Coupling Methodology

All algorithms for the simulation of fluid-structure interaction problems fall into one of two major categories: Monolithic algorithms solve the equations governing the flow field and those governing the structural deformation simultaneously as a single set of equations [7, 21]. Partitioned algorithms employ dedicated solvers for each field which are coupled via a suitable interface. The monolithic method en­sures that the mutually dependent solutions in each field are always on the same time level, which eliminates the issue of synchronising individual solvers for a con­servative solution. In practise, though, this method has one significant disadvantage, which has limited its acceptance: A monolithic coupled solver generally has to be developed completely from scratch, whereas with a partitioned approach one can employ pre-existing single-field solvers and benefit from the developments of spe­cialised research groups. Ideally, the necessary coupling interface should be suf­ficiently modular to allow the replacement of a single-field solver either with an updated version or with an entirely different implementation. The aeroelastic code package conceived at LFM/CATS, henceforth denoted as “Aeroelastic Coupling Module” (ACM) [9, 24, 25], is based on this rationale. The ACM allows the modu­lar coupling of arbitrary flow and structural solvers with only minor code changes. Both steady simulations with a staggered (Block-Gauss-Seidel) algorithm and un­steady simulations with weak and strong temporal coupling schemes are possible. Pursuant to the partitioned approach followed here, the ACM serves as the interface between the dedicated single-field solvers for the flow field and for the structural deformation, as is shown in Fig. 1. The ACM carries out the synchronisation of the solvers by initiating iteratively their respective calls.

Apart from the synchronisation of the single-field solvers, the ACM also per­forms the spatial coupling, i. e. the projection of loads from the wetted surface to the structure and in reverse direction the projection of the structural deformations to the wetted surface. The projection methods available are tailored to reduced struc­tural models. These are beneficial especially during unsteady simulations because of their smaller number of degrees of freedom and thus lower requirements of com­putational resources. With reduced structural models the geometries of the wetted surface and of the structural model coincide only in parts or not at all. This is es­pecially true for beam models which do not even share the same dimensionality as the wetted surface. Also with more detailed models like shell models, in many cases one does not want to represent the complete structure. When modelling a wing, of­ten only the wing box is taken into consideration. The high lift devices and other components which do not contribute significantly to the overall structural stiffness are disregarded. In both examples there are “gaps” between the wetted surface and the structural model which have to be bridged by the projection algorithm.

Because of these gaps between wetted surface and structure, forces have to be projected from the wetted surface to the structure instead of surface stresses. Latter do not possess an effective direction required in this case. As to increase the mod­ularity of the ACM, the aerodynamic surface forces are calculated already inside the flow solver and passed on to the ACM. They should be derived in a consistent manner from the discrete distribution of surface stresses [12]. The ACM receives a cloud of load incidence points with associated force vectors and returns a cloud of surface coordinates representing the deformed wetted surface. Thus, the ACM can be coupled with structured and unstructured flow solvers alike since it is independ­ent of the manner in which points are associated with surface cells. As a side note, the number of the load incidence points and their position in the undeformed wet­ted surface do not have to be identical with the surface points defining the wetted surface.

To summarise above statements, the aeroelastic coupling with the ACM com­prises three steps:

1. From the pressure and surface stress distribution on the wetted surface, discrete force vectors are determined by the flow solver.

2. These surface forces are projected from the wetted surface to the nodes of the structural model.

3. The structural deformations resulting from the projected load distribution are projected back onto the wetted surface.

Each of these steps may contribute to the total error of the coupled simulation scheme. The second and third ones are closely related, though: The projection of (generalised) forces from the wetted surface to the structure can conveniently be ex­pressed as a matrix-vector product with a force projection matrix Pf, and likewise the projection of (generalised) deformations can be expressed with a deformation projection matrix PU:

The conservativity of the projection method is assured if Pu = PTF, which can be shown via the principle of virtual work [12]:

SWcfD = FCfd ^UCFD SWcsD = FcSd ^UCSD

= FCFD PU ^UCSD = (PFFCFD)T ^uCSD (2)

= FCFD PF ^UCSD

Consequently, the same projection method has to be used during the projection of forces as during the projection of deformations.

A fourth step, external to the ACM, involves the deformation of the CFD volume mesh in order to accommodate the deformed wetted surface. Mesh deformation methods generally depend on the formulation of the flow solver employed, and their associated error sources are not investigated here.