Yuri Eremin

IV – Discrete sources method in acoustic theory

This chapter presents the mathematical justification of the discrete sources method (DSM). Only the exterior Dirichlet, Neumann and impedance boundary value problems are discussed in detail as the basic concepts are fully represented in these cases. A description of the smoothing procedure developed by Yasuura and Ikuno is included in analysis. In addition to a strict mathematical model practical aspect of the method is also considered, such as the correct choice of the support of discrete sources, the determination of the amplitudes by a stable numerical algorithm, and the estimation of the accuracy of the results. In particular, it will show that the convergence and the stability of the method depend on the position of the singularities of the scattered field. The chapter proceeds to analyze the acoustic scattering by axisymmetric particles. It will show that the use of distributed spherical wave functions located on the particle symmetry axis reduces the scattering problem to a sequence of one-dimensional approximation problems relative to the Fourier harmonics of the fields. Finally, the chapter ends with a presentation of a method for correlating the position of the support of discrete sources with the singularities of the scattered field.

Null-field method in acoustic theory

This chapter presents the fundaments of the null-field method (NFM) for solving the Dirichlet and Neumann boundary-value problems. The chapter begins by showing that the scattering problem reduces to the approximation problem of the surface densities by convergent sequences. It then presents convergent projection methods for the general null-field equations. Next, it will investigate the conventional null-field method with discrete sources. The foundations of the method include convergence analysis following Ramms treatment and derivation of sufficient conditions that guarantee the convergence of the approximate solution. The conclusion of this analysis is that the null-field method converges if the systems of expansion and testing functions form a Riesz basis in “L 2 (S) .” Finally, it presents the equivalence between the null-field method and the auxiliary current method.

The scalar Helmholtz equation

This chapter is devoted to presenting the foundations of obstacle scattering problems for time-harmonic acoustic waves. The chapter provides brief discussion of the physical background of the scattering problem, and then formulates the boundary-value problems for the Helmholtz equation. It will synthetically recall the basic concepts as they were presented by Colton and Kress. However, some of the details are left in the analysis. In this context, the technical proof is not repeated for the jump relations and the regularity properties for single- and double-layer potentials with continuous densities. However, Laxs theorem will be presented that enables to extend the jump relations from the case of continuous densities to square integrable densities. It then establishes some properties of surface potentials vanishing in sets of R 3 . These results play a significant role in the completeness analysis. Discussing the Green representation theorems will help in deriving some estimates of the solutions. The chapter concludes by analyzing the general null-field equation for the exterior Dirichlet and Neumann problems. In particular, it will establish the existence and uniqueness of the solutions and will prove the equivalence of the null-field equations with some boundary integral equations.

Elements of functional analysis

This chapter discusses the fundamental results of functional analysis. It firstly presents the notion of a Hilbert space and discusses some basic properties of the orthogonal projection operator. It then introduces the concept of closeness and completeness of a system of elements that belong to a Hilbert space. The completeness of the system of elementary sources is a necessary condition for the solution of scattering problems in the framework of the discrete sources method. After this discussion, the chapter briefly presents the notions of Schauder and Riesz bases. This concept will be used to analyze the convergence of the null-field method. A convergent projection scheme is constructed by appealing on the fundamental theorem of discrete approximation. The chapter concludes by analyzing projection methods for a linear operator “A” acting from a Hilbert space “H” onto a Hilbert space G, and for the operator equation.

X – Null-field method in electromagnetic theory

This chapter discusses the analysis of electromagnetic scattering using the null-field method (NFM) with discrete sources. The chapter focuses on the exterior and the transmission boundary-value problems. It begins by constructing projection methods for the general null-field equations. Next, it presents the conventional null-field method with discrete sources for solving the exterior Maxwell problem. The efficiency of the conventional null-field method, in comparison to other projection methods, is investigated from a computational point of view. The chapter discusses the use of Tikhonov regularization and presents some numerical results. It considers the conventional null-field method for solving the transmission problem. For the sake of completeness the different formulations of the method that serves as input for our computer simulations is listed. The chapter then proceeds to establishing the expression of the transition matrix and investigate general constraints of the transition matrix such as symmetry and unitarity. It will demonstrate that the use of distributed sources rather than a localized source improves the symmetry of the transition matrix. Finally, it presents some computer simulations. These include scattering by particles with large size parameters, particles with extreme geometries, cubs, ellipsoids, and rough particles.

Systems of functions in acoustic theory

This chapter discusses the analysis of complete and linear independent systems of functions for the Helmholtz equation. As complete systems of functions, the systems of discrete sources will be discussed in the chapter. There is a close relation between the properties of the fields of discrete sources and the structure of their support. In particular, if the supports are chosen as a point, a straight line, or a surface, then the corresponding systems of functions are the localized spherical wave functions, the distributed spherical wave functions and the distributed point sources, respectively. The chapter begins by presenting some basic results on the completeness of localized spherical wave functions. In order to preserve the completeness at irregular frequencies, linear combinations of regular functions and their normal derivatives on the particle surface will be used. It then proceeds to describe a scheme for complete systems construction, using radiating solutions to the Helmholtz equation. In particular, the completeness of distributed radiating spherical wave functions will be discussed. After that, it will provide a similar scheme using entire solutions to the Helmholtz equations. The next section of the chapter is concerned with the completeness of point sources. The systems of functions with singularities distributed on closed and open auxiliary surfaces are described. The last section of this chapter deals with the linear independence of these systems.

Projection methods in electromagnetic theory

This chapter describes the use of the fundamental theorem of discrete approximation to construct projection schemes for a category of variational problems in the Hilbert spaces. The results of these problems are then used to construct approximate solutions to the boundary-value problems for the Maxwell equations.

Systems of functions in electromagnetic theory

The chapter discusses the analysis of complete and linear independent systems of functions for the Maxwell equations. The analysis begins by presenting some fundamental results on the completeness of the localized spherical vector wave functions. The completeness properties of the systems of discrete sources are of primary interest as they provide a means for approximating the exact solutions to the scattering problems. To preserve the completeness at irregular frequencies, linear combinations of these functions are considered. Next, the completeness properties of the systems of distributed sources are analyzed. The analysis is based on the addition theorem for spherical wave and vector wave functions. The next section of the chapter is concerned with the completeness of the system of vector Mie potentials with singularities distributed on auxiliary closed and open surfaces. These functions are suitable for analyzing the scattering by particles without rotational symmetry. The last section of this chapter deals with the linear independence of these systems.

Discrete sources method in electromagnetic theory

Publisher Summary This chapter introduces the basic concepts of the discrete sources method (DSM) for solving electromagnetic scattering problems. In the acoustic case, the electromagnetic scattering problem reduces to the approximation problem of the boundary value of the incident field in the “L 2” -norm. The technical aspect of the acoustic case is not repeated. Analysis is mainly concentrated on the construction of convergent approximations using the fundamental theorem of discrete approximation. For the impedance boundary-value problem, a somewhat different approach called the “D-matrix” method is presented, as the matrix of the corresponding linear system of equations is dissipative. Also, the dissipativity, the convergence of the approximate solution and the solvability of the linear system of equations using the conservation law of energy is established. Special attention is paid to the discrete sources method with distributed vector multipoles. The mathematical foundation of the method is accompanied by the results of computer simulations. The chapter also discusses the numerical experiments including comparison with other methods and scattering analysis of concave particles, and clusters of particles.