FE/BE coupling for an acoustic fluid–structure interaction problem. Residual a posteriori error estimates

Abstract

SUMMARYIn this paper we develop an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid-structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time-harmonic vibration. Our methods combine integral equations for the exterior fluid and finite element methods for the elastic structure. It is well-known that due to the reduction of the boundary value problem to boundary integral equations the solution is not unique in general. However, due to superposition of various potentials, we consider a boundary integral equation which is uniquely solvable and which avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures are considered; one is based on the non-symmetric formulation and the other one is based on a symmetric formulation. For both formulations we derive reliable residual a posteriori error estimates. From the estimators we compute local error indicators which allow us to develop an adaptive mesh refinement strategy. For the two dimensional case we perform an adaptive algorithm on triangles and for the three dimensional case we use hanging nodes on hexahedrons. Numerical experiments underline our theoretical results. Copyright c 2000 John Wiley & Sons, Ltd.key words: Coupling of finite elements and boundary elements, fluid-structure interaction problem, residual a posteriori error estimator, adaptive algorithm.

Boundary Integral Equation Methods for the Two-Dimensional Fluid-Solid Interaction Problem

“…One popular method to overcome the difficulty that the acoustic scattered wave propagates in an unbounded domain is known as the Dirichlet-to-Neumann (DtN) method ( [9,28,29]), that is, the original transmission problem is reduced to a boundary value problem by introducing a DtN mapping defined on an artificial boundary enclosing the elastic body inside. Another conventional numerical method is the coupling of the boundary element method (BEM) and the finite element method (FEM) ( [7,8,[10][11][12][13][14]24]). Precisely, the BEM and FEM are employed for solving fields of the exterior acoustic wave and the interior elastic wave, respectively.…”

Local active control for an exterior fluid‐structure interaction problem

“…Now, we introduce the procedure for finding z op using the linearity of the problem (10). Let (u, + 0 p) ∈ H 1 (Ω) × H 1∕2 (Γ) be the solution of problem (10). We take…”

“…Kress found that taking α = i / k where k is the wave number of the harmonic system is an optimal value to minimize the condition number of the discrete system coming from the boundary integral equations. Bielak et al presented numerical implementations of different discrete weak problems obtained using both procedures, while Dominguez et al used two of the formulations presented by Bielak to develop a posteriori error estimators and adaptive schemes.…”

“…An earlier procedure, given by Brakhage and Werner [6], represents the solution in the exterior region as a linear combination of a single-layer and a double-layer potential evaluated at a density function ; that is,with the coupling constant again required to have a nonvanishing imaginary part.Kress [7] found that taking = i/k where k is the wave number of the harmonic system is an optimal value to minimize the condition number of the discrete system coming from the boundary integral equations. Bielak et al [8] presented numerical implementations of different discrete weak problems obtained using both procedures, while Dominguez et al [9,10] used two of the formulations presented by Bielak to develop a posteriori error estimators and adaptive schemes.Depending on the chosen representation for the pressure p, both procedures could result in larger discrete systems than the usual fe/be coupling schemes, or in the construction of additional matrices induced from boundary integral operators, which correspond to full matrices with a high computational cost [11]. Furthermore, if the procedure of Brakhage and Werner is used, apart from the calculation of the displacement field u in the solid, the density function on the interface must be calculated in order to later calculate the pressure field and its normal derivative on the interface and then, by using the representation, calculate the pressure at any point on the external domain.In this work, we do not consider the representation formula proposed by Brakhage and Werner [6] or by Burton and Miller [4]; instead, we work with the standard representation formula, which is characterized by the present critical frequencies.…”

A BIE-Based DtN-FEM for Fluid-Solid Interaction Problems

“…Recently, several numerical methods have been studied for the solution of the fluid-solid interaction problem including the boundary integral equation (BIE) method [31,39] and its coupling with the finite element method (FEM) [8,9,16,27,33]. For the coupling scheme, a popular way is to use the BIE methods to solve the acoustic problem outside the obstacle while FEM is employed for the approximation of the interior elastic wave.…”