Application of the boundary integral equation method to Reissner's plate model

Numerical solution of Reissner's model for the bending of elastic plates by an advanced boundary integral equation method is outlined. Construction of fundamental solutions and computational strategy is fully described. Implementation of the resulting one-dimensional model is illustrated by two test problems, including comparison to analytical solutions.     

A boundary integral equation method for the two-dimensional diffusion equation subject to a non-local condition

A boundary integral equation method is proposed for the numerical solution of the two-dimensional diffusion equation subject to a non-local condition. The non-local condition is in the form of a double integral giving the specification of mass in a region which is a subset of the solution domain. A specific test problem is solved using the method.     

The singular function boundary integral method for a two-dimensional fracture problem

The singular function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.     

Application of boundary integral method to elastoplastic analysis of V-notched beams

The boundary integral equation method was applied in the solution of the plane elastoplastic problems. The use of this method was illustrated by obtaining stress and strain distributions for a number of specimens with a single edge notch and subjected to pure bending. The boundary integral equation method reduced the non-homogeneous biharmonic equation to two coupled Fredholm-type integral equations. These integral equations were replaced by a system of simultaneous algebraic equations and solved numerically in conjunction with the method of successive elastic solutions.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

The boundary element method applied to orthotropic shear deformable plates

This work presents a formulation for thick plates following Mindlin theory. The fundamental solution takes into account an assumed displacement distribution on the thickness, and was derived by means of Hormander operator and the Radon transform. To compute the inverse Radon transform of the fundamental solution, some numerical integrals need to be computed. How these integrations are carried out is a key point in the performance of the boundary element code. Two approaches to integrate fundamental solutions are discussed. Integral equations are obtained using Betti's reciprocal theorem. Domain integrals are exactly transformed into boundary integrals by the radial integration technique.     

Application of local boundary integral equation method into micropolar elasticity

A new meshless method for solving boundary value problems in micropolar elasticity is presented. The method is based on the local boundary integral equation (LBIE) method with the moving least squares approximation of physical quantities. Randomly scattered nodes are utilized for interpolation of field data. Every node is surrounded by a simple surface centered at the collocation point in the LBIE method. On the surface of subdomains the LBIEs are written. Fundamental solutions corresponding to uncoupled governing equations are derived. To eliminate the traction vector in the LBIE, the modified fundamental solution is introduced.     

Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.     

Hydrodynamics of rising bubbles and drops

Conclusion An analysis of the problem concerning the interaction of moving single formations (bubbles and drops) with the surrounding carrying medium shows the possibility of describing real physical processes within the frameworks of the models of ideal and viscous liquids. Generalization of experimental and theoretical data is made by constructing charts of flow modes. A successful selection of the dimensionless numbersR andR as coordinates made it possible to arrange in an ordered fashion in the diagram all of the media by means of the parameter M, with the data for each specific medium being located on a certain straight line whose slope is determined by M. Vast computational data both of the present author and of other investigators for the region of small and intermediate values of dimensionless parameters, converted into theR ,R functions, made it possible to fully display the region inaccessible for investigation when the effect of all the parameters is significant and their number cannot be reduced. The use of these very parameters for correlating the data of the problem of steady rise of a drop made it possible to simply demonstrate the influence both of the presence of the medium itself and its motion on the drop rise. Comparison of the data with the problem of bubble rise can be made by simple superposition of diagrams. The logical clarity of the parametersR , which is equal to the ratio of the equivalent radiusa of a bubble (drop) to the capillary constant of the surrounding medium , allowed the determination of a certain critical sizea after which waves appear on the surface at the back side of the bubble (drop) when the value ofa is higher than 2-3. Another form of instability of steady rise is associated with the formation of a stagnant zone and with the separation of flow from the surface of a bubble or drop characteristic for surrounding media with small values of M. The use of the parametersR andR will definitely significantly simplify the analysis of more complex problems, for example, the problems of the motion of bubbles and drops in tubes with liquid and also the change in the external conditions (reduced or increased gravitation).Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 1, pp. 93–123, January, 1994.     

Time marching analysis of boundary integral equations in two-dimensional elastodynamics

The present paper is an improvement of the existing time-domain formulation for solution of the boudary-initial problems in two-dimensional elastodynamics by the boundary element method. Two different time-marching schemes are applied to these problems and the boundary integral equations are made free of any singularities.     

Application of new fast multipole boundary integral equation method to crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large scale problems. This paper discusses an application of the new version of FMM to three-dimensional boundary integral equation method (BIEM) for crack problems for the Laplace equation. The boundary integral equation is discretised with collocation method. The resulting algebraic equation is solved with generalised minimum residual method (GMRES). The numerical results show that the new version of FMM is more efficient than the original FMM.     

Added mass computation by the boundary integral method

Computational techniques for the treatment of fluid-structure interaction effects by discrete boundary integral methods are examinde. Attention is focused on the computation of the added mass matrix by finite element methods for a structure submerged in an infinite, inviscid, incompressible fluid. A general computational procedure is presented that is based upon a variational approach involving the assumption of constant source strength over each surface element. This is followed by an analysis of the discretization error for a spherical body that is then used to develop a hierarchy of computational schemes. These schemes are than evaluated numerically in terms of ‘fluid boundary modes’ for a submerged spherical surface. One scheme has been found to be surprisingly accurate in relation to its computational demands.     

An application of the boundary integral equation method to dynamic fracture mechanics

In this paper the direct boundary integral equation method is applied to dynamic fracture mechanics, and the computational results are compared with experimental values. The comparison shows that the authors' computation is successful.     

Laser interference method of determining the parameters of gas bubbles

The fundamentals of the laser interference method of determining the parameters of gas bubbles in liquids and solids are considered. The role of polarization in the development of interference bands is analyzed. Features in the realization of an experimental plant and a technique for performing experiments are presented.     

Buckling analysis of shear deformable plates by boundary element method

In this paper, the derivation and numerical implementation of boundary integral equations for the buckling analysis of shear deformable plates are presented. Plate buckling equations are derived as a standard eigenvalue problem. The formulation is formed by coupling boundary element formulations of shear deformable plate and two dimensional plane stress elasticity. The eigenvalue problem of plate buckling yields the critical load factor and buckling modes. The domain integrals which appear in this formulation are treated in two different ways: initially the integrals are evaluated using constant cells, and next, they are transformed into equivalent boundary integrals using the dual reciprocity method (DRM). Several examples with different geometry, loading and boundary conditions are presented to demonstrate the accuracy of the formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient method for the two-dimensional elastostatic boundary element method

In order to obtain accurate results and to reduce computation time, we have proposed in this paper a new strategic method, where quadratic elements are used at the corner points and linear elements at the points off the corner points. A computer program using this method has been developed and applied to several problems of various shapes. The usefulness of this method was illustrated by the application results.     

Buckling analysis of shear deformable shallow shells by the boundary element method

In this work a boundary element (BE) formulation for buckling problem of shear deformable shallow shells is presented. A set of five boundary integral equations are obtained by coupling two-dimensional plane stress elasticity with shear deformable plate bending (Reissner). The domain integrals appearing in the formulation (due to the curvature and due to the domain load) are transferred into equivalent boundary integrals. The BE formulation is presented as an eigenvalue problem, to provide direct evaluation of critical load factors and buckling modes. Several examples are presented. The BE results for a cylindrical shallow shell with different curvatures are compared with other numerical solutions and good agreements are obtained.     

Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large‐scale problems. This paper discusses an application of FMM to three‐dimensional boundary integral equation method for elastostatic crack problems. The boundary integral equation for many crack problems is discretized with FMM and Galerkin's method. The resulting algebraic equation is solved with generalized minimum residual method (GMRES). The numerical results show that FMM is more efficient than conventional methods when the number of unknowns is more than about 1200 and, therefore, can be useful in large‐scale analyses of fracture mechanics. Copyright © 2001 John Wiley & Sons, Ltd.