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.     

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

The finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second‐order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non‐symmetrical BEM formulations. The method connects non‐matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non‐symmetrical flexibility equations with a Bi‐CGstab iterative algorithm. Scalability issues of nsBETI in BEM–BEM and combined BEM–FEM coupled problems are also investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Iterative coupling algorithms for large multidomain problems with the boundary element method

A new parallel Robin-Robin adaptive iterative coupling algorithm with dynamic relaxation parameters is proposed for the boundary element method (BEM), and relaxation parameters are derived for other existing iterative coupling algorithms. The performances of the new algorithm and of the modified existing algorithms are investigated in terms of convergence properties with respect to the number of subdomains, mesh density, interface mesh conformity, and BEM element types. Results show that the number of subdomains and the refinement level of the mesh are the two dominant factors affecting the performances of the considered algorithms. The proposed parallel Robin-Robin algorithm shows the best overall convergence behavior for the tested large problems, thanks to its effectiveness in handling complex boundary conditions and large number of subdomains, thus resulting to be very promising for efficient parallel BEM computing and large coupling problems. Source code is available at https://github.com/BinWang0213/PyBEM2D .     

3D multidomain BEM for solving the Laplace equation

An efficient 3D multidomain boundary element method (BEM) for solving problems governed by the Laplace equation is presented. Integral boundary equations are discretized using mixed boundary elements. The field function is interpolated using a continuous linear function while its derivative in a normal direction is interpolated using a discontinuous constant function over surface boundary elements. Using a multidomain approach, also known as the subdomain technique, sparse system matrices similar to the finite element method (FEM) are obtained. Interface boundary conditions between subdomains leads to an over-determined system matrix, which is solved using a fast iterative linear least square solver. The accuracy and robustness of the developed numerical algorithm is presented on a scalar diffusion problem using simple cube geometry and various types of meshes. Efficiency is demonstrated with potential flow around the complex geometry of a fighter airplane using tetrahedral mesh with over 100,000 subdomains on a personal computer.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

2.5D coupled BEM–FEM used to model fluid and solid scattering wave

We describe a hybrid method to study fluid and solid interaction problems in the frequency domain. The numerical method is based on subdomain decomposition. The BEM is used to model unbounded solid mediums, whereas the confined subdomains, both fluid and solid, are represented by the FEM. The analysis is carried out by superposing two‐and‐a‐half dimension (2.5D) problems for different longitudinal wave numbers. A novel 2.5D FEM formulation for inviscid fluids is proposed, which include the energy lost at the fluid boundary enclosure. The fluid and solid subdomains are coupled, and appropriate boundary conditions are imposed at the interfaces. The proposed technique is verified from analytical solutions. A cylindrical cavity located in an unbounded solid medium excited by a dilatational point source is studied. Computed results are in good agreement with the analytical solution. Later, noise and vibration in a concrete tunnel due to an internal pressure load is analysed with the proposed methodology. Results show that tunnel and soil displacements increase with the load speed, as did the air pressure inside the tunnel, according with the travelling ranges defined by the wave propagation velocities in each medium. Copyright © 2014 John Wiley & Sons, Ltd.     

Modal acoustic transfer vector approach in a FEM–BEM vibro-acoustic analysis

The aim of the present work is to set up an integrated approach for an automobile vibro-acoustic analysis, useful to assess, visualise and compare vibro-acoustic performance to pre-determined design targets, while identifying and quantifying the forces and sound sources responsible for the current behaviour. Such design approach, based on experimental and numerical procedures, enables the prediction of noise emissions and the correlation with the structural vibration source.Vibro-acoustic prediction in the low- to mid-frequency range is generally performed through finite element method (FEM) or boundary element method (BEM) but in this work a combined usage of the two methodologies is adopted: FEM is used for the structural dynamics and BEM for the acoustic problem resolution. The BE methodology adopted is based on an indirect formulation and on a variational solution scheme.The adopted FEM–BEM approach takes advantage of the Modal Acoustic Transfer Vector algorithm that is particularly useful when big problems are to be analysed. The comparison between numerical and experimental results enables an assessment of the accuracy level.     

3D multidomain BEM for a Poisson equation

This paper deals with the efficient 3D multidomain boundary element method (BEM) for solving a Poisson equation. The integral boundary equation is discretized using linear mixed boundary elements. Sparse system matrices similar to the finite element method are obtained, using a multidomain approach, also known as the ‘subdomain technique’. Interface boundary conditions between subdomains lead to an overdetermined system matrix, which is solved using a fast iterative linear least square solver. The accuracy, efficiency and robustness of the developed numerical algorithm are presented using cube and sphere geometry, where the comparison with the competitive BEM is performed. The efficiency is demonstrated using a mesh with over 200,000 hexahedral volume elements on a personal computer with 1 GB memory.     

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     

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.     

Adaptive modeling of damage growth using a coupled FEM/BEM approach

We propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non‐linear damage growth is accurately accounted by a local FEM mesh. An integral‐type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non‐linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

基于拓扑优化的结构加强筋布局降噪方法研究

基于变密度拓扑优化方法,提出了通过优化加强筋布局来降低谐振结构辐射声功率的策略。优化中将加强筋单元的伪密度作为设计变量,约束为加强筋质量上限,所用寻优算法为移动渐近线(MMA)系列算法中的MMA-GC-MMA混合算法。对结构的振动响应使用有限元方法求解,对声辐射使用边界元方法求解,并在此基础上分析了目标函数对设计变量的灵敏度。以加筋箱体为例进行优化,验证了所提方法在降噪设计中的可行性和有效性,并对结果进行了讨论。     

Dynamic analysis of large-scale SSI systems for layered unbounded media via a parallelized coupled finite-element/boundary-element/scaled boundary finite-element model

An algorithm for a parallelized coupled model based on finite element method (FEM), boundary element method (BEM), and scaled boundary FEM (SBFEM) for harmonic and transient dynamic response of large-scale 2D structures embedded in or on layered soil media is presented. The BEM and SBFEM are used for modelling the dynamic response of the unbounded media. The standard FEM is used for modelling the finite region and the embedded structure. The objective of the development of this parallelized coupled model is to use the power of high performance computing, and to take into account the advantages and evade the disadvantages of the above mentioned numerical methods for modelling of the unbounded media in soil-structure interaction (SSI) systems. The development of the parallel algorithm for this model is essential for solving arbitrarily shaped large-scale SSI problems, which cannot be solved within reasonable elapsed times by a serial algorithm. The efficiency of the proposed parallel algorithm and the validity of the coupled model are shown by means of three numerical examples, indicating the excellent accuracy and applicability of the parallel algorithm with considerable time-savings in large-scale problems.     

On coupling of reproducing kernel particle method and boundary element method

In this paper a new coupling method to merge the reproducing kernel particle method (RKPM) and the boundary element method (BEM) is proposed. In this formulation, the whole problem domain will first be divided into two subdomains in which the RKPM and the BEM are applied respectively. A simple and direct coupling procedure is then applied to preserve the compatibility of the solution along the interface of the two subdomains. Unlike other coupling procedures suggested previously, the present coupling procedure neither requires modification of the RKPM and the BEM formulations nor the use of finite elements along the interface boundary between the two subdomains. The validity and efficiency of the coupling procedure are demonstrated by using it to solve five benchmark elastostatic stress analysis problems. In addition, a simple analysis of the a priori convergence rate of the coupling procedure is given. Preliminary assessment of the convergence characteristics of the coupling procedure is done by carrying out uniform refinements on the benchmark problems.     

Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification

In this paper, the Boundary Element Method (BEM) for 3-D elastostatic problems is studied for the analysis of shell or shell-like structures. It is shown that the conventional boundary integral equation (CBIE) for 3-D elasticity does not degenerate when applied to shell-like structures, contrary to the case when it is applied to crack-like problems where it does degenerate due to the closeness of the two crack surfaces. The treatment of the nearly singular integrals, which is a crucial step in the applications of BIEs to thin shapes, is presented in detail. To verify the theory, numerical examples of spherical and ellipsoidal vessels are presented using the BEM approach developed in this paper. It is found that the system of equations using the CBIE is well conditioned for all the thickness studied for the vessels. The advantages, disadvantages and potential applications of the proposed BEM approach to shell-like structures, as compared with the FEM regarding modelling and accuracy, are discussed in the last section. Applications of this BEM approach to shell-like structures with non-uniform thickness, stiffeners and layers will be reported in a subsequent paper. © 1998 John Wiley & Sons, Ltd.     

无网格FEM-BEM法计算舱室空间声传递函数

对于舱室空间低频声场计算问题,通常采用有限元法(FEM)或者边界元法(BEM),但是当封闭空间内部存在其他物体,又需要考虑外部激励对内场的影响时,直接采用单一的前述方法不能满足建模计算要求,而且这些基于网格的方法在前处理、光顺化处理等方面都存在不足。针对这些问题,将有限元和边界元法结合,并将其无网格化,这对于封闭环境的声场预测是一种新的尝试。首先推导了无网格FEM-BEM计算模型,对微分方程的离散、形函数构造等细节进行了详细阐述,然后以实际算例对提出的方法进行了验证,结果表明,本文的无网格FEM-BEM方法和SYSNOISE的计算结果保持一致;进一步与实测结果的对比表明,该方法在低频率范围内,各测点平均声压级相对误差在5.26%以内。这说明,该方法不仅适用于复杂问题的计算,同时还具有良好的计算精度。     

A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method

This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). In this approach, the material properties and coordinates of an unknown region boundary are taken as the optimization variables, and the sensitivity coefficients are computed by the Complex-Variable-Differentiation Method (CVDM). Due to the use of CVDM, the sensitivity coefficients can be accurately determined in a way that is as simple to use as the Finite Difference Method (FDM) and an inverse analysis for a complex composite structure can be easily performed through a similar procedure to the direct computation. Although basic integral equations are presented for heat conduction problems, the application of the proposed algorithm to other problems, such as elastic problems, is straightforward. Two numerical examples are given to demonstrate the potential of the proposed approach.     

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.     

声学-结构设计灵敏度分析

声学 -结构设计灵敏度分析揭示了结构振动引起的辐射声压与结构设计变量之间的关系。分别用有限元法和边界元法计算结构设计灵敏度和声学灵敏度。将两个灵敏度结合得到最终的声学 -结构设计灵敏度。在边界元计算中 ,采用退化元处理奇异积分问题 ,对特征频率不唯一问题采用CHIEF方法处理。以脉动球和箱体为例 ,验证了算法的可行性和精确性。     

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.