无域积分的弹塑性边界元法的非线性互补方法

该文引入非线性互补方法来求解边界元法的弹塑性问题,其中方程组由内部点应力方程和反映塑性本构定律的互补函数形成。涉及的域积分采用径向积分法转化为边界积分。通过受内压的厚壁圆筒的应力、位移和荷载-位移情况表明了该算法的精度。     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Elastodynamic problems by meshless local integral method: Analytical formulation

In this paper, analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Transient meshless boundary element method for prediction of chloride diffusion in concrete with time dependent nonlinear coefficients

Chloride-induced corrosion of steel reinforcements has been identified as one of the main causes of deterioration of concrete structures. A feasible numerical method is required to predict chloride penetration in concrete structures. A transient meshless boundary element method is proposed to predict chloride diffusion in concrete with time dependent nonlinear coefficient. Taking Green's function as the weighted function, the weighted residue method is adopted to transform the diffusion equation into equivalent integral equations. By the coupling of radial integral method and radial basis function approximation, the domain integrals in equivalent control equations are transformed into boundary integrals. Following the general procedure of boundary element meshing and traditional finite difference method, a set of nonlinear algebraic equations are constructed and are eventually solved with the modified Newtonian iterative method. Several numerical examples are provided to demonstrate the effectiveness and efficiency of the developed model. A comparison of the simulated chloride concentration with the corresponding reported experimental data in a real marine structure indicates the high accuracy and advantage of the time dependent coefficient and nonlinear model over the conventional constant coefficient model.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999     

The radial integration method for evaluation of domain integrals with boundary-only discretization

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

Numerical evaluation of arbitrary singular domain integrals based on radial integration method

In this paper, a new approach is presented for the numerical evaluation of arbitrary singular domain integrals based on the radial integration method. The transformation from domain integrals to boundary integrals and the analytical elimination of singularities can be accomplished by expressing the non-singular part of the integration kernels as polynomials of the distance r and using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some numerical examples are provided to verify the correctness and robustness of the presented method.     

Boundary element analysis in thermoelasticity with and without internal cells

In this paper, a set of internal stress integral equations is derived for solving thermoelastic problems. A jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations. The strongly singular domain integral is then regularized using a semi‐analytical technique. To avoid the requirement of discretizing the domain into internal cells, domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method (RIM). Two numerical examples for 2D and 3D, respectively, are presented to verify the derived formulations. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Using analytical expressions in radial integration BEM for variable coefficient heat conduction problems

In this paper, a new approach using analytical expressions in the radial integration boundary element method (RIBEM) is presented for solving variable coefficient heat conduction problems. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. The fourth-order spline RBF is employed to approximate unknowns appearing in domain integrals arising from the varying heat conductivity. The radial integration method is utilized to convert domain integrals to the boundary resulting in a pure boundary discretization algorithm. Numerical examples are given to demonstrate the efficiency of the presented approach.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

The use of Stokes' fundamental solution for the boundary only element formulation of the three-dimensional Navier–Stokes equations for moderate Reynolds numbers

This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.     

Complex variable boundary integral method for linear viscoelasticity: Part I—basic formulations

The basic formulations (direct and indirect) of the complex variable boundary integral method for linear viscoelasticity are presented. Complex variable temporal integral equations for the formulations are obtained for viscoelastic solids whose behavior in shear is governed by a Boltzmann model while the bulk behavior is purely elastic. The functions involved in the integral equations are the time-dependent complex boundary tractions and displacements for the direct approach and the unknown time-dependent complex density functions for the indirect approaches. The temporal integral equations give the displacements and stresses at a point inside a viscoelastic region in terms of time convolution and space integrals over the boundary of this region. The equations are valid for the boundaries of arbitrary shapes provided that these boundaries are sufficiently smooth. Complex variable temporal boundary equations are obtained by taking the inner point to the boundary. Numerical treatment of spatial and time convolution integrals involved in the boundary equations is discussed.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.     

A two-dimensional time-domain boundary element method for dynamic crack problems in anisotropic solids

A time-domain boundary element method (BEM) for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented in this paper. Strongly singular displacement boundary integral equations (DBIEs) are applied on the external boundary of the cracked body while hypersingular traction boundary integral equations (TBIEs) are used on the crack-faces. The present time-domain method uses the quadrature formula of Lubich for approximating the convolution integrals and a collocation method for the spatial discretization of the time-domain boundary integral equations. Strongly singular and hypersingular integrals are dealt with by a regularization technique based on a suitable variable change. Discontinuous quadratic quarter-point elements are implemented at the crack-tips to capture the local square-root-behavior of the crack-opening-displacements properly. Numerical examples for computing the dynamic stress intensity factors are presented and discussed to demonstrate the accuracy and the efficiency of the present method.