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.     

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.     

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.     

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.     

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.     

A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity

In this paper, a new and simple boundary‐domain integral equation is presented for heat conduction problems with heat generation and non‐homogeneous thermal conductivity. Since a normalized temperature is introduced to formulate the integral equation, temperature gradients are not involved in the domain integrals. The Green's function for the Laplace equation is used and, therefore, the derived integral equation has a unified form for different heat generations and thermal conductivities. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the normalized temperature using a series of basis functions and polynomials in global co‐ordinates. Numerical examples are given to demonstrate the robustness of the presented method. Copyright © 2005 John Wiley & Sons, Ltd.     

A nonlinear complementarity approach for elastoplastic problems by BEM without internal cells

A nonlinear complementarity approach is presented to solve elastoplastic problems by the boundary element method, in which the equations are formulated by stress equations and complementarity function obtained from the plasticity constitutive law. The domain integrals involved are transformed into boundary integrals by radial integration method, using compactly supported radial basis functions. Two numerical examples demonstrate the algorithm’s applicability and effectiveness.     

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.     

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。     

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     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

Dual reciprocity BEM without matrix inversion for transient heat conduction

Presence of domain integrals in the formulation of the boundary element method dramatically decreases the efficiency of this technique. Dual reciprocity boundary element method (DRBEM) is one of the most popular methods to convert domain integrals into a series of boundary integrals. This is done at the expense of generating some additional matrices and inverting one of them. The latter feature makes the DRBEM inefficient for large-scale problems. This paper describes simple means of avoiding matrix inversion for transient heat transfer problems with arbitrary set of boundary conditions. The technique is also directly applicable to other phenomena (acoustic wave propagation, elastodynamics). For the boundary conditions of Neumann and Robin type, the proposed technique produces exactly the same results as the standard approach. In the presence of Dirichlet conditions, a lower bound on the time step has been detected in the backward difference time stepping procedure. The approach has been tested on some transient heat conduction benchmark problems and accurate results have been obtained.     

Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions

A new method for the boundary element analysis of unsteady heat conduction problems involving non-homogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented. Nonlinear terms are converted to a fictitious heat source and implemented in the present formulation. The domain integrals are efficiently treated by the recently introduced Cartesian transformation method. Similar to the dual reciprocity method, some internal grid points are considered for the treatment of the domain integrals. In the present method, unlike the dual reciprocity method, there is no need to find particular solution for the shape functions in the interpolation computations and the form of the shape functions can be arbitrary and sufficiently complicated. In the present method, at each time step the temperature at boundary nodes and some internal grid points is computed and used as pseudo-initial values for the next time step. Most of the generated matrices are constant at all time steps and computations can be carried out fast. An example with different forms of heat sources is presented to show the efficiency and accuracy of the proposed method.     

The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary

This paper presents a new boundary element approach to transform domain integrals into equivalent boundary integrals. The technique, called the Multiple Reciprocity Method, is applied to 2-D elasticity problems and operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions. Numerical examples are presented to demonstrate the accuracy and efficiency of the method.     

Dual reciprocity boundary element method using compactly supported radial basis functions for 3D linear elasticity with body forces

A new computational model by integrating the boundary element method and the compactly supported radial basis functions (CSRBF) is developed for three-dimensional (3D) linear elasticity with the presence of body forces. The corresponding displacement and stress particular solution kernels across the supported radius in the CSRBF are obtained for inhomogeneous term interpolation. Subsequently, the classical dual reciprocity boundary element method, in which the domain integrals due to the presence of body forces are transferred into equivalent boundary integrals, is formulated by introducing locally supported displacement and stress particular solution kernels for solving the inhomogeneous 3D linear elastic system. Finally, several examples are presented to demonstrate the accuracy and efficiency of the present method.     

Calculation of domain integrals of two dimensional boundary element method

In this paper, a new method is applied to deal with domain integrals of boundary element method (BEM). In fact we focus to convert the domain integrals into boundary integrals for non-homogenous Laplace, Helmholtz and advection diffusion equations in two dimensional BEM. The transformation presented in this paper is based on divergence theorem. In addition, we prove the efficiency of method mathematically when the domain integrals are weakly singular. Numerical results are presented to verify the validity of this method for different geometries. Numerical implementation is done for the constant BEM, which can be implemented easily. To verify the new scheme, some test problems have been designed at end of the paper. The numerical results generally show that the new scheme has good accuracy with regards to other popular schemes.     

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.     

Interface integral BEM for solving multi-medium heat conduction problems

In this paper, a new and simple boundary element method, called interface integral boundary element method (IIBEM), is presented for solving heat conduction problems consisting of multiple media. In the method, the boundary integral equation is derived by a degeneration technique from domain integrals involved in varying heat conductivity problems into interface integrals in multi-medium problems. The main feature of the presented technique is that only a single boundary integral equation is used to solve heat conduction problems with different material properties. The effect of nonhomogeneity between adjacent materials is embodied in the interface integrals including the material property difference between the two adjacent materials. Comparing with conventional multi-domain boundary integral equation techniques, the presented method is more efficient in computational time, data preparing, and program coding. Numerical examples are given to verify the correctness of the presented technique.     

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

An improved procedure for solving transient heat conduction problems using the boundary element method

In employing the boundary element method to solve transient heat conduction problems, domain integrals need to be calculated. These integrals can be calculated either by directly discretizing the domain, or indirectly by utilizing a time-marching scheme which requires the time integrations to be evaluated from the initial time. Although the second approach overcomes the need for domain discretization, it has the disadvantage of requiring large storage and CPU time for increasing number of time steps. This paper is concerned with a procedure which approximates the domain integrals without the need for domain discretization. The time-marching scheme is employed so that the domain integrals can be calculated at a particular time with a known weighting and temperature distribution. These integrals can then be utilized to approximate similar domain integrals with a different weighting. It is shown that the method proposed dramatically reduces the storage requirements and CPU time, even for a small number of time divisions.