6 research outputs found
Recommended from our members
FE/BE coupling for an acoustic fluid-structure interaction problem. Residual a posteriori error estimates
This is the author's accepted manuscript. The final published article is available from the link below. Copyright © 2011 John Wiley & Sons, Ltd.In this paper, we developed 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 combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well-known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two-dimensional case we performed an adaptive algorithm on triangles, and for the three-dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results.DFG German Research Foundatio
A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem
We introduce and analyze a fully-mixed finite element method for a fluid-solid
interaction problem in 2D. The model consists of an elastic body which is subject to a
given incident wave that travels in the fluid surrounding it. Actually, the fluid is
supposed to occupy an annular region, and hence a Robin boundary condition imitating the
behavior of the scattered field at infinity is imposed on its exterior boundary, which is
located far from the obstacle. The media are governed by the elastodynamic and acoustic
equations in time-harmonic regime, respectively, and the transmission conditions are given
by the equilibrium of forces and the equality of the corresponding normal displacements.
We first apply dual-mixed approaches in both domains, and then employ the governing
equations to eliminate the displacement u of the solid and the pressure p
of the fluid. In addition, since both transmission conditions become essential, they are
enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the
Cauchy stress tensor and the rotation of the solid, together with the gradient of
p and the traces of u and p on the boundary of the
fluid, constitute the unknowns of the coupled problem. Next, we show that suitable
decompositions of the spaces to which the stress and the gradient of p
belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative
for analyzing the solvability of the resulting continuous formulation. The unknowns of the
solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms
of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and
continuous piecewise linear functions on the boundary. Then, the analysis of the discrete
method relies on a stable decomposition of the corresponding finite element spaces and
also on a classical result on projection methods for Fredholm operators of index zero.
Finally, some numerical results illustrating the theory are presented