## 1. Introduction

Commercial transport aircraft often fly above open waters and, thus, have to prove the safe landing on water. This process is also referred to as ditching and is part of the certification specifications by the European Aviation Safety Agency (EASA) and the Federal Aviation Administration (FAA) (see [

1,

2]). Those regulations aim for minimizing the risk of injury for the occupants of the aircraft during the impact phase as well as allowing a safe evacuation in the subsequent floatation phase. This requires an accurate prediction of the structural behaviour of the aircraft during the impact phase as well the damage and its influence on the floatation capabilities.

A common practice to certify aircraft for ditching is to compare the new structural design with existing aircraft configurations [

3]. As this approach impedes the development of novel, unconventional aircraft configurations, contemporary aircraft design relies on alternative methods to analyse ditching. Experimental investigations of ditching are limited to low horizontal speed (e.g., helicopters [

4]) or are conducted on model scale [

5,

6] where they suffer from scaling effects, particularly with respect to the accurate representation of hydrodynamic effects [

7]. Consequently, there is a motivation to employ numerical tools to simulate ditching. Those tools can be broadly classified into

high-fidelity and

low-fidelity approaches. High-fidelity methods are based upon general computational mechanics and usually employ particles or a mesh to discretize the aircraft structure and the surrounding fluid. Conversely, low-fidelity methods discretize structure and fluid in a simplified manner and are based on analytical or semi-analytical formulations (see e.g., [

8,

9]). Those tools are often used in the context of a one-way coupling approach, where the hydrodynamic loads are calculated by simplified models and are subsequently applied to a high-resolution finite element (FE) model. In summary, it can be stated that, due to the impracticability of full-scale tests, the limitations of model scale experiments as well as the simplifications in numerical approaches, there is a demand for advanced, high-fidelity simulation methods.

The development of such methods is an active field of current research. Many publications focus on the simulation of ditching events at zero or low horizontal speed (see e.g., [

4,

10,

11]). This condition is normally met during the ditching of helicopters for which results of full-scale experiments are available and are used to validate simulations [

4]. Conversely, the ditching of fixed-wing aircraft involves a significant horizontal speed and, therefore, hydrodynamic effects such as cavitation, ventilation and the formation of bow waves and spray jets need to be considered [

3,

12]. These physical effects present additional numerical challenges compared to helicopter ditching.

Furthermore, the large hydrodynamic loads arising in the impact zone usually rise and fall within milliseconds and therefore require small time step sizes in the fluid simulation [

3]. Those loads result in local structural deformation and possibly failure. Moreover, failure occurring in fuselage frames usually occurs locally as well. If modelled using a high-resolution FE discretization, this may lead to large differences in model scale [

13].

As the flexibility of the structure has a significant effect on the hydrodynamic loads acting on the structure during the ditching event, fluid-structure interaction (FSI) needs to be considered [

14]. Due to the specific requirements of the individual solvers, monolithic coupled solvers are not practical. Instead, a partitioned coupling approach is required.

The process of ditching is accompanied by a strong added-mass effect of the fluid on the structure [

15]. This gives rise to numerical instabilities in the coupled solution, particularly if fluid and structural densities are comparable, the structure is slender and incompressible flow is assumed [

16,

17,

18]. Iterative (implicit) coupling methods have been widely adopted to simulate strongly coupled problems with a pronounced added-mass effect (see e.g., [

19,

20]). However, those methods may also suffer from stability issues and increased computational effort [

18]. It is therefore crucial to employ an efficient partitioned procedure in order to make highly resolved ditching simulations feasible.

Current high-fidelity methods usually employ the FE method to simulate the structural response of the aircraft. Conversely, the fluid domain typically is discretized by using either a mesh-free approach (see e.g., [

21]) or a hybrid approach. The latter combines classical Lagrangian FE methods with smoothed-particle hydrodynamics (SPH) to alleviate the problem of growing computational effort with larger fluid domains (see e.g., [

3,

6,

22]). The advantage of such particle methods is that they eliminate the issue of mesh deformation in the fluid domain. However, cavitation is only accounted for in a simplified manner by cutting off the pressure as soon as it falls below a defined minimum value [

3,

23]. Hence, both phase changes and the presence of vapour are not considered, although those effects may considerably influence the hydrodynamic loads of the fluid on the structure [

8,

24]. This can be remedied by using the finite volume (FV) method for which cavitation models are already available and tested (see e.g.,

interPhaseChangeFoam solver in

OpenFOAM 5).

The focus of this study is twofold: a suitable fluid-structure coupling framework is presented and validated using the two-dimensional drop test of a rigid cylinder and a deformable cylindrical shell. Furthermore, the process of developing a multiscale model, which simulates the failure of fuselage frames, is illustrated. The paper is structured accordingly: the next section introduces the numerical framework as well as an implicit fixed-point coupling scheme. In

Section 3.1, this scheme is employed to simulate the validation problems mentioned above. The elastic validation problem will additionally be simulated using a simple staggered coupling scheme.

Section 3.2 deals with the development of a multiscale structural model of the aircraft fuselage. The particular challenge is to accurately model structural failure while limiting the computational effort to a manageable amount. A summary along with conclusions are given in

Section 4.

## 2. Numerical Simulation Framework

This study employs the coupling environment

ifls, which has already been used to model various multi-physics problems (see e.g., [

25,

26,

27]).

Ifls follows a modular approach that allows to couple two or more black-box solvers in a partitioned way. The usage of the interpreted, high-level programming language

Python facilitates a flexible and easy implementation of the desired coupling scheme. Time-critical operations are performed using compiled code as well as the

NumPy library.

Figure 1 shows the individual processing units of

ifls. Simulation steering and solver communication are performed by the

control code and the

co-processes, respectively. Each of those units is run in a separate thread.

The control code implements the selected partitioned coupling scheme and sends control commands to the co-processes. Those commands provide direct control over the respective solvers and employ a predefined set of instructions that includes, for example, solving a time step or defining and obtaining boundary conditions. Additional instructions for sending and receiving coupling quantities enable access to the coupling data.

NumPy is used to store those data and perform arithmetic operations, thus circumventing the speed limitations of Python. If coupling data is exchanged between different surface meshes, the send and receive instructions implicitly invoke appropriate interpolation routines, which are implemented in

converter objects and employ methods from the

visualisation toolkit (VTK) (The visualization toolkit is Open-Source software, which can be downloaded from

https://www.vtk.org.) As the interpolation is computationally more expensive, all interpolation filters are implemented in

C++. Another advantage of the usage of VTK is that it facilitates the easy visualization of the simulation results.

Each co-process implements the instruction set used by the control code where basic features common to all co-processes, for example, inter-thread communication and quantity array management, are separately implemented in a generic parent class. Communication between a co-process and the corresponding solver can be achieved in different ways. In this case, a socket-based approach is employed, which has superior performance compared to a simple file-based data exchange. Further information on the coupling environment

ifls as well as its underlying principles can be found in Reference [

28].

For the simulation of the fluid domain, the finite-volume solver

interDyMFoam, which is part of the

OpenFOAM 5 toolbox, is employed. It solves the incompressible Navier-Stokes equations and is able to model multiphase flow by means of the

Volume of Fluid (VOF) method. The fluid equations are solved in the

Arbitrary Lagrangian-Eulerian (ALE) form to account for mesh movement in the convective term. The communication between

ifls and

OpenFOAM is achieved by using a socket-based data exchange as presented in Reference [

29]. The structure is simulated with the implicit

Abaqus solver assuming linear elastic material behaviour.

The general partitioned FSI problem can be formulated as follows. The computational domain

$\mathsf{\Omega}$ is divided into the structural domain

${\mathsf{\Omega}}_{S}$ and the fluid domain

${\mathsf{\Omega}}_{F}$ including their respective boundaries

$\partial {\mathsf{\Omega}}_{S}$and

$\partial {\mathsf{\Omega}}_{F}$. Both domains share the common boundary

${\Gamma}_{i}\stackrel{\mathrm{def}}{=}\partial {\mathsf{\Omega}}_{S}\cap \partial {\mathsf{\Omega}}_{F}$, referred to as the FSI interface, on which the coupling values are exchanged. On the fluid side of the interface, the Dirichlet boundary condition

is applied, where

${\mathit{v}}_{\mathbf{f}}$ and

${\mathit{v}}_{\mathbf{s}}$ represent the fluid and structural velocity vector. Equation (1) corresponds to a no-slip velocity boundary condition in the fluid domain.

Conversely, the fluid forces are applied as a Neumann boundary condition to the structural domain:

The quantities ${\mathit{\sigma}}_{\mathbf{f}}$ and ${\mathit{\sigma}}_{\mathbf{s}}$ represent the fluid and structural stress tensor, respectively, whereas $\mathit{n}$ denotes the unit normal vector.

Ditching exhibits typical characteristics of a strongly coupled FSI problem. For that reason, the implicit fixed-point iteration procedure summarized in

Figure 2 is employed. The variables

${\mathit{u}}_{\mathbf{\Gamma}}$ and

${\mathit{v}}_{\mathbf{\Gamma}}$ contain the nodal displacements and velocities at the interface

${\mathsf{\Gamma}}_{\mathrm{i}}$, whereas

${\mathit{p}}_{\mathbf{\Gamma}}$ represents the nodal forces. The operator

$\mathit{S}$ describes the solution in the corresponding computational domain. The mapping of the coupling values at the FSI interface from fluid to structure and vice versa is expressed by

${\mathit{T}}_{\mathbf{\Gamma},\mathbf{fs}}$ and

${\mathit{T}}_{\mathbf{\Gamma},\mathbf{sf}}$. The subscripts

$\mathrm{s}$ and

$\mathrm{f}$ denote the structural and fluid domain. The index

$n$ represents the current time increment, whereas

$i$ stands for the current iteration.

The fixed-point scheme is usually suitable for strongly coupled problems, although it requires several equilibrium iterations per time step. To accelerate the convergence, the

Aitken relaxation method is used to calculate the relaxation factor

$\omega $ [

30]. Convergence is checked at the end of each iteration by the following displacement-based criterion:

where the constant value

$C$ ensures that the denominator does not become zero.

## 4. Conclusions and Outlook

In this study, a fluid-structure coupling framework suitable for ditching simulations is presented and validated. The partitioning of the computational domain is done by the standard Dirichlet-Neumann approach. It is shown that this framework works well for two-dimensional cases with rigid and deformable structures if an implicit fixed-point iteration scheme is used. However, in the deformable case, this scheme proves to be inefficient for the iterative coupling scheme and even becomes unstable if an explicit structural solver is used together with a simple staggered coupling approach. Those results indicate that carefully designed coupling procedures need to be developed to increase computational efficiency and facilitate the usage of explicit structural solvers in the simulation of ditching problems. In this context, a simple staggered coupling scheme would be favourable but certainly requires partitioning approaches other than standard Dirichlet-Neumann boundary conditions. Possible options include operator-splitting techniques (e.g., [

40]), mixed boundary conditions (e.g., [

41,

42]) or the weak introduction of coupling conditions (e.g., [

43]). For full-scale ditching simulations, which include the failure of the structure, the time scales of fluid and structural simulation are expected to differ significantly. Hence, sub-cycling of the structural solver will likely be necessary to keep the computational effort at a manageable amount.

Furthermore, as soon as horizontal speed is introduced and mesh deformation becomes an issue, the overset grid technique is a viable option to allow large movements of the fluid-structure boundary while at the same time preserving the quality of the fluid mesh [

44].

The second key point of this study pertains the development of a multiscale structural model of the aircraft fuselage. It is shown that a structural model that on the one hand can capture the overall dynamics of the aircraft and, on the other hand, directly simulate structural damage is not feasible. Due to severe nonlinearities in the structure, explicit time integration is needed. However, simply refining damage relevant regions would lead to impracticable short time steps and very poor overall computational efficiency. A method is presented how the highly localized damage behaviour of aeronautical structures can be incorporated in full-size aircraft models while retaining a model size that can be handled by today‘s computers. This is done by reduced order models that account for the correct kinematics of structural damage and the combination of a GFEM model with a locally refined section of interest. An iterative process is proposed how relevant regions, where reduced order damage models should be placed, can be identified.