Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

The coupling of boundary integral and finite element methods for the nonstationary exterior flow

In this article, we represent a new numerical method for solving the nonstationary Navier–Stokes equations in an unbounded domain. The technique consists of coupling the boundary integral and the finite element method. The variational formulation and the well-posedness of the coupling method are obtained. The convergence and optimal error estimates for the approximate solution are provided. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 549–565, 1998     

On the coupling of boundary integral and finite element methods with nonlinear transmission conditions

We apply the coupling of boundary integral and finite element methods to study the weak solvability of certain exterior boundary value problems with nonlinear transmission conditions. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane coupled with the Laplace equation in the corresponding exterior domain. The flux jump across the common nonlinear-linear interface is unknown and assumed to depend nonlinearly on the Dirichlet data. We establish the associated variational formulation in an operator equation setting and provide existence, uniqueness and approximation results.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

Strong stability for a fluid–structure interaction model

In this paper we consider the question of stabilization of a fluid–structure model that describes the interaction between a 3‐D incompressible fluid and a 2‐D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with inclusion of the shear stress, which the fluid exerts on the plate. We show that the energy associated with the model decays strongly when the interface is equipped with a locally supported dissipative mechanism. Our main tool is an abstract resolvent criterion due to Tomilov. Copyright © 2008 John Wiley & Sons, Ltd.     

Some formulations coupling finite element and integral equation methods for Helmholtz equation and electromagnetism

Summary. In this paper a study of the coupling between integral equations and finite element methods is presented for two problems of propagation in frequency domain. It is shown that these problems can be viewed as multidomain problems and treated by the mean of the Schur complement technique. The complement coming from the integral equation part is expressed with the integral operators of the scattering theory. This allows to predict the behaviour of the Schur method, either primal or dual, as far as its convergence speed is concerned. Furthermore, the difference of behaviour between electromagnetism and acoustics from this point of view is explained. Received October 21, 1993 / Revised version received February 28, 1994     

Coupling of boundary integral equation and finite element methods for transmission problems in acoustics

Numerical Algorithms - In this paper, we propose a coupling of finite element method (FEM) and boundary integral equation (BIE) method for solving acoustic transmission problems in two dimensions....     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Unified finite element discretizations of coupled Darcy–Stokes flow

In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods proposed here utilize the same finite element spaces on the entire domain. In particular, we show how the coupled problem can be solved by using standard Stokes elements like the MINI element or the Taylor–Hood element in the entire domain. Furthermore, for all the methods the handling of the interface conditions are straightforward. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On mixed finite element methods for linear elastodynamics

Summary We construct and analyze finite element methods for approximating the equations of linear elastodynamics, using mixed elements for the discretization of the spatial variables. We consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth-order in time fully discrete approximations.L ² optimal-order error estimates are proved for the approximations of displacement and stress.Work supported in part by the Hellenic State Scholarship Foundation     

A new coupling of mixed finite element and boundary element methods for an exterior Helmholtz problem in the plane

This paper deals with the scattering of time harmonic electromagnetic waves by an infinitely long cylinder containing a non-homogeneous conducting medium. More precisely, we study the transverse magnetic field that solves an interface problem holding between the cross section of the cylinder and the exterior two-dimensional free space. We apply a dual-mixed variational formulation in the obstacle coupled with a boundary integral equation method in the unbounded homogeneous space. A Fredholm alternative is utilized to prove that this continuous formulation is well posed. We define the corresponding discrete scheme by using the lowest order rotated Raviart-Thomas finite elements for the magnetic field and spectral elements for the boundary unknown. Then, we show that the resulting Galerkin scheme is uniquely solvable and convergent, and prove optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments. This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program, and by the Ministerio de Educación y Ciencia of Spain, through the project No. MTM2004-05417.     

The h-p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains

Summary. This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain and with constant coefficients in the exterior domain . This procedure uses the variational formulation of the differential equation in and involves integral operators on the interface between and . The finite elements are used to obtain approximate solutions of the differential equation in and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements. Received February 10, 1996 / Revised version received April 4, 1997     

Finite element method to fluid–solid interaction problems with unbounded periodic interfaces

Consider a time‐harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This article is concerned with a variational approach to the fluid–solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasiperiodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet‐to‐Neumann mappings is proposed and the convergence analysis is performed. The Dirichlet‐to‐Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water–brass and water–brass–water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 5–35, 2016     

Non‐local approach in mathematical problems of fluid–structure interaction

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯₁⊂ℝ³ while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯₂=ℝ³\Ω₁ which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω₁. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity

Summary. We consider a finite-element-in-space, and quadrature-in-time-discretization of a compressible linear quasistatic viscoelasticity problem. The spatial discretization uses a discontinous Galerkin finite element method based on polynomials of degree r—termed DG(r)—and the time discretization uses a trapezoidal-rectangle rule approximation to the Volterra (history) integral. Both semi- and fully-discrete a priori error estimates are derived without recourse to Gronwall's inequality, and therefore the error bounds do not show exponential growth in time. Moreover, the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution (e.g. when using the standard ^k polynomial basis on simplicies, or tensor product polynomials, ^k, on quadrilaterals). When this is not the case (e.g. using ^k on quadri-laterals) the convergence rate is suboptimal in r but remains optimal in h. We also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the DG(r) approximation. Mathematics Subject Classification (2000):65N36Shaw and Whiteman would like to acknowledge the support of the US Army Research Office, Grant #DAAD19-00-1-0421, and the UK EPSRC, Grant #GR/R10844/01. Whiteman would also like to acknowledge support from TICAM in the form of Visiting Research Fellowships.     

Error analysis of finite element methods for mildly nonlinear variational inequalities

The finite element method is applied through the use of a variational inequality to an obstacle problem involving nonhomogeneous boundary data. For piecewise linear conforming trial functions energy norm error bounds are derived.     

Heat transfer analyses by finite element and boundary element methods. A bibliography (1997–1998)

This bibliography contains references to papers, conference proceedings and theses/dissertations dealing with finite element and boundary element analyses of heat transfer problems that were published in 1997–1998.     

Adaptive space-time finite element methods for dynamic linear thermoelasticity

This article focuses on goal oriented error control for dynamic linear thermoelastic problems. To this end, we present a space-time formulation of this problem class. The corresponding space-time Galerkin discretization is the basis for the derivation of the error estimator using the dual weighted residual (DWR) method. A numerical example substantiates the accuracy and efficiency of the presented approach. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)