Full Numerical Computations Research Articles

Tangential contacts of three-dimensional power-law graded elastic solids: A general theory and application to partial slip

New technologies in additive manufacturing have enabled the cost-effective production of gradient materials with desired multifunctional properties. In mechanical engineering, such functionally graded materials (FGMs) are often used within mechanical joints, as they ensure higher strength as well as wear resistance and prevent fretting fatigue. However, due to the variations in their composition and structure, theoretical predictions of their contact-mechanical behavior are significantly complicated. In this work, a general theory for solving tangential contacts between 3D power-law graded elastic solids of arbitrary geometry is presented. For multiple contacts, such as those occurring between two nominally flat but rough half-spaces, the well-known Ciavarella–Jäger theorem is established accompanied by a discussion of tangential coupling. Nevertheless, the focus of the work is on axisymmetric single contacts under arbitrary unidirectional tangential loading, for which closed-form analytical solutions are derived based on the Mossakovskii–Jäger procedure. In comparison to the results of common approximate methods, the solutions include the non-axisymmetric components of tangential displacements, which are indispensable for the accurate determination of the relative slip components and thus the surface density of frictional energy dissipation in the partial slip regime. Although a simplified approach is used for the calculation of the dissipated energy density, the results in the limiting case of homogeneous material are in excellent agreement with those from a full numerical computation. As an application example, the complete solutions for the tangential contact of parabolically shaped power-law graded elastic solids in the partial slip regime are derived and the influence of the material gradient as well as Poisson’s ratio on the surface density of dissipated energy is investigated.

A contribution to the mathematical theory of diffraction: a note on double fourier integrals

SummaryWe consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of $u$. We conclude the article by validating this theory against full numerical computations for two specific examples.

Near-BPS baby Skyrmions with Gaussian tails

We consider the baby Skyrme model in a physically motivated limit of reaching the restricted or BPS baby Skyrme model, which is a model that enjoys area-preserving diffeomorphism invariance. The perturbation consists of the kinetic Dirichlet term with a small coefficient ϵ as well as the standard pion mass term, with coefficient upepsilon {m}_1^2 . The pions remain lighter than the soliton for any ϵ and therefore the model is physically acceptable, even in the ϵ → 0 limit. The version of the BPS baby Skyrme model we use has BPS solutions with Gaussian tails. We perform full numerical computations in the ϵ → 0 limit and even reach the strict ϵ = 0 case, finding new nontrivial BPS solutions, for which we do not yet know the analytic form.

Gain-forming effect on optical signal-to-noise ratio in Raman fiber amplifier

Abstract We have introduced the comprehensive optical-to-noise ratio (OSNR) to investigate simultaneously the effects of all three major linear noise sources in distributed Raman fiber amplifier (DRFA). The total noise power contains double Rayleigh backscattering (DRB) noise, amplified spontaneous emission (ASE) noise, and relative intensity noise (RIN). Also, we have introduced an approximated gain profile approach, which helps in evaluating the OSNR for higher-order pumping (HOP) configuration for backward, forward, and bidirectional DRFA schemes. Finally, by applying the dynamic and static full-numerical computations on a system of eight wavelength division multiplexing (WDM) channels (for amplifier lengths of 70, 140, and 210 km), we demonstrated that for RIN frequencies greater than 100 kHz, the higher-order bidirectional pumping scheme exceeds the backward schemes by 12 dB.

Analysis of thermodynamic problems with the Lambert W function

We present an analysis of two problems in thermodynamics in terms of the Lambert W function, including the mean-field approximation of the Ising model, and Bose–Einstein condensation. Both problems are well known to exhibit the critical behavior of phase transition. Standard treatment of the problems involves numerical or graphical solutions. Utilizing justified simplifying approximations, we find a closed-form mean-field solution for the Ising model in terms of the special W function. With the same special function, we present an analysis of Bose–Einstein condensation, allowing approximate quantitative determination of the dependence of the chemical potential on temperature without full numerical computation. The analysis helps to facilitate understanding and to gain insight on these processes involving phase transitions in a straightforward manner.

Low-temperature tunneling of CH3 quantum rotor in van der Waals solids

Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polynomials are compared to full numerical computations and tested against experimental EPR lineshape simulations. In particular, the X-band of methyl radical was displaying partial anisotropy averaging even at lowest temperatures. EPR lineshape simulations involving rotational dynamics were applied for the accurate determination of the potential barrier and the tunneling frequency. Tunneling frequency, as the splitting between the A and E torsional levels by the presence of a periodic C3 model potential with periodic boundary conditions, was computed and related to the EPR-lineshape alteration. The corresponding C2 rotary tunneling about the in-plane axes of methyl was also studied while both the C2 and C3 rotations were compared with the rotation of deuteriated methyl radical.

An Analytical and Numerical Estimation of the Effective Thermal Conductivity of Complex Metal Frame Core Structures

An analytical and numerical study was conducted for estimation of the effective thermal conductivities of curved metal frame core structures, which can replace metal foams, in views of their advantages over the metal foams for both load bearing and heat dissipation. The trajectory of the frame ligament and its cross-sectional area were allowed to vary arbitrarily in the three-dimensional (3D) space. The analytical formula obtained by extending the formula previously proposed by Bai et al. (2017, “A General Expression for the Stagnant Thermal Conductivity of Stochastic and Periodic Structures,” ASME J. Heat Transfer, 140(5), p. 052001) was examined by comparing it with the numerical results directly obtained from full 3D numerical computations. An air layer partially filled with a collection of coiled circular rods was treated both analytically and numerically. Furthermore, the effect of lattice nodes on the effective thermal conductivity was investigated by introducing an analytical model with the lattice ligaments merging together at one nodal point. The analytical expressions thus derived for the lattice structures with nodes were applied to tetrahedral structure and octet-truss structure to find their effective thermal conductivities, which are found to agree closely with the 3D numerical results. Thus, the present analytical expressions can be used to customize the structure to meet its desired thermal performance.

Numerical Study on the Propagation of Premixed Flames in Confined Narrow Disc-Shape Chambers

ABSTRACTThe laminar flame propagation in narrow confined disc-shape chambers of millimeter scale is investigated, concerning the flame front displacement speed and the influences from the changes of the gap width and radius. An asymptotic analysis treating the flame front as a discontinuity successfully predicts the velocity profiles of the flame front and illustrates the importance of heat release parameter and laminar flame speed of the fuel. The unsteady flame propagation process is simulated by a full numerical computation which employs the n-butane/air mixture as the reactant. The numerical simulation emphasizes the sensitivity of spatial scale to the flame propagation in narrow spaces, indicating that the acceleration rate and the maximum speed of the flame have a non-monotonic relation with the gap width, of which the optimal value is about 0.8 mm. Faster flame propagation is favored for larger gap radius due to stronger flame straining, but it is restricted by the gap width.

Transition to blow-up in a reaction–diffusion model with localized spike solutions

For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

Compact Model for Ferroelectric Negative Capacitance Transistor With MFIS Structure

We present a physics-based compact model for a ferroelectric negative capacitance FET (NCFET) with a metal–ferroelectric–insulator–semiconductor (MFIS) structure. The model is computationally efficient, and it accurately calculates the gate charge density as a function of the applied voltages. For the first time, an explicit expression for the channel current in bulk NCFET is also deduced taking into account the spatial variation of ferroelectric polarization in the longitudinal direction. Using current continuity condition in the channel, we find that different regions of the ferroelectric may operate in a positive or a negative capacitance state depending on the external biases. The model captures the impact of ferroelectric thickness scaling and variation in the ferroelectric material parameters, and has been validated against the implicit approach involving full numerical computations as well as experimental data. We also compare the device characteristics of the MFIS structure with those of the metal–ferroelectric–metal–insulator–semiconductor structure.

Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.

Electroosmosis in a finite cylindrical pore: simple models of end effects.

A theoretical model of electroosmosis through a circular pore of radius a that traverses a membrane of thickness h is investigated. Both the cylindrical surface of the pore and the outer surfaces of the membrane are charged. When h ≫ a, end effects are negligible, and the results of full numerical computations of electroosmosis in an infinite pore agree with theory. When h = 0, end effects dominate, and computations again agree with analysis. For intermediate values of h/a, an approximate analysis that combines these two limiting cases captures the main features of computational results when the Debye length κ–1 is small compared with the pore radius a. However, the approximate analysis fails when κ–1 ≫ a, when the charge cloud due to the charged cylindrical walls of the pore spills out of the ends of the pore, and the electroosmotic flow is reduced. When this spilling out is included in the analysis, agreement with computation is restored.

COMPUTATIONAL AND ASYMPTOTIC METHODS IN AEROACOUSTICS WITH APPLICATIONS

In this article the computational and asymptotic methods used in aeroacoustics are reviewed. In particular, two different aeroacoustic applications are demonstrated.In the first problem we investigate the first and second order asymptotic predictions of the thickness and loading noise of a subsonic B-bladed helicopter rotor in the far field and compare the SPL noise results with those of full numerical computations. The results of the second order asymptotic formula seem to be in better agreement with full numerical computations than the first order asymptotic formula. In the second problem, the effect of acoustic wave propagation in transonic nozzle flow is investigated by solving the unsteady quasi-one-dimensional transonic nozzle equations in conservative form using high order computational aeroacoustic schemes, where a novel non-reflecting boundary condition is implemented in addition to the standard non-reflecting boundary condition using characteristics. Excellent agreement with the exact solution is obtained in each case.

Buckling of Long Thin Plates under Residual Stresses with Application to Strip Rolling

We present a simplified numerical method which can be used to predict efficiently the response of long thin plates under effects of residual stresses induced by production process such as rolling or continuous annealing. The principle consists in assuming harmonic buckling mode along the sheet length, and we consider Koiter-Budiansky post-buckling theory to compute the stress-deflection curve. In this way, only the width of the sheet has to be discretized by 1D finite elements. The size and shape of the flatness defects can be predicted efficiently and for a large number of cases. Various types of residual stresses and loadings can be accounted for. In particular, we will see the influence of the global traction on the buckling and post-buckling behavior. The numerical results are compared with experimental data and full numerical computations

Recovering multiple small inclusions in a three-dimensional domain using a single measurement

We present and test an effective algorithm to recover multiple small heterogeneities inside arbitrary three-dimensional bounded regions. In applications of electrical impedance tomography, such heterogeneities might represent tumours inside the body, weaknesses or cracks in a solid material or subterranean mineral deposits. The algorithm determines the location of all heterogeneities using a single boundary measurement. The only non-linear step in the algorithm is to find roots of a single polynomial of degree . The efficacy of this method is demonstrated by full numerical computation of the three-dimensional forward problem. In addition, the methods employed can give information regarding the shape of irregular heterogeneities.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: Three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that ϵ ∝ h/l ≪ 1, we show that, to leading order in ϵ, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in ϵ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, ϵ, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

A Refinement of Asymptotic Predictions and Full Numerical Solution of Helicopter Rotor Noise in the Far Field

The asymptotic analysis of Parry and Crighton [1] for propeller noise in the far field, which is based on Hanson's formulation [2] of the FW-H equation, is refined to second order by Laplace's method [3] for evaluating integrals, accounting for second order contributions near the blade tip for loading and thickness noise. The full numerical solution of Hanson's integrals for both thickness and loading noise is also presented. In particular, the theory is applied to a four-bladed helicopter rotor with tip Mach numbers ranging between 0.5 and 0.7. The aerodynamic loading in this case is obtained using a 3D compressible code based on finite volume method with intensified grid density near the blade tip. The far field angular SPL noise distributions of a helicopter rotor in hover show that the present second order asymptotic formula is in better agreement with full numerical computations than that of the first order formula, especially for thickness noise.

Smoke-Ring Solutions of Gierer–Meinhardt System in $\mathbb{R}^{3}$

We consider the steady states of the Gierer–Meinhardt system on all of $\mathbb{R}^{3}$: $\varepsilon^{2}\Delta a-a+\frac{a^{p}}{h^{q}}=0$, $\Delta h-h+\frac{a^{m}}{h^{s}}=0$ with an additional restriction $q=p-1$. In the limit $\varepsilon\rightarrow0$, we use formal asymptotics to construct a solution whose activator component a concentrates on a circle. Under the additional constraints $p>1$, $m>0$, and $1<m-s<3$, we find that such a solution exists and is unique. The radius of the circle of concentration is given explicitly in terms of certain integrals. Full numerical computations are shown to support the analytical results.

The Unsteady Flow of a Weakly Compressible Fluid in a Thin Porous Layer I: Two-Dimensional Theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a two-dimensional reservoir in an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting or extracting fluid. Numerical solution of this problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l. This is a situation which occurs frequently in the application to oil reservoir recovery. Under the assumption that $\epsilon=h/l\ll1$, we show that the pressure field varies only in the horizontal direction away from the wells (the outer region). We construct two-term asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive analytical expressions for all significant process quantities. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the reservoir, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighborhood of wells and away from wells.

Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen–Cahn System

We consider the singular limit of a perturbed Allen–Cahn model on a bounded two-dimensional domain: $\left\{\begin{array}{@{}ll@{}} u_t = \varepsilon^2 \Delta u - 2 (u - \varepsilon a) (u^2 - 1), & x \in \Omega \subset \mathbb{R}^2 \ \partial_n u = 0, & x \in \partial \Omega \end{array} \right.$ where $\varepsilon$ is a small parameter and a is an $O(1)$ quantity. We study equilibrium solutions that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the time-dependent PDE as well as of the associated two-dimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.