MATHEMATICAL MODELING OF WAVE PHENOMENA: 2nd Conference on Mathematical Modeling of Wave Phenomena

Using a zipper algorithm to find a conformal map for a channel with smooth boundary
View Description Hide DescriptionThe so called geodesic algorithm, which is one of the zipper algorithms for conformal mappings, is combined with a Schwarz‐Christoffel mapping, in its original or in a modified form, to produce a conformal mapping function between the upper half‐plane and an arbitrary channel with smooth boundary and parallel walls at the end.

Impulsive spherical‐wave reflection against a planar absorptive and dispersive Dirichlet‐to‐Neumann boundary — An extension of the modified Cagniard method
View Description Hide DescriptionA closed‐form analytic time‐domain expression is obtained for the scalar wave function associated with the reflection of a point‐source excited impulsive spherical wave by a planar boundary with absorptive and dispersive properties. The physical properties of the boundary are modeled as a local Dirichlet‐to‐Neumann boundary condition in the form of a time‐convolution integral, the kernel function in which (denoted as the boundary’s time‐domain admittance) meets the conditions for linear, time‐invariant, causal, passive behavior. Parametrizations of the boundary admittance function are put forward that have the property of showing up explicitly, and in a relatively simple manner, in the expression for the reflected wave field. The partial fraction representation of the time Laplace‐transform domain boundary admittance and the plane‐wave admittance representation are shown to serve the purpose. The expression for the reflected wave field is constructed through an appropriate combination of the modified Cagniard method for analyzing wave propagation in layered media, Lerch’s uniqueness theorem of the unilateral Laplace transformation, and the Schouten‐Van der Pol theorem pertaining to a change of transform parameter in this transformation. The result can serve as a benchmark solution to the modeling of transient wave reflection against absorptive and dispersive boundaries in more complicated geometries where numerical methods are the tool of analysis. The obtained analytical expressions show that, in the configuration under consideration, no true surface waves occur along the boundary.

Helmholtz Path Integrals
View Description Hide DescriptionThe multidimensional, scalar Helmholtz equation of mathematical physics is addressed. Rather than pursuing traditional approaches for the representation and computation of the fundamental solution, path integral representations, originating in quantum physics, are considered. Constructions focusing on the global, two‐way nature of the Helmholtz equation, such as the Feynman/Fradkin, Feynman/Garrod, and Feynman/DeWitt‐Morette representations, are reviewed, in addition to the complementary phase space constructions based on the exact, well‐posed, one‐way reformulation of the Helmholtz equation. Exact, Feynman/Kac, stochastic representations are also briefly addressed. These complementary path integral approaches provide an effective means of highlighting the underlying physics in the solution representation, and, subsequently, exploiting this more transparent structure in natural computational algorithms.

A Fast Method for the Creeping Ray Contribution to Scattering Problems
View Description Hide DescriptionCreeping rays can give an important contribution to the solution of medium to high frequency scattering problems. They are generated at the shadow lines of the illuminated scatterer by grazing incident rays and propagate along geodesics on the scatterer surface, continuously shedding diffracted rays in their tangential direction. In this paper we show how the ray propagation problem can be formulated as a partial differential equation (PDE) in a three‐dimensional phase space. To solve the PDE we use a fast marching method. The PDE solution contains information about all possible creeping rays. This information includes the phase and amplitude of the field, which are extracted by a fast postprocessing. Computationally the cost of solving the PDE is less than tracing all rays individually by solving a system of ordinary differential equations(ODE). We consider an application to monostatic radar cross section problems where creeping rays from all illumination angles must be computed.

Indoor Propagation Simulation using FEM for Short‐Range Wireless Communication Systems Operating at 2.4 GHz
View Description Hide DescriptionIn this paper we investigate the wave propagation effects of a short‐range wireless device, like the Bluetooth, in an indoor office environment. Specifically, we investigate line of sight (LOS) and non line of sight (NLOS) propagation scenarios and verify the theoretical statistics of the fading using real measurements and finite element method (FEM) simulations of the environment. The simulations are also used for assessing the impact on propagation when doors are opened or closed. In addition, we also investigate the improvement in performance resulting from receive diversity gain of a system employing multiple receive antennas with various combining techniques.

On the efficiency of antennas
View Description Hide DescriptionThe fundamental limits of the power efficiency of an antenna are explored. The antenna is confined in a sphere and all of the currents are assumed to run in a material with a conductivity that is a function of the radial coordinate. The analysis is based on the expansion of the electromagnetic fields in terms of vector spherical harmonics. Explicit expressions for the limits of efficiency are derived for different types of antennas.

Scattering of an electromagnetic plane wave from a curved cylindrical conductor
View Description Hide DescriptionIn this paper, the overall objective is to investigate a method to calculate the so‐called equivalent surface currents, arisen from an incoming high frequency plane wave, on a long and slightly curved cylindrical conductor, such as a wire. One application for this calculation method is the study of how radar waves are affected by wires mounted near a radar antenna. The paper also includes a brief discussion on how this surface current can be used to calculate the scattered electric field from the conductor.

Efficient Calculations of Antenna Radiation Patterns Using the Fast Fourier Transform
View Description Hide DescriptionTime‐efficient calculations of radiated electromagnetic fields are desirable in many antenna applications. The specific problem analyzed in this paper concerns the development of antenna radomes. To find the optimal configuration, a large number of simulations of virtual antennas with small changes are required. A fast and reliable calculation method is needed for such simulations.
In this paper we study the possibility to calculate the far‐field with the Discrete Fourier Transform, which enables usage of the Fast Fourier Transform algorithm. An accurate and unique representation of the far‐field with this method requires special attention of the sampling interval of the current distribution.
Our experience is that this Fourier method is about one thousand times faster than standard practise in industry.

Extrapolation and Modelling of Method of Moments Currents on a PEC Surface
View Description Hide DescriptionWe present a current‐based approach to high frequency approximation techniques. Our goal is to numerically model the behaviour of electromagnetic and surface current vector fields at medium‐range or high frequency, using information extracted from lower frequency solutions.

Parameter Dependence of Flow Acoustic Interaction
View Description Hide DescriptionThe possibility to predict flow acoustic coupling at a sharp edge of an area expansion in a flow duct is explored, starting from the analytical model proposed by Nilsson and Boij. In the model, the vortex shedding is treated as infinitely thin, causing an instability for all frequencies. For small and large values of the Strouhal number, the acoustic field is well defined and can be treated separately from the remaining flow field. For intermediate Strouhal numbers, this classification into acoustic and non‐acoustic waves is more complex. In particular, a hydrodynamic wave and higher order, non‐propagating acoustic waves are changing properties. The possibility to use the amount of deviation from the asymptotic behaviour as a measure of the strength of the interaction at the edge is explored. It is shown that the interaction is strongly dependent on the area ratio at the duct expansion. Further studies would aim to a more elaborate investigation of the implications of the wave number parameter dependence. The objective would be to better predict the circumstances for strong interaction, in order to design less noisy ventilation and exhaust systems and to enhance dissipation effects.

A numerical method for time dependent acoustic scattering problems involving smart obstacles and incoming waves of small wavelengths
View Description Hide DescriptionIn this paper we propose a highly parallelizable numerical method for time dependent acoustic scattering problems involving realistic smart obstacles hit by incoming waves having wavelengths small compared with the characteristic dimension of the obstacles. A smart obstacle is an obstacle that when hit by an incoming wave tries to pursue a goal circulating on its boundary a pressure current. In particular we consider obstacles whose goal is to be undetectable and we refer to them as furtive obstacles. These scattering problems are modelled as optimal control problems for the wave equation. We validate the method proposed to solve the optimal control problem considered on some test problems where a “smart” simplified version of the NASA space shuttle is hit by incoming waves with small wavelengths compared to its characteristic dimension. That is we consider test problems with ratio between the characteristic dimension of the obstacle and wavelength of the time harmonic component of the incoming wave up to approximately one hundred. The website: http://www.econ.univpm.it/recchioni/w14 contains animations and virtual reality applications showing some numerical experiments relative to the problems studied in this paper.

Sound propagation over inhomogeneous ground including a sound velocity profile
View Description Hide DescriptionThe atmospheric profile whose sound speed varies linearly with height is simple in concept, but leads to complications when solving for the sound pressure. Its effects are commonly approximated by a similar profile whose squared refractive index is a linear function of height. In this paper, the validity of the approximation has been examined for sound propagation above an impedance ground and a computational approximation is given. The method relies upon a family of radiation boundary conditions for the wave equation derived by truncating a summation function approximation of a corresponding plane wave reflection coefficient representation. It is demonstrated how these boundary conditions can be formulated in terms of a finite element approach. Numerical examples illustrate results with four coefficients included at the upper fictitious boundary conditions for the case of sound propagation above a grass‐strip and sound propagation over the grass‐strip with an atmospheric profile.

Parabolic Equation Techniques for Range‐Dependent Seismo‐Acoustics
View Description Hide DescriptionSome recent progress in the development of parabolic equation techniques for range‐dependent seismo‐acoustics is discussed. An improved formulation makes it possible to handle arbitrary stratification and all wave types. Range dependence can be handled with greater accuracy by using approaches based on rotating coordinates and single scattering. One of the ways the parabolic solutions are being tested is through comparison with data.

An Adaptive Multigrid FE‐Method Application to Elasto‐Plastic Wave Propagation
View Description Hide DescriptionThe solution of linear systems of equations is the most time‐consuming part in large‐scale implicit FE computations of wave propagation problems. Traditionally, direct solvers have been used but recent developments of iterative solvers and precondition techniques may impose a change. In particular, preconditioning by multigrid seems to be favorable for finite element (FE) applications since it has a natural interpretation and substantially improves the rate of convergence of the conventional iterative solvers.
The multigrid preconditioner uses a sequence of grids on which the fine‐grid residual forces are prolongated to coarser grids and computed corrections are interpolated back to the fine grid such that the fine‐grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time and at low memory requirements.
The proposed adaptive multigrid FE‐method combines adaptivity and multigrid solution strategy in a sophisticated manner. The sequence of computational grids is successively refined (adapted) and generated according to the guidance of a posteriori error estimates until the solution fulfils a predefined accuracy specification. In contrast to standard adaptive procedures where rejected grids are deleted, the adaptive multigrid algorithm uses previous solution and generated grids to speed up the solution process.
A refinement strategy based on element splitting and introduction of hanging nodes requires special care since the constraint equations of the hanging nodes are incorporated in the system by usage of Lagrange multipliers. The approach leads to indefinite systems and hence a special preconditioner that enforces the constraint equations to be fulfilled while iterating has been developed.
The paper presents results using the adaptive multigrid procedure on an elasto‐plastic wave propagation problem.

Selective mode excitation in elastic waveguides by piezoceramic actuators
View Description Hide DescriptionThe problem of a single required mode (or several modes) excitation in a plate‐like structure by a system of thin flexible piezoelectric patches is considered. The algorithm for obtaining proper driving fields is developed in the context of a 2D mathematical model (strip patches on a layer), which takes strictly into account the patch‐structure contact interaction, the mutual influence of the actuators and the multimode character of traveling waves propagation in the substructure. Numerical results illustrate the high level of selectivity provided by the method proposed.

Real natural frequencies and resonance wave localization in an elastic waveguide with a horizontal crack
View Description Hide DescriptionPreviously developed semi‐analytical methods for evaluating elastodynamic diffraction by a horizontal crack in a layer are used to analyze the mechanisms of normal mode transmission, reflection and blocking. It is shown that resonance blocking is accompanied by energy localization near the crack. The dependence of transmission and reflection coefficients on frequency and the size and depth of the crack are analyzed, along with spectral points. The present paper focuses on a search for pure real spectral points and the respective eigenforms of the motion. Some examples of pure real resonance frequencies have been found. The associated eigenforms exhibit strong localization of oscillation near the crack at the resonance frequencies.

The diffraction of a plane wave by a 2D traction free isotropic wedge
View Description Hide DescriptionWedge diffraction is a well‐known problem in applied mathematics. The currently favoured semi‐analytical scheme is to reduce the original elastodynamic equations supplemented with boundary, radiation and tip conditions first to a system of functional equations and then to a system of algebraic and integral equations. The latter are to be solved numerically on a line in a complex plane; the solution is to be extended first to a strip centred on this line and then to the whole complex plane. We have investigated the latest contribution to the body of wedge literature provided by Budaev and Bogy, who followed Malyuzhinets and represented solutions of the elastodynamic equations using the Sommefeld integral transform. Our main achievement has been the clarification of the status of the unknown constants in the functional equations which emerge on substituting such transforms into boundary conditions, taking into account the tip conditions and applying the Nullification Theorem for the Sommerfeld Integrals. As the result we have developed a new numerical schedule for solution of the corresponding singular integral equations. This has now been implemented in a user‐friendly code 2DWeC to compute Sommerfeld amplitudes which satisfy the functional equations, exhibit the correct behavior at infinity, possess correct singularities and do not possess parasitic singularities in the regions of interest. We argue that by implication the corresponding inverse transforms satisfy the original equations as well as boundary, radiation and tip conditions and thus constitute the solution of the original diffraction problem.

Periodic orbits in scattering from elastic voids
View Description Hide DescriptionThe scattering determinant for the scattering of waves from several obstacles is considered in the case of elastic solids with voids. The multi‐scattering determinant displays contributions from periodic ray‐splitting orbits. A discussion of the weights of such orbits is presented.

Standing and propagating waves in cubically nonlinear media
View Description Hide DescriptionThe paper has three parts. In the first part a cubically nonlinear equation is derived for a transverse finite‐amplitude wave in an isotropic solid. The cubic nonlinearity is expressed in terms of elastic constants. In the second part a simplified approach for a resonator filled by a cubically nonlinear medium results in functional equations. The frequency response shows the dependence of the amplitude of the resonance on the difference between one of the resonator’s eigenfrequencies and the driving frequency. The frequency response curves are plotted for different values of the dissipation and differ very much for quadratic and cubic nonlinearities. In the third part a propagating N‐wave is studied, which fulfils a modified Burgers’ equation with a cubic nonlinearity. Approximate solutions to this equation are found for new parts of the wave profile.