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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(|x−xCj|) based on the Euclidean distance between the control points position

xjand x:

f(x) =

Nc

X

j=1

γjφ(|x−xCj|) + 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|) = exp−a|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 10−3to 10−6for 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∗=ρ∞u2

∞a3

D, where D is the plate

stiﬀness, Eh3

12(1−ν2). 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=ρ∞a

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 ×10−3m.

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

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66