A dual-reciprocity method for acoustic radiation in a subsonic non-uniform flow

In this paper, the dual-reciprocity boundary-element method is used to model acoustic radiation in a subsonic non-uniform-flow field. The boundary-integral formulation is based on a direct boundary-integral equation developed very recently by the authors for acoustic radiation in a subsonic uniform flow. All the terms due to the non-uniform-flow effect are taken to the right-hand side and treated as source terms. The source terms result in a domain integral in the standard boundary-integral formulation. The dual-reciprocity method is then used to transform the resulting domain integral into boundary integrals. Numerical tests show reasonably good agreement with an analytical soution for a pulsating sphere submerged in a potential-flow field.     

A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media

A dual-reciprocity boundary element method is proposed for the numerical solution of a two-dimensional boundary value problem (BVP) governed by an elliptic partial differential equation with variable coefficients. The BVP under consideration has applications in a wide range of engineering problems of practical interest, such as in the calculation of antiplane stresses or temperature in non-homogeneous anisotropic media. The proposed numerical method is applied to solve specific test problems.     

A dual-reciprocity boundary element approach for axisymmetric nonlinear time-dependent heat conduction in a nonhomogeneous solid

A dual-reciprocity boundary element method is presented for the numerical solution of a nonsteady axisymmetric heat conduction problem involving a nonhomogeneous solid with temperature dependent properties. It is applied to solve some specific problems including one which involves the laser heating of a cylindrical solid.     

Non-steady state heat conduction across an imperfect interface: A dual-reciprocity boundary element approach

The two-dimensional problem of determining the time-dependent temperature in a bimaterial with a homogeneously imperfect interface is considered. A temperature jump which is proportional to the thermal heat flux is assumed across the imperfect interface. Through the use of the corresponding steady-state Green's function for the imperfect interface, a dual-reciprocity boundary element method is derived for the numerical solution of the problem under consideration. To assess the validity and accuracy of the proposed method of solution, some specific problems are solved.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

On the choice of interpolation functions used in the dual-reciprocity boundary-element method

The dual-reciprocity boundary-element method is a very powerful technique for solving general elliptic equations of the type ²u=b. In this method, a series of interpolation functions is used to approximate b in order to convert the associated domain integral, which it is necessary to evaluate in a traditional boundary-element analysis, into boundary integrals only. Hence the choice of interpolation functions has direct effects on the numerical results. According to Partridge and Brebbia, the adoption of a comparatively simple form of interpolation function gives the best results. Unfortunately, when b contains partial derivatives of the unknown function u(x, y), the adoption of such a type of interpolation function inevitably leads to the creation of singularities on all boundary and internal nodes used in a dual-reciprocity boundary-element analysis, as was pointed out by Zhu and Zhang in 1992. To avoid this problem, a functional transformation, which applies only to linear governing equations, can be employed to eliminate these derivative terms and thus to obtain better numerical results. In this paper, two new interpolation functions are proposed and examined; they are proven to be generally applicable and satisfactory.     

A time-stepping DRBEM for the transient magneto-thermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid

The main objective of this paper is to study the transient magneto-thermo-visco-elastic stresses in a non-homogeneous anisotropic solid placed in a constant primary magnetic field acting in the direction of the z-axis and rotating about it with a constant angular velocity. The system of fundamental equations is solved by means of a dual-reciprocity boundary element method (DRBEM). In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and thermal stress components. The validity of DRBEM is examined by considering a magneto-thermo-visco-elastic solid occupies a rectangular region and good agreement is obtained with the results obtained by other methods. The results obtained are presented graphically to show the effect of inhomogeneity on the displacement components and thermal stress components.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

The boundary element method for the solution of Stokes equations in two-dimensional domains

A boundary element method for the solution of Stokes equations governing creeping flow or Stokes flow in the interior of an arbitrary two-dimensional domain is presented. A procedure for introducing pressure data on the boundary of the domain is also included and the integral coefficients of the resulting linear algebraic equations are evaluated analytically. Calculations are performed in a circular domain using a variety of different boundary conditions, including a combination of the fluid velocity and the pressure. Results are presented both on the boundary and inside the solution domain in order to illustrate that the boundary element method developed here provides an efficient technique, in terms of accuracy and convergence, to investigate Stokes flow numerically.     

Boundary element formulations for Mindlin plate on an elastic foundation with dynamic load

Boundary element method (BEM) for a shear deformable plate (Reissner/Mindline's theories) resting on an elastic foundation subjected to dynamic load is presented. Formulations for both Winkler and Pasternak foundations are presented. The boundary element formulation in Laplace domain is presented together with complete expressions for the internal point kernels (i.e. fundamental solutions). Quadratic isoparameteric boundary elements are used to discretise the boundary of plate domain. Time domain variables are obtained by the Durbin's inversion method from transform domain. Numerical examples are presented to demonstrate the accuracy of the boundary element method and the comparisons are made with other numerical technique.     

Free vibration analysis of anisotropic material structures using the boundary element method

This paper presents a formulation for the analysis of free vibration in anisotropic structures using the boundary element method. The fundamental solution for elastostatic is used and the inertial terms are treated as body forces providing domain integrals. The dual reciprocity boundary element method is used to reduce domain integrals to boundary integrals. Mode shapes and natural frequencies for free vibration of orthotropic structures are obtained and compared with finite element results showing good agreement.     

Thermoelastic stress field in a piecewise homogeneous domain under non‐uniform temperature using a coupled boundary and finite element method

This study concerns the development of a coupled finite element–boundary element analysis method for the solution of thermoelastic stresses in a domain composed of dissimilar materials with geometric discontinuities. The continuity of displacement and traction components is enforced directly along the interfaces between different material regions of the domain. The presence of material and geometric discontinuities are included in the formulation explicitly. The unknown interface traction components are expressed in terms of unknown interface displacement components by using the boundary element method for each material region of the domain. Enforcing the continuity conditions leads to a final system of equations containing unknown interface displacement components only. With the solution of interface displacement components, each region has a complete set of boundary conditions, thus leading to the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, stresses from specific BEM regions are first expressed in terms of interface displacements, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of FEM regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Interval boundary element method in the presence of uncertain boundary conditions,integration errors,and truncation errors

In engineering, most governing partial differential equations for physical systems are solved using finite element or finite difference methods. Applications of interval methods have been explored in finite element analysis to model systems with parametric uncertainties and to account for the impact of truncation error on the solutions. An alternative to the finite element method is the boundary element method. The boundary element method uses singular functions to reduce the dimension of the domain by transforming the domain variables to boundary variables. In this work, interval methods are developed to enhance the boundary element method for considering causes of imprecision such as uncertain boundary conditions, truncation error, and integration error. Examples are presented to illustrate the effectiveness and potential of an interval approach in the boundary element method.     

Arch dam–fluid interactions: By fem–bem and substructure concept

Arch dams can be conveniently analysed by the finite element method. For dam–fluid interaction problems, the fluid domain may be more conveniently handled by the boundary element method as a substructure first before connecting to the dam substructure. The added-mass matrix calculated from the fluid domain is symmetrized and lumped first so that the banded and symmetrical characteristics of the finite element method are retained. In the boundary element formulation, a mirror image method and quadratic elements are used for computational efficiency and accuracy. The strong singular terms are handled by using a solution which satisfies the governing equation and the free surface boundary condition. Infinite boundary conditions at the upstream of the reservoir can be reasonably approximated from the fundamental solution with accurate results, if the interior pressure distribution in the fluid domain is neglected. Numerical solutions on hydrodynamic pressure distribution and the natural frequencies of the dam–reservoir system with various water levels are obtained and compared with available analytical and experiment results.     

A boundary/domain element method for analysis of building raft foundations

In this paper, a new boundary/domain element method is developed to analyse plates resting on elastic foundations. The developed formulation is then used in analysing building raft foundations. For more practical representation, the considered raft plate is treated as thick plate with free edge boundary conditions. The soil or the elastic foundation is represented as continuous media (follows the Winkler assumption). The boundary element method is employed to model the raft plate; whereas the soil is modelled using constant domain cells or elements. Therefore, in the present formulation both the domain and the boundary of the raft plate are discretized. The associate soil domain integral is replaced by equivalent boundary integrals along each cell contour. The necessary matrix implementation of such formulation is carried out and explained in details. The main advantage of the present formulation is the ability of analysing rafts on non-homogenous soils. Two examples are presented including raft on non-homogenous soil and raft for practical building applications. The results are compared with those obtained from other finite element and alternative boundary element methods to verify the validity and accuracy of the present formulation.     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.     

An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation

A new hybrid algorithm for automatically generating either an all-quadrilateral or an all-triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight-line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All-triangular element meshes are generated using a boundary contraction procedure. All-quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon-splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.     

无界区域上Stokes问题的自然边界元与Mini元耦合法

奉文讨论用自然边界元与mini元耦合法求解描述平面尢界区域上不可压缩粘滞低速流动的定常Stokes问题.首先以圆为人工边界,利用自然边界归化将原问题转化为耦合变分问题,并证明该变分问题的存在唯一性,然后在人工边界上采用分段线性边界元,在有界区域上应用mini元分别进行离散化,合成总刚度矩阵,从而建立耦合法的线性方程组,最后,证明其收敛性和误差估计,并通过数值实验以表现该方法的实际有效性及其理论分析的正确性.     

Boundary element frequency domain formulation for dynamic analysis of Mindlin plates

A new boundary element formulation for analysis of shear deformable plates subjected to dynamic loading is presented. Fundamental solutions for the Mindlin plate theory are derived in the Laplace transform domain. The characteristics of the three flextural waves are studied in the time domain. It is shown that the new fundamental solutions exhibit the same strong singularity as in the static case. Two numerical examples are presented to demonstrate the accuracy of the boundary element method and comparisons are made with the finite element method. Copyright © 2006 John Wiley & Sons, Ltd.