Further improvement to the stability of the coupling BEM/FEM scheme for 2-D elastodynamic problems

 The stability of the coupling BEM/FEM scheme as applied in 2-D elastodynamic problems is studied further. The linear θ method, which is more stable than the standard BEM scheme (Mansur, 1983), makes the coupling BEM/FEM scheme more stable also. But due to the inter-influence of these two kinds of numerical methods – FEM and BEM, the coupling of BEM and FEM is less stable than both BEM algorithm and FEM algorithm. Unlike in FEM, any error in BEM scheme will affect all the later results, which makes the BEM procedure tends to be less stable. And even more, a procedure that is stable when only BEM is used can be unstable in the coupling BEM/FEM procedure due to the effect of the oscillations by FEM. The procedure used in this paper is to reduce such kind of oscillations caused by the FEM scheme, so that it will not cause instability problem to the BEM scheme and further maintain the stability of the coupling BEM/FEM scheme. Although little numerical damping will be introduced by the new method, the stability is greatly improved and the computer time is nearly the same. Received 13 October 1999     

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.     

Finite element modelling and experimental investigation of single point incremental forming process of aluminum sheets: influence of process parameters on punch force monitoring and on mechanical and geometrical quality of parts

New trends in sheet metal forming are rapidly developing and several new forming processes have been proposed to accomplish the goals of flexibility and cost reduction. Among them, Incremental CNC sheet forming operations (ISF) are a relatively new sheet metal forming processes for small batch production and prototyping. In single point incremental forming (SPIF), the final shape of the component is obtained by the CNC relative movements of a simple and small punch which deform a clamped blank into the desired shape and which appear quite promising. No other dies are required than the ones used in any conventional sheet metal forming processes. As it is well known, the design of a mechanical component requires some decisions about the mechanical resistance and geometrical quality of the parts and the product has to be manufactured with a careful definition of the process set up. The use of computers in manufacturing has enabled the development of several new sheet metal forming processes, which are based upon older technologies. Although standard sheet metal forming processes are strongly controlled, new processes like single point incremental sheet forming can be improved. The SPIF concept allows to increase flexibility and to reduce set up costs. Such a process has a negative effect on the shape accuracy by initiating undesired rigid movement and sheet thinning. In the paper, the applicability of the numerical technique and the experimental test program to incremental forming of sheet metal is examined. Concerning the numerical simulation, a static implicit finite element code ABAQUS/Standard is used. These two techniques emphasize the necessity to control some process parameters to improve the final product quality. The reported approaches were mainly focused on the influence of four process parameters on the punch force trends generated in this forming process, the thickness and the equivalent plastic deformation distribution within the whole volume of the workpiece: the initial sheet thickness, the wall angle, the workpiece geometry and the nature of tool path contours controlled through CNC programming. The tool forces required to deform plastically the sheet around the contact area are discussed. The effect of the blank thickness and the tool path on the punch load and the deformation behaviour is also examined with respect to several tool paths. Furthermore, the force acting on the traveling tool is also evaluated. Similar to the sheet thickness, the effect of wall angle and part geometry on the load evolution, the distribution of calculated equivalent plastic strain and the variation of sheet thickness strain are also discussed. Experimental and numerical results obtained allow having a better knowledge of mechanical and geometrical responses from different parts manufactured by SPIF with the aim to improve their accuracy. It is also concluded that the numerical simulation might be exploited for optimization of the incremental forming process of sheet metal.     

A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis

This paper presents a procedure for coupling general finite element models with three‐dimensional bodies modelled by the Boundary Element Method (BEM). Shells, plates and frames are modelled by the Finite Element Method (FEM) and coupled to the BEM domain directly or by means of rigid blocks. The coupling is used for the analysis of buildings connected to half‐space by means of rigid footings, piles or plates in bending and other problems where combinations of different types of sub‐domains are required, composite domains for instance. Several numerical examples are analysed to demonstrate the robustness and accuracy of the proposed scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Local buckling of thin-walled structures by the boundary element method

In this work a multi-region boundary element formulation for linear local buckling analysis of assembled plate and shallow shell structures is presented. The assembly is divided into sub-regions. In each sub-region, the formulation is formed by coupling boundary element formulations of shear deformable plate bending and two-dimensional plane stress elasticity. Domain integrals appearing in the formulation (due to the curvature and due to the domain load) are transformed into equivalent boundary integrals. Membrane stresses at discrete domain points of each sub-region (plate or shallow shell) in the assembly are obtained from the prebuckling state, resulting in a set of linear buckling equations in terms of the buckling deflection and the buckling load factor. Buckling equation is presented as a standard eigenvalue problem. Results are compared with FEM solutions and it is shown that good accuracy can be achieved with the present multi-region BEM formulation.     

Symmetric collocation BEM/FEM coupling procedure for 2-D dynamic structural–acoustic interaction problems

 This paper presents a symmetric collocation BEM (SCBEM)/FEM coupling procedure applicable to 2-D time domain structural–acoustic interaction problems. The use of symmetry for BEM not only saves memory storage but also enables the employment of efficient symmetric equation solvers, especially for BEM/FEM coupling procedure. Compared with symmetric Galerkin BEM (SGBEM) where double boundary integration should be carried out, SCBEM can reduce significantly the computing cost. Two numerical examples are included to illustrate the effectiveness and accuracy of the proposed method. Received: 2 November 2001 / Accepted: 27 May 2002     

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.     

A least squares procedure for the evaluation of multiple generalized stress intensity factors at 2D multimaterial corners by BEM

Detailed characterization of linear elastic stress states at corners and crack tips requires knowledge of the stress singularity orders, the characteristic angular functions and the generalized stress intensity factors (GSIF). Typically a high accuracy is found in the literature for the evaluation of the stress singularity orders and characteristic angular functions (numerically computed from analytical expressions in most cases). Nevertheless, GSIF values, evaluated by means of a numerical model using FEM or BEM and usually by postprocessing the results, are often reported with a lower level of confidence. A robust procedure is presented in this work for the evaluation of the GSIF at multimaterial corners. The procedure is based on a simple least squares technique involving stresses and/or displacements, computed by BEM, at the neighborhood of the corner tip. A careful verification of the robustness and accuracy of the procedure using a few benchmark problems in the literature has been carried out. Applications of the procedure developed to the evaluation of GSIFs appearing at corners in metal-composite adhesive lap joints are presented.     

Investigation of the effect of different variables on strength of adhesive joints

In this article, the tensile strength of different adhesive bonded joints under a tensile load was analyzed numerically. The effects of certain parameters, including the bonding length and bonding ratio, were investigated. For this reason, the epoxy adhesive was used. Joints were prepared with aluminum materials. The stress analyses were employed using the Finite Element Method (FEM). ANSYS (v.14.0.1) FEM tool was utilized to investigate the stress distribution characteristics of aluminum lap joint under tensile loading. Numerical results were found to be quite reasonable. The numerical results show that the influences of variations are very notable when the equivalent stresses are between 18 MPa and 20 MPa.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

An overlapping domain decomposition approach for coupling the finite and boundary element methods

An overlapping iterative domain decomposition approach for the coupling of the finite element method (FEM) and the boundary element method (BEM) is presented in this paper. In this proposed method, the domain of the original problem is subdivided into a FEM sub-domain and a BEM sub-domain, such that the two sub-domains partially overlap over a common region. The common region is modeled by both methods. A brief discussion on the existing iterative coupling methods and their limitations are given in the first part of this paper. In the second part, the proposed overlapping method is described and the convergence conditions are presented. Two numerical examples are given to demonstrate the capability of the proposed method for handling cases where the Neumann boundary conditions are specified on the entire external boundary of the FEM or BEM sub-domains.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

复合材料边界元素法研究

边界元法是一种迅速发展的数值解法.本文研究了在复合材料层合板中运用平面二次边界元法的有关问题.首先,讨论了将层合板等效为一般正交各向异性单层板的方法;其次,给出了偏轴情况下的基本解,研究了直线型二次边界元偏轴影响系数的解析计算以及域内应力表达式;最后对算例进行了计算,所得结果与相应的解析解和有限元相符甚好.     

Comparison of mixed and isoparametric boundary elements in time domain poroelasticity

A poroelastodynamic Boundary Element (BE) formulation in time domain based on the Convolution Quadrature Method (CQM) is used to model wave propagation phenomena. In the conventional BEM implementation, the same shape functions are applied to all state variables.Motivated by the improvements due to mixed elements in FEM, i.e. the shape function for the pore pressure is chosen one degree lower than for the displacement, such elements have been added to the BEM implementation in both two dimensional (2-d) and three dimensional (3-d) formulations. A study about the influence of the mixed shape functions to the quality of numerical results and the stability of the time-stepping scheme is presented. The mixed elements increase the numerical cost significantly and the results are inconclusive as to whether they improve the CQM based BEM. Therefore, they are only recommended for special cases, in particular the incompressible model in 2-d, where they tend to a significant reduction of the lower stability limit.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

Three-Dimensional BEM and FEM Submodelling in a Cracked FML Full Scale Aeronautic Panel

This paper concerns the numerical characterization of the fatigue strength of a flat stiffened panel, designed as a fiber metal laminate (FML) and made of Aluminum alloy and Fiber Glass FRP. The panel is full scale and was tested (in a previous work) under fatigue biaxial loads, applied by means of a multi-axial fatigue machine: an initial through the thickness notch was created in the panel and the aforementioned biaxial fatigue load applied, causing a crack initiation and propagation in the Aluminum layers. Moreover, (still in a previous work), the fatigue test was simulated by the Dual Boundary Element Method (DBEM) in a bidimensional approach. Now, in order to validate the assumptions made in the aforementioned DBEM approach and concerning the delamination area size and the fiber integrity during crack propagation, three-dimensional BEM and FEM submodelling analyses are realized. Due to the lack of experimental data on the delamination area size (normally increasing as the crack propagates), such area is calculated by iterative three-dimensional BEM or FEM analyses, considering the inter-laminar stresses and a delamination criterion. Such three-dimensional analyses, but in particular the FEM proposed model, can also provide insights into the fiber rupture problem. These DBEM-BEM or DBEM-FEM approaches aims at providing a general purpose evaluation tool for a better understanding of the fatigue resistance of FML panels, providing a deeper insight into the role of fiber stiffness and of delamination extension on the stress intensity factors.     

Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.