An efficient boundary element method for two-dimensional transient wave propagation problems

The boundary element method (BEM) has been recognized by its unique feature of requiring neither internal cells nor their associated domain integrals in the computation. The method preserves its elegance for transient problems by means of a certain time-stepping scheme that initiates the time integration always from the initial time. Unfortunately, this time-marching scheme becomes rather difficult to apply, because the computation time and storage requirement grow dramatically with the increasing number of time steps. This paper shows that a reduction of one half of the computation time as well as the storage requirement can be achieved by an efficient truncation scheme for two-dimensional transient wave propagation problems. In particular, a guiding parameter for the determination of the truncation limit is proposed, and the overall measure of the error with respect to the truncation guide parameter is established.     

The completed double layer boundary integral equation method for two-dimensional Stokes flow

Power and Miranda (1987) explained how integral equations ofthe second kind can be obtained for general exterior three-dimensionalStokes flows. They observed that, although the double layerrepresentation that leads to an integral equation of the secondkind coming from the jump property of its velocity field acrossthe density carrying surface can represent only those flow fieldsthat correspond to a force and torque free surface, the representationmay be completed by adding terms that give arbitrary total forceand torque in suitable linear combinations, precisely a Stokesletand a Rotlet located in the interior of the three-dimensionalparticles. Karrila and Kim (1989) called Power and Miranda'snew method the completed double layer boundary integral equationmethod, since it involves the idea of completing the deficientrange of the double layer operator. The main objective of thispaper is to extend Power and Miranda's completed method to theproblem of multiple cylinders in twodimensional bounded andunbounded domains. This extension is not trivial, owing to theunbounded behaviour at infinity of the fundamental solutionof the Stokes equation in two dimensions and the associatedparadoxes arising from this unbounded behaviour.     

Numerical method for solving a two-dimensional electrical impedance tomography problem in the case of measurements on part of the outer boundary

The two-dimensional electrical impedance tomography problem is considered in the case of a piecewise constant electrical conductivity. The task is to determine the unknown boundary separating the regions with different conductivity values, which are known. Input information is the electric field measured on a portion of the outer boundary of the domain. A numerical method for solving the problem is proposed, and numerical results are presented.     

反演声波阻尼系数的另一种方法

提出了一种方法,利用正则化方法和积分方程,由散射波的近场数据反演时间调和声波阻尼系数.给出了该方法收敛性的证明及数值例子,算法与数值例子表明这种方法不仅简单而且很有效.     

Excitation of a wave field in a triangular domain with impedance boundary conditions

The Helmholtz equation in a closed domain that is an equilateral triangle with inhomogeneous impedance boundary conditions is considered. A functional equation in which the unknown function is the Fourier-image of a wave field on the boundary of the domain is constructed. This functional equation is solved for the case of homogeneous boundary conditions (the problem on eigenvalues), as well as for the case of inhomogeneous boundary conditions in the absence of the resonance. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 300–318. Translated by A. V. Shanin.     

A boundary-only treatment by singular boundary method for two-dimensional inhomogeneous problems

In this study, an effective singular boundary method (SBM) in conjunction with the recursive multiple reciprocity method (MRM) is developed and validated for inhomogeneous problems. It avoids the inner nodes or domain discretizations to evaluate the particular solution, and preserves the boundary-only property of the SBM. Rather than using only polyharmonic operators in the traditional MRM, a recursive MRM is proposed to annihilate source terms with different partial differential operators recursively. Nevertheless, high-order fundamental solutions are involved in the recursive MRM. The absence of the origin intensity factors of higher order fundamental solutions is a major bottleneck in applying the SBM. In order to remedy this difficulty, the origin intensity factors of higher order fundamental solutions are derived with simple formulas. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.     

Viscous flow through corrugated tube by boundary element method

Summary The creeping flow of a Newtonian fluid through a sinusoidally-corrugated tube is solved by the Boundary Element Method. Agreement with another numerical method is noted. In addition, it is shown that previous perturbation theory is valid only when the corrugation amplitude is small (<0.3a) and the wavelength of the corrugation is large (>3a), wherea is the mean radius of the tube.

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Zusammenfassung Das Problem der schleichenden Bewegung eines Newton'schen Fluids durch ein Rohr mit sinusförmig gewellter Wand wird mit Hilfe der Boundary Element-Methode gelöst. Übereinstimmung mit einer anderen numerischen Methode wird festestellt. Zudem wird gezeigt, daß eine früher gefundene Störungstheorie nur gültig ist wenn die Wellenamplitude klein (<0.3a) und die Wellenlänge groß (>3a) ist (a=mittlerer Rohrradius).

     

A boundary element procedure for transient wave propagations in two-dimensional isotropic elastic media

An integral formulation of the elastodynamic equations is presented and discretized to develope a numerical solution procedure. Constant space and linear time dependent interpolation functions are implemented. The expression obtained, the boundary element equations can be solved using a time-stepping scheme. An example is given for comparison with known solutions.     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.     

Analytical integration and exact geometrical representation in the two-dimensional elastostatic boundary element method

Two ways of improving the accuracy of results in the boundary element method are considered. Since the geometries of many problems of practical interest are created from straight lines and circular arcs, errors caused by representing such geometries approximately using quadratic shape functions can be removed using exact geometrical representations for straight and circular boundaries. Besides, exact geometrical representations enable exact analytical integrations for some situations, thereby eliminating the errors caused by approximate numerical integration. The results of some simple test problems show that the use of exact representation of straight and circular geometries, and analytical integration in the situations where this is possible, offers worthwhile benefits in the boundary element analysis of two-dimensional elastostatics problems.     

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.     

Heat transfer analysis of unsteady boundary layer flow by homotopy analysis method

This paper aims to present complete analytic solution to the unsteady heat transfer flow of an incompressible viscous fluid over a permeable plane wall. The flow is started due to an impulsively stretching porous plate. Homotopy analysis method (HAM) has been used to get accurate and complete analytic solution. The solution is uniformly valid for all time τ ∈ [0, ∞) throughout the spatial domain η ∈ [0, ∞). The accuracy of the present results is shown by giving a comparison between the present results and the results already present in the literature. This comparison proves the validity and accuracy of our present results. Finally, the effects of different parameters on temperature distribution are discussed through graphs.     

Boundary element solution of the two-dimensional groundwater flow equation with stochastic free surface boundary condition

An innovative approach to the approximate solution of stochastic partial differential equations in groundwater flow is presented. The method uses a formulation of the Ito's lemma in Hilbert spaces to derive partial differential equations satisfying the moments of the solution process. Since the moments equations are deterministic, they could be solved by any analytical or numerical method existing in the literature. This permits the analysis and solution of stochastic partial differential equations occurring in two-dimensional or three-dimensional domains of any geometrical shape. The method is tested for the first time in the present paper through a practical application in a sandy phreatic aquifer at the Chalk River Nuclear Laboratories, Ontario, Canada. The equation solved is the two-dimensional LaPlace equation with a dynamic, randomly perturbed, free surface boundary condition. The moments equations are derived and solved by using the boundary integral equation method. A comparison is made with a previous analytical solution obtained by applying the randomly forced one-dimensional Boussinesq equation, and some observations on modeling procedures are given.     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.