Iterative coupling of BE and FE methods in elastostatics

Iterative domain decomposition coupling is one of the recent approaches for combining the boundary element method (BEM) and the finite element method (FEM). The domain of the original problem is subdivided into two sub-domains, which are separately modeled by the FEM and BEM. Successive renewal of the variables on the interface of the two sub-domains is performed through an iterative procedure to reach the final convergence. In this paper, we investigate the iterative method. We also establish the convergence conditions. A simple numerical example is given to elaborate on the effect of different factors such as initial guess, boundary conditions, and geometrical and material properties of the sub-domains on solution convergence.     

Non-linear elastodynamic analysis by the BEM: an approach based on the iterative coupling of the D-BEM and TD-BEM formulations

The present paper is concerned with the development of a scheme based on iterative coupling of two boundary element formulations to obtain time-domain numerical solution of dynamic non-linear problems. The domain is divided into two sub-domains: the sub-domain that presents non-linear behaviour is modelled by the D-BEM formulation (D: domain) whereas the sub-domain that behaves elastically is modelled by the TD-BEM formulation (TD, time-domain). The solution of the problem is obtained independently in each sub-domain and the variables at common interfaces are computed iteratively. Two examples are presented, in order to verify the potentialities of the proposed methodology.     

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.     

Frequency domain analysis of interacting acoustic–elastodynamic models taking into account optimized iterative coupling of different numerical methods

In this work, interacting acoustic–elastodynamic models are analyzed by means of an optimized iterative coupling algorithm. In this iterative coupling procedure, each acoustic/elastodynamic sub-domain of the model is solved independently, and the variables at the common interfaces of the sub-domains are successively renewed, until convergence is achieved. A relaxation parameter is introduced in order to ensure and/or speed up the convergence of the iterative analysis, and an expression to compute optimal values for the relaxation parameter is presented. Several numerical methods are considered to discretize the acoustic and elastodynamic sub-domains of the coupled model, and the performance of these different methodologies, in the coupled analysis, is discussed. In this context, the boundary element method and the method of fundamental solutions are applied to model the acoustic sub-domains, whereas the finite element method, the collocation method and the meshless local Petrov–Galerkin method are applied to model the elastodynamic sub-domains. Independent discretizations of the acoustic/elastodynamic sub-domains are allowed, being no matching nodes required along the common interfaces. At the end of the paper, numerical examples are presented, illustrating the performance and potentialities of the adopted procedures.     

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.     

Acoustic modelling by BEM–FEM coupling procedures taking into account explicit and implicit multi‐domain decomposition techniques

The numerical modelling of interacting acoustic media by boundary element method–finite element method (BEM–FEM) coupling procedures is discussed here, taking into account time‐domain approaches. In this study, the global model is divided into different sub‐domains and each sub‐domain is analysed independently (considering BEM or FEM discretizations): the interaction between the different sub‐domains of the global model is accomplished by interface procedures. Numerical formulations based on FEM explicit and implicit time‐marching schemes are discussed, resulting in direct and optimized iterative BEM–FEM coupling techniques. A multi‐level time‐step algorithm is considered in order to improve the flexibility, accuracy and stability (especially when conditionally stable time‐marching procedures are employed) of the coupled analysis. At the end of the paper, numerical examples are presented, illustrating the potentialities and robustness of the proposed methodologies. Copyright © 2008 John Wiley & Sons, Ltd.     

Domain decomposition boundary element method with overlapping sub-domains

A numerical approach based on the domain decomposition boundary element method (BEM) with overlapping sub-domains has been developed. The approach simplifies the assembly of the equations arising from the BEM sub-domain methods, reduces the size of the system matrix, produces a closed system of equations when continuous elements are used, and reduces any problems arising from near-singular or singular integrals which otherwise may appear in the integral equations. The overlapping numerical approach is tested on three different problems, i.e., the Poisson equation, and a one-dimensional and two-dimensional convection–diffusion problems. The approach is implemented in combination with the dual reciprocity method (DRM) with two different radial basis functions (RBFs), though the approach is general and can be applied with other BEM formulations. The results are compared with the previous results obtained using the dual reciprocity method–multi domain (DRM–MD) approach, showing comparable accuracy and convergence.     

Definition of two-dimensional condensation via BEM,using the Glaser method approach

This work describes an iterative technique for the definition of condensation across two-dimensional elements via the boundary element method (BEM). Initially, the BEM is used to calculate the steady-state conduction of heat and vapour diffusion, discretizing only the boundary materials' discontinuities. Then, a small sub-domain is defined, where condensation develops; that is, where the vapour pressure exceeds the vapour saturation pressure. Using the Glaser approach, the vapour pressure is equalised to the vapour saturation pressure, and then the vapour equilibrium is redefined by means of the BEM solution. This process is repeated until all sub-domains where vapour pressure exceeds vapour saturation pressure are eliminated.The method is first implemented and validated by applying it to a simple one-dimensional hygrothermal problem, for which the solution is calculated analytically. The applicability of the proposed method is then illustrated by computing the two-dimensional condensation across a T shaped element, for different boundary conditions.     

Numerical modelling of acoustic–elastodynamic coupled problems by stabilized boundary element techniques

In this work, an efficient, flexible, accurate and stable algorithm to numerically model interacting acoustic–elastodynamic sub-domains is described. Stabilized time-domain boundary element techniques are considered to discretize each sub-domain of the model and proper numerical expressions on acoustic–elastodynamic interfaces are presented. Moreover, stabilized iterative coupling procedures are adopted and different time and space sub-domain discretizations are allowed, improving the robustness and versatility of the methodology. At the end of the paper, numerical results are presented, illustrating the potentialities of the proposed formulation.     

模拟纤维增强复合材料的相似子域边界元法

根据夹杂相积分区域的相似性,提出了相似子域边界元法求解方案。把含随机分布夹杂相的固体归结为对一个含有内边界条件复连通域问题的求解,与传统的有限元和边界元分域解法相比,显著地提高了计算效率。应用相似子域边界元法,对含有随机分布圆形和椭圆形夹杂相的固体材料进行了大量数值计算,并把夹杂相与基体材料之间从理想粘结扩展到带有界面层的情况。这些计算为相应纤维增强复合材料宏观等效力学特性研究提供了有效的数值模拟方法。      

Coupling the BEM/TBEM and the MFS for the numerical simulation of acoustic wave propagation

The coupling of the boundary element method (BEM)/the traction boundary element method (TBEM) and the method of fundamental solutions (MFS) is proposed for the transient analysis of acoustic wave propagation problems in the presence of multi-inclusions to overcome the limitations posed by each method. The full domain is divided into sub-domains which are modeled using the BEM/TBEM and the MFS, and the sub-domains are coupled with the imposition of the required boundary conditions. The accuracy of the proposed algorithms, using different combinations of BEM/TBEM and MFS, is verified by comparing the solutions against reference solutions. The applicability of the proposed method is shown by simulating the acoustic behavior of a rigid acoustic screen in the vicinity of a dome and by computing the acoustic attenuation provided by a fluid-filled thin inclusion separating two railway tracks in an underground train station.     

A coupling of multi‐zone curved Galerkin BEM with finite elements for independently modelled sub‐domains with non‐matching nodes in elasticity

When the different parts of a structure are modelled independently by BEM or FEM methods, it is sometimes necessary to put the parts together without remeshing of the nodes along the part interfaces. Frequently the nodes do not match along the interface. In this work, the symmetric Galerkin multi‐zone curved boundary element is a fully symmetric formulation and is the method used for the boundary element part. For BEM–FEM coupling it is then necessary to interpolate the tractions in‐between the non‐matching nodes for the FEM part. Finally, the coupling is achieved by transforming the finite element domains to equivalent boundary element domains in a block symmetric formulation. This system is then coupled with a boundary element domain with non‐matching nodes in‐between. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

Coupling of the BEM with the MFS for the numerical simulation of frequency domain 2-D elastic wave propagation in the presence of elastic inclusions and cracks

This paper proposes a coupling formulation between the boundary element method (BEM displacement and TBEM traction formulations) and the method of fundamental solutions (MFS) for the transient analysis of elastic wave propagation in the presence of multiple elastic inclusions to overcome the specific limitations of each of these methods. The full domain of the original problem is divided into sub-domains, which are handled separately by the BEM or the MFS. The coupling is enforced by imposing the required boundary conditions.The accuracy, efficiency and stability of the proposed algorithms, using different combinations of BEM and MFS, are verified by comparing the solutions against reference solutions. The computational efficiency of the proposed coupling formulation is illustrated by computing the CPU time and the error at high frequencies.The potential of the proposed procedures is illustrated by simulating the propagation of elastic waves in the vicinity of an empty crack, with null thickness placed close to an elastic inclusion.     

FE/FMBE coupling to model fluid–structure interaction

In this paper, a finite element (FE)/fast multipole boundary element (FMBE)‐coupling method is presented for modeling fluid–structure interaction problems numerically. Vibrating structures are assumed to consist of elastic or sound absorbing materials. An FE method (FEM) is used for this part of the solution. This structural sub‐domain is embedded in a homogeneous fluid. The case where the boundary of the structural sub‐domain has a very complex geometry is of special interest. In this case, the BE method (BEM) is a more suitable numerical tool than FEM to account for the sound propagation in the homogeneous fluid. The efficiency of the BEM is increased by using FMBEM. The BE‐surface mesh required is directly generated by the FE‐mesh used to discretize the structural sub‐domain and the absorbing material. This FE/FMBE‐coupling method makes it possible to predict the effects of arbitrarily shaped absorbing materials and vibrating structures on the sound field in the surrounding fluid numerically. The coupling method proposed is used to study the acoustic behavior of the lining of an anechoic chamber and that of an entire anechoic chamber in the low‐frequency range. The numerical results obtained are compared with the experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

A sub-domain RPIM with condensation of degree of freedom

A sub-domain radial point interpolation method is proposed to simulate problem of linear elasticity. In present method, the problem domain is firstly divided into sub-domains with arbitrary shape, and then, nodes without connectivity are imbedded in every sub-domain. The local variational weak formulation is established over sub-domains, in which nodes within the sub-domain are used for approximation. Local discrete equations of weak form are simplified by condensation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on sub-domains. Compatibility of displacement in adjacent sub-domains and convergence of present method are discussed. And displacements and its gradient are continuous in the entire problem domain. In contrast to an early formulation of RPIM based on Galerkin weak form, which is proposed by Liu and coworkers, certain modifications are presented to increase its computational efficiency in this paper. Numerical examples show that computational efficiency of present method is higher than that of standard RPIM based on Galerkin weak form, and good accuracy, high convergence can also be obtained.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis

A coupled BEM–FEM methodology is presented for 3D wave propagation and soil–structure interaction analysis in the direct time domain. The employed boundary element method (BEM) uses a new generation of the Stokes fundamental solutions that utilize the B-Spline family of polynomials. A standard finite element methodology for dynamic analysis along with direct integration in time is coupled to the BEM through a staggered solution approach. Each method provides initial conditions to the other at the beginning of each time step. Formulation and computational aspects of the proposed coupling scheme are discussed. A number of numerical examples are presented for the validation and demonstration of the general nature of the proposed methodology.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.