Equivalent Boundary Integral Equation Research Articles

The ∂̄-Neumann problem and boundary integral equations

In this paper, we find an equivalent boundary integral equation to the classical [Formula: see text]-Neumann problem. The new equation contains an equivalent regularity to the global regularity of the [Formula: see text]-Neumann problem. We also use the integral equations to observe a new-found rigidity property of the [Formula: see text]-Neumann problem on the unit ball in [Formula: see text].

On a mathematical model of dynamics of the elastic wedge-shaped medium with radiating defect

In the paper the mixed boundary value problem of antiplane vibrations is considered in the elastic wedge-shaped medium containing the radiating defect J2. Radiating generators are assumed to be located on defect boundaries and on the interval J1 of the wedge free boundary as well. The problem of reconstructing the wave field in the whole wedgeshaped region with its boundary is stated. A number of problems of analyzing acoustic emission signals by radiating defect are reduced to the problem considered in connections with using non-destructive testing elements of the technological equipment under exploitation. The problem in question is reduced to studying the solvability problems of the equivalent boundary integral equation system both for stress saltus on the defect J2and contact stresses on the interval J1 of the upper plane of the wedge.

Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

Abstract The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces ℍ p s ⁢ ( Ω α ) {\mathbb{H}^{s}_{p}(\Omega_{\alpha})} , s > 1 p {s>\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis 𝕎 p s - 1 / p ⁢ ( ℝ + ) {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})} , which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii 𝕎 p r ⁢ ( ℝ + ) {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential ℍ p r ⁢ ( ℝ + ) {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

Boundary elements with mesh refinements for the wave equation

The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as equivalent boundary integral equations in time domain, study the regularity properties of their solutions and the numerical approximation. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates known for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet and Neumann problems, as well as the Dirichlet-to-Neumann operator and applications to the sound emission of tires.

On the existence of mosaic-skeleton approximations for discrete analogues of integral operators

Exterior three-dimensional Dirichlet problems for the Laplace and Helmholtz equations are considered. By applying methods of potential theory, they are reduced to equivalent Fredholm boundary integral equations of the first kind, for which discrete analogues, i.e., systems of linear algebraic equations (SLAEs) are constructed. The existence of mosaic-skeleton approximations for the matrices of the indicated systems is proved. These approximations make it possible to reduce the computational complexity of an iterative solution of the SLAEs. Numerical experiments estimating the capabilities of the proposed approach are described.

APPLICATION OF THE METHOD OF BOUNDARY INTEGRAL EQUATIONS FOR NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY IN AXIALLY SYMMETRIC DOMAIN

The article considers the non-stationary initial-boundary problem of thermal conductivity in axially symmetric domain in Minkowski space, formulated as equivalent boundary integral equation. Using the representation of the solution in the form of a Fourier series expansion, the problem is reformulated as an infinite system of two-dimensional singular integral equations regarding expansion coefficients. The paper presents and investigates the explicit form for fundamental solutions used in the integral representation of the solution in the domain and on the border. The obtained results can be used in the construction of efficient numerical boundary element method for estimation of structures behavior under the influence of intense thermal loads in real-time.

Some research of mapped radiation modes and its application in analyzing the radiation surface of vibrating structures

Acoustic radiation modes divide the velocity patterns into groups with effective and ineffective radiation efficiencies. It suggests a promising way in the computation of sound power and near field acoustic holography. However, obtaining the radiation modes based on eigenvalue analysis suffers from issues of CPU time and memory, especially for large-scale problems. Based on the idea of equivalent source method and boundary integral equation, it is mathematically and numerically proved that the spherical basis functions is a set of linearly independent patterns on an arbitrary surface. They are termed as mapped radiation modes. The analytical sound power of a structure vibrating in its mapped radiation modes is obtained. An efficient and accurate method to compute the sound power based on the mapped radiation modes is proposed. Based on the relationship between the mapped radiation modes of a vibrating structure and its field sound pressure distribution patterns on a sphere, an improved near field acoustic holography is also developed. It does not need the inverse process but requires microphones to locate at the integral points on a sphere. Numerical examples are presented to validate advantages of applying mapped radiation modes in the sound power evaluation and near field acoustic holography.

A wavelets method for plane elasticity problem

A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.

A symmetric boundary element model for the analysis of Kirchhoff plates

The paper deals with the formulation and implementation of a new symmetric boundary element model for the analysis of Kirchhoff plates. The transversal displacement and normal slope boundary integral equations, usually adopted in the standard boundary element analysis, are considered together with bending moment, twisting moment and equivalent shear boundary integral equations. These equations are weighted by considering distributed sources related to the kinematic and static variables in the virtual-work sense. Moreover, particular attention is paid to the discretization of the boundary variables by shape functions selected in order to ensure continuity over the boundary and symmetry for the matrix system. The evaluation of the highly singular boundary integrals for overlapped integration domains is performed in closed form using a limit approach which provides self-contributions as limit values of non-singular terms. The corner effects and their treatment in the numerical procedure are also discussed. Various numerical examples for plates having different boundary conditions illustrate the performance of the model.

Boundary element modelling to solve the Grad–Shafranov equation as an axisymmetric problem

The Grad–Shafranov equation describes the magnetic flux distribution of plasma in an axisymmetric system like a tokamak-type nuclear fusion device. The equation is transformed into an equivalent boundary integral equation by expanding the inhomogeneous term related to the plasma current into a polynomial. In the present research, the singularity of the fundamental solution, which consists of two elliptic integrals, and the properties of singular integrals have been minutely investigated. The discontinuous quadratic boundary elements have been introduced to give an accurate solution with a small number of boundary elements. Ampere's circuital law has been applied to estimate the total plasma current from the boundary integral of the poloidal field, and based on this idea, a new scheme to calculate the eigenvalue has also been proposed.

Equivalent boundary integral equations with indirect variables for plane elasticity problems

The exact form of the exterior problem for plane elasticity problems was produced and fully proved by the variational principle. Based on this, the equivalent boundary integral equations (EBIE) with direct variables, which are equivalent to the original boundary value problem, were deduced rigorously. The conventionally prevailing boundary integral equation with direct variables was discussed thoroughly by some examples and it is shown that the previous results are not EBIE.

Galerkin trigonometric wavelet methods for the natural boundary integral equations

A simply supported boundary value problem of biharmonic equations in the unit disk is reduced into an equivalent second kind natural boundary integral equation (NBIE) with hypersingular kernel (in the sense of Hadamard finite part). Trigonometric Hermite interpolatory wavelets introduced by Quak [Math. Comput. 65 (1996) 683–722] as trial functions are used to its Galerkin discretization with 2 J+1 degrees of freedom on the boundary. It is proved that the stiffness matrix is a block diagonal matrix and its diagonal elements are some symmetric and block circulant submatrices. The simple computational formulae of the entries in stiffness matrix are obtained. These show that we only need to compute 2(2 J + J+1) elements of a 2 J+2 ×2 J+2 stiffness matrix. The error estimates for the approximation solutions are established. Finally, numerical examples are given.

Equivalent Boundary Integral Equations with indirect unknowns for thin elastic plate bending theory

Equivalent Boundary Integral Equations (EBIE) with indirect unknowns for thin elastic plate bending theory, which is equivalent to the original boundary value problem, is established rigorously by mathematical technique of non-analytic continuation and is fully proved by means of the variational principle. The previous three kinds of boundary integral equations with indirect unknowns are discussed thoroughly and it is shown that all previous results are not EBIE.

An equivalent boundary integral formulation for bending problems of thin plates

For the bending problems of simply supported thin plates, the governing bi-harmonic equation can be decomposed into two independent Poisson equations based upon Kirchhoff theory. The conventional boundary integral formulations based on this decomposition technique have been proved to yield non-equivalent solutions compared with the boundary value problem. In this paper, a boundary integral formulation called an equivalent boundary integral equation has been deduced from Poisson equations, and it is also numerically shown that the problem of solution non-equivalence of the conventional boundary integral equation does not exist in the equivalent boundary integral equation.

The boundary contour method based on the equivalent boundary integral equation for 2-D linear elasticity

The conventional boundary integral equation in two dimensions is non-equivalent to its corresponding boundary value problem when the scale in the fundamental solution reaches its degenerate scale values. An equivalent boundary integral equation was recently derived. This equation has the same solution as the boundary value problem of differential equations. This paper presents the boundary contour method based on the equivalent boundary integral equation for two-dimensional linear elasticity. The method requires only numerical evaluation of potential functions and gives correct equivalent results to the boundary value problem of differential equations in two dimensions. Numerical results are presented for some examples. The present approach is shown to give excellent results in illustrative examples. Meanwhile, the traction results from the BCM based on the conventional displacement boundary integral equation are incorrect.

Numerical conformal mapping based on the generalised conjugation operator

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators.

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld-Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations.