Application of a boundary integral equation to prediction of freezing fronts in soil

A boundary integral equation formulation based on the complex Cauchy integral theorem is applied to two-dimensional soil-water phase change problems encountered in algid soils. The model assumes that potential theory applies in the estimation of heat flux along a freezing front of differential thickness and that quasi-steady-state temperatures occur along the problem domain boundary. Application of the boundary integral formulation to two-dimensional problems results in predicted locations of the freezing front which are highly accurate. Although the proposed formulation is based on the Cauchy integral theorem, similar models may be developed based on other forms of integration equation methods.     

非经典阻尼分布参数系统复振型叠加方法

附加减震装置的一维杆或剪切梁模型属于非连续的非经典阻尼分布参数系统.对于它的动力分析,通常是建立分段的运动方程,然后利用各段动力反应的实振型叠加形式和连续条件进行动力计算.这是一种实模态综合方法,尽管它可以求得近似的动力反应,但反映不出阻尼对整体系统动力特性的影响.为了考虑附加减震装置引起的阻尼和刚度非连续性,基于广义...     

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.     

COMPLEX VARIABLE BOUNDARY ELEMENT METHOD FOR TORSION OF COMPOSITE SHAFTS

The problem of torsion of composite shafts consisting of a cylindrical matrix surrounding a finite number of inclusions is solved by using the complex variable boundary element method. The method consists in reducing the problem to the solution of a singular integral equation in terms of an analytic function of a complex variable using the Cauchy integral. The resulting integral equation is then solved numerically by discretizing the boundaries into segments called complex boundary elements and replacing the analytic function on the boundaries by interpolating function. Numerical examples are given for a square shaft with a circular inclusion, and for an elliptic shaft with two elliptic inclusions. © 1997 by John Wiley & Sons, Ltd.     

Determination of groundwater flownets, fluxes, velocities, and travel times using the complex variable boundary element method

The complex variable boundary element method, CVBEM, employs the Cauchy integral with any complex variable (e.g., complex potential, complex flux, or complex velocity) to solve boundary value problems. The CVBEM formulation is consistent with the primal and dual solutions of the boundary integral equation, as well as the analytic element method. The resulting problem is overdetermined because two boundary conditions can be specified at each node. Ordinary least squares provides a unique solution that minimizes boundary specification errors. Flownets are obtained by noting that the position of fluid–stream potential intersections can be found by exchanging potential and position in the Cauchy integral, which enhances the determination of travel times along streamlines. Three regional groundwater flow problems are used to illustrate the CVBEM approach, the original problem as defined by Tóth, plus two related problems as described by Domenico and Paliauskas, and by Nawalany.     

Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.     

A boundary element method for plane isotropic elastic media using complex variable technique

A system of singular integral equations is formulated based on the theory of complex variables with Cauchy kernels for the general problem of plane isotropic elastostatics. The integral equations are represented over the image of problems in multiply-connected regions. A numerical scheme is developed by introducing suitable complex polynomial functions for a discretized boundary curve and integrations are performed exactly for any arbitrary curved boundaries using complex contour integration. This reduces to an explicit set of complex linear algebraic equations with no need for numerical integrations. The major advantage of this technique is that numerical formulations is carried out in the complex plane and does not involve real variables which depend on are length. This yields highly accurate results in the presence of strong boundary curvature with steep stress gradients. Further, this formulation does not have boundary layer effects so that accurate stresses are obtained at any interior points in contrast to previous formulations where the accuracy deteriorates near the boundary points.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

Dual boundary integral equation formulation in plane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in plane elasticity using a complex variable. Four kinds of BIE are studied, and they are: (1) the first complex variable BIE for the interior region, (2) the second complex variable BIE for the interior region, (3) the first complex variable BIE for the exterior region, and (4) the second complex variable BIE for the exterior region. Using the Somigliana identity and letting the domain point approach a boundary point, the first complex variable BIE is obtained. Displacement versus traction operator is suggested. Using this operator and letting the domain point approach a boundary point, the second complex variable BIE is obtained. When the domain point approaches a boundary point, all limit processes are performed exactly through the generalized Sokhotski–Plemelj’s formulae. For the exterior problems, two degenerate boundary cases, the curved crack and the deformable curved line, are studied. Particularly, for the degenerate boundary case, or the shrinking curved crack case, four kinds of BIE are obtained.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

A complex variable boundary element method for solving plane and plate problems of elasticity

We consider the complex variable boundary element approximation of biharmonic problem on a smooth domain with various boundary conditions. Based on the Vekua's complex integral representation of the analytic function, a new boundary integral equation is formulated. The density function appearing in the integral equation is determined directly by using the boundary element method. Some plane and plate examples are presented, and the results of the numerical solutions are accurate everywhere in the solid, including the regions near the boundary.

The approach presented is only suitable for bounded simply connected regions.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.     

Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method

This paper presents a new set of boundary integral equations for three dimensional acoustic shape sensitivity analysis based on the direct differentiation method. A linear combination of the derived equations is used to avoid the fictitious eigenfrequency problem associated with the conventional boundary integral equation method when solving exterior acoustic problems. The strongly singular and hypersingular boundary integrals contained in the equations are evaluated as the Cauchy principal values and Hadamard finite parts for constant element discretization without using any regularization technique in this study. The present boundary integral equations are more efficient to use than the usual ones based on any other singularity subtraction technique and can be applied to the fast multipole boundary element method more readily and efficiently. The effectiveness and accuracy of the present equations are demonstrated through some numerical examples.     

Mathematical boundary integral equation analysis of an embedded shell under dynamic excitations

A boundary integral equation method is presented for the analysis of a thin cylindrical shell embedded in an elastic half-space under axisymmetric excitations. By virtue of a set of ring-load Green's functions for the shell and a group of dynamic fundamental solutions for the semi-infinite medium, the structure–medium interaction problem of wave propagation is shown to be reducible to a set of coupled boundary integral equations. Through the analysis of an auxiliary pair of Cauchy integral equations, the singularities of the contact stress distributions arc rendered explicit. With a direct incorporation of such analytical features into the formulation, an effective computational procedure is developed which involves an interpolation of regular functions only. Typical results for the dynamic contact load distributions, displacements, and complex compliance functions are included as illustrations. In addition to furnishing quantities of direct engineering interest, this treatment is apt to be useful as a foundation for further rigorous as well as approximate developments for various related physical problems and boundary integral methods.     

Boundary element solution of 3-D wave scatter problems in a poroelastic medium

This study details the development of boundary integral equations suitable for treating problems involving the scatter of a plane harmonic wave by an inclusion embedded in an infinite poroelastic medium. The pore pressure-solid displacement form of the harmonic equations of motion are developed from the form of the equations originally presented by Biot. Fundamental solutions and a generalized reciprocal work relation are developed, and these are used to formulate a solution representation in terms of an integral over the inclusion surface. The corresponding boundary integral equations are developed in a form that is integrable in the usual sense, eliminating the need to evaluate Cauchy principal value integrals. These boundary integral equations are discretized and implemented into a boundary element computer program. The so-called forbidden frequency problem which causes computational difficulties in boundary integral treatments of wave scatter in elastic and acoustic media is shown to be absent in the poroelastic case. Numerical results obtained from the boundary element program are compared with analytical results for some test problems, and the program appears to produce accurate results at moderate frequencies.     

Initial strain effects in multilayer composite laminates

A boundary integral formulation for the analysis of stress fields induced in composite laminates by initial strains, such as may be due to temperature changes and moisture absorption is presented. The study is formulated on the basis of the theory of generalized orthotropic thermo-elasticity and the governing integral equations are directly deduced through the generalized reciprocity theorem. A suitable expression of the problem fundamental solutions is given for use in computations. The resulting linear system of algebraic equations is obtained by the boundary element method and stress interlaminar distributions in the boundary-layer are calculated by using a boundary only discretization. The approach is general and it does not require a priori assumptions. Numerical results are presented to show the potential of the proposed approach.     

Analysis of the fiber-matrix cylindrical model with a circumferential crack

An axisymmetrical fiber-matrix cylindrical model with a circumferential crack in the matrix of finite diameter is formulated within elastostatic scope. The problem is considered by means of integral transforms and a singular integral equation with a dominant generalized Cauchy kernel is obtained. Following the numerical solution technique developed by Erdogan, Gupta and Cook, the singular integral equation is reduced to a system of linear equations. By solving the linear equations, stress intensity factors associated with the crack length and the material properties are calculated and discussed. The solutions presented in this study are found to be general, including the solutions as special cases of the present formulation for a homogeneous solid cylindrical bar and a thick-walled shell with an outer circumferential crack. This revised version was published online in August 2006 with corrections to the Cover Date.     

A successive boundary element model for investigation of sloshing frequencies in axisymmetric multi baffled containers

This study presents a developed successive Boundary Element Method to determine the symmetric and antisymmetric sloshing natural frequencies and mode shapes for multi baffled axisymmetric containers with arbitrary geometries. The developed fluid model is based on the Laplace equation and Green's theorem. The governing equations of fluid dynamic and free surface boundary condition are also applied to proposed model. A zoning method is presented to model arbitrary arrangement of baffles in multi baffled axisymmetric tanks. The influence of each zone on neighboring zones is applied by introducing interface influence matrix which correlates the velocity potential of interfaces to their flux. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. The proposed method has a considerable effect on decreasing computational cost and a good accuracy in determining the sloshing natural frequencies. The obtained results for different types of container based on the application of the presented study are validated in comparison with the literature and very good agreement is achieved. Finally, the effect of baffle parameters on the sloshing natural frequencies was investigated and some conclusions are outlined.