A comparison of FE–BE coupling schemes for large‐scale problems with fluid–structure interaction

To predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well‐known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton–Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated. Copyright © 2008 John Wiley & Sons, Ltd.     

Interface‐reduction for the Craig–Bampton and Rubin method applied to FE–BE coupling with a large fluid–structure interface

Component mode‐based model‐order reduction (MOR) methods like the Craig–Bampton method or the Rubin method are known to be limited to structures with small coupling interfaces. This paper investigates two interface‐reduction methods for application of MOR to systems with large coupling interfaces: for the Craig–Bampton method a direct reduction method based on strain energy considerations is investigated. Additionally, for the Rubin method an iterative reduction scheme is proposed, which incrementally constructs the reduction basis. Hereby, attachment modes are tested if they sufficiently enlarge the spanned subspace of the current reduction basis. If so, the m‐orthogonal part is used to augment the basis. The methods are applied to FE–BE coupled systems in order to predict the vibro‐acoustic behavior of structures, which are partly immersed in water. Hereby, a strong coupling scheme is employed, since for dense fluids the feedback of the acoustic pressure onto the structure is not negligible. For two example structures, the efficiency of the reduction methods with respect to numerical effort, memory consumption and computation time is compared with the exact full‐order solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Structural‐acoustic coupling on non‐conforming meshes with quadratic shape functions

Fully coupled finite element/boundary element models are a popular choice when modelling structures that are submerged in heavy fluids. To achieve coupling of subdomains with non‐conforming discretizations at their common interface, the coupling conditions are usually formulated in a weak sense. The coupling matrices are evaluated by integrating products of piecewise polynomials on independent meshes. The case of interfacing elements with linear shape functions on unrelated meshes has been well covered in the literature. This paper presents a solution to the problem of evaluating the coupling matrix for interfacing elements with quadratic shape functions on unrelated meshes. The isoparametric finite elements have eight nodes (Serendipity) and the discontinuous boundary elements have nine nodes (Lagrange). Results using linear and quadratic shape functions on conforming and non‐conforming meshes are compared for an example of a fluid‐loaded point‐excited sphere. It is shown that the coupling error decreases when quadratic shape functions are used. Copyright © 2012 John Wiley & Sons, Ltd.     

A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces

Interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities. The conformal decomposition FEM decomposes multimaterial elements of a non‐conforming background mesh into sub‐elements that conform to material interfaces captured using a level set method. As the interface evolves, interfacial nodes move, and background nodes may change material. The present work describes approaches for handling moving interfaces in the context of the conformal decomposition FEM for both weakly and strongly discontinuous fields. Dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first‐order and second‐order time integration methods. The moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one‐dimensional and two‐dimensional test problems, while achieving the expected second‐order error convergence rate in space and time. Copyright © 2014 John Wiley & Sons, Ltd.     

非对称周期结构中耦合波的传播特性

为了揭示周期结构中纵向波和弯曲波的耦合作用,设计了对称和非对称周期结构。考虑子结构中的纵向和弯曲耦合运动,利用导纳法和传递矩阵法,得到了周期单元的传递方程。由于结构中存在多种波的耦合作用,在求解周期单元的传播系数时将出现变态矩阵,采用波型分组法,求得了周期结构中多种波型的传播系数。推导了半无限长和有限长周期结构在纵向力、横向力和弯矩作用下的动态响应。数值计算结果表明,对称周期结构中纵向波和弯曲波的带隙结构相互独立;非对称周期结构中纵向波和弯曲波的耦合明显改变了两种波的带隙结构,只有在两种波阻带重叠的频段内结构上的振动响应才存在衰减。     

T型耦合板结构振动特性研究

本文以T型耦合板为研究对象,在同时考虑面内振动和面外振动条件下采用改进傅立叶级数方法（Improved Fourier Series Method,IFSM）对其自由振动特性进行了计算分析。板结构的面内振动和面外振动位移函数表示为改进傅立叶级数形式,并引入正弦傅立叶级数以解决边界的不连续或跳跃现象。将位移函数的级数展开系数作为广义坐标,采用Rayleigh-Ritz方法对其进行求解。通过对不同边界条件及耦合连接情况下T型板自由振动特性进行计算,并将之与有限元法结果相比较,验证了本文方法的正确性和有效性,为耦合板结构的振动控制提供可靠的理论依据。     

FEM and BEM coupling in elastostatics using localized Lagrange multipliers

The equations produced by the finite and boundary element methods in structural mechanics are expressed in different variables and cannot be linked without modifications. Conventional coupling methods of these numerical techniques have traditionally been based on the use of classical Lagrange multipliers making a direct connection of the two solids through their common interfaces using matching meshes and altering the formulation of one of the methods to make it compatible with the other. In this work, a discrete surface called frame is interposed between the connected subdomains to approximate their common interface displacements, it is treated using a finite element discretization and connected to each substructure using localized Lagrange multipliers collocated at the interface nodes. This methodology facilitates the connection of non‐matching finite and boundary element meshes avoiding modifications to the numerical methods used and providing a partitioned formulation, which preserves software modularity. Copyright © 2006 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 .     

空间桁架结构动力刚化有限元分析

作高速大范围运动的弹性体,由于运动和变形的耦合将产生动力刚化现象,传统的动力学理论难以计及这种影响。本文在有限元方法中首次引入了单元耦合形函数(阵),以此将单元弹性位移表示成为单元结点位移的二阶小量形式。利用几何非线性的应变—位移关系式,在小变形假设条件下确定了单元耦合形函数。在此基础上,根据Kane方程,运用模态坐标压缩,并通过适当的线性化处理,得到了一致线性化的动力学方程。编制了空间桁架结构动力刚化有限元分析程序,仿真算例证明了理论和算法的正确性。     

A posteriori error estimates for the Johnson-Nédélec FEM-BEM coupling

Only very recently, Sayas [The validity of Johnson-Nédélec's BEM-FEM coupling on polygonal interfaces. SIAM J Numer Anal 2009;47:3451-63] proved that the Johnson-Nédélec one-equation approach from [On the coupling of boundary integral and finite element methods. Math Comput 1980;35:1063-79] provides a stable coupling of finite element method (FEM) and boundary element method (BEM). In our work, we now adapt the analytical results for different a posteriori error estimates developed for the symmetric FEM-BEM coupling to the Johnson-Nédélec coupling. More precisely, we analyze the weighted-residual error estimator, the two-level error estimator, and different versions of (h−h/2)-based error estimators. In numerical experiments, we use these estimators to steer h-adaptive algorithms, and compare the effectivity of the different approaches.     

Selective damping method for the weak‐Arlequin coupling of molecular dynamics and finite element method

An artificial damping force is introduced in the weak coupling between the molecular dynamics (MD) and finite element (FE) models, to reduce the reflection of the high‐frequency motion that cannot be transmitted from the MD domain to the FE domain. We take advantage of the orthogonal property of the decomposed velocity in the weak coupling method and apply the damping force only to the high‐frequency part, therefore minimizing its effect on the low‐frequency part, which can be transmitted into the FE domain. The effectiveness of the damping method will be demonstrated by 1D numerical examples with linear force field applied to the atomistic model. In addition, we emphasize the importance of using the Arlequin energy interpolation, which is usually ignored in the weak coupling literature. Non‐uniform rational basis spline functions have been used to interpolate the MD data for the weak coupling method, and the influence of changing the number and order of basis functions on the interpolation accuracy has been investigated numerically. For this work, we restrict our discussion to mechanical problems only, involving only mechanical energy terms (e.g., strain potential and kinetic energy). Copyright © 2013 John Wiley & Sons, Ltd.     

A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily‐shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re‐stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet‐ and Neumann‐type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two‐dimensional linear scalar‐ and vector‐valued elliptic problems are investigated by studying the convergence behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

DOMAIN DECOMPOSITION WITH BEM AND FEM

The Finite Element Method and the Boundary Element Method are two different structure analysis methods with a totally different numerical character. Therefore, it makes no sense to couple these two methods pointwise at the interface. In contrast to a lot of coupling strategies in the past, in this paper a method is constructed where we have coupling of the two different methods in a weak form. As a result we can analyse the given structure with two different grids independent of each other. On this account, we see that the big advantage of the proposed method is in its ablity to couple BEM and FEM. The construction of a robust and reliable numerical algorithm depends on the adaptive control of symmetry and definiteness of the coupling matrix. Therefore, we use an iterative method for solving the boundary integral equation by expanding the Calderon projector in a Neumann series. Numerical results show the preciseness and efficiency of the method. © 1997 John Willey & 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.     

A monolithic,mortar‐based interface coupling and solution scheme for finite element simulations of lithium‐ion cells

This work introduces a novel, mortar‐based coupling scheme for electrode‐electrolyte interfaces in 3‐dimensional finite element models for lithium‐ion cells and similar electrochemical systems. The coupling scheme incorporates the widely applied Butler‐Volmer charge transfer kinetics, but conceptually also works for other interface equations. Unlike conventional approaches, the coupling scheme allows flexible mesh generation for the electrode and electrolyte phases with nonmatching meshes at electrode‐electrolyte interfaces. As a result, the desired spatial mesh resolution in each phase and the resulting computational effort can be easily controlled, leading to improved efficiency. All governing equations are solved in a monolithic fashion as a holistic, unified system of linear equations for computational robustness and performance reasons. Consistency and optimal convergence behavior of the coupling scheme are demonstrated in elementary numerical tests, and the discharge of two different realistic lithium‐ion cells, each consisting of an anode, a cathode, and an electrolyte, is also simulated. One of the two cells involves about 1.35 million degrees of freedom and very complex microstructural geometries obtained from X‐ray tomography data. For validation purposes, characteristic numerical results from the literature are reproduced, and the coupling scheme is shown to require considerably fewer degrees of freedom than a standard discretization with matching interface meshes to achieve a similar level of accuracy.     

Prediction of material coupling effect on structural damping of composite beams and blades

The present paper investigates the effect of material coupling on static and modal characteristics of composite structures. Incorporation of stiffness and damping coupling terms into a beam formulation yields equivalent section stiffness and damping properties. Building upon the damping mechanics, an extended beam finite element is developed capable of providing the stiffness and damping matrices of the structure. Validation cases on beams and blades demonstrate the importance of all stiffness and damping terms. Numerical results validate the predicted effect of material coupling on static characteristics of composite box-section beams. The effect of the full coupling damping matrices on modal frequencies and structural modal damping of composite beams is investigated. Box-section beams and small blade models with various ply angle laminations at the girder segments are considered. Finally, the developed finite element is applied to the prediction of the modal characteristics of a 19 m realistic wind-turbine model blade.     

Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

We introduce a new class of explicit coupling schemes for the numerical solution of fluid‐structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566–593, 2013, Numer. Math., 123(1):21–65, 2013] to the case of the coupling with thick‐walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi‐discrete level, using a lumped‐mass approximation in the structure. The fluid–solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameter‐free, Robin–Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

Magnetoelectric coupling in ferroelectromagnets with antiferroelectric and antiferromagnetic orders

This paper reports a theoretical prediction of magnetoelectric-coupling-induced ferromagnetic ordering behavior in ferroelectromagnets with antiferroelectrically and antiferromagnetically ordered components. A Monte-Carlo algorithm based on Janssen model with an intrinsic magnetoelectric coupling between electric polarization and Ising spin is developed. A mean-field approach based on the Heisenberg model is proposed. The simulation reveals that both weak ferromagnetic order and weak ferroelectric order below certain temperature can be activated by the magnetoelectric coupling and by applying external electrical field. The mean-field approach confirms the weak-ferromagnetic phase transitions revealed by the simulation.     

Coupling techniques in boundary-combined methods for solving elliptic problems with singularities

Three coupling strategies in matching the Ritz-Galerkin method and the finite element method are introduced for general elliptic equations, and useful numerical techniques are provided. Numerical experiments have been carried out for solving the typical, singular Motz problem, which shows that optimal convergence rates of numerical solutions can be achieved by using the combined methods and techniques provided in this paper.