Exact Linear Solutions Research Articles

Thermal-Diffusive Processes of an Electron-Hole Non-Local Semiconductor Model with Variable Thermal Conductivity and Hall Current Effect

In this work, a novel model is presented that describes thermal diffusion processes through non-local semiconductor materials. The material under study is subjected to the influence of a strong magnetic field, which creates a Hall current. Interference between the excited electrons and the excited holes of a non-local semiconductor that had been exposed to temperature was present, and thermal conductivity depending on changes in graduated temperature were accounted for. The governing equations are written in a dimensionless form in one dimension (1D) where the thermal conductivity is taken as a function of temperature through electronic and elastic deformation (ED and ED) processes. Laplace transforms in one dimension with initial conditions were used to convert partial differential equations to arrive at exact formulas of solutions. To obtain the exact linear solutions, some boundary conditions taken on the free surface of the non-local semiconductor were used. Using numerical methods of inverse Laplace transforms, the complete solutions of the physical quantities under study were obtained. To further understand how various variables (thermal memory, variable thermal conductivity, and Hall current) affect the non-local semiconductor, numerical physical fields were simulated, and are graphically depicted, and discussed herein.

Discrete kinetic eigenmode spectra of electron plasma oscillations in weakly collisional plasma: A numerical study

It has been demonstrated that in the presence of weak collisions, described by the Lenard-Bernstein (LB) collision operator, the Landau-damped solutions become true eigenmodes of the system and constitute a complete set [C.-S. Ng et al., Phys. Rev. Lett. 83, 1974 (1999) and C. S. Ng et al., Phys. Rev. Lett. 96, 065002 (2004)]. We present numerical results from an Eulerian Vlasov code that incorporates the Lenard-Bernstein collision operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456 (1958)]. The effect of collisions on the numerical recursion phenomenon seen in Vlasov codes is discussed. The code is benchmarked against exact linear eigenmode solutions in the presence of weak collisions, and a spectrum of Landau-damped solutions is determined within the limits of numerical resolution. Tests of the orthogonality and the completeness relation are presented.

FINITE ELEMENT SOLUTION OF MULTI-SCALE TRANSPORT PROBLEMS USING THE LEAST SQUARES-BASED BUBBLE FUNCTION ENRICHMENT

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

The Frontogenetical Forcing of Secondary Circulations. Part II: Properties of Q Vectors in Exact Linear Solutions

Abstract An exact solution of the primitive equations (PEs) and the corresponding exact solutions of the alternative balance (AB), geostrophic momentum (GM), and quasigeostrophic (QG) equations are presented. The PE solution illustrates how the temperature and horizontal vorticity fields evolve in a linear horizontal flow with constant deformation and vertical vorticity when the initial temperature field is also linear, as well as how ageostrophic circulations are produced. The other exact solutions show the errors produced by the various approximations and confirm that the AB solution is more accurate than the GM one and that the QG solution is almost always the most inexact. The utility of the Q vector and similar vectors is examined for each solution. The PE solution verifies that in a hyperbolic wind field (i) the isotherms eventually parallel the outflow axis, (ii) the ageostrophic circulation ultimately becomes normal to the outflow axis, (iii) thermal-wind balance becomes established in the direction normal to the isotherms, and (iv) the rotational component of the vector frontogenetical function decays.

Extensions of the Ferry shear wave model for active linear and nonlinear microrheology

The classical oscillatory shear wave model of Ferry et al. [J.D. Ferry, W.M. Sawyer, J.N. Ashworth, Behavior of concentrated polymer solutions under periodic stresses, J. Polym. Sci. 2 (1947) 593–611] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [J.C. Crocker, M.T. Valentine, E.R. Weeks, T. Gisler, P.D. Kaplan, A.G. Yodh, D.A. Wietz, Phys. Rev. Lett. 85 (2000) 888–891]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior.

Enriched multi-point flux approximation for general grids

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.

Thermospheric responses to gravity waves: Influences of increasing viscosity and thermal diffusivity

A gravity wave anelastic dispersion relation is derived that includes molecular viscosity and thermal diffusivity to explore the damping of high‐frequency gravity waves in the thermosphere. The time dependence of the wave amplitudes and general ray trace equations are also derived. In the special case that the thermal structure is isothermal and the Prandtl number (Pr) equals 1, exact linear solutions are obtained. For high‐frequency gravity waves with ωIr/N ≪ 1 an upward propagating gravity wave dissipates at an altitude given by ≃ z1 + H ln(ωIr/2H∣m∣3ν1), where H is the density scale height, N is the buoyancy frequency, ν1 is the viscosity at z = z1, and ωIr and m are the gravity wave intrinsic frequency and vertical wave number, respectively. Thus high‐frequency gravity waves with large vertical wavelengths dissipate at the highest altitudes, resulting in momentum and energy inputs extending to very high altitudes. We find that the vertical wavelength of a gravity wave with an initially large vertical wavelength decreases significantly by the time it dissipates just below where it begins to reflect. The effect of diffusion on a gravity wave is similar to the effect of shear in the sense that as the molecular viscosity and thermal diffusivity increase due to decreasing background density, the intrinsic frequency plus mν/H decreases and the vertical wave number increases in order to satisfy the dispersion relation for Pr = 1. We also briefly explore the results with different Prandtl numbers using numerical ray tracing. Gravity waves in a Pr = 0.7 environment dissipate just a few kilometers below those in a Pr = 1 environment when H = 7 km, showing the utility of the analytic, Pr = 1 solutions.

Transient wave propagation of isotropic plates using a higher-order plate theory

Transient wave propagation of isotropic thin plates using a higher-order plate theory is presented in this paper. The aim of this investigation is to assess the applicability of the higher-order plate theory in describing wave behavior of isotropic plates at higher frequencies. Both extensional and flexural waves are considered. A complete discussion of dispersion of isotropic plates is first investigated. All the wave modes and wave behavior for each mode in the low and high-frequency ranges are provided in detail. Using the dispersion relation and integral transforms, exact integral solutions for an isotropic plate subjected to pure impulse load and a number of wave excitations based on the higher-order theory are obtained and asymptotic solutions which highlight the physics of waves are also presented. The axisymmetric three-dimensional analytical solutions of linear wave equations are also presented for comparison. Results show that the higher-order theory can predict the wave behavior closely with exact linear wave solutions at higher frequencies.

Numerical Solution Procedures for Nonlinear Elastic Curved Rods Using the Theory of a Cosserat Point

The numerical solution of problems of curved rods can be formulated using rod elements developed within the context of the theory of a Cosserat point. Although the general theory is valid for curved rods, the constitutive coefficients have been determined by comparison with exact linear solutions only for straight beams. The objective of this paper is to explore the accuracy of the predictions of the Cosserat theory for curved rods by comparison with exact solutions. Specifically, these problems include: linearized axisymmetric deformation of a circular ring loaded with internal and external pressures; nonlinear axisymmetric inversion of a circular ring; and linearized pure bending of a section of a circular ring. In all cases, the Cosserat theory performs well with no modifications of the constitutive constants, even in the limit of reasonably thick rods. Also, it is shown that the Cosserat theory does not exhibit shear locking in the limit of thin rods.

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

The existing statically exact beam finite element (FE) based on the first order shear deformation theory (FSDT) is used to study the geometric nonlinear effects on static and dynamic responses in isotropic, composite and functionally graded material (FGM) beams. In this beam element, the exact solution of the static part of the governing differential equations is used to construct the interpolating polynomials for the element stiffness and mass matrix formulation. The rotary inertia is also taken into account while formulating the mass matrix. Consequently, the stiffness matrix is statically exact and the mass matrix is much more accurate compared to any other existing FEs. These two aspects make the element devoid of shear locking and an efficient instrument for analysing wave propagation problems, wherein refined mesh is essential. A total Lagrangian formulation of the linear beam element is employed for large displacement and large rotation analysis. The superconvergent property of the beam element is shown by comparing the convergence rate of this beam element to the reduced integrated FSDT beam element formed using the linear shape functions. Numerical examples deal with the static and wave propagation problems to highlight the nonlinear effects. In the static case, both Von-Karman strain tensor and Green’s strain tensor is used, whereas, for the wave propagation studies only the Von-Karman strains are used. Power-law variations of the material property distribution is used for the FGM beam. Both high and low frequency pulse loading is employed to bring out the effect of nonlinearity on the transient response.

A sector Fourier p-element for free vibration analysis of sectorial membranes

A sector Fourier p-element for free vibration analysis of sectorial membranes is presented. A major feature of this element is that it can describe the geometry of a sectorial membrane exactly and is therefore suitable to use for this type of membrane. The element transverse displacement is described by a fixed number of linear polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element’s nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. Different numbers of trigonometric terms are allowed in radial and circumferential directions. Results are obtained for a number of simply supported sectorial membranes and comparisons are made with exact and sector linear finite element solutions. The results clearly show that the solutions converge rapidly from above to the exact values as the number of trigonometric terms is increased and highly accurate values are obtained with the use of a very few terms. The results also show that the sector Fourier p-element produces a much higher accuracy than the sector linear finite element with fewer system degrees of freedom.

Finite Element and Plate Theory Modeling of Acoustic Emission Waveforms

A comparison was made between two approaches to predict acoustic emission waveforms in thin plates. A normal mode solution method for Mindlin plate theory was used to predict the response of the flexural plate mode to a point source, step-function load, applied on the plate surface. The second approach used a dynamic finite element method to model the problem using equations of motion based on exact linear elasticity. Calculations were made using properties for both isotropic (aluminum) and anisotropic (unidirectional graphite/epoxy composite) materials. For simulations of anisotropic plates, propagation along multiple directions was evaluated. In general, agreement between the two theoretical approaches was good. Discrepancies in the waveforms at longer times were caused by differences in reflections from the lateral plate boundaries. These differences resulted from the fact that the two methods used different boundary conditions. At shorter times in the signals, before reflections, the slight discrepancies in the waveforms were attributed to limitations of Mindlin plate theory, which is an approximate plate theory. The advantages of the finite element method are that it used the exact linear elasticity solutions, and that it can be used to model real source conditions and complicated, finite specimen geometries as well as thick plates. These advantages come at a cost of increased computational difficulty, requiring lengthy calculations on workstations or supercomputers. The Mindlin plat theory solutions, meanwhile, can be quickly generated on personal computers. Specimens with finite geometry can also be modeled. However, only limited simple geometries such as circular or rectangular plates can easily be accommodated with the normal mode solution technique. Likewise, very limited source configurations can be modeled and plate theory is applicable only to thin plates.

Dynamics of slender viscous dielectric liquid bridges subjected to axial AC fields

An analysis of slender axisymmetric liquid bridges is performed on the basis of one-dimensional models recently derived, and generalized here to include the effect of dielectric forces at the interface. The natural frequencies and stability criteria in the absence of gravity are obtained. In the inviscid case, results are compared with the known exact linear solutions of the corresponding three-dimensional problem.

Theory of Second Harmonic Generation in Periodic Ferroelectric Materials

Second harmonic generation is investigated in SmC* liquid crystals as an example of periodic ferroelectric materials. The electric field is expanded in a Fourier series having the same periodicity as the ferroelectric material. If the nonlinear dielectric susceptibility is ignored, the Fourier components, or the amplitudes of the electric field in various modes, satisfy a five term vector recursion relation and the solutions are obtained by iteration for small tilt angles. The exact linear solutions are obtained for normal incidence and become identical with those for a cholesteric liquid crystal if the tilt angle becomes 90°. In the lowest order approximation, it is shown that the second harmonic field is given by a product of the linear solution multiplied by the leading term of the Fourier series which is calculated using the slowly varying envelope approximation.

An Energy Criterion for the Onset of Shear Localization in Thermal Viscoplastic Materials, Part II: Applications and Implications

Necessary and/or sufficient conditions for the onset of shear flow localization in thermal viscoplastic materials were derived by Shawki (1994) through the analysis of a one-dimensional model of dynamic simple shear with velocity-controlled boundaries. In the former work, an energy-based viewpoint of localization allows for the association of the positive rate of change of the total kinetic energy of the absolute perturbations with the onset of shear strain localization. Here, we derive explicit conditions for the onset of shear localization without the need to obtain exact linear solutions. Explicit localization conditions are then applied to a number of empirical constitutive descriptions of material response. The effects of inertia, heat conduction, thermal softening, strain hardening, and strain rate sensitivity as regards the onset of shear localization are thoroughly discussed. Furthermore, the validity of conclusions based on a constant homogeneous solution is quantified together with the validity of conclusions based on a quasi-static approximation. Moreover, we examine the inadequacy of the notion of asymptotic stability for shear localization in the presence of heat conduction effects. A critical wavelength threshold is derived, for a general description of material response, below which perturbations will not grow. This threshold may also serve as a lower bound for shear band thickness.

ASYMPTOTIC ANALYSIS AND SPECTRUM OF THREE ANYONS

The spectrum of anyons confined in a harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions has been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of a nonlinearly interpolating spectrum. To be specific we analyze the eigenvalue equation in various asymptotic regions for the three-anyon problem.

The interaction of deep-water gravity waves and an annular current: linear theory

The interaction of linear, steady, axisymmetric deep-water gravity waves with preexisting large-scale annular currents has been investigated. Waves originating inside the annulus as well as waves approaching the annulus from the outside were studied. Exact linear ray solutions were obtained and involve two non-dimensional parameters, a radius-angle parameter and a velocity parameter. For opposing currents the linear solutions also allow the derivation of radii at which the waves are blocked, reflected at a linear caustic or stopped by the current. Various examples of rays interacting with an annular current are presented to illustrate aspects of the solutions obtained. In particular, the behaviour of the ray solutions at blocking, reflection and stopping is investigated. Linear ray theory is shown to fail at caustics and caustic solutions are briefly discussed.

Overshooting Thunderstorm Cloud Top Dynamics as Approximated by a Linear Lagrangian Parcel Model with Analytic Exact Solutions

This Paper Presents results for a linear Lagrangian entraining parcel model of an overshooting thunderstorm cloud top. The model is based on that of Adler and Mack, but differs in that it achieves analytic, exact solutions (for constant stratospheric lapse rates), by representing mixing with Rayleigh damping instead of nonlinearly. These exact linear solutions are forced additively by the initial parcel rise rate, hydrometeor drag, and initial temperature deviation (if any). Without drag or an initial temperature deviation, an initially rising parcel decelerates, cools below the ambient temperature, sinks and then becomes warmer than the environment. Depending on mixing strength, the parcel approaches a stratospheric equilibrium height at the ambient temperature or fallS back through the tropopause during the first oscillation. With drag, the parcel eventually returns to the tropopause. If the parcel remains in the stratosphere sufficiently long, it approaches a constant rate of descent at a constant temperature excess over ambient temperature. Both limiting parameters are directly proportional to drag. The descent rate decreases with ambient stability and increases with mixing strength, and vice versa for the temperature excess (over a suitable range of input parameters). A parcel initially warmer (colder) than the environment undergoes slightly delayed (advanced) oscillations, with a higher (lower) peak overshoot and minor increases (decreases) in both maximum and minimum temperatures, but without other material alterations. In some respects, the linear parcel model performs similarly to the Alder–Mack simulation (AM). Strong mixing favors a close-in warm point and a large cold-high offset, neither of which is present with weak mixing. Increased stability lowers the cloud summit and cold point while amplifying the cloud-top thermal couplet. Increased drag lowers the cloud summit, cold point, and warm point without appreciably changing the amplitude. Due to the contrasting mixing formulations, there are also several significant differences between our model results and those of AM under conditions as analogous as possible. With strong mixing, large cold-high offsets require both an inversion and rapid initial ascent in AM, but can occur in our model without either feature. A downslope warm point requires an inversion in AM, but not in our model. Cold-warm couplets show greater amplitude in our model than in AM, especially if mixing is strong. Parcels overshoot farther in our model than in AM, having undergone weaker mixing during vigorous early ascent. In our model, a parcel can approach a stratospheric equilibrium level only if there is no drag, whereas in AM some parcels with weak drag appear to approach an equilibrium level while superimposing several slowly damping oscillations on gradual overall descent.

Linear Hyperbolic Model for Alluvial Channels

The exact linear solutions for the sediment transport and bed form evolution for one‐dimensional sediment‐water two‐phase motion have been obtained using the St. Venant shallow‐water equations with the assumption of quasisteady flow. These solutions are applicable to alluvial channels of infinite length, initially at equilibrium followed by an arbitrary forcing function of either sediment transport or bed elevation imposed as an upstream boundary condition. The solutions have been used to predict aggradation in a channel due to constant overloading. Comparison of the results with available experimental data and with the solution obtained from a parabolic model is satisfactory. The present theory is significant conceptually since it provides valuable insight into the physical phenomenon as well as into the mathematical behavior of the solutions.