Résumé pour le 1er colloque du GDR interactions fluide-structure - 26-27 sept.
2005. Implicit Partitioned Fluid-Structure Interaction Coupling. Michael ...
Résumé pour le 1er colloque du GDR interactions fluide-structure - 26-27 sept. 2005
Implicit Partitioned Fluid-Structure Interaction Coupling Michael SCHÄFER, Holger LANGE, Marcus HECK Department of Numerical Methods in Mechanical Engineering Technische Universität Darmstadt, Petersenstr. 30, 64287 Darmstadt, Germany
[email protected]
1
Introduction
Coupled fluid-solid problems, which are characterized by the interaction of fluid forces and structural deformations occur in many applications in industry and science. For a realistic numerical simulation of such kind of problems one of the crucial issues is the algorithmic realization of the coupling mechanisms. These can be invoked at different levels in diverse manners within the numerical scheme resulting either in a more weakly or more strongly coupled solution procedure (e.g. [1]). In the present paper we consider an implicit partitioned solution approach, which tries to combine the advantages of weakly and strongly coupled schemes in a complementary way. For each time step the (implicit) solution procedure consists in the application of different nested iteration processes for linearization, pressure-velocity coupling, and linear system solving, which are linked by a special predictor-correction iteration for the fluid-structure coupling (following [2]). Within the predictorcorrector iteration an underrelaxation is employed to stabilize the solution procedure. In addition, a multigrid technique for accelerating the computations for the fluid part is involved (e.g. [3]). The method is realized on the basis of the finite-volume flow solver FASTEST, the finite-element structural solver FEAP, and the quasi-standard coupling interface MpCCI. On the one hand the considered approach gives a great deal of flexibility due to its modularity, and on the other hand the implicit predictor-correction scheme ensures a strong numerical coupling which can be controlled by the underrelaxation. By considering a representative test case the scheme is investigated with respect to its functionality and robustness. In particular, the influences of the underrelaxation and the choice of convergence criteria on the convergence properties of the scheme are discussed.
2
Governing Equations
We consider a problem domain Ω consisting of a fluid part Ω f and a solid part Ωs , which regarding the shape as well as the location of fluid and solid parts can be arbitrary. For the fluid domain part Ωf we assume a flow of an incompressible Newtonian fluid. In this case the basic conservation equations governing transport of mass, momentum and energy for a fluid control volume Vf with surface Sf are given by: Z Z d ρf dVf + ρf (vj − vjg )nj dSf = 0 , (1) dt Vf Sf Z Z Z d ρf vi dVf + [ρf vj (vi − vig )nj − Tij ] dSf = ρf ffi dVf , (2) dt Vf Sf Vf ¶ Z Z µ Z d ∂θ g ρf cpf θ dVf + ρf cpf (vi − vi )θ − κf ni dSf = ρf qf dVf , (3) dt ∂xi Vf
Sf
Vf
Résumé pour le 1er colloque du GDR interactions fluide-structure - 26-27 sept. 2005
where vi is the velocity vector with respect to Cartesian coordinates x i , t is the time, θ is the temperature, ρf is the fluid density, κf is the thermal conductivity, cpf is the specific heat at constant pressure, ffi are external body forces (e.g. buoyancy forces) and q f are external heat sources. vig is the velocity with which Sf may move (grid velocity) due to displacements of solid parts. The stress tensor T ij for incompressible Newtonian fluids is defined by Tij = µf
µ
∂vj ∂vi + ∂xi ∂xj
¶
− pδij
(4)
with the pressure p and the dynamic viscosity µf . The model equations for the solid domain Ωs may take rather different forms depending on the concrete problem and coupling mechanisms involved. The models range from a simple rigid body motion without any deformation of the solid up to strongly nonlinear (physically and/or geometrically) deformations. The basic balance equations for momentum and energy for the solid domain Ω s can be written as ∂σij ρs u ¨i − = ρs fsi , ∂xj µ ¶ ∂ ∂θ ˙ αs θ − κs = ρs qs − βs ²˙kk , ∂xi ∂xi
(5) (6)
where ui is the displacement, σij denotes the Cauchy stress tensor, εij is a suitable strain tensor, ρs is the density of the solid, κs its heat conductivity, fsi are external volume forces acting on the solid (e.g. gravitational forces) and qs are external heat sources. αs and βs are coefficients describing the heat capacity and the thermal expansion of the solid, respectively. The solid model equations are completed by a suitable problem dependent constitutive equation relating the stresses with strains and temperature: σij = Wij (ε, θ) . (7) The problem formulation has to be closed by prescribing suitable boundary and interface conditions. On solid and fluid boundaries Γs and Γf standard conditions as for individual solid and fluid problems can be prescribed. For the velocities and the stresses on a fluid-solid interface Γ i we have the conditions vi = u˙ bi and σij nj = Tij nj , (8) where u˙ bi is the velocity of the interface. In addition, the temperatures as well as the heat fluxes have to be identical on Γi .
3 Numerical Techniques For the discretization of the problem domain we follow a block-structuring technique. Domains with different fluid or solid parts can be handled straightforwardly by assigning the different parts to different blocks. Solid blocks are treated by the finite-element solver FEAP (see [7]). For the fluid blocks the parallel multigrid finite-volume flow solver FASTEST is employed (see [9, 6]). Both solvers involve second-order spatial discretizations and fully implicit second-order time discretizations. For the fluid-structure coupling an implicit partitioned approach is employed. In Fig. 1 a schematic view of the iteration process, which is performed for each time step, is given. After the initializations the flow field is determined in the actual flow geometry. From this the friction and pressure forces on the interacting walls are computed, which are passed to the structural solver as boundary conditions. The structural solver computes the deformations, with which then the fluid mesh is modified, before the flow solver is started again.
Résumé pour le 1er colloque du GDR interactions fluide-structure - 26-27 sept. 2005
Initialization
Computation of flow field (finite volumes)
Computation of modified mesh (interpolation)
grid
Computation of wall forces Fw
t=t+∆t
FSI iterations
p,v i ,T
No
Computation of deformations (finite elements)
ui
Convergence
Yes
Figure 1: Flow chart of coupled solution procedure For the mesh deformation in the actual implementation a simple linear interpolation scheme is employed, which gives satisfactory results, if the deformations are not too large. In the fluid solver a discrete form of the space conservation law Z Z d dV = vjg nj dS (9) dt Vf
Sf
is taken into account in order to compute the additional convective fluxes in Eqs. (1)-(3). This is done via the swept volumes δVc of the control volume faces for which one has the relation (see [1]): X δV n c
c
∆tn
=
Vfn − Vfn−1 X g = (vj nj Sf )nc , ∆tn c
(10)
where the summation index c runs over the faces of the control volume, the index n denotes the time level tn and ∆tn is the time step size. This way interface displacements enters the fluid problem part in a manner strictly ensuring mass conservation. The FSI iteration loop is repeated until a convergence criterion is reached, which is defined by the change of the mean displacements:
