Null Condition for Semilinear Wave Equation with Variable Coefficients

∗The author was partially supported by M.U.R.S.T. Progr. Nazionale “Problemi Non Lineari...”

On Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear PDEs: Case of Elliptic Simplest Equation

2010 Mathematics Subject Classification: 74J30, 34L30.

Solutions of Fractional Diffusion-Wave Equations in Terms of H-functions

MSC 2010: 35R11, 42A38, 26A33, 33E12

Inverse space-dependent force problems for the wave equation

The determination of the displacement and the space-dependent force acting on a vibrating structure from measured final or time-average displacement observation is thoroughly investigated. Several aspects related to the existence and uniqueness of a solution of the linear but ill-posed inverse force problems are highlighted. After that, in order to capture the solution a variational formulation is proposed and the gradient of the least-squares functional that is minimized is rigorously and explicitly derived. Numerical results obtained using the Landweber method and the conjugate gradient method are presented and discussed illustrating the convergence of the iterative procedures for exact input data. Furthermore, for noisy data the semi-convergence phenomenon appears, as expected, and stability is restored by stopping the iterations according to the discrepancy principle criterion once the residual becomes close to the amount of noise. The present investigation will be significant to researchers concerned with wave propagation and control of vibrating structures.

Numerical calculation of diffraction coefficients from building edges using FDTD method

Finite Difference Time Domain (FDTD) Method and software are applied to obtain diffraction waves from modulated Gaussian plane wave illumination for right angle wedges and Fast Fourier Transform (FFT) is used to get diffraction coefficients in a wideband in the illuminated lit region. Theta and Phi polarization in 3-dimensional, TM and TE polarization in 2-dimensional cases are considered respectively for soft and hard diffraction coefficients. Results using FDTD method of perfect electric conductor (PEC) wedge are compared with asymptotic expressions from Uniform Theory of Diffraction (UTD). Extend the PEC wedges to some homogenous conducting and dielectric building materials for diffraction coefficients that are not available analytically in practical conditions. ^

Determining polarization dependent diffraction coefficients from building edges using FDTD method

Electromagnetic waves in suburban environment encounter multiple obstructions that shadow the signal. These waves are scattered and random in polarization. They take multiple paths that add as vectors at the portable device. Buildings have vertical and horizontal edges. Diffraction from edges has polarization dependent characteristics. In practical case, a signal transmitted from a vertically polarized high antenna will result in a significant fraction of total power in the horizontal polarization at the street level. Signal reception can be improved whenever there is a probability of receiving the signal in at least two independent ways or branches. The Finite-Difference Time-Domain (FDTD) method was applied to obtain the two and three-dimensional dyadic diffraction coefficients (soft and hard) of right-angle perfect electric conductor (PEC) wedges illuminated by a plane wave. The FDTD results were in good agreement with the asymptotic solutions obtained using Uniform Theory of Diffraction (UTD). Further, a material wedge replaced the PEC wedge and the dyadic diffraction coefficient for the same was obtained.

Study of the calibration method of pressure-velocity probes and its application in a field of progressive plane waves

This dissertation presents a calibration procedure for a pressure velocity probe. The dissertation is divided into four main chapters. The first chapter is divided into six main sections. In the firsts two, the wave equation in fluids and the velocity of sound in gases are calculated, the third section contains a general solution of the wave equation in the case of plane acoustic waves. Section four and five report the definition of the acoustic impedance and admittance, and the practical units the sound level is measured with, i.e. the decibel scale. Finally, the last section of the chapter is about the theory linked to the frequency analysis of a sound wave and includes the analysis of sound in bands and the discrete Fourier analysis, with the definition of some important functions. The second chapter describes different reference field calibration procedures that are used to calibrate the P-V probes, between them the progressive plane wave method, which is that has been used in this work. Finally, the last section of the chapter contains a description of the working principles of the two transducers that have been used, with a focus on the velocity one. The third chapter of the dissertation is devoted to the explanation of the calibration set up and the instruments used for the data acquisition and analysis. Since software routines were extremely important, this chapter includes a dedicated section on them and the proprietary routines most used are thoroughly explained. Finally, there is the description of the work that has been done, which is identified with three different phases, where the data acquired and the results obtained are presented. All the graphs and data reported were obtained through the Matlab® routine. As for the last chapter, it briefly presents all the work that has been done as well as an excursus on a new probe and on the way the procedure implemented in this dissertation could be applied in the case of a general field.

Efficient Computation of Electromagnetic Waves in Hydrocarbon Exploration Using the Improved Numerical Mode Matching (NMM) Method

In this study, we developed and improved the numerical mode matching (NMM) method which has previously been shown to be a fast and robust semi-analytical solver to investigate the propagation of electromagnetic (EM) waves in an isotropic layered medium. The applicable models, such as cylindrical waveguide, optical fiber, and borehole with earth geological formation, are generally modeled as an axisymmetric structure which is an orthogonal-plano-cylindrically layered (OPCL) medium consisting of materials stratified planarly and layered concentrically in the orthogonal directions.

In this report, several important improvements have been made to extend applications of this efficient solver to the anisotropic OCPL medium. The formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application to more general materials. The perfectly matched layer (PML) is incorporated along the radial direction as an absorbing boundary condition (ABC) to make the NMM method more accurate and efficient for wave diffusion problems in unbounded media and applicable to scattering problems with lossless media. We manipulate the weak form of Maxwell's equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem which is ignored by all previous researchers. The spectral element method (SEM) is introduced to more efficiently compute the eigenmodes of higher accuracy with less unknowns, achieving a faster mode matching procedure between different horizontal layers. We also prove the relationship of the field between opposite mode indices for different types of excitations, which can reduce the computational time by half. The formulas for computing EM fields excited by an electric or magnetic dipole located at any position with an arbitrary orientation are deduced. And the excitation are generalized to line and surface current sources which can extend the application of NMM to the simulations of controlled source electromagnetic techniques. Numerical simulations have demonstrated the efficiency and accuracy of this method.

Finally, the improved numerical mode matching (NMM) method is introduced to efficiently compute the electromagnetic response of the induction tool from orthogonal transverse hydraulic fractures in open or cased boreholes in hydrocarbon exploration. The hydraulic fracture is modeled as a slim circular disk which is symmetric with respect to the borehole axis and filled with electrically conductive or magnetic proppant. The NMM solver is first validated by comparing the normalized secondary field with experimental measurements and a commercial software. Then we analyze quantitatively the induction response sensitivity of the fracture with different parameters, such as length, conductivity and permeability of the filled proppant, to evaluate the effectiveness of the induction logging tool for fracture detection and mapping. Casings with different thicknesses, conductivities and permeabilities are modeled together with the fractures in boreholes to investigate their effects for fracture detection. It reveals that the normalized secondary field will not be weakened at low frequencies, ensuring the induction tool is still applicable for fracture detection, though the attenuation of electromagnetic field through the casing is significant. A hybrid approach combining the NMM method and BCGS-FFT solver based integral equation has been proposed to efficiently simulate the open or cased borehole with tilted fractures which is a non-axisymmetric model.

On thermodynamics in the primary power conversion of oscillating water column wave energy converters

The paper presents an investigation to the thermodynamics of the air flow in the air chamber for the oscillating water column wave energy converters, in which the oscillating water surface in the water column pressurizes or de-pressurises the air in the chamber. To study the thermodynamics and the compressibility of the air in the chamber, a method is developed in this research: the power take-off is replaced with an accepted semi-empirical relationship between the air flow rate and the oscillating water column chamber pressure, and the thermodynamic process is simplified as an isentropic process. This facilitates the use of a direct expression for the work done on the power take-off by the flowing air and the generation of a single differential equation that defines the thermodynamic process occurring inside the air chamber. Solving the differential equation, the chamber pressure can be obtained if the interior water surface motion is known or the chamber volume (thus the interior water surface motion) if the chamber pressure is known. As a result, the effects of the air compressibility can be studied. Examples given in the paper have shown the compressibility, and its effects on the power losses for large oscillating water column devices.

Primary wave energy conversions of oscillating water columns

This paper presents a study on the numerical simulation of the primary wave energy conversion in the oscillating water column (OWC) wave energy converters (WECs). The new proposed numerical approach consists of three major components: potential flow analysis for the conventional hydrodynamic parameters, such as added mass, damping coefficients, restoring force coefficients and wave excitations; the thermodynamic analysis of the air in the air chamber, which is under the assumptions of the given power take-off characteristics and an isentropic process of air flow. In the formulation, the air compressibility and its effects have been included; and a time-domain analysis by combining the linear potential flow and the thermodynamics of the air flow in the chamber, in which the hydrodynamics and thermodynamics/aerodynamics have been coupled together by the force generated by the pressurised and de-pressurised air in the air chamber, which in turn has effects on the motions of the structure and the internal water surface. As an example, the new developed approach has been applied to a fixed OWC device. The comparisons of the measured data and the simulation results show the new method is very capable of predicting the performance of the OWC devices.

A New Method for Modeling Free Surface Flows and Fluid-structure Interaction with Ocean Applications

The computational modeling of ocean waves and ocean-faring devices poses numerous challenges. Among these are the need to stably and accurately represent both the fluid-fluid interface between water and air as well as the fluid-structure interfaces arising between solid devices and one or more fluids. As techniques are developed to stably and accurately balance the interactions between fluid and structural solvers at these boundaries, a similarly pressing challenge is the development of algorithms that are massively scalable and capable of performing large-scale three-dimensional simulations on reasonable time scales. This dissertation introduces two separate methods for approaching this problem, with the first focusing on the development of sophisticated fluid-fluid interface representations and the second focusing primarily on scalability and extensibility to higher-order methods.

We begin by introducing the narrow-band gradient-augmented level set method (GALSM) for incompressible multiphase Navier-Stokes flow. This is the first use of the high-order GALSM for a fluid flow application, and its reliability and accuracy in modeling ocean environments is tested extensively. The method demonstrates numerous advantages over the traditional level set method, among these a heightened conservation of fluid volume and the representation of subgrid structures.

Next, we present a finite-volume algorithm for solving the incompressible Euler equations in two and three dimensions in the presence of a flow-driven free surface and a dynamic rigid body. In this development, the chief concerns are efficiency, scalability, and extensibility (to higher-order and truly conservative methods). These priorities informed a number of important choices: The air phase is substituted by a pressure boundary condition in order to greatly reduce the size of the computational domain, a cut-cell finite-volume approach is chosen in order to minimize fluid volume loss and open the door to higher-order methods, and adaptive mesh refinement (AMR) is employed to focus computational effort and make large-scale 3D simulations possible. This algorithm is shown to produce robust and accurate results that are well-suited for the study of ocean waves and the development of wave energy conversion (WEC) devices.

Analysis of irregular microstrip structures using a full wave MoM scheme

[EN]This article presents the results obtained in the analysis of irregular microstrip structures using a full wave method of moments scheme. The irregular microstrip structures are divided into rectangular subdomains. The EFIE is discretized an solved over the subdomains using a Galerkin type scheme. Base and weight functions are piece wise sinusoidals (PWS) or triangular. Delta gap voltage generators are used as sources]. Green functions are computed using a freely available library developed by our research group. All the calculations are carried out in the so called ”spatial domain” so there is no need of using regular grids during the discretization process.

Numerical Simulation Of Ultrasonic Wave Propagation In Elastically Anisotropic Media

The ultrasonic non-destructive testing of components may encounter considerable difficulties to interpret some inspections results mainly in anisotropic crystalline structures. A numerical method for the simulation of elastic wave propagation in homogeneous elastically anisotropic media, based on the general finite element approach, is used to help this interpretation. The successful modeling of elastic field associated with NDE is based on the generation of a realistic pulsed ultrasonic wave, which is launched from a piezoelectric transducer into the material under inspection. The values of elastic constants are great interest information that provide the application of equations analytical models, until small and medium complexity problems through programs of numerical analysis as finite elements and/or boundary elements. The aim of this work is the comparison between the results of numerical solution of an ultrasonic wave, which is obtained from transient excitation pulse that can be specified by either force or displacement variation across the aperture of the transducer, and the results obtained from a experiment that was realized in an aluminum block in the IEN Ultrasonic Laboratory. The wave propagation can be simulated using all the characteristics of the material used in the experiment evaluation associated to boundary conditions and from these results, the comparison can be made.

Wind wave prediction in the Caspian Sea

In this thesis, wind wave prediction and analysis in the Southern Caspian Sea are surveyed. Because of very much importance and application of this matter in reducing vital and financial damages or marine activities, such as monitoring marine pollution, designing marine structure, shipping, fishing, offshore industry, tourism and etc, gave attention by some marine activities. In this study are used the Caspian Sea topography data that are extracted from the Caspian Sea Hydrography map of Iran Armed Forces Geographical Organization and the I 0 meter wind field data that are extracted from the transmitted GTS synoptic data of regional centers to Forecasting Center of Iran Meteorological Organization for wave prediction and is used the 20012 wave are recorded by the oil company's buoy that was located at distance 28 Kilometers from Neka shore for wave analysis. The results of this research are as follows: - Because of disagreement between the prediction results of SMB method in the Caspian sea and wave data of the Anzali and Neka buoys. The SMB method isn't able to Predict wave characteristics in the Southern Caspian Sea. - Because of good relativity agreement between the WAM model output in the Caspian Sea and wave data of the Anzali buoy. The WAM model is able to predict wave characteristics in the southern Caspian Sea with high relativity accuracy. The extreme wave height distribution function for fitting to the Southern Caspian Sea wave data is obtained by determining free parameters of Poisson-Gumbel function through moment method. These parameters are as below: A=2.41, B=0.33. The maximum relative error between the estimated 4-year return value of the Southern Caspian Sea significant wave height by above function with the wave data of Neka buoy is about %35. The 100-year return value of the Southern Caspian Sea significant height wave is about 4.97 meter. The maximum relative error between the estimated 4-year return value of the Southern Caspian Sea significant wave height by statistical model of peak over threshold with the wave data of Neka buoy is about %2.28. The parametric relation for fitting to the Southern Caspian Sea frequency spectra is obtained by determining free parameters of the Strekalov, Massel and Krylov etal_ multipeak spectra through mathematical method. These parameters are as below: A = 2.9 B=26.26, C=0.0016 m=0.19 and n=3.69. The maximum relative error between calculated free parameters of the Southern Caspian Sea multipeak spectrum with the proposed free parameters of double-peaked spectrum by Massel and Strekalov on the experimental data from the Caspian Sea is about 36.1 % in spectrum energetic part and is about 74M% in spectrum high frequency part. The peak over threshold waverose of the Southern Caspian Sea shows that maximum occurrence probability of wave height is relevant to waves with 2-2.5 meters wave fhe error sources in the statistical analysis are mainly due to: l) the missing wave data in 2 years duration through battery discharge of Neka buoy. 2) the deportation %15 of significant height annual mean in single year than long period average value that is caused by lack of adequate measurement on oceanic waves, and the error sources in the spectral analysis are mainly due to above- mentioned items and low accurate of the proposed free parameters of double-peaked spectrum on the experimental data from the Caspian Sea.

Wave and turbulence measurements at a tidal energy site

This work presents the analysis of wave and turbulence measurements collected at a tidal energy site. A new method is introduced to produce more consistent and rigorous estimations of the velocity fluctuations power spectral densities. An analytical function is further proposed to fit the observed spectra and could be input to the numerical models predicting power production and structural loading on tidal turbines. Another new approach is developed to correct for the effect of the Doppler noise on the high frequencies power spectral densities. The analysis of velocity time series combining wave and turbulent contributions demonstrates that the turbulent motions are coherent throughout the water column, rendering the wave coherence-based methods not applicable to our dataset. To avoid this problem, an alternative approach relying on the pressure data collected by the ADCP is introduced and shows appreciable improvement in the wave-turbulence separation.