Parabolic Modeling Research Articles

Analysis of Doppler Effects in Underwater Acoustic Channels using Parabolic Expansion Modeling

Underwater communication systems play an important role in understanding various phenomena that take place within our vast oceans. They can be used as an integral tool in countless applications ranging from environmental monitoring to gathering of oceanographic data, marine archaeology, and search and rescue missions. Acoustic Communication is the viable solution for communication in highly attenuating underwater environment. However, these systems pose a number of challenges for reliable data transmission. Nonnegligible Doppler Effect emerges as a major factor. In order to support reliable high data rate communication, understanding the channel behavior is required. As sea trials are expensive, simulators are required to study the channel behavior. Modeling this channel involves solution to wave equations and validation with experimental data for that portion of the sea. Parabolic expansion model is a wave theory based acoustic channel model. This model applies Pade coefficients and Fourier coefficients as expansion functions to solve the wave equations. This work attempts to characterize the impact of Doppler Effect in the underwater acoustic channel using parabolic expansion models.

Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems

This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage $ \lambda $. On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage $ \lambda $ and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.

Concerted cycloaddition reactions: a study using the parabolic model and quantum chemical modeling

Concerted cycloaddition reactions were studied by the method of intersecting parabolas (M3IP) and quantum chemical calculations. Experimental data were processed within the framework of the M3IP method and an algorithm for calculating the activation energies (E) and rate constants (k) for reactions from the enthalpies of reactions was developed. The parameters E and k for twelve cycloaddition reactions not studied previously were calculated. Factors affecting the activation energies were established and evaluated; these include the enthalpy of reaction, substituents, and the molecular structure of reactants. Quantum chemical modeling and topological analysis of transition states (TS) of six concerted cycloaddition reactions were performed. Depending on structure of the starting olefins, the TS of reactions can have either a symmetric or asymmetric geometry. This influences their electronic structures, the energies of chemical bonds, and the activation energies of reactions. A comparison of the activation energy values obtained from the M3IP and DFT(B3lyp/6-311++G** ) calculations revealed good agreement between them.

Parabolic Equation Modeling of Propagation over Terrain Using Digital Elevation Model

The parabolic equation method based on digital elevation model (DEM) is applied on propagation predictions over irregular terrains. Starting from a parabolic approximation to the Helmholtz equation, a wide-angle parabolic equation is deduced under the assumption of forward propagation and the split-step Fourier transform algorithm is used to solve it. The application of DEM is extended to the Cartesian coordinate system and expected to provide a precise representation of a three-dimensional surface with high efficiency. In order to validate the accuracy, a perfectly conducting Gaussian terrain profile is simulated and the results are compared with the shift map. As a consequence, a good agreement is observed. Besides, another example is given to provide a theoretical basis and reference for DEM selection. The simulation results demonstrate that the prediction errors will be obvious only when the resolution of the DEM used is much larger than the range step in the PE method.

Asymptotic behavior for a fourth-order parabolic equation modeling thin film growth

The main goal of this work is to study a fourth-order parabolic equation with nonstandard growth conditions. For non-positive initial energy J(u0)≤0, by using the concavity method, we establish that weak solutions blow up in finite time. On the other hand, we also derive the blow-up time estimate and the asymptotic behavior of weak solutions.

Three-dimensional modeling in global acoustic propagation

The ocean is nearly transparent for acoustic propagation at low frequencies (<100 Hz), leading to the detection of signals (seismic events, volcanoes, and man-made signals) at distances as large as the ocean basin. Historically, basin acoustic modeling has neglected out-of-plane effects and has been performed with the model computed in the range/depth plane for multiple radials following geodesics (Nx2D). Both oceanographic and bathymetric features can lead to out-of-plane effects. In this paper, a summary of computational approaches to this problem will be presented, including vertical-mode, horizontal ray hybrid approaches, and full-3D Parabolic Equation modeling. Out-of-plane effects include refraction and diffraction—which have different effects as well as different approaches to modeling. Experiments where 3D propagation effects were significant will be presented within this context, including Perth-Bermuda (1960), the Heard Island Feasibility Test (1993), and a recent seismic tomography test off the coast of Japan (2015). Three physics mechanisms will be addressed: horizontal deflection due to mesoscale eddies and fronts, reflection from islands (refraction), and diffraction behind bathymetric edges.

An analysis of the oscillatory movements of the suspension ropes for an elevator with parabolic speed modeling and continuous deceleration

In order to insure passengers’ security and smooth functioning, elevators benefit from speed modeling. The paper aims to compare the evolution of the kinematic parameters (acceleration, speed, amplitude) of the oscillatory movements performed by the suspension ropes of the elevator cabin, considering speed variation at start and stop realized using S – curves generated from arches of parabolic curves. Usually, the functioning cycle of an elevator ends with a short period of constant speed traveling, in order to insure levelling of the cabin. We have already made a comparative analysis of the oscillatory movements of the suspension ropes, considering three varieties of speed modeling: with trapezoidal variation of acceleration, with parabolic variation of acceleration and with cosinusoidal variation of acceleration. The paper aims to analyze the possibility to remove the levelling period, and the impact of this upon the kinematic parameters of the oscillatory movements performed by the suspension ropes. We will take into consideration two possibilities of deceleration: with levelling step and with continuous variation of speed. In both cases, for comparison, the same duration of the acceleration period and deceleration period will be considered.

Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nabla\Delta u-{\bf g}(\nabla u)\right)=0$. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function ${\bf g}$ is continuous and the initial datum $u_0$ lies in $C^1(\mathbb{R}^N)$ with $\nabla u_0(x)$ being uniformly continuous and bounded on $\mathbb{R}^N$ and that the global existence assertion also holds true if we assume that ${\bf g}$ is locally Lipschitz and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha$ for some $c>0, \alpha\in (2, 3)$, $\sup_{\mathbb{R}^N}|\nabla u_0|<\infty$, and the norm of $u_0$ in the space $L^{\frac{(\alpha-1)N}{3-\alpha}}(\mathbb{R}^N) $ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that ${\bf g}$ is continuous and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha+c$ for some $c, \alpha\in (0,\infty)$. Our investigations reveal that if $\alpha\leq 1$ we have global existence of a weak solution, while if $1<\alpha<\frac{N^2+2N+4}{N^2}$ only a local existence theorem can be established. Our method here is based upon a new interpolation inequality, which may be of interest in its own right.

Acoustic characterization of a wave energy converter

As progress toward the commercial deployment of wave energy converters accelerates, it is important to ensure that these renewable energy systems do not have unintended, adverse environmental consequences. While the sound from wave energy converters is unlikely to cause acoustic injury to marine animals, it may affect their behavior. Here, we present measurements from a point-absorber wave energy converter at the U.S. Navy Wave Energy Test Site in Kaneohe Bay, HI. Measurements of wave converter sound are obtained for a range of sea states using a combination of free-drifting near-surface measurements and stationary bottom packages. The relative effectiveness of these systems are contrasted and the unique challenges associated with acoustic measurements at energetic sites discussed. For example, fixed measurements are found to be substantially contaminated by flow-noise (non-propagating sound) during long-period ocean swell, while free-drifting measurements require significant post-processing to avoid convolving flow-noise or self-noise with wave converter sound. Preliminary results of parabolic equation modeling is also presented and used to interpret spatially distributed measurements.

Lateral diffraction of underwater sounds between Japan and Chile: experimental results and modeling

One hundred shallow underwater explosions on a 300-km profile along an isoline of 500 m sea depth above a slope zone of off the coast of Japan are received at a water-column hydrophone triplet with 2-km intervals in Chile. The over 16,000-km propagations from the northwest to the southeast Pacific Ocean are classified into two groups. One is from the southern explosions. The data show high sound pressure levels and normal travel times. On the other hand, data from the northern ones reveal 6.5 dB lower sound pressure levels and 6.6 s travel time delays on average. Some islands and seamounts in the central Pacific are located on the halfway from the northern explosions. They affect received sound pressure levels and arrival times, but do not block completely, based on the observation. Modeling of this propagation path, using in-plane 2D modeling, shows nearly complete blockage of the sound by the Northern Hawaiian Island chain. Three dimensional parabolic equation modeling shows that this shadow is filled in from diffraction caused by the Midway Islands. Modeling results quantitatively match the measurements for both energy level and travel time.

ON THE OPTIMAL CONTROL OF A PROBLEM OF ENVIRONMENTAL POLLUTION

This article is studied the optimal control of distributed parameter systems applied to an environmental pollution problem. The model consists of a differential equation partial parabolic modeling of a pollutant transport in a fluid. The model is considered the speed with which the pollutant spreads in the environment and degradation that suffers the contaminant by the presence of a factor biological inhibitor, which breaks the contaminant at a rate that is not dependent on space and time. Using the method of Lagrange multipliers is possible to prove the existence solving the problem of control and obtaining optimality conditions for optimal control.

Insensitizing controls for a phase field system

In this paper, a nonlinear parabolic system modeling phase field phenomena is considered. This system consists of two coupled parabolic equations, the first one describes the temperature of the material and the second one describes a phase field function. Under small perturbations of the initial data, we study the existence of controls insensitizing the phase field function and acting only on the temperature equation. This problem is equivalent to the null controllability of a parabolic system, which is studied by means of duality arguments, Carleman estimates, and fixed point theorems.

On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth

Abstract In this paper, we study the existence and uniqueness of global weak solution, the regularity of the solutions and the existence of global attractor for a fourth order parabolic equation modeling epitaxial thin film growth with Neumann boundary conditions in two space dimensions.

Ultra‐long‐range hydroacoustic observations of submarine volcanic activity at Monowai, Kermadec Arc

AbstractMonowai is an active submarine volcanic center in the Kermadec Arc, Southwest Pacific Ocean. During May 2011, it erupted over a period of 5 days, with explosive activity directly linked to the generation of seismoacoustic T phases. We show, using cross‐correlation and time‐difference‐of‐arrival techniques, that the eruption is detected as far as Ascension Island, equatorial South Atlantic Ocean, where a bottom moored hydrophone array is operated as part of the International Monitoring System of the Comprehensive Nuclear‐Test‐Ban Treaty Organization. Hydroacoustic phases from the volcanic center must therefore have propagated through the Sound Fixing and Ranging channel in the South Pacific and South Atlantic Oceans, a source‐receiver distance of ~15,800 km. We believe this to be the furthest documented range of a naturally occurring underwater signal above 1 Hz. Our findings, which are consistent with observations at regional broadband stations and long‐range, acoustic parabolic equation modeling, have implications for submarine volcano monitoring.

Three-dimensional parabolic equation modeling of mesoscale eddy deflection.

The impact of mesoscale oceanography, including ocean fronts and eddies, on global scale low-frequency acoustics is examined using a fully three-dimensional parabolic equation model. The narrowband acoustic signal, for frequencies from 2 to 16 Hz, is simulated from a seismic event on the Kerguellen Plateau in the South Indian Ocean to an array of receivers south of Ascension Island in the South Atlantic, a distance of 9100 km. The path was chosen for its relevance to seismic detections from the HA10 Ascension Island station of the International Monitoring System, for its lack of bathymetric interaction, and for the dynamic oceanography encountered as the sound passes the Cape of Good Hope. The acoustic field was propagated through two years (1992 and 1993) of the eddy-permitting ocean state estimation ECCO2 (Estimating the Circulation and Climate of the Ocean, Phase II) system. The range of deflection of the back-azimuth was 1.8° with a root-mean-square of 0.34°. The refraction due to mesoscale oceanography could therefore have significant impacts upon localization of distant low-frequency sources, such as seismic or nuclear test events.

Solutions of fourth-order parabolic equation modeling thin film growth

In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method [1–3] we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namely C0([0,T];Lp(Ω)) where p=nα2−α with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are Lp–Lq-estimates, regularization property of the linear part of e−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L 1 norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

An equivalent source model for simulating high intensity focused ultrasound fields using a nonlinear parabolic equation

The nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation is commonly used for simulation of nonlinear focused fields generated by medical ultrasound transducers. Axially symmetric codes can be easily implemented on personal computers with simulation times on the order of tens of minutes even for the most extensive simulations of shock formation conditions. However, model accuracy can be limited for high intensity focused ultrasound sources with large convergence angles. In previous studies, it was shown that nonlinear parabolic modeling can provide good matching with measurements and full diffraction modeling in the focal region for both single element piston sources and multi-element arrays. Boundary conditions for the modeling were adjusted by varying the aperture of an equivalent planar source with a uniform pressure magnitude and a phase distribution for parabolic focusing. With this approach, there were slight discrepancies in the lateral pressure distributions in the focal plane. Here, it is proposed that an additional variation in the position of the equivalent source enables matching both axial and lateral distributions of the real HIFU sources with higher accuracy. Several examples are presented to compare modeling and measurements of the fields generated by laboratory and clinical HIFU transducers. [Work supported by RSF 14-12-00974 and NIH EB007643.]

Some properties of solutions for a parabolic equation modeling epitaxial thin film growth

In this paper, we study a fourth-order parabolic equation with Dirichlet boundary which arises in epitaxial growth of nanoscale thin films. We proved the equation with condition possesses a global attractor in , which attracts any bounded subset of in the -norm. In addition to this, we consider the equation under the conditions is blow-up.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient $L^2$ norm.