Integral method for the calculation of incompressible two-dimensional transitional boundary layers

The method proposed here considers the mean flow in the transition zone as a linear combination of the laminar and turbulent boundary layer in proportions determined by the transitional intermittency, the component flows being calculated by approximate integral methods. The intermittency distribution adopted takes into account the possibility of subtransitions within the zone in the presence of strong pressure gradients. A new nondimensional spot formation rate, whose value depends on the pressure gradient, is utilized to estimate the extent of the transition zone. Onset location is determined by a correlation that takes into account freestream turbulence and facility-specific residual disturbances in test data. Extensive comparisons with available experimental results in strong pressure gradients show that the proposed method performs at least as well as differential models, in many cases better, and is always faster.

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Impedance of foundations using the Boundary Integral Equation Method

Dynamic soil-structure interaction has been for a long time one of the most fascinating areas for the engineering profession. The building of large alternating machines and their effects on surrounding structures as well as on their own functional behavior, provided the initial impetus; a large amount of experimental research was done,and the results of the Russian and German groups were especially worthwhile. Analytical results by Reissner and Sehkter were reexamined by Quinlan, Sung, et. al., and finally Veletsos presented the first set of reliable results. Since then, the modeling of the homogeneous, elastic halfspace as a equivalent set of springs and dashpots has become an everyday tool in soil engineering practice, especially after the appearance of the fast Fourier transportation algorithm, which makes possible the treatment of the frequency-dependent characteristics of the equivalent elements in a unified fashion with the general method of analysis of the structure. Extensions to the viscoelastic case, as well as to embedded foundations and complicated geometries, have been presented by various authors. In general, they used the finite element method with the well known problems of geometric truncations and the subsequent use of absorbing boundaries. The properties of boundary integral equation methods are, in our opinion, specially well suited to this problem, and several of the previous results have confirmed our opinion. In what follows we present the general features related to steady-state elastodynamics and a series of results showing the splendid results that the BIEM provided. Especially interesting are the outputs obtained through the use of the so-called singular elements, whose description is incorporated at the end of the paper. The reduction in time spent by the computer and the small number of elements needed to simulate realistically the global properties of the halfspace make this procedure one of the most interesting applications of the BIEM.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

Recovering boundary data in planar heat conduction using boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in [2], where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical examples are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

Eigenstrain boundary integral equations with local Eshelby matrix for stress analysis of ellipsoidal particles

Aiming at the large scale numerical simulation of particle reinforced materials, the concept of local Eshelby matrix has been introduced into the computational model of the eigenstrain boundary integral equation (BIE) to solve the problem of interactions among particles. The local Eshelby matrix can be considered as an extension of the concepts of Eshelby tensor and the equivalent inclusion in numerical form. Taking the subdomain boundary element method as the control, three-dimensional stress analyses are carried out for some ellipsoidal particles in full space with the proposed computational model. Through the numerical examples, it is verified not only the correctness and feasibility but also the high efficiency of the present model with the corresponding solution procedure, showing the potential of solving the problem of large scale numerical simulation of particle reinforced materials.

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated.

Interaction of linear waves with infinitely long horizontal cylinders studied by boundary element method

The two-dimensional problems concerning the interaction of linear water waves with cylinders of arbitrary shape in two-layer deep water are investigated by use of the Boundary Integral Equation method (BIEM). Simpler new expressions for the Green functions are derived, and verified by comparison of results obtained by BIEM with these by an analytical method. Examined are the radiation and scattering of linear waves by two typical configurations of cylinders in two-layer deep water. Hydrodynamic behaviors including hydrodynamic coefficients, wave forces, reflection and transmission coefficients and energies are analyzed in detail, and some interesting physical phenomena are observed.