Conference PaperPDF Available

Framework for Fluid-Structure Interaction Simulations with UZEN and PreCICE: Simulations procedure and Validation

Authors:

Abstract and Figures

The paper illustrates a new framework developed to face with Fluid Structure Interaction phenomena in a partitioned approach. The CIRA multi-block structured flow solver for unsteady RANS equations, UZEN, was updated and tightly coupled with an open-source FEM code CalculiX. The solvers are glued in space and time through an open source library preCICE, to deliver exchanging data. preCICE manages the communications, loads mapping and time coupling. Motivation of the work is the simulation of unsteady aerodynamic problems strongly dependent upon structural behaviour, like flexible aircraft, rotor-craft, counter-rotating rotors, etc. As validation tests, the results of 2D and 3D panel flutter response at supersonic velocity are illustrated. The results are compared in terms of Limit Cycle Oscillation amplitude and frequency of panel flutter with data available in literature.
Content may be subject to copyright.
Framework for Fluid-Structure Interaction Simulations with UZEN and PreCICE: Simulations
Procedure and Validation
D. Cinquegrana and P. L. Vitagliano
VIII International Conference on Computational Methods for Coupled Problems in Science and Engineering
COUPLED PROBLEMS 2019
E. O˜nate, M. Papadrakakis and B. Schrefler (Eds)
FRAMEWORK FOR FLUID-STRUCTURE INTERACTION
SIMULATIONS WITH UZEN AND PRECICE:
SIMULATIONS PROCEDURE AND VALIDATION
Davide Cinquegrana, Pier Luigi Vitagliano
Italian Aerospace Research Centre (CIRA)
Department of Fluid Mechanics, Computational Fluid Dynamics Lab.
Via Maiorisi, 80143, Capua, Italy
e-mail: d.cinquegrana@cira.it, web page: http://www.cira.it/
Italian Aerospace Research Centre (CIRA)
Department of Fluid Mechanics, Computational Fluid Dynamics Lab.
Via Maiorisi, 80143, Capua, Italy
e-mail: p.vitagliano@cira.it, web page: http://www.cira.it
Key words: FSI, load transfer, time coupling, panel flutter
Abstract.
The paper illustrates a new framework developed to face with Fluid Structure In-
teraction phenomena in a partitioned approach. The CIRA multi-block structured flow
solver[1] for unsteady RANS equations, UZEN, was updated and tightly coupled with
an open-source FEM code CalculiX[2]. The solvers are glued in space and time through
an open source library, preCICE[3] to deliver exchanging data. preCICE manages the
communications, loads mapping and time coupling.
Motivation of the work is the simulation of unsteady aerodynamic problems strongly
dependent upon structural behaviour, like flexible aircraft, rotor-craft, counter-rotating
rotors, etc.
As validation tests, the results of 2D and 3D panel flutter response at supersonic
velocity are illustrated. The results are compared in terms of Limit Cycle Oscillation
amplitude and frequency of panel flutter with data available in literature.
1 INTRODUCTION
In the present work, a multi-block structured flow solver[4] for unsteady RANS equa-
tions has been coupled with a structural solver within the software environment managed
by the open source library preCICE [3], in order to perform fluid-structure interaction
simulations.
47
Davide Cinquegrana, Pier L. Vitagliano
preCICE (Precise Code Interaction Coupling Environment) is a coupling library for
partitioned multi-physics simulations. The software offers methods for transient equation
coupling, communication means, and data mapping schemes.
An adapter that delivers exchanging data from CFD solver to preCICE library has
been developed, that translates data from CFD to the interfaces, and receives data from
FEM solver through preCICE. The loads transfer, or space coupling, is afforded within the
library with different techniques, such as Nearest-Neighbour or RBF; the time coupling
and convergence check are also managed by the schemes available in the library.
Motivation of the work is the simulation of unsteady aerodynamic problems strongly
dependent from structural behaviour, like flexible aircraft, rotor-craft, counter-rotating
rotors, or in the framework of airplane weight minimization, where flow induced vibrations
influence stability and durability of aircraft[5, 6].
Structural solver, already interfaced within the preCICE framework, is the open source
FEM code Calculix[7].
The choice of the preCICE framework was driven by several reasons: the open source
environment (no costs and limitations stemming from licences), possibility of testing both
explicit and implicit fluid-structure interaction, several interpolation and exchange meth-
ods for forces and deformations already available and implemented. Finally the library
has been developed with the aim of allowing efficient usage of massively parallel comput-
ing architectures[8]. At the same time, the FEM solver, Calculix, is already coupled with
an official adapter. The package is able to perform linear and non-linear analysis of static
and dynamic problems.
As far as we know, this is the only CFD system based upon structured multi-block
meshes fully integrated with a structural dynamic solver for unsteady simulations with
dynamic meshes.
To validate the FSI framework, 2D and 3D inviscid panel flutter at supersonic speed
is investigated. The problem of instability of plates in gas flow has been studied and
reported in literature. In supersonic flow, the instability has the oscillatory nature known
as flutter, that can be distinguished in coupled or single-mode flutter[9]. Several time-
domain simulations of non-linear panel flutter were performed from transonic to supersonic
flows by several authors [10, 11, 9, 12, 13]. From their results at supersonic Mach number
flows we should expect high frequency Limit Cycle Oscillations (LCO).
Since it was discovered, inter alia in the 60s at NASA on the Atlas-Centaur and Saturn
V rocket, considerable effort was spent to investigate the impact on panel flutter of several
structural and aerodynamic parameters. Due to the great cost and complexity involved in
supersonic wind tunnel tests, panel flutter has always been studied mainly via mathemat-
ical and numerical modelling. Focusing on the aerodynamics side of numerical approach
(i.e. the kind of solvers coupled to structural ones to model the aeroelastic phenomena),
in the decade going from 50s to 70s theoretical studies have considered different aerody-
namic modelling: from the linear piston-theory, a quasi-steady relation between pressure
and panel deflection valid for high supersonic Mach numbers in Von Karman’s plate the-
48
Davide Cinquegrana, Pier L. Vitagliano
ory to the unsteady linearized potential flow theory, and to the unsteady ’shear flow’
theory developed by Dowell[14] that takes in account the viscous effects in the boundary
layer on the aeroelastic behavior.
Recent works have introduced FEM and CFD solvers to investigate panel flutter in
high subsonic and low supersonic flows. Euler equations and FEM were coupled to study
non-linear Karman plate equations [10], Navier-Stokes equations to take in account the
unsteady viscous laminar boundary layer were introduced in flutter studies by Gordinier
and Visbal [11] for the solution of non-linear Karman plate equations again. Hashimoto
[12] involved RANS computations to take in account the effects of a turbulent boundary
layer, confirming that the boundary layer has a stabilizing effect on the flutter. Also Alder
[15] focused on the effects of the turbulent boundary layer defining the stability limits and
post flutter characteristics of a 2D and 3D simply supported panel.
Recently, Hejranfar[16] employed a second-order central-difference cell-vertex finite vol-
ume method to study 2D panel flutter in inviscid compressible flow: unsteady Euler
equations governing fluid flows in the arbitrary Lagrangian-Eulerian form and the large
deformation of the solid structure is considered to be governed by the Cauchy equations
formulated in the total Lagrangian form. In his study, on the FEM solver side, the panel
is modelled with plane strain element, where also the thickness is modelled in order to
compute the stress distribution.
In this paper the results of two and three-dimensional analyses of panel flutter are
shown, after presenting the two solvers and the way they are coupled by means of preCICE
library. The 2D panel flutter analysis is suited to set up the numerical approaches to face
with the problem, verifying time and space convergence of the solutions. Finally, the
results of the flutter of square and rectangular 3D plates are compared with literature
data.
2 Aerodynamic Solver
ZEN is a multi-block structured flow solver developed at CIRA for the U-RANS equa-
tions with classical ALE formulation[17, 1]. The spatial discretization is based upon cell-
centred finite volumes discretization, with second and fourth order artificial dissipation.
The unsteady computations are carried out by using an implicit second order backward
difference method together with dual time stepping toward steady state for each physical
time step.
CIRA has recently developed a system for flow simulation[4] which allows the non-
conformal block to block coupling (i.e. sliding mesh) and dynamic mesh on block base
(i.e. some specific blocks in the flow field can be deformed and updated at each time
step). Mesh is updated outside the flow solver, which makes possible to iterate with other
systems to compute, in a segregated approach, structural deformation, body dynamics
and possibly other physical phenomena. Dynamic meshes are implemented by following
the 3 step backward implicit time scheme in such a way to satisfy the discrete geometrical
conservation law (DGCL[6]). The updated mesh is reloaded at each time step, which
49
Davide Cinquegrana, Pier L. Vitagliano
allows to post-process surface deformations due to structural dynamics and, possibly,
body dynamics outside the flow solver. It is possible to apply dynamic mesh modifications
only in some blocks, to save computational time. The code is adapted to face with FSI
problem: in detail, to allows a strong-coupling, the current time step calculations can
be repeated (and the updated mesh reloaded), when required, under control of external
routines that check for suitable convergence criteria. This will described in details in the
following sections.
The flow simulation system communicates by delivering local forces on specific mesh
surfaces, as specified in a set-up file, and it is capable to re-mesh the flow domain starting
from a set of updated geometric entities like surfaces, curves and vertices, by following
specific directives. Geometric entities can be specified or modified by control points.
The adapter code developed for ZEN code is designed as a stand alone software that
synchronizes communications with preCICE library, sending and receiving data to and
from interface grids. It will be described in the following sections.
3 Structural Solver
CalculiX is a free/open-source (GPL) Finite Element package, developed at the MTU
Aero Engines, currently coupled with preCICE with an official adapter in order to face
with Fluid-Structure Interaction (structure part), Conjugate Heat Transfer (solid part).
Through its adapter CalculiX can write displacement, temperature, heat flux, sink tem-
perature, heat transfer coefficient and can read force, temperature, heat Flux, sink Tem-
perature, heat transfer coefficient. According to the type of analysis, different solvers
are available: linear and non-linear, implicit and explicit solver (CCX), written by Guido
Dhondt[7, 2]. Those solvers are implemented in C and Fortran modules.
The implicit solver uses incomplete Cholesky pre-conditioning and the iterative solver
by Rank and Ruecker [7], which is based on the algorithms by Schwarz [2]. The equation
of motion is integrated in time using the α-method developed by Hilber, Hughes and
Taylor[7]. This implicit scheme is unconditionally stable and second-order accurate when
the αparameter lies in the interval [-1/3,0], in order to controls the high frequency
dissipation: α=0 corresponds to the classical Newmark method inducing no dissipation
at all, while α=-1/3 corresponds to maximum dissipation[2].
In this work, Calculix is employed to carry out non-linear analysis of structural dynamic
problems. A ready-to use adapter able to communicate through preCICE is already
available[18].
4 FLUID - STRUCTURE COUPLING: preCICE
This section describes the way CFD and FEM solvers are coupled in a partitioned
approach. The aim is to achieve convergence towards solution in every time step by
executing each solver independently.
In the context of a partitioned approach, crucial aspects are the load and deformations

Davide Cinquegrana, Pier L. Vitagliano
transfer over the interfaces shared by the different computational domains, and the time
coupling of the different solutions, in order to ensure convergence at each time iteration.
Momentum and energy exchanged between the two sub-systems have to be conserved,
otherwise the spurious work introduced leads to instabilities divergence of the solution.
The introduction or removal of spurious energy by the interface scheme may affect the
overall stability properties of the aeroelastic system [19]. Hence, the time synchronization
and the transfer of information at interface for non-matching space discretization influence
the stability of the algorithm.
The coupling between the two solvers is managed by the open-source preCICE library.
The library treats the numerical methods for equation’s coupling among the different
solvers involved in the multi-physics simulations.
From the structural solver side, the architecture illustrated by Rush[3] is adopted, i.e.
the Calculix solver has its adapter that exchanges data with CFD solver through preCICE
library. From the CFD solver side, a black-box adapter coupled with a bash script that is
able to manage the timing processes and the ZEN-interface with preCICE was developed.
The adapter interface developed in this framework is designed as a stand-alone code,
not fully integrated within the CFD solver. Only the communication toward the FEM di-
rection is managed by preCICE: this is configured with a point-to-point based on TCP/IP
socket. The communications between CFD and preCICE direction, are based on file-
transfer.
4.1 Time-coupling
A coupling scheme describes the logical execution order of two participants In the
partitioned approach it is possible to distinguish between a weakly (explicit) and fully
coupled (implicit) schemes: the former solves the fluid and solid sub-domains in a stag-
gered fashion without convergence or residual checks, and the stability of this procedure
is dependent on density ratio (structure vs fluid), temporal discretization precision order,
fluid velocity and flow compressibility[20]. The latter approach foresees that both solvers
are executed multiple times until convergence criteria are satisfied at the end of each time
step. To this family belong the Block Gauss-Seidel (BGS), the Newton and quasi-Newton
strategies.
In the preCICE library either explicit and implicit coupling schemes are available, with
the possibility to apply different convergence criteria.
Furthermore, either serial or parallel executions can be selected. Serial refers to the
staggered execution of one participant after the other. Parallel refers to the simultaneous
execution of both participants.
The strong coupling approaches transform the coupling conditions into Fixed Point
Formulation or Equation (FPE). The kind of FPE used determines the execution sequence
of the two solvers. One is the staggered execution (see Figure 1), while two parallel
approaches introduced by preCICE developers are showed in figure 2, both allowing for
the simultaneous execution of fluid and structure solver.
51
Davide Cinquegrana, Pier L. Vitagliano
Figure 1: Staggered - Implicit scheme
A key role in time coupling is played by the post-processing technique to be applied at
data exchanged between the segregated solvers. A pure implicit coupling without post-
processing corresponds to a simple fixed-point iteration, which still has the same stability
issues as an explicit coupling. A postprocessing techniques is needed in order to stabilize
and accelerate the fixed-point iteration.
In preCICE, three different types of post-processing can be configured: constant (con-
stant under-relaxation), Aitken (adaptive under-relaxation), and various quasi-Newton
variants (IQN-ILS also known as Anderson acceleration, IQN-IMVJ or generalized Broy-
den ).
4.2 Load Transfer
In the frame of partitioned approach, the meshes of the different solvers are not con-
forming at the fluid structure domain interface: they differ in the refinement and gaps
and or overlap can also be present. Then a projection of the variables valued with the
different solvers at interface should be implemented.
Several authors treat those approaches, showing pro’s and con’s[21, 22, 23, 24, 25, 26,
27]. There are several criteria which such a data exchange or coupling method ideally
should satisfy. The most important are: (i) global conservation of energy over the inter-
face, (ii) global conservation of loads over the interface, (iii) accuracy, (iv) conservation
of the accuracy order of the coupled solvers and (v) efficiency, which is defined as a ratio
(a) Parallel - Implicit scheme (b) Vectorial system
Figure 2: Flow Chart of Parallel Approaches for fixed point problem solution[8]
52
Davide Cinquegrana, Pier L. Vitagliano
between accuracy and computational costs.
According to the main hypotheses on which the transfer operator is built, it is possible
to follow a conservative or consistent approach: in the former, the energy is conserved
when transferring displacement, pressure and viscous forces over the interface, and it is
based on the global conservation of Virtual Work over the interface. In the consistent
approach, the constant displacement and constant pressure are exactly interpolated over
the interface. In this case the energy conservation is not guaranteed, however can be
shown that the error in work transferred can be reduced refining the meshes[23].
A number of methods able to transfer data from grid to grid are available, they are
classified in groups: Point-to-Point, Point-to-element and Virtual Surface Method.
In this paper, focus is given on Radial Basis Function (RBF) method, belongs the class
of point-to-point schemes, also known as multivariate transfer technique, they are widely
adopted in the frame of FSI([28, 29, 30, 31, 24]. Those are mesh-less methods allow to
couple structural and fluid domains by reducing them to pure point information. A clear
advantage of this technique is the fact that the information about discretization schemes
and geometrical typologies are not required.
The value of fat a generic location x, is obtained as a weighted sum of radial basis
functions φ(|xxCj|) based on the Euclidean distance between the control points position
xjand x:
f(x) =
Nc
X
j=1
γjφ(|xxCj|) + q(x) (1)
The interpolation condition is also imposed at the control points:
f(xCj) = fCjfor j = 1, . . . , Nc(2)
in order to guarantee the positive definiteness of the RBF problem, the following con-
dition Nc
X
j=1
γjp(xCj) = 0 (3)
should hold for any polynomial pwith degree less or equal than the degree of q.
The basis functions investigated in this work, are Thin Plate Spline and Gaussian
spline:
Thin Plate (TPS):
φ(|x|) = |x|2log |x|(4)
Gaussian Spline:
φ(|x|) = expa|x|2(5)
where ais a shape parameter.
53
Davide Cinquegrana, Pier L. Vitagliano
Radial Basis Functions guarantee a consistent interpolation, but energy is not pre-
served when loads are transferred over the interface. On the other hand, by adopting a
conservative interpolations, the consistency is lost[23]. Moreover, the basis functions need
a tuning for the proper setting of the shape parameters.
4.3 CFD ZEN INTERFACE
The CFD ZEN code is coupled with the FEM solver Calculix through the preCICE
library. From the FEM side, CalculiX exchanges data with preCICE through a devoted
interface[3, 8, 18]. In this section the way the ZEN solver exchanges data with preCICE
is described.
The adapter is designed in a less intrusive way, from the point of view of the ZEN
solver, since the adapter is totally self-standing and very few coding was made within the
solver.
Figure 3 illustrates conceptually how the FSI analysis starts and how data are ex-
changed during the calculation between the ZEN CFD solver and the ZEN adapter: the
latter sends data to preCICE.
The ZEN adapter is coded in cpp language. It is designed to share the CFD variables
(i.e. the non-dimensional loads), available after each time step, with the preCICE library.
The simulation starts by launching a shell script that manages the library initializa-
tion step. The initialization sets up communications between the adapters to preCICE
(from both sides), in order to assign labels, allocate pointers and memory for deforming
grids nodes. Moreover, the interface reads information about the face numbering for the
exchange of local loads, those have to match the structural mesh.
As the CFD code solves non-dimensional equations, reference dimensions are required
in order transform local and global forces, i.e. free-stream static pressure and static
temperature, time step and scale length. When launched, the CFD adapter employs the
information about free-stream values, to calculate the dimensional loads.
In order to explain how the FSI simulation works, it is convenient to describe first
how the CFD solver ZEN performs a flow simulation with dynamic meshes. When a
flow simulation with dynamic meshes is running, ZEN requires an updated mesh at each
new time step, together with a flag that informs the flow solver whether the time step
is completed or the updated mesh has to replace the current one without advancing in
time. The latter case happens when implicit coupling occurs and iterative loops are exe-
cuted with other models. Handling the mesh updating outside the flow solver has several
advantages. Among them it allows different models to interact with the configuration
geometry (like structural dynamics, body dynamics, configuration changes driven by time
laws or any kind of control systems, and so on) and it makes possible to check convergence
on both aerodynamic forces and configuration shape by using procedures external to the
flow solver. Figure 4 highlights the main tasks performed during a time step. The ZEN
flow solver reads a GRID file and writes output files containing (non dimensional) local
and global aerodynamic forces, then it waits for an updated GRID. Meanwhile the ZEN
54
Davide Cinquegrana, Pier L. Vitagliano
Figure 3: Global chart
adapter reads the aerodynamic forces, computes dimensional loads and sends them to the
FEM solver CalculiX through preCICE library. The FEM solver computes displacements,
preCICE routines checks loads and displacements for convergence and returns displace-
ments to the adapter, together with information whether to advance in time or to repeat
the computation to improve residuals. The adapter writes output files containing the up-
dated shape of configuration surfaces, required to make a new GRID, and the FSI.LOOP
flag to inform ZEN about the time step convergence. As the updated surfaces are ready,
the procedure to update the dynamic grid produces an updated GRID. The CFD solver
checks whether to advance the time step or repeat it by reading the FSI.LOOP flag, then
it reads the updated GRID and computes aerodynamic forces again. In figure 4 is also
shown how another model which requires aerodynamic forces and computes configuration
movements could fit into the procedure, by communicating information to the procedure
to update the GRID. More details about the tasks are given in the next paragraph.
5 RESULTS
In this section the output of the 2D panel flutter investigation will be shown, aimed to
explore the preCICE capability and to find a better tuning to obtain reliable results in
an efficient way. The investigation has regarded the load transfer methods, time-coupling
and post-processing scheme, and convergence threshold. Finally, 3D panel flutter results
are compared with literature data.
5.1 Numerical Settings
In order to set the numerical methods discussed in previous sections, preCICE can be
configured at runtime via an xml file (precice-config.xml), where the solvers, the fluid and
solid interfaces exchanging data, the kind of data (i.e. scalars, vectors) are defined and
55
Davide Cinquegrana, Pier L. Vitagliano
Figure 4: Detailed procedure chart
tagged.
The nature of the problem under investigation requires a strongly coupled approach, as
noted also from [11], hence an implicit coupling was chosen. To stabilize and accelerate the
fixed-point iteration several post-processing schemes were tested, such as Aitken and the
two Quasi Newton variants: IQN-ILS also known as Anderson acceleration, IQN-IMVJ
or generalized Broyden.
For implicit coupling scheme, preCICE employs a convergence measure that can be
relative or absolute. Here a relative convergence threshold values for displacements and
forces is adopted. The FEM solution is taken as reference: all data relative to FEM
are post-processed and the convergence is valued on forces and displacement elaborated
from the FEM side. The effects of threshold are investigated for forces and displacement
ranging from 103to 106for both displacement and forces.
Other parameters in the coupling schemes and postprocessing have to be set, such as
starting relaxation parameters applied to delivered forces and or displacements, maximum
number of iterations and extrapolation order. The maximum number of sub-iterations
within an implicit coupling loop was set to 30.
On fluid side, in order to compare the results with other authors without the uncer-
tainties stemming from boundary layer thickness and turbulence model effects, the flow
is solved with Euler equations.
Even if the non-linear FEM solver allows subcycling within each iteration by changing
the time step size to achieve convergence of loads, the current Calculix adapter cannot
56
Davide Cinquegrana, Pier L. Vitagliano
manage sub-cycling. Hence the user-defined initial time step is not changed.
5.2 2D Panel Flutter
Here we investigate how the amplitude and the frequency of the LCO changes, accord-
ing to the critical dynamic pressure, λ, defined as λ=ρu2
a3
D, where D is the plate
stiffness, Eh3
12(1ν2). Hence, by changing the Young modulus of the plate, we have obtained
a set of λvalues at which the analysis are carried out. The mass ratio, µ, calculated as
µ=ρa
ρmh, is kept constant during the simulations and equal to 0.1. The thickness ratio,
h/a is equal to 0.002. Free stream Mach number is set to M= 1.2, furthermore pressure
and static temperature are fixed and equal to 25000 Pa and 223 K. The panel is pinned
at ends, the length is a = 0.5 (m) with uniform thickness h= 1.00 ×103m.
The CFD grid, shown in Figure 5, is made of a single block with three different grid
density that will be tested to verify their influence on the LCO calculations showed in the
following. The coarser grid, L1, has 105x80 cells, the medium grid L2, 169x80 cells and
the finest grid, L3, has 297x80 cells. The flexible interface surface that defines the panel
is hence sampled respectively with 64, 128 and 256 cells.
The FEM model of the panel is made of quadratic plane-strain elements with 8 nodes,
that would be able to model an infinite panel in the cross-wise direction, in fact, as stated
by Calculix[2] manual, they are used to model a slice of a very long structure.
For 2D calculations, the FEM element chosen is a second order Plain - Strain one
(CPE8, see in [2]).
x/a
z/a
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
0
1
2
(a) CFD computational domain
(b) FEM panel discretization
Figure 5: CFD grid
Dependency on spatial resolution and time step, from CFD side, is evaluated by con-
sidering as a term of comparison the maximum vertical panel displacement, localized at
75% of plate length, normalized with panel thickness, w/h, that identifies the normalized
57
Davide Cinquegrana, Pier L. Vitagliano
DTF
w/h
0 0.05 0.1 0.15
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
L1 grid
L2 grid
L3 grid
M = 1.2
λ = 100
µ = 0.1
x/a = 0.75
a/b = 0
a/h = 2.0e-03
pi
(a) LCO amplitude convergence
DTF
frequency [Hz]
0 0.05 0.1 0.15
130
132
134
136
138
140
142
L1 grid
L2 grid
L3 grid
M = 1.2
λ = 100
µ = 0.1
x/a = 0.75
a/b = 0
a/h = 2.0e-03
pinned edge panel
(b) LCO frequency convergence
Figure 6: solutions at different time-step size
LCO amplitude. The physical time step of the CFD solver is varied from 0.1 to 0.01 (plus
0.005 for L2 grid), while the grid density changes according to the discretization of the
panel, from 64 to 256 CFD grid points.
Figure 6 (a) resume the spatial and temporal study, taking in account both the effects
of time and grid refinement and showing the maximum displacement of the point at 0.75a
versus timestep sampling, for L1, L2 and L3 grid levels: the maximum displacement
decreases as timestep is reduced, and slightly increases with grid level refinement.
Figure 6 (b) is referred to the LCO frequencies. The range of frequencies is within 5%
with respect to the finest timestep frequency solution on L2 grid level.
In the following, the solution with timestep 0.025 is considered the reference, in order
to compare CIRA results with those available in literature.
Figure 7 shows the flowfield pressure coefficient contours in the expansion phase φ= 90,
in which are noticeable expansion fans at leading and trailing edges; and compression
phase, φ= 270, with presence of shock waves on leading and trailing edges. In this phase
the panel showed a deformation in the form of its first natural mode.
In figure 8 are shown both the deformation an the relative pressure coefficient distri-
bution in the expansion and compression stages.
The two solvers start together the integration from free-stream values, and an initial
vertical displacement is assigned to the flat panel, equal to δw =w0sin2(πx
L) where w0,
is a constant suitable to produce an initial perturbation. The choice of the modulus of
this perturbations influence the time needed to reach the oscillating period of the panel.
Even if it can be convenient to simulate the minimum transient time to reach LCO, the
modulus of displacements has an influence on the LCO amplitude, as stated by [9]. Figure
9 shows the effects of modulus w0on the final LCO.
58
Davide Cinquegrana, Pier L. Vitagliano
0.017
0.023
0.020
0.003
0.003
0.027
0.007
-0.003
0.013
-0.003
0.000
0.003
-0.007
0.003
0.000
-0.023
-0.013
0.000
-0.017
0.007
-0.013
-0.003
-0.043
-0.010
-0.030
-0.020
-0.013
0.020
0.010
0.017
0.000
x/a
z/a
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Cp
0.050
0.040
0.030
0.020
0.010
0.000
-0.010
-0.020
-0.030
-0.040
-0.050
(a) φ= 90, minimum deflection
-0.023
-0.023
-0.023
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.040
-0.020
0.003
0.010
-0.013
x/a
z/a
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Cp
0.050
0.040
0.030
0.020
0.010
0.000
-0.010
-0.020
-0.030
-0.040
-0.050
(b) φ= 270, maximum deflection
Figure 7: Pressure Coefficient contours at minimum and maximum deflection, for dt=0.025
The higher w0, the shorter time needed to reach a stable LCO. This is not the only
effect since the modulus of w/h maximum displacement at 75% of panel length starts to
change, up to reach a value of 1.556 with the highest initial perturbation.
Figure 10 shows the time-history, the phase portrait and the spectrum of a point at 75%
of panel length, for increasing λ. The LCO becomes wider as the λgrows, but the other
noticeable phenomena is that it becomes non-symmetric with larger λ. Such phenomena
was evidenced also by Shishaeva[9], even if the analysis was relative to increasing Mach
number. The progressive switch from symmetric to non-symmetric LCO is due to the
appearance of the second mode, whose frequency yields increase of plate velocity: the
flutter is changing from non-resonant to resonant LCO[9]. This is evident the phase-
velocity portrait: in the Figure 10 a, it is slightly non-elliptic due to the triple-harmonic,
evidenced on the relative frequency spectra on the right. As pressure grows, the 2nd
mode amplitude grows as well because to the internal resonance, the velocity increases
on phase-velocity portraits, and the relative importance of the 2nd peak of frequency also
increases, as it is shown in the spectra on the right.
The results are compared in terms of flutter amplitude and frequency with some data
available in literature: figure 16 (a) shows that the maximum displacement foreseen at
0.75a panel point is aligned with recent works that are based on coupling of inviscid CFD
codes on fluid side, in the case of pinned panel. Compared to Alder[15], the instabilty of
post-flutter is anticipated to lower λ. Figure 16 (b) is relative to LCO frequency: in this
case the trend is similar to other data, even if a shift toward higher frequencies can be
noted.
After that physical results have been shown, some numerical considerations about the
time coupling techniques follow. The average number of coupling iterations, k, shown
59
Davide Cinquegrana, Pier L. Vitagliano
x/a
Cp
w/h
0 0.5 1 1.5
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08 -2
0
2
4
6
8
φ = 270°
x/a
Cp
w/h
0 0.5 1 1.5
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08 -2
0
2
4
6
8
φ
Figure 8: Panel shape and Pressure Coefficient during flutter at M = 1.2, λ= 100
TF[-]
w/h @ 0.75a
0 2 4 6 8 10
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 (w/h start)MAX = 3.5e-1
(w/h start)MAX = 3.5e-2
(w/h start)MAX = 3.5e-3
(w/h start)MAX =
(a) CFD computational domain
TF[-]
w/h @ 0.75a
0 50 100 150 200
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(w/h start)MAX = 3.5e-1
(w/h start)MAX = 3.5e-2
(w/h start)MAX = 3.5e-3
(w/h start)MAX =
(b) FEM panel discretization
Figure 9: Effect of w0modulus on panel flutter LCO. DTF = 0.025, level L2 grid
in table 1, are relative to results obtained employing a parallel-implicit coupling with
Aitken adaptive relaxation, chosen as post-processing technique, that should stabilize
and accelerate the fixed point iteration.
From table 1 can be summarized that, by reducing the timestep, the average amount
of sub-iterations decreases, and, when a finer CFD grid is employed, the sub-iterations

Davide Cinquegrana, Pier L. Vitagliano
Tf
w/h
0 100 200 300
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Frequency [Hz]
0 200 400 600 800
0
2000
4000
6000
8000
10000
12000
w/h
dw/dt [m/s]
-2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1
(a) λ=75
Tf
w/h
0 100 200 300
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
0 200 400 600 800
0
2000
4000
6000
8000
10000
12000
w/h
dw/dt [m/s]
-3 -2 -1 0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
(b) λ=100
Tf
w/h
0 100 200 300
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
0 200 400 600 800
0
2000
4000
6000
8000
10000
12000
14000
w/h
dw/dt [m/s]
-3 -2 -1 0 1 2 3
-2
-1.5
-1
-0.
(c) λ=150
Tf
w/h
0 100 200 300
-3
-2
-1
0
1
2
3
4
Frequency [Hz]
0 200 400 600 800
0
2000
4000
6000
8000
10000
12000
14000
w/h
dw/dt [m/s]
-4 -3 -2 -1 0 1 2 3 4
-2
-1.5
-1
-0.5
(d) λ=200
Figure 10: LCO, Phase portrait and Spectra of reference point at 0.75a, M=1.2, for different λ, pinned
plate.
15
61
Davide Cinquegrana, Pier L. Vitagliano
Non-Dimensional Pressure, λ
LCO Amplitude, (w/h)max
0 100 200 300 400 500 600
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Dowell
Alder
CIRA -pin
Xiaomin-pin
Visball-pin
Hejranfar-pin
M = 1.2
x/a = 0.7
(a) LCO amplitude
Non-Dimensional Pressure, λ
ω/ω0
0 100 2 00 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Visball
Xiaomin
Hejranfar-pinned
Alder
CIRA - F-pin-FI
M = 1.2
x/a = 0.75
a/b = 0
µ
(b) LCO frequency
Figure 11: LCO amplitude, pinned panel
needed to reach convergence increases.
The time-coupling methods and post-processing techniques are investigated in the fol-
lowing, in order to verify whether they have an influence on computational costs and/or
the quality of the solutions. The post-processing techniques compared are Aitken and the
two Quasi Newton variants, IQN-ILS and IQN-IMVJ. Aitken was tested also in the stag-
gered coupling, since it is known from literature that it performs better than in parallel
coupling.
Table 2 shows that IQN-ILS performs better of other settings in terms of computational
costs.
Not reported in the table is also an analysis carried out with a Staggered explicit cou-
pling, and Aitken post-processing. As expected, this analysis diverges, after 450 timesteps.
Other numerical aspects influence the average number of coupling iterations. First
of all, the average number of iterations grows by decreasing the measure of convergence
tolerance on the data exchanged by the coupled solvers. When an implicit parallel coupling
DTF/Grid Lev. L1 L2 L3
0.005 - 4.02 -
0.01 3.97 4.84 -
0.025 4.00 5.25 -
0.05 6.16 6.25 6.57
0.075 7.64 7.74 -
0.1 11.35 11.86 -
Table 1: Average number of coupling iterations for different DTF and Grid Level
62
Davide Cinquegrana, Pier L. Vitagliano
is adopted, as in those runs, the relative tolerance of forces and displacement is checked
for the convergence within the time-step.
From the table 3 can be noted that, as expected, when the tolerance is small, the
iterations needed increases.
Other parameter that depends on the post-processing coupling and that can influence
the results, is the initial value of the relaxation factor (common to Aitken and Quasi
Newton methods): a larger value of ω(0.1) has produced a longer convergence history
than smaller values (i.e. ω= 0.01 ).
In the two Quasi Newton variants, a relevant factor is the timestep-reused parameter,
that limits the previous iterations used to generate the data basis for Jacobian estimation.
To set this, some preliminary evaluations were needed.
5.3 3D Panel Flutter
The 3D panel test constitutes a more significant test case for the validation of the FSI
environment.
The FEM model of a square panel, a/b = 1, is a 20x20x2 solid elements (C3D20R). The
CFD grid has 64x64 cells on panel, and 112x112x80 in stream-wise, span-wise and normal
direction respectively. Furthermore, a rectangular plate a/b = 0.5 is also analysed at the
same conditions. The square plate results are relative to clamped and pinned boundary
conditions (all four edge), while the rectangular plate is clamped at all four edges.
The free-stream flow conditions and the structural plate properties are the same as the
2D simulations: Mach 1.2 and lambda=250,300 and 350.
Figure 12 (a) and (b) show the surface pressure contours for square and rectangular
plate, respectively, at maximum upward deflection. Both are symmetric with respect to
the centerline at y/b = 0.
A sequence of compression, expansion and finally a shock wave can be noted in both
cases, with larger gradient in the a/b = 0.5 case.
Figure 13 (a) and (b) show, for square and rectangular plate respectively, the normal-
ized displacements, w/h, on the panel at λ= 300, when it is at the maximum upward
deflection. The results are symmetric with respect to the centerline. The peak of the
deflection is found at about x/a = 0.66: those results are comparable to the ones shown
in [11] for the case of the squared plate.
Coupling k
Stag-Impl Aitken 4.05
Par-Impl Aitken 6.25
Par-Impl IQN-ILS 3.27
Par-Impl IQN-IMVJ 4.83
Table 2: Average number of coupling iterations per coupling type (DTF=0.05, L2 grid; point at
x=0.75a.)
63
Davide Cinquegrana, Pier L. Vitagliano
ǫforces;ǫdisplacements k
1e-3;1e-3 2.94
1e-4;1e-4 4.60
1e-5;1e-3 3.27
1e-5;1e-5 7.16
1e-6;1e-6 15.76
Table 3: Average number of coupling iterations: tolerance effects (DTF=0.05, L2 grid; point at x=0.75a.)
-0.035
-0.035
-0.025
-0.025
-0.025
-0.025
-0.015
-0.015
-0.015
-0.015
-0.015
-0.015
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.015
0.015
0.015
0.015
0.0
25
x/a
y/b
0 0.25 0.5 0.75 1
-0.5
-0.25
(a) a/b = 1
-0.11
0.09
0.04
0
-0.03
-0.07
-0.1
-0.07
-0.04
-0.02
0
0.01
0.02
0.03
0.04
0.03
0.02
0.01
x/a
y/b
0 0.25 0.5 0.75 1
-0.5
-0.25
0
0.
(b) a/b = 0.5
Figure 12: Surface pressure contours for (M= 1.2, λ = 300) at maximum upward deflection
Figure 14 and 15 show the time history, phase portrait and frequency for the 3D panel
at λ= 300, with a/b = 1 and a/b = 0.5, respectively. The reference point considered is
the same of previous figures at 0.75 of x/a: the three dimensional effects are evident in
phase portraits: when a/b = 0.5 the portrait is quite similar to the 2D results shows in
fig. 10.
Figures 16 (a) and (b) show the LCO amplitude and frequency versus dynamic pressure
of a point at y= 0.5band x= 0.75aof the square plate, with clamped and pinned
edges. The calculations here performed show good agreement with literature, especially
at high pressure levels. The fundamental linear frequency for a 3D panel is valued by
ω0=π2(1 + AR2)qD
ρsha4[11], where AR =a
b.
Finally, some considerations from the numerical point of view. In the 3D simulations,
the Par-Impl IQN-ILS time coupling is chosen. The average number of coupling iterations
for the square plate is k= 3.1, while for the rectangular plate k= 2.9. A difference
between the two cases is the average amount of iterations that the CFD solver has to run
in order to reach convergence in the time step: it is lower for the case with a/b = 1.0 (72)
64
Davide Cinquegrana, Pier L. Vitagliano
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.0495514
0.198206
0.198206
0.198206
0.198206
0.198206
0.198206
0.198206
0.198206
0.34686
0.34686
0.34686
0.34686
0.34686
0.34686
0.34686
0.495514
0.495514
0.495514
0.495514
0.495514
0.495514
0.495514
0.644169
0.644169
0.644169
0.644169
0.644169
0.644169
0.792823
0.792823
0.792823
0.792823
0.792823
0.941477
0.941477
0.941477
0.941477
0.941477
1.09013
1.09013
1.09013
1.09013
1.23879
1.23879
1.23879
1.38744
1.38744
1.53609
x/a
y/b
0 0.25 0.5 0.75 1
-0.5
-0.25
0
0.2
(a) a/b = 1
3.32254
2.53603
1.99175
1.44746
1.08461
0.54032
0.177463
3.4695
3.08032
x/a
y/b
0 0.25 0.5 0.75 1
-0.5
-0.25
0
0
(b) a/b = 0.5
Figure 13: Surface deflection contours for (M= 1.2, λ = 300) at maximum upward deflection
than for the case a/b = 0.5 (99).
6 CONCLUSIONS
This report has described the development and validation of a new framework able to
face with FSI problems. Partitioned approach is followed to take advantage of already
existing codes: in house CFD solver, and open source FEM. An adapeter was developed,
by using the preCICE library, that ’glues’ the CFD code with FEM solver, Calculix.
Classical FSI problem of 2D and 3D panel flutter at supersonic speed have been simulated
with strongly-coupled partitioned approach and the results are compared with available
literature showing good agreement.
REFERENCES
[1] Marongiu C, Catalano P, A. M. e. a., “U-ZEN: a computational tool solving U-
RANS equations for industrial unsteady applications,” 34th AIAA fluid dynamics
conference, Portland OR, No. AIAA Paper 2004-2345, 2004.
[2] Dhondt, G., CalculiX CrunchiX USER’S MANUAL version 2.13 , 08 2017.
[3] Bungartz, H.-J., Lindner, F., Gatzhammer, B., Mehl, M., Scheufele, K., Shukaev, A.,
and Uekermann, B., “preCICE – A Fully Parallel Library for Multi-Physics Surface
Coupling,” Computers and Fluids, Vol. 141, 2016, pp. 250—-258.
[4] Vitagliano, P. L., Minervino, M., Quagliarella, D., and Catalano, P., “A conservative
sliding mesh coupling procedure for U-RANS flow simulations,” Aircraft Engineering
and Aerospace Technology, Vol. 88, No. 1, 2016, pp. 151–158.
65
Davide Cinquegrana, Pier L. Vitagliano
Frequency [Hz]
0 200 40 0 600 800
0
2000
4000
6000
w/h
dw/dt [m/s]
-2 -1 0 1 2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Tf
w/h
0 100 200 300
-2
-1.5
-1
-0.5
0
0.5
1
Figure 14: LCO, phase portrait and spectra of point at 0.75a,0.5b, λ= 300, a/b = 1, clamped edges
[5] Farhat, C. and Geuzaine, P., “Design and analysis of robust ALE time-integrators
for the solution of unsteady flow problems on moving grids,” Computer Methods in
Applied Mechanics and Engineering, Vol. 193, No. 39, 2004, pp. 4073 – 4095, The
Arbitrary Lagrangian-Eulerian Formulation.
[6] Farhat, C., Lesoinne, M., and LeTallec, P., “A conservative algorithm for exchanging
aerodynamic and elastodynamic data in aeroelastic systems,” Aerospace Sciences
Meetings, American Institute of Aeronautics and Astronautics, Jan. 1998, pp. –.
[7] Dhondt, G., The Finite Element Method for Three-Dimensional Thermomechanical
Applications, Wiley, 2004.
[8] Uekermann, B., Partitioned Fluid-Structure Interaction on Massively Parallel Sys-
tems, Dissertation, Institut f¨ur Informatik, Technische Universit¨at M¨unchen, Oct.
2016.
[9] Shishaeva, A., Vedeneev, V., and Aksenov, A., “Nonlinear single-mode and multi-
mode panel flutter oscillations at low supersonic speeds,” Journal of Fluids and Struc-
tures, Vol. 56, 2015, pp. 205 – 223.
[10] DAVIS, G. and BENDIKSEN, O., “Transonic panel flutter,” Structures, Structural
Dynamics, and Materials and Co-located Conferences, American Institute of Aero-
nautics and Astronautics, April 1993, pp. –.
[11] GORDNIER, R. and VISBAL, M., “DEVELOPMENT OF A THREE-
DIMENSIONAL VISCOUS AEROELASTIC SOLVER FOR NONLINEAR PANEL
FLUTTER,” Journal of Fluids and Structures, Vol. 16, No. 4, 2002, pp. 497 – 527.
66

Supplementary resource (1)

ResearchGate has not been able to resolve any citations for this publication.
Conference Paper
Full-text available
The purpose of this paper is to assess the capabilities of the newly developed tool U-ZEN, a code solving the Unsteady Reynolds Averaged Navier-Stokes equations. The activity is being performed within a CIRA project focusing on Flow Control; the specific goal for the CFD task is to simulate the unsteady flow phenomena due to particular fluid dynamic conditions (Mach, Reynolds, !) or to time dependent boundary conditions (e.g. flow control devices) around complex 3D configurations. The CIRA steady state flow solver ZEN (Zonal Euler Navier-Stokes) has been extended to treat unsteady problems by introducing a time discretization scheme, namely the Dual Time Stepping (DTS) technique. The validation is performed simulating unsteady flows around bluff obstacles or airfoils/wings at high angles of attack which are available experimental test cases.
Article
In the emerging field of multi-physics simulations, we often face the challenge to establish new connections between physical fields, to add additional aspects to existing models, or to exchange a solver for one of the involved physical fields. If in such cases a fast prototyping of a coupled simulation environment is required, a partitioned setup using existing codes for each physical field is the optimal choice. As accurate models require also accurate numerics, multi-physics simulations typically use very high grid resolutions and, accordingly, are run on massively parallel computers. Here, we face the challenge to combine flexibility with parallel scalability and hardware efficiency. In this paper, we present the coupling tool preCICE which offers the complete coupling functionality required for a fast development of a multi-physics environment using existing, possibly black-box solvers. We hereby restrict ourselves to bidirectional surface coupling which is too expensive to be done via file communication, but in contrast to volume coupling still a candidate for distributed memory parallelism between the involved solvers. The paper gives an overview of the numerical functionalities implemented in preCICE as well as the user interfaces, i.e., the application programming interface and configuration options. Our numerical examples and the list of different open-source and commercial codes that have already been used with preCICE in coupled simulations show the high flexibility, the correctness, and the high performance and parallel scalability of coupled simulations with preCICE as the coupling unit.
Article
Purpose – This paper aims to simulate unsteady flows with surfaces in relative motion using a multi-block structured flow solver. Design/methodology/approach – A procedure for simulating unsteady flows with surfaces in relative motion was developed, based upon a multi-block structured U-RANS flow solver1. Meshes produced in zones of the flow field with different rotation speed are connected by sliding boundaries. The procedure developed guarantees that the flux conservation properties of the original scheme are maintained across the sliding boundaries during the rotation at every time step. Findings – The solver turns out to be very efficient, allowing computation in scalar mode with single core processors as well as in parallel. It was tested by simulating the unsteady flow on a propfan configuration with two counter-rotating rotors. The comparison of results and performances with respect to an existing commercial flow solver (unstructured) is reported. Originality/value – This paper fulfils an identified need to allow for efficient unsteady flow computations (structured solver) with different bodies in relative motion.
Article
IntroductionThe General Framework of PlasticityThree-dimensional Single Surface ViscoplasticityThree-dimensional Multisurface Viscoplasticity: The Cailletaud Single Crystal ModelAnisotropic Elasticity with a von Mises–Type Yield Surface
Article
A new three-dimensional (3-D) viscous aeroelastic solver for nonlinear panel flutter is developed in this paper. A well-validated full Navier–Stokes code is coupled with a finite-difference procedure for the von Karman plate equations. A subiteration strategy is employed to eliminate lagging errors between the fluid and structural solvers. This approach eliminates the need for the development of a specialized, tightly coupled algorithm for the fluid/structure interaction problem. The new computational scheme is applied to the solution of inviscid two-dimensional panel flutter problems for subsonic and supersonic Mach numbers. Supersonic results are shown to be consistent with the work of previous researchers. Multiple solutions at subsonic Mach numbers are discussed. Viscous effects are shown to raise the flutter dynamic pressure for the supersonic case. For the subsonic viscous case, a different type of flutter behavior occurs for the downward deflected solution with oscillations occurring about a mean deflected position of the panel. This flutter phenomenon results from a true fluid/structure interaction between the flexible panel and the viscous flow above the surface. Initial computations have also been performed for inviscid, 3-D panel flutter for both supersonic and subsonic Mach numbers.
CalculiX CrunchiX USER'S MANUAL version 2.13
  • G Dhondt
Dhondt, G., CalculiX CrunchiX USER'S MANUAL version 2.13, 08 2017.
A conservative algorithm for exchanging aerodynamic and elastodynamic data in aeroelastic systems
  • C Farhat
  • M Lesoinne
  • P Letallec
Farhat, C., Lesoinne, M., and LeTallec, P., "A conservative algorithm for exchanging aerodynamic and elastodynamic data in aeroelastic systems," Aerospace Sciences Meetings, American Institute of Aeronautics and Astronautics, Jan. 1998, pp. -.
Transonic panel flutter
  • G Davis
  • O Bendiksen
DAVIS, G. and BENDIKSEN, O., "Transonic panel flutter," Structures, Structural Dynamics, and Materials and Co-located Conferences, American Institute of Aeronautics and Astronautics, April 1993, pp. -.