Kernel Of Integral Equation Research Articles

Three-body breakup within the fully discretized Faddeev equations

A novel approach is developed to find the three-body breakup amplitudes and cross sections within the modified Faddeev equation framework. The method is based on the lattice-like discretization of the three-body continuum with a three-body stationary wave-packet basis in momentum space. The approach makes it possible to simplify drastically all the three- and few-body breakup calculations due to discrete wave-packet representations for the few-body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels. As a result, the few-body breakup can be treated as a particular case of multi-channel scattering in which part of the channels represents the true few-body continuum states. As an illustration for the novel approach, an accurate calculations for the three-body breakup process $n+d\to n+n+p$ with non-local and local $NN$ interactions are calculated. The results obtained reproduce nicely the benchmark calculation results using the traditional Faddeev scheme which requires much more tedious and time-consuming calculations.

Faster convergence for MoM analysis of resonant circular arrays of cylindrical dipoles

Method-of-moments analyses of resonant circular arrays of cylindrical dipoles present unique and particular difficulties. A previous publication overcame the major difficulties by using a set of improved kernels in the usual Hallen-type integral equations and by applying an appropriate convergence acceleration method. Based on previous works dealing with the isolated cylindrical dipole, proposed is a better method of calculating the constant C (m) appearing in the aforementioned equations; this results both in improved accuracy and in faster overall convergence of the authors' moment method.

Formulation and selected applications of a regularized elastodynamics integral equation approach to large scale scattering problems involving inhomogeneous objects

Elements are presented of the formulation and selected applications of the combined surface and volume integral-equation approach for finding pressure, displacement, and traction fields in composite objects consisting of piece-wise homogeneous regions characterized by different Lame's material parameters, as well as of highly inhomogeneous regions. The proposed elastodynamics surface/volume-based integral equations, in which the unknowns are volumetric pressure or displacement variables and surface displacement and traction fields on the interfaces, involve only weakly singular integral-equation kernels with at most 1/r singularity. The proposed method is found to provide well-convergent and accurate solutions for a large spectrum of problems, in particular those involving high-contrast material interfaces. The proposed method's accuracy and its capability (achieved through the (non-lossy FFT-based) Adaptive Integral Method matrix compression) of handling large-scale problems involving several millions of unknowns, is demonstrated through comparison with analytically constructed solution for a layered material sphere and in selected numerical simulations of propagation of acousto-elastic waves in human head and inner-ear region through air- and bone-conduction pathways. The proposed approach can be used in assessing effectiveness of noise-protection devices. This work is supported by a grant from the Air Force Office of Scientific Research.

A note on the integral equation for the Wilson loop in D = 4 superconformal Yang–Mills theory

We propose an alternative method to study the saddle point equation in the strong coupling limit for the Wilson loop in D = 4 super-Yang–Mills with an SU(N) gauge group and 2N hypermultiplets. This method is based on an approximation of the integral equation kernel which allows us to solve the simplified problem exactly. To determine the accuracy of this approximation, we compare our results with those obtained recently by Passerini and Zarembo. Although less precise, this simpler approach provides an explicit expression for the density of eigenvalues that is used to derive the planar free energy.

Green quasifunction method for free vibration of clamped thin plates

The Green quasifunction method is employed to solve the free vibration problem of clamped thin plates. A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of the free vibration problem of clamped thin plates is reduced to Fredholm integral equation of the second kind by Green formula. Irregularity of the kernel of integral equation is overcome by choosing a suitable form of the normalized boundary equation. Two examples demonstrate the validity of the present method. Comparison with both the series solution and ANSYS finite-element solution shows fine agreement. The present method is a novel and effective mathematical one.

A Multigrid Framework for $S_n$ Discretizations of the Boltzmann Transport Equation

The space-angle multiple/semicoarsening multigrid methods of [B. Lee, SIAM J. Sci. Comput., 31 (2010), pp. 4744--4773] for $S_n$ discretizations of the Boltzmann transport equation were developed assuming rediscretizations of the streaming-collision operators on the coarser spatial grids. These rediscretized operators are directly used in the kernels of the spatial coarse-level scalar flux integral equations. Since the computational procedure of these methods is essentially performed on these integral equations, it appears that the multigrid procedure is applied directly to the finest level scalar flux integral equation rather than to the finest level $S_n$ system. However, this procedure is actually implicitly applied to the finest level $S_n$ system. It would be beneficial to relate the multiple/semicoarsening methods directly to the full $S_n$ system or, better yet, to develop a general multigrid framework for the full $S_n$ system, of which the multiple/semicoarsening methods are particular cases. This general framework can help guide the development of multigrid algorithms for more general $S_n$ discretization settings, such as when spatial rediscretization of the integral equation kernel cannot be applied. In this latter case, applying an operator-dependent Galerkin coarsening procedure to the integral operator is difficult because the kernel is dense and often given only by its action, i.e., numerically sweeping over all the streaming-collision operators. In this paper, we develop and analyze a multigrid framework that is applied directly to the $S_n$ system and encompasses the multiple/semicoarsening methods. We further relate this framework to a multigrid method that directly coarsens the integral operator using a Galerkin coarsening procedure. A key component in developing this framework is to relate the smoothing of the multiple/semicoarsening methods to an approximate Schur complement smoothing on the full $S_n$ system. In this paper, this relation is established, and the smoothing and approximation properties are derived for the multigrid procedure induced by this framework.

Method-of-Moments Analysis of Resonant Circular Arrays of Cylindrical Dipoles

We perform a method-of-moments (MoM) analysis of a circular array of cylindrical dipoles. The array is known from earlier theoretical and experimental studies to possess very narrow resonances. The earlier theoretical studies were carried out using the “two-term theory.” The present paper is a direct continuation of a recent work showing that the problem possesses unique and particular difficulties. The main difficulties are overcome herein using a set of improved kernels in the usual Hallen-type integral equations (these kernels had been developed in previous works, and were successfully incorporated into the aforementioned two-term theory analyses). We make a detailed comparison of our MoM results to two-term theory results and, also, to the earlier experimental results.

Multilevel Adaptive Cross Approximation (MLACA)

The Multilevel Adaptive Cross Approximation (MLACA) is proposed as a fast method to accelerate the matrix-vector products in the iterative solution of the linear system that results from the discretization of electromagnetic Integral Equations (IE) with the Method of Moments (MoM). The single level ACA, already described in the literature, is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements. The main qualities of ACA are maintained: it is purely algebraic and independent of the integral equation kernel Green's function as long as it produces pseudo-rank-deficient matrix blocks. The algorithm is presented in such a way that it can be easily implemented on top of an existing MoM code with most commonly used Green's functions.

A general method for finding the exact solution of linear Volterra integral equations of the second kind

In this paper, a general method is given for the solution of linear Volterra integral equations of the second kind, which is based on the action of the operator defined by the kernel of the integral equation on a suitable basis for the corresponding function spaces. The necessary conditions for using this method are so weak that extends its applicability. The solved examples show the strength of this method.

About the chiral symmetry breaking in QED3

The problem of the chiral symmetry breaking in QED 3 is considered by solving the Schwinger–Dyson equation for the fermion propagator in the ladder approximation using the Landau gauge for the photon propagator. Within the framework of the indicated approximation, different simplifications that allow expressions for the fermion mass function to be retrieved in an explicit form are analyzed. The results obtained are compared with the data of numerical analysis. It appears that the neglect of higher Gegenbauer harmonics in the kernel of the initial integral equation for the fermion mass function influences the dynamic mass value and the asymptotics of the mass function only weakly. On the other hand, it is established that the conclusion about a complicated structure of the fermion vacuum of the massive phase is an artifact of linearization of the Schwinger–Dyson equation kernel: consideration of the kernel nonlinearity yields a simple massive phase structure of the fermion vacuum.

APPLICATION OF BOUNDARY ELEMENT METHOD FOR 3D WAVE SCATTERING IN CYLINDERS

A boundary element method (BEM) is presented to study 3D wave scattering by cracks in a cylinder. Green's functions needed in the kernel of boundary integral equations in BEM are derived with the help of guided wave functions. Guided wave modes in the cylinder are obtained by a semi-analytical finite element (SAFE) method. Green's functions are constructed numerically by superposition of guided wave modes. In this method, the cylinder is discretized in the radial direction into several coaxial circular cylinders (sub-cylinders) and the radial dependence of the displacement in each sub-cylinder is approximated by quadratic interpolation polynomials. A numerical procedure is used here to accurately calculate the Cauchy's principal value (CPV) and weakly singular integrals. The multi-domain technique is employed here to model the crack surface. Numerical results are presented to show the effectiveness of the proposed solution.

A BEM-based domain meshless method for the analysis of Mindlin plates with general boundary conditions

In this paper, a BEM-based domain meshless method is developed for the analysis of moderately thick plates modeled by Mindlin’s theory which permits the satisfaction of three physical conditions along the plate boundary. The presented method is achieved using the concept of the analog equation of Katsikadelis. According to this concept, the original governing differential equations are replaced by three uncoupled Poisson’s equations with fictitious sources under the same boundary conditions. The fictitious sources are established using a technique based on BEM and approximated by radial basis functions series. The solution of the actual problem is obtained from the known integral representation of the potential problem. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The presented method has the advantages of the BEM in the sense that the discretization and integration are performed only on the boundary, and consequently Mindlin plates with general boundary conditions can be analyzed without difficulty. To illustrate the effectiveness, applicability as well as accuracy of the method, numerical results of various example problems are presented.

Green quasifunction method for free vibration of simply-supported trapezoidal shallow spherical shell on Winkler foundation

The idea of Green quasifunction method is clarified in detail by considering a free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation. A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, the irregularity of the kernel of integral equations is avoided. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.

3 차원 시간영역 근사비선형 2 차경계요소법에 의한 선체의 대진폭 운동 및 파랑하중 계산

A three-dimensional time-domain calculation method is of crucial importance in prediction of the motions and wave loads of a ship advancing in a severe irregular sea. The exact solution of the free surface wave-ship interaction problem is very complicated because of the essentially nonlinear boundary conditions. In this paper, an approximate body nonlinear approach based on the three-dimensional time-domain forward-speed free-surface Green function has been presented. The Froude-Krylov force and the hydrostatic restoring force are calculated over the instantaneous wetted surface of the ship while the forces due to the radiation and scattering potentials over the mean wetted surface. The time-domain radiation and scattering potentials have been obtained from a time invariant kernel of integral equations for the potentials which are discretized according to the second-order boundary element method (Hong and Hong 2008). The diffraction impulse-response functions of the Wigley seakeeping model advancing in transient head waves at various Froude numbers have been presented. A simulation of coupled heave-pitch motion of a long rectangular barge advancing in regular head waves of large amplitude has been carried out. Comparisons between the linear and the approximate body nonlinear numerical results of motions and wave loads of the barge at a nonzero Froude number have been made.

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.

Toward the Application of Three-Dimensional Approach to Few-body Atomic Bound States

The first step toward the application of an effective non partial wave (PW) numerical approach to few-body atomic bound states has been taken. The two-body transition amplitude which appears in the kernel of three-dimensional Faddeev-Yakubovsky integral equations is calculated as function of two-body Jacobi momentum vectors, i.e. as a function of the magnitude of initial and final momentum vectors and the angle between them. For numerical calculation the realistic interatomic interactions HFDHE2, HFD-B, LM2M2 and TTY are used. The angular and momentum dependence of the fully off-shell transition amplitude is studied at negative energies. It has been numerically shown that, similar to the nuclear case, the transition amplitude exhibits a characteristic angular behavior in the vicinity of 4He dimer pole.

Simulation of Near-Ground Long-Distance Radiowave Propagation Over Terrain Using NystrÖm Method With Phase Extraction Technique and FMM-Acceleration

A 2D surface integral equation-based NystrOumlm solver in which a phase extraction technique is utilized to reduce the number of surface unknowns is described. As forward scattering is the dominant mechanism for the near-ground wave propagation scenario, the associated rapidly-varying phase components of the integral equation kernel and solution unknowns are deduced and isolated in advance and subsequently built into the solver. It is shown that by applying this method, when combining with an adaptive surface segmentation routine, as few as one to two average unknowns per wavelength is adequate in obtaining accurate solutions. This significantly reduces the memory storage and computational expense for the simulation of long-distance propagation effects. The efficiency of this method is further improved by incorporating it into the framework of the fast multipole scheme. The full details of the algorithm are discussed, along with performance comparisons of the new solver to a regular NystrOumlm solver for terrain surfaces in terms of solution convergence.

Closed-Form Green's Function Representations in Cylindrically Stratified Media for Method of Moments Applications

Closed-form Green's function (CFGF) representations for cylindrically stratified media, which can be used as the kernel of an electric field integral equation, are developed. The developed CFGF representations can safely be used in a method of moments solution procedure, as they are valid for almost all possible source and field points that lie on the same radial distance from the axis of the cylinder (such as the air-dielectric and dielectric-dielectric interfaces) including the axial line (rho = rho' and phi = phi' ), which has not been available before. In the course of obtaining these expressions, the conventional spectral domain Green's function representations are rewritten in a different form so that: i) we can attack the axial line problem and ii) the method can handle electrically large cylinders. Available acceleration techniques that exist in the literature are implemented to perform the summation over the cylindrical eigenmodes efficiently. Lastly, the resulting expressions are transformed to the spatial domain using the discrete complex image method with the help of the generalized pencil of function method, where a modified two-level approach is used. Numerical results are presented in the form of mutual coupling between two current modes to assess the accuracy of the final spatial domain CFGF representations.

Contact problems for a two-layer elastic wedge

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex.

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.