The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C ¹ extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy. This work was partially supported by the United States Department of Energy (DOE) grants DE-FG03-97ER14778 and DE-FG03-97ER25332. This financial support does not constitute an endorsement by the DOE of the views expressed in this paper.     

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.     

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.     

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     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.     

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

The boundary integral equation that results from the application of the reciprocity theorem to non‐linear or non‐homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh‐based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply‐connected geometries can become a costly task, especially in three‐dimensional analyses. In this paper, a method that allows a mesh‐based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier–Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell‐based domain integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

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.     

A Galerkin boundary element formulation with dual reciprocity for elastodynamics

In this paper, a Galerkin boundary integral equation method for two‐dimensional elastodynamic problems is presented. The formulation makes use of a static fundamental solution to weight the dynamic equilibrium equations. Following the Galerkin approach, the equations are weighted again with the interpolation functions used in the discretization of the unknowns. For the numerical integration, a regularization process is followed to deal with the integrands containing strong singularities. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is also presented in the context of the Galerkin formulation. Finally, the Hubolt integration scheme was used for the time‐marching process. Several numerical examples are presented to demonstrate the accuracy of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

Dual reciprocity hybrid boundary node method for acoustic eigenvalue problems

The hybrid boundary node method (HBNM) is a truly meshless method, and elements are not required for either interpolation or integration. The method, however, can only be used for solving homogeneous problems. For the inhomogeneous problem, the domain integration is inevitable. This paper applied the dual reciprocity hybrid boundary node method (DRHBNM), which is composed by the HBNM and the dual reciprocity method (DRM) for solving acoustic eigenvalue problems. In this method, the solution is composed of two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by HBNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of HBNM. The moving least squares (MLS) is employed to approximate the boundary variables, while the domain variables are interpolated by the fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The Q–R algorithm and Householder algorithm are applied for solving the eigenvalues of the transformed matrix. The parameters that influence the performance of DRHBNM are studied through numerical examples. Numerical results show that high convergence rates and high accuracy are achievable.     

A regular variational boundary model for free vibrations of magneto-electro-elastic structures

In this paper a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented. The method is based on a hybrid variational principle expressed in terms of generalized magneto-electro-elastic variables. The domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magneto-electro-elastic problem, whereas the boundary variables are expressed in terms of nodal values. The variational principle coupled with the proposed discretization scheme leads to the calculation of frequency-independent and symmetric generalized stiffness and mass matrices. The generalized stiffness matrix is computed in terms of boundary integrals of regular kernels only. On the other hand, to achieve meaningful computational advantages, the domain integral defining the generalized mass matrix is reduced to the boundary through the use of the dual reciprocity method, although this implies the loss of symmetry. A purely boundary model is then obtained for the computation of the structural operators. The model can be directly used into standard assembly procedures for the analysis of non-homogeneous and layered structures. Additionally, the proposed approach presents some features that place it in the framework of the weak form meshless methods. Indeed, only a set of scattered points is actually needed for the variable interpolation, while a global background boundary mesh is only used for the integration of the influence coefficients. The results obtained show good agreement with those available in the literature proving the effectiveness of the proposed approach.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Three-dimensional transient heat conduction analysis by Laplace transformation and multiple reciprocity boundary face method

In this paper, a new multiple reciprocity formulation is developed to solve the transient heat conduction problem. The time dependence of the problem is removed temporarily from the equations by the Laplace transform. The new formulation is derived from the modified Helmholtz equation in Laplace space (LS), in which the higher order fundamental solutions of this equation are firstly derived and used in multiple reciprocity method (MRM). Using the new formulation, the domain integrals can be converted into boundary integrals and several non-integral terms. Thus the main advantage of the boundary integral equations (BIE) method, avoiding the domain discretization, is fully preserved. The convergence speed of these higher order fundamental solutions is high, thus the infinite series of boundary integrals can be truncated by a small number of terms. To get accurate results in the real space with better efficiency, the Gaver-Wynn-Rho method is employed. And to integrate the geometrical modeling and the thermal analysis into a uniform platform, our method is implemented based on the framework of the boundary face method (BFM). Numerical examples show that our method is very efficient for transient heat conduction computation. The obtained results are accurate at both internal and boundary points. Our method outperforms most existing methods, especially concerning the results at early time steps.     

An overlapping domain decomposition dual reciprocity method

We present a domain decomposition boundary integral equation method for two-dimensional partial differential equations. The overlapping Schwarz method is employed to improve the dual reciprocity boundary element method. The resulting algorithm turns out to be more accurate than a non-overlapping approach previously proposed. Some numerical results showing the improved accuracy and efficiency of the methods are given.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

A boundary element method for analysis of thermoelastic deformations in materials with temperature dependent properties

In this paper, the boundary element method (BEM) for solving quasi‐static uncoupled thermoelasticity problems in materials with temperature dependent properties is presented. The domain integral term, in the integral representation of the governing equation, is transformed to an equivalent boundary integral by means of the dual reciprocity method (DRM). The required particular solutions are derived and outlined. The method ensures numerically efficient analysis of thermoelastic deformations in an arbitrary geometry and loading conditions. The validity and the high accuracy of the formulation is demonstrated considering a series of examples. In all numerical tests, calculation results are compared with analytical and/or finite element method (FEM) solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient implementation of the radial basis integral equation method

An efficient and accurate implementation of the meshless radial basis integral equation method (RBIEM) is proposed. The proposed implementation does not involve discretization of the subdomains’ boundaries. By avoiding the boundary discretization, it was hypothesised that a significant source of error in the numerical scheme is avoided. The proposed numerical scheme was tested on two problems governed by the Poisson and Helmholtz equations. The test problems were selected such that the spatial gradients of the solutions were high to examine the robustness of the numerical scheme. The dual reciprocity method (DRM) and the cell integration technique were used to treat the domain integrals arising from the source terms in the partial differential equations. The results showed that the proposed implementation is more accurate and more robust than the previously suggested implementation of the RBIEM. Though the CPU time usage of the proposed scheme is lower, the difference to the previously proposed scheme is not significant. The proposed scheme is easier to implement, since the task of keeping track of boundary elements and boundary nodes is not needed. The proposed implementation of the RBIEM is promising and opens up possibilities for efficient implementation in three-dimensional problems. This is currently under investigation.