Formulation of Fluid-Structure-Interaction by Hybrid ... - CiteSeerX

Gaul, L.; Wagner, M.

Formulation of Fluid-Structure-Interaction by Hybrid Boundary Integral Method For the computation of an acoustic eld generated by a vibrating structure, it is necessary to solve the coupled system of eld equations. The Boundary Element Method can be eectively used for the discretization of both eld boundaries. A shortcoming of some BEM formulations are unsymmetric matrices. This avoids the overall symmetry of the coupled uid-structure system, resulting in a loss of eciency for the solver. Thus, a symmetric Boundary Element Formulation is derived from hybrid variational principles. Hamilton's principle is extended to a multi eld variational principle by decoupling the eld variables in the domain from those on the boundary, leading to a symmetric stiness matrix in a frequency domain formulation. A coupling procedure is introduced by using Hybrid BEM for the acoustic eld as well. A symmetric system of coupled uid-structure equations is obtained.

1. Introduction Coupling of dierent discretization methods is a means to increase computational eciency for multi- eld problems, such as uid-structure interaction. Advantages can be gained by combining Finite-Element-Methods (FEM) with Boundary-Element-Methods (BEM). For the sake of computational eciency the system matrices of both methods should be symmetric. Common BEM, such as the collocation and the Galerkin method [1], [2] lead to nonsymmetric matrices so that symmetric solvers can no longer be used. The Bubnov-Galerkin BEM [3] gives symmetric matrices but the double integration over spatial shape functions requires considerable numerical eort. An alternative approach emerges from variational principles of solid mechanics and leads to symmetry by construction. The principle of minimum potential energy [4] only contains the displacement as unknown eld. By relaxing continuity between the elds on the boundary and those in the domain a multi- eld variational principle has been formulated. Emerging from this, Hybrid Boundary Element Methods (HBEM) have been developed for elastostatics, namely the Hybrid Displacement Model [5]. The generalization of this principle from elastostatics to elastodynamics based on Hamilton's principle was introduced in [6] and applied to the stationary vibrations of a beam in [7]. In what follows the HBEM for the Lame-Navier equations of elastodynamics and the Helmholtz equation for an acoustic medium are presented. After the derivation of the two formulations, coupling conditions for uid-structure interaction are established. They lead to a symmetric system of coupled equations for uid-structure interaction.

2. Hamilton's Principle in Spatial Elastodynamics The HBEM is derived from Hamilton's principle. It states that the solution of a boundary value problem is achieved with the displacement vector ui which minimizes the Lagrangian (ui ) given for a linear-elastic domain with boundary @ = ? as (ui ) =

Zt1 " Z

t0

#

1 [ u_ u_ ?  " + 2 b u ] d + Z t u d? dt ) Min.; i i i i ij ij i i 2

(1)

?t

with Dirichlet boundary conditions ui = ui on ?u , incorporating the kinetic energy, the potential energy, the work of body forces bi and the given tractions ti on the boundary ?t . In equation (1) ij and "ij are the stress- and strain-tensor respectively, both depending on the displacement ui via kinematic and constitutive relations. Thus (1) is a one- eld principle. More exibility is gained by introducing two dierent displacement elds. One is ui in the domain and the second is u~i on the boundary ?. To maintain continuity between the domain and the boundary eld, a compatibility condition holds: u ~i

= ui on ? :

(2)

If the compatibility condition (2) is enforced in Hamilton's principle only in a weak sense weighted by Lagrange

multipliers the extended three- eld variational principle reads HD (ui ; u~i ; i ) =

Zt1 " Z

t0

1 [  u_ u_ ?  " + 2 b u ] d + Z i i ij ij i i 2

i (ui ? u~i )d? +

?

#

Z

ti ui d?

dt ) Stat. ; (3)

?t

with Dirichlet boundary conditions ui = ui on ?u .

3. Hybrid Variational Principle in the Frequency Domain In the following only time-harmonic motion is considered, thus the eld variables ui , u~i and t~i can be separated in space and time harmonic functions, where only the real part of the complex ansatz yields the physical result, e.g.

=