are interrelated by a constant multiplicative factor. Two elastic nonlinearities appear at fourth order, with shear stress scaling as ω4, and are also interrelated by a constant multiplicative factor. These nonlinear measures are decoupled from the linear material functions G′ and G″ because they depend on different expansion coefficients. Experimental measurements are presented for all four asymptotic shear material functions using a linear and well-entangled homopolymer of polyisoprene. The experimental observations are consistent with both the predicted frequency scaling and the predicted interrelations in the terminal regime. Signs and magnitudes cannot be universally predicted, leaving these as free parameters that depend on the specifics of the material microstructure or constitutive model. These general results explain previous observations involving different materials and constitutive models, and provide an important reference for future experiments, analytical results, and numerical computations of these rheological fingerprints.

Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.

The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang etal. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yangand Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.

Phenomenological theory of phase diagrams. The case of Landau’s potentials with $$\mathcal{L} = 43m$$

Phase diagrams of the crystals undergoing phase transitions induced by irreducible representations with the $$\mathcal{L} = 43m$$ symmetry group have been constructed within the phenomenological theory of the second-order phase transitions. The equations of state of all phases induced by symmetry have been received, and the general conditions of their thermodynamic stability have been formulated. For the fourth-order expansion of thermodynamic potential in terms of the order parameter components, exhaustive results have been obtained: equations of phases stability boundaries and all possible types of phase diagrams have been found analytically. For the eighth-order expansion of the potential the variety of phase diagrams has been analyzed using computer calculations. For the first time the possibility of existence of phase diagrams described by this potential and containing one or several triple points, three-phase and four-phase regions is shown. Examples of the crystals undergoing phase transitions with considered symmetry of the order parameter have been given.

Explicit solutions for the modeling of laminated composite plates with arbitrary stacking sequences

In this paper, a method to compute explicit solutions for laminated plate with arbitrary stacking sequences is presented. This technique is based on the construction of an a posteriori Reduced-Order Model using the so-called Proper Generalized Decomposition. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x,y, the transverse coordinate z and the orientation of each ply θi. This choice yields to an iterative process that consists of solving a 2D and some 1D problems successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used. The functions of θi are discretized with linear interpolations. Mechanical tests with different numbers of layers are performed to show the accuracy of the method.

Three crossing limit cycles in planar piecewise linear systems with saddle-focus type

This paper presents an analysis on the appearance of limit cycles in planar Filippov system with two linear subsystems separated by a straight line. Under the restriction that the orbits with points in the sliding and escaping regions are not considered, we provide firstly a topologically equivalent canonical form of saddle-focus dynamic with five parameters by using some convenient transformations of variables and parameters. Then, based on a very available fourth-order series expansion of the return map near an invisible parabolic type tangency point, we show that three crossing limit cycles surrounding the sliding set can be bifurcated from generic codimensionthree singularities of planar discontinuous saddle-focus system. Our work improves and extends some existing results of other researchers.

Laplacian-Level Kinetic Energy Approximations Based on the Fourth-Order Gradient Expansion: Global Assessment and Application to the Subsystem Formulation of Density Functional Theory.

We tested Laplacian-level meta-generalized gradient approximation (meta-GGA) noninteracting kinetic energy functionals based on the fourth-order gradient expansion (GE4). We considered several well-known Laplacian-level meta-GGAs from the literature (bare GE4, modified GE4, and the MGGA functional of Perdew and Constantin (Phys. Rev. B 2007,75, 155109)), as well as two newly designed Laplacian-level kinetic energy functionals (L0.4 and L0.6). First, a general assessment of the different functionals is performed to test them for model systems (one-electron densities, Hooke's atom, and different jellium systems) and atomic and molecular kinetic energies as well as for their behavior with respect to density-scaling transformations. Finally, we assessed, for the first time, the performance of the different functionals for subsystem density functional theory (DFT) calculations on noncovalently interacting systems. We found that the different Laplacian-level meta-GGA kinetic functionals may improve the description of different properties of electronic systems, but no clear overall advantage is found over the best GGA functionals. Concerning the subsystem DFT calculations, the here-proposed L0.4 and L0.6 kinetic energy functionals are competitive with state-of-the-art GGAs, whereas all other Laplacian-level functionals fail badly. The performance of the Laplacian-level functionals is rationalized thanks to a two-dimensional reduced-gradient and reduced-Laplacian decomposition of the nonadditive kinetic energy density.

Shell finite element based on the Proper Generalized Decomposition for the modeling of cylindrical composite structures

The introduction of the Proper Generalized Decomposition (PGD) is presented for the layer-wise modeling of heterogeneous cylindrical shells. The displacement field is approximated as a sum of separated functions of the in-plane coordinates and the transverse coordinate. This choice yields to an iterative process that consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used. The approach is assessed through mechanical tests for thin/thick and deep/shallow laminated cylindrical shells. Both convergence rate and accuracy are discussed.

Accurate modeling of high order spatial dispersion of wire medium

The wire medium consisting of an array of parallel thin metallic wires was previously studied by using an effective medium with spatial dispersion. In this paper, the validity of conventional effective model was examined analytically and numerically by studying a canonical structure of the wire medium. It is noted that the conventional model fails for high transversal spatial harmonics, which consequently results in discrepancy in the scattering between the effective model and the physical structure. In this study, we propose a new effective model to include higher order spatial dispersions: instead of the second-order expansion, the proposed dispersion equation is based on the fourth-order expansion of the dispersion equation of the photonic states. Compared with the 3D full-wave simulation results of the wire medium, the proposed model has demonstrated significant improvement in numerical accuracy in characterizing the EM behavior in this type of metamaterials.

Nonequilibrium dynamical mean-field theory based on weak-coupling perturbation expansions: Application to dynamical symmetry breaking in the Hubbard model

We discuss the general formalism and validity of weak-coupling perturbation theory as an impurity solver for nonequilibrium dynamical mean-field theory. The method is implemented and tested in the Hubbard model, using expansions up to fourth order for the paramagnetic phase at half filling and third order for the antiferromagnetic and paramagnetic phase away from half filling. We explore various types of weak-coupling expansions and examine the accuracy and applicability of the methods for equilibrium and nonequilibrium problems. We find that in most cases an expansion of local self-energy diagrams including all the tadpole diagrams with respect to the Weiss Green's function (bare-diagram expansion) gives more accurate results than other schemes such as self-consistent perturbation theory using the fully interacting Green's function (bold-diagram expansion). In the paramagnetic phase at half filling, the fourth-order bare expansion improves the result of the second-order expansion in the weak-coupling regime, while both expansions suddenly fail at some intermediate interaction strength. The higher-order bare perturbation is especially advantageous in the antiferromagnetic phase near half filling. We use the third-order bare perturbation expansion within the nonequilibrium dynamical mean-field theory to study dynamical symmetry breaking from the paramagnetic to the antiferromagnetic phase induced by an interaction ramp in the Hubbard model. The results show that the order parameter, after an initial exponential growth, exhibits an amplitude oscillation around a nonthermal value followed by a slow drift toward the thermal value. The transient dynamics seems to be governed by a nonthermal critical point, associated with a nonthermal universality class, which is distinct from the conventional Ginzburg-Landau theory.

Third harmonic shear horizontal and Rayleigh Lamb waves in weakly nonlinear plates

The third order harmonic generation (third harmonics as well as cubic sum and difference harmonics) due to the cubic interaction of two collimated elastic waves in a homogeneous, isotropic, weakly nonlinear plate is investigated by using a fourth order expansion of strain energy density to formulate the nonlinear boundary problems. Waves with both shear horizontal (SH) and Rayleigh Lamb (RL) nature are considered as primary or tertiary wave fields. The non-zero power flux condition is evaluated using characteristic parity matrices of the cubic nonlinear forcing terms and third order harmonic mode shapes. Results indicate that waves with either SH or RL nature receive power flux from a specific pattern of primary mode interaction. Further analytical evaluation of the synchronism condition enables identification of primary SH and RL modes that are able to generate cumulative third harmonics. The primary SH modes are shown to be holo-internal-resonant with third harmonic SH fields. This simply means that all points on the primary dispersion curves are internally resonant with third harmonics, which is not the case for second harmonics. Such flexibility will be advantageous for laboratory and field measurements.

Free-energy power expansion for orientationally ordered phases: Energy and entropy

We propose an approach to the description of orientational phase transitions that utilizes the specific features of orientational energy and entropy. The approach is applied to building a model for nematic phases in materials with nonpolar parallelepiped-type molecules with symmetry D(2h). The model operates with complex order parameters, generalizes the Landau-de Gennes theory, and predicts the existence of a biaxial nematic phase for the fourth-order expansion of free energy.

Proper Generalized Decomposition and layer-wise approach for the modeling of composite plate structures

In the framework of the design of laminated and sandwich structures, the computation of local quantities needs a layer-wise approach. But, the computational cost of such approach increases with the number of layers. In this work, the introduction of the Proper Generalized Decomposition (PGD) is presented for the layer-wise modeling of heterogeneous structures in order to reduce the number of unknowns. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x,y and the transverse coordinate z. This choice yields to an iterative process that consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used.After a preliminary study to show the relevance of the present approach, mechanical tests for thin to thick laminated and sandwich plates with various boundary conditions are presented. The results are compared with elasticity reference solutions.

Relating the Progeny Production Curve to the Speed of an Epidemic

The dependence of the initial infection rate, r, on the basic reproductive number, R(0), and the temporal moments of the progeny production curve are examined. A solution to the linearized Kermack-McKendrick equation is presented and used to analyze a variety of theoretical models of pathogen reproduction. The solution yields a relation between r and the basic reproductive number, R(0); the mean time between pathogen generations, μ; and the standard deviation about this mean, σ. A transformation using the dimensionless variables rμ and rσ is introduced, which maps the solution onto a one-dimensional curve. An approximation for the value of r in terms of R(0) and the first four temporal moments of the reproductive curve is derived. This allows direct comparison of epidemics resulting from theoretical models with those generated using experimentally obtained reproduction curves. For epidemics characterized by a value of rμ < 5, the value of r is well determined (<2%) by this fourth-order expansion regardless of the functional form of the reproduction curve.

Dynamical Mean-Field Theory on the Basis of the Fourth-Order Perturbation Expansion

Adopting a fourth-order perturbation expansion as a solver for the impurity problem, we develop a dynamical mean-field theory for the Hubbard model. Zero temperature Mott transition is discussed by calculating a quasiparticle weight z in the infinite dimensional Hubbard model at half filling. We obtain fairly good agreement with the result on the basis of the numerical renormalization group as an impurity solver, revising the result based on the second-order perturbation expansion as the impurity solver.

Assessment of a composite beam finite element based on the proper generalized decomposition

In this paper, a finite element based on the Proper Generalized Decomposition (PGD) is presented for the analysis of bi-dimensional laminated beams. The displacement field is approximated as a sum of separated functions of x (axial coordinate) and z (transverse coordinate). This choice yields to an iterative process that consists of computing a product of two one-dimensional functions at each iteration. The capability and the behavior of the PGD approach are shown on isotropic beam with different slenderness ratios. A second and fourth-order expansion with respect to the thickness are considered. Mechanical tests for thin/thick laminated and sandwich beams are presented in order to evaluate the two approaches. They are compared with elasticity and 2D finite element reference solutions.

Long-time dynamics and stationary nonequilibrium of an optically driven strongly confined quantum dot coupled to phonons

We study the long-time dynamics and the stationary nonequilibrium state of an optically driven quantum dot coupled to acoustic phonons using numerically exact real-time path integrals and a fourth-order correlation expansion. By exploring wide ranges of temperatures, carrier-phonon and carrier-light coupling strengths, we characterize the stationary nonequilibrium state and compare the exact solution to known, approximatively derived results. It is found that analytical calculations tend to overestimate the influence of the carrier-phonon coupling, particularly at low temperatures and in the weak-coupling regime. The possibility of controlling the dot occupation in the stationary nonequilibrium by varying the laser detuning is discussed. A comparison between the numerical methods identifies the range of validity of the correlation expansion, which in the long-time limit is found to be surprisingly wide.

Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates

This paper compares and evaluates various plate finite elements to analyse the static response of thick and thin plates subjected to different loading and boundary conditions. Plate elements are based on different assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner-Mindlin), refined (Reddy and Kant), and other higher-order displacement fields are implemented up to fourth-order expansion. The Unified Formulation UF by the first author is used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The MITC4 shear-locking free type formulation is used for the FE approximation. Accuracy of a given plate element is established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates are implemented and compared using displacement and stress variables for various plate problems. Reduced models that are able to detect the 3D solution are built and a Best Plate Diagram (BPD) is introduced to give guidelines for the construction of plate theories based on a given accuracy and number of terms. It is concluded that the UF is a valuable tool to establish, for a given plate problem, the most accurate FE able to furnish results within a certain accuracy range. This allows us to obtain guidelines and recommendations in building refined elements in the bending analysis of plates for various geometries, loadings, and boundary conditions.

Second-order and fluctuation-induced first-order phase transitions with functional renormalization group equations

We investigate phase transitions in scalar field theories using the functional renormalization group (RG) equation. We analyze a system with $U(2)\times U(2)$ symmetry, in which there is a parameter $\lambda_2$ that controls the strength of the first-order phase transition driven by fluctuations. In the limit of $\lambda_2\to0$, the $U(2)\times U(2)$ theory is reduced to an $O(8)$ scalar theory that exhibits a second-order phase transition in three dimensions. We develop a new insight for the understanding of the fluctuation-induced first-order phase transition as a smooth continuation from the standard RG flow in the $O(8)$ system. In our view from the RG flow diagram on coupling parameter space, the region that favors the first-order transition emerges from the unphysical region to the physical one as $\lambda_2$ increases from zero. We give this interpretation based on the Taylor expansion of the functional RG equations up to the fourth order in terms of the field, which encompasses the $\epsilon$-expansion results. We compare results from the expansion and from the full numerical calculation and find that the fourth-order expansion is only of qualitative use and that the sixth-order expansion improves the quantitative agreement.

Accuracy of refined finite elements for laminated plate analysis

This paper compares and evaluates various plate finite elements to analyze the static response of laminated plate when varying the thickness ratio, orthotropic ratio and the stacking sequence of the lay-out. Plate elements were created from different theoretical assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner–Mindlin), known refined (Reddy, Pandya, and Kant), and other higher-order displacement fields were then implemented up-to fourth-order expansion. Following this the Carrera Unified Formulation was used to derive finite element matrices in terms of fundamental nuclei which consist of 3 × 3 arrays. The accuracy of a given plate element was established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates were implemented and compared using displacement and stress variables for various plate problems. In this paper concluding results are presented as number and type of required displacement variables vs. accuracy on a given stress/displacement parameter. These diagrams are labelled as Best Plate Theories BPT curves, which differ by changing geometry, material data and lay-out as well as loading and boundary conditions.

Field-Induced Phase Transitions in Relaxor Ferroelectrics

The Landau free energy expansion as a function of three-dimensional polarization for perovskite relaxor ferroelectrics is examined to find effects of large electric fields in causing transitions from one ferroelectric state to another. Our examination of field-induced effects includes a fourth order expansion to obtain the Landau parameters for the observed temperature- and field-induced rhombohedral (R) tetragonal (T) transitions in an (001)-cut 0.70Pb(Mg1/3Mn2/3)O3-0.30PbTiO3 (PMN-30%PT) crystal. These parameters enabled us to predict the appearance of an orthorhombic (O) phase in (110)-cut PMN-30%PT crystals under an applied field. These results show that also for field-induced effects, the Landau expansion shows qualitative agreement with experiment for relaxor ferroelectrics.