On the coupling of boundary integral and finite element methods

We prove some error estimates for a procedure obtained by combining the boundary integral method and the usual finite element method. This work was carried out while Franco Brezzi was visiting the Department of Computer Science at Chalmers Institute of Technology during September 1977.     

An investigation of boundary element methods for the exterior acoustic problem

This paper is concerned with the efficient determination of acoustic fields around arbitrary-shaped finite structures in an infinite three-dimensional acoustic medium. The boundary integral formulation due to Burton and Miller is used to overcome the non-existence and non-uniqueness problems associated with classical integral equation formulations of this problem. A class of numerical approximation schemes is developed and the results of applying these schemes to a number of test problems are discussed and compared. The choice of the parameters of the method is critically considered. In particular, the gains to be made, in terms of solution accuracy, by using higher-order approximations to the unknown boundary function, rather than the commonly employed piecewise constant representation, are examined in view of their additional computational costs.     

Coupled algebraic multigrid methods for the Oseen problem

We provide a concept combining techniques known from geometric multigrid methods for saddle point problems (such as smoothing iterations of Braess- or Vanka-type) and from algebraic multigrid (AMG) methods for scalar problems (such as the construction of coarse levels) to a coupled algebraic multigrid solver. Coupled here is meant in contrast to methods, where pressure and velocity equations are iteratively decoupled (pressure correction methods) and standard AMG is used for the solution of the resulting scalar problems. To prove the efficiency of our solver experimentally, it is applied to finite element discretizations of real life industrial problems.     

Analysis of finite element methods for the Brinkman problem

The parameter dependent Brinkman problem, covering a field of problems from the Darcy equations to the Stokes problem, is studied. A mathematical framework is introduced for analyzing the problem. Using this uniform a priori and a posteriori estimates for two families of finite element methods are proved. Nitsche’s method for imposing boundary conditions is discussed.     

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations.The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

Mixed finite element method and high-order local artificial boundary conditions for exterior problems of elliptic equation

The mixed finite element method is used to solve the exterior elliptic problem with high-order local artificial boundary conditions. New unknowns are introduced to reduce the order of the derivatives to two. This leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary. Error estimates are obtained for some local artificial boundary conditions with prescribed order. Numerical examples are presented and the results demonstrate the effectiveness of this method.     

A comparison of the finite element and finite difference methods for the analysis of steady two dimensional heat conduction problems

A comparison between the finite difference method and the finite element method for solving the linear Two-dimensional heat conduction equation is presented. In all areas except computer core storage, the finite element method is demonstrated to be superior to the finite difference method for this type of problem.     

A nonconforming finite element method for a singularly perturbed boundary value problem

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh ^1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.     

The coupled finite element and boundary integral analysis of ocean wave loading: A versatile tool

The numerical methods available for the solution of linear ocean wave diffraction and radiation problems are reviewed. The versatile nature of a coupled finite element and boundary integral formulation (BIE) is demonstrated. The important features, advantages and disadvantages of the method are highlighted. It is concluded that this formulation provides a powerful tool for the solution of a wide range of linear inviscid fluid/structure interaction problems.     

A-posteriori local error estimates for some finite and boundary element methods

In this paper some a-posteriori local error estimates for finite and boundary element methods are presented. These results can be efficiently applied to adaptive procedures of FEM and BEM.     

Influence surfaces by boundary element/least square methods coupling

This work presents a new application for calculating the influence surfaces of transverse displacements, directional derivatives and bending moments for generic bridge decks. A plate bending boundary element method formulation is coupled with the application of a continuous field surface derived by the least square procedure. This original BE formulation permits calculating influence surfaces of plates with polygonal, curved or circular geometry, and several transverse load conditions. The proposal allows future analysis of building floors and single and continuous bridge trusses connected to longitudinal and transversal girders. Numerical examples are presented to demonstrate the potential of the present formulation and the results are compared with analytical values and with usual vehicular loads.     

A coupled formulation of finite and boundary element methods in two-dimensional elasticity

A symmetric stiffness formulation based on a boundary element method is studied for the structural analysis of a shear wall, with or without cutouts. To satisfy compatibility requirements with finite beam elements and to avoid problems due to the eventual discontinuities of the traction vector, different interpolation schemes are adopted to approximate the boundary displacements and tractions. A set of boundary integral equations is obtained with the collocation points on the boundary, which are selected by the error minimization technique proposed in this paper, and the stiffness matrix is formulated with those equations and symmetric coupling techniques of finite and boundary element methods. The newly developed plane stress element can have the openings in its interior domain and can be easily linked with the finite beam/column elements.     

All-floating coupled data-sparse boundary and interface-concentrated finite element tearing and interconnecting methods

Efficient and robust tearing and interconnecting solvers for large scale systems of coupled boundary and finite element domain decomposition equations are the main topic of this paper. In order to reduce the complexity of the finite element part from ${\mathcal{O}}((H/h)^{d})$ to ${\mathcal{O}}((H/h)^{d-1})$ , we use an interface-concentrated hp finite element approximation. The complexity of the boundary element part is reduced by data-sparse approximations of the boundary element matrices. Finally, we arrive at a parallel solver whose complexity behaves like ${\mathcal{O}}((H/h)^{d-1})$ up to some polylogarithmic factor, where H, h, and d denote the usual scaling parameters of the subdomains, the minimal discretization parameter of the subdomain boundaries, and the spatial dimension, respectively.     

Fully discrete potential-based finite element methods for a transient eddy current problem

Fully discrete potential-based finite element methods called methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7. This work was supported by Postech BSRI Research Fund-2009, National Basic Research Program of China (2008CB425701), NSFC under the grant 10671025 and the Key Project of Chinese Ministry of Education (No. 107018).     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.