Indirect Boundary Integral Method Research Articles

Effect of an Inflating/Deflating Magma Chamber on a Nearby Earthquake Fault Using the Indirect Boundary Integral Method

Effect of an Inflating/Deflating Magma Chamber on a Nearby Earthquake Fault Using the Indirect Boundary Integral Method

The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

New integral representations in the linear theory of viscoelastic materials with voids

We investigate the two basic internal BVPs related to the linear theory of viscoelasticity for Kelvin-Voigt materials with voids by means of the potential theory. By using an indirect boundary integral method, we represent the solution of the first (second) BVP of steady vibrations in terms of a simple (double) layer elastopotential. The representations achieved are different from the previously known ones. Our approach hinges on the theory of reducible operators and on the theory of differential forms.

113 間接仮想境界積分法による円柱の曲げ解析(材料力学I)

113 間接仮想境界積分法による円柱の曲げ解析(材料力学I)

ON AN INTEGRAL EQUATION OF THE FIRST KIND ARISING IN THE THEORY OF COSSERAT

In this paper we study an integral equation of the first kind concerning an indirect boundary integral method for the Dirichlet problem in the theory of Cosserat continuum. Our method hinges on the theory of reducible operators and on the theory of differential forms.

Motion of a solid particle in a shear flow along a porous slab

AbstractThe flow field around a solid particle moving in a shear flow along a porous slab is obtained by solving the coupled Stokes–Darcy problem with the Beavers and Joseph slip boundary condition on the slab interfaces. The solution involves the Green’s function of this coupled problem, which is given here. It is shown that the classical boundary integral method using this Green’s function is inappropriate because of supplementary contributions due to the slip on the slab interfaces. An ‘indirect boundary integral method’ is therefore proposed, in which the unknown density on the particle surface is not the actual stress, but yet allows calculation of the force and torque on the particle. Various results are provided for the normalized force and torque, namely friction factors, on the particle. The cases of a sphere and an ellipsoid are considered. It is shown that the relationships between friction coefficients (torque due to rotation and force due to translation) that are classical for a no-slip plane do not apply here. This difference is exhibited. Finally, results for the velocity of a freely moving particle in a linear and a quadratic shear flow are presented, for both a sphere and an ellipsoid.

Boundary integral method for Stokes flow past a porous body

AbstractIn this paper we obtain an indirect boundary integral method in order to prove existence and uniqueness of the classical solution to a boundary value problem for the Stokes–Brinkman‐coupled system, which describes an unbounded Stokes flow past a porous body in terms of Brinkman's model. Therefore, one assumes that the flow inside the body is governed by the continuity and Brinkman equations. Some asymptotic results in both cases of large and, respectively, of low permeability are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.

Mixed boundary integral methods for Helmholtz transmission problems

In this paper we propose a hybrid between direct and indirect boundary integral methods to solve a transmission problem for the Helmholtz equation in Lipschitz and smooth domains. We present an exhaustive abstract study of the numerical approximation of the resulting system of boundary integral equations by means of Galerkin methods. Some particular examples of convergent schemes in the smooth case in two dimensions are given. Finally, we extend the results to a thermal scattering problem in a half plane with several obstacles and provide numerical results that illustrate the accuracy of our methods depending on the regularity of the interface.

Time domain three-dimensional fully nonlinear computations of steady body–wave interaction problem

Three-dimensional fully nonlinear waves generated by moving disturbances with steady forward speed without motions are solved using a mixed Eulerian–Lagrangian method in terms of an indirect boundary integral method and a Runge–Kutta time marching approach which integrates the fully nonlinear free surface boundary conditions with respect to time. A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The computational window maintains the computational domain and tracks the free surface profile by a node-shifting scheme applied within it. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window. Numerical computations are applied to free surface waves generated by Wigley and Series 60 hulls for the steady problem. The present numerical results are presented and compared with existing linear theory, experimental measurements, and other numerical nonlinear computations. The comparisons show satisfactory agreements for these hydrodynamic problems.

An indirect boundary integral method for an oscillatory Stokesflow problem

The purpose of this paper is to present an indirect boundary integral method for the oscillatory Stokes flow provided by the translational oscillations of two rigid spheres in an incompressible Newtonian fluid of infinite expanse.

THE INDIRECT BOUNDARY INTEGRAL METHOD FOR CURVED CRACKS IN PLANE ELASTICITY

For curved crack problems in plane elasticity, subjected to the traction conditions on the crack faces, we present a system of boundary integral equations. The procedure is based on the indirect boundary integral method in terms of real variables. For efficient mathematical analysis, we decompose the singular kernel into the Cauchy singular part and the regular one. As a result, solvability of the presented system is proved and availability of the present approach is shown by the numerical example of a circular arc crack.

An indirect boundary integral method for a Stokes flow problem

In this paper we obtain an indirect boundary integral method for the problem of the Stokes flow generated by the slow motion of a cylindrical solid particle in a viscous incompressible fluid bounded by two plane walls. The problem is formulated exactly as a Fredholm integral equation of the second kind. We also prove the existence and uniqueness result of solution of this equation that is used as the basis of a numerical model that uses a standard boundary element method. The numerical results are presented in the case of a circular obstacle when it translates parallel or normal to the walls.

COMPUTATION OF TRANSIENT NONLINEAR SHIP WAVES USING AN ADAPTIVE ALGORITHM

An indirect boundary integral method is used to solve transient nonlinear ship wave problems. A resulting mixed boundary value problem is solved at each time-step using a mixed Eulerian– Lagrangian time integration technique. Two dynamic node allocation techniques, which basically distribute nodes on an ever changing body surface, are presented. Both two-sided hyperbolic tangent and variational grid generation algorithms are developed and compared on station curves. A ship hull form is generated in parametric space using a B-spline surface representation. Two-sided hyperbolic tangent and variational adaptive curve grid-generation methods are then applied on the hull station curves to generate effective node placement. The numerical algorithm, in the first method, used two stretching parameters. In the second method, a conservative form of the parametric variational Euler–Lagrange equations is used the perform an adaptive gridding on each station. The resulting unsymmetrical influence coefficient matrix is solved using both a restarted version of GMRES based on the modified Gram–Schmidt procedure and a line Jacobi method based on LU decomposition. The convergence rates of both matrix iteration techniques are improved with specially devised preconditioners. Numerical examples of node placements on typical hull cross-sections using both techniques are discussed and fully nonlinear ship wave patterns and wave resistance computations are presented.

A new velocity-vorticity boundary integral formulation for Navier-Stokes equations

In the present work, an indirect boundary integral method for the numerical solution of Navier–Stokes equations formulated in velocity–vorticity dependent variables is proposed. This wholly integral approach, based on Helmholtz's decomposition, deals directly with the vorticity field and gives emphasis to the establishment of appropriate boundary conditions for the vorticity transport equation. The coupling between the vorticity and the vortical velocity fields is expressed by an iterative procedure. The present analysis shows the usefulness of an integral formulation not only in providing a potentially more efficient computational tool, but also in giving a better understanding to the physics of the phenomenon. Copyright © 2000 John Wiley & Sons, Ltd.

Spatial variation of coda wave amplitude in media with a localized heterogeneous region

Using an indirect boundary integral method, seismograms are computed for elastic media with localized heterogeneities. Models are two-dimensional homogeneous full spaces (P and SV waves) with many circular cavities as heterogeneities or scatterers. Heterogeneities are localized within a depth range, forming a relatively thin random layer. Seismograms are obtained for receivers at depth 0 km (surface) and for several focal depths, with sources radiating S waves isotropically. Seismograms are composed of the direct S wave and all the possible scattered waves by the heterogeneities, exhibiting late arrivals or coda waves. The coda wave amplitude, or coda energy level, and its duration vary for events with different focal depths. As the focal depth increases and the source gets closer to the layer of localized heterogeneities, the coda level becomes small. When the source is within the heterogeneous layer, however, the coda level becomes larger than for a case of the source either above or below the heterogeneous layer. This local enhancement of coda takes place clearly only in the frequency range for which the scattering is the most effective, that is, when the non-dimensional frequency kd takes values from 2 to 3, where k is the wavenumber and d is the size of each heterogeneity. Such enhancement of coda is not observed when the density of cavities, or strength of the heterogeneities, is reduced. Coda level becomes locally large for a source within the heterogeneous layer only in the case that the heterogeneities are strong enough to excite multiply scattered waves, as compared with the singly scattered ones. Robinson (1987) studied the temporal variation of coda-duration magnitude relative to event magnitude based on P and S wave amplitudes using the seismic network of the Wellington, New Zealand, region. He found that coda-duration magnitude relative to amplitude magnitude decreases with focal depth and becomes large locally for events in the depth range from 65 and 75 km. Our synthetic seismograms explain his results well, implying that there must be a region of localized, strong heterogeneity at depths around 70 km. The effective size of the heterogeneous region may be of about several kilometers because the observation was performed with 1 Hz seismometers. This localized heterogeneous layer is probably associated with the subducting Pacific plate underneath the Wellington region.

A modified boundary integral method on open arcs in the plane

For the Dirichlet problem on open acrs in an infinite plane, a modified numerical scheme using the indirect boundary integral method is presented. We suggest a single layer potential, added by a constant, as a solution of the problem. Then the solution for the density function of the induced boundary integral equation gives the potential which satisfies all the conditions of the original problem. Approximation to the unknown density function is fulfilled by using the collocation method with the Chebyshev polynomials of the first kind. The procedure given in this work reduces the number of steps for computation of the method introduced by Atkinson and Sloan [1]. Experimental results of some examples are given to show the convergence of our method, which is the same as that of the literature [1].

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIFs (stress intensity factors) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral term in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIFs using only one SIF solution, without any modification for the crack geometries symmetric in the two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential cracks or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straightforward with only one SIF solution. The necessary information in the analysis is two reference SIFs. The analysis results using several examples verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.

Three‐dimensional response of a layered cylindrical valley embedded in a layered half‐space

AbstractAn indirect boundary integral method to obtain the three‐dimensional response of an infinitely long, layered, viscoelastic valley of arbitrary cross‐section embedded in a layered viscoelastic half‐space is presented. The valley is excited by homogeneous plane waves impinging at an oblique angle with respect to the axis of the valley. The method and associated computer programs are tested by comparison with available results in the limiting two‐dimensional case of incidence normal to the axis of the valley. Additional comparisons with previous three‐dimensional results obtained by a hybrid finite element‐boundary integral method for cylindrical valleys subjected to obliquely incident waves show large differences. However, the results obtained here for an infinitely long valley appear to be in some agreement with earlier results for an elongated prolate semi‐ellipsoidal valley and with results obtained by a discrete wavenumber boundary element approach. An extensive bibliography on the dynamic response of valleys is also presented.

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

AbstractThis paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.