coupled fluid-structural solver cfd incompressible flow solver has been coupled with a fea code to...
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Coupled Fluid-Structural SolverCoupled Fluid-Structural Solver
•CFD incompressible flow solver has been coupled with a FEA code to analyze dynamic fluid-structure coupling phenomena
•CFD solver uses pressure splitting and orthogonal subgrid subscale stabilization to prevent non-physical behavior
•Structural solver suitable for large deformation analysis, including wrinkling of structural membranes
•All coding has been done in C++ inside the KRATOS object-oriented development framework for multiphysics analysis
•KRATOS allows for completely modular code design and ease of incorporation of new capabilities
KRATOS•An environment for implementing innovative computational methods
•Under continual development at CIMNE specifically to address coupled problems
•Based on Object Oriented Approach using C++
•Features a Python-Based programmable input
•Featured coupling strategies:
•STRONG COUPLING “SAFE” but often computationally expensive, requires iterative solving strategy
•LOOSE COUPLING is often considered “UNSAFE”, computational efficiency is potentially very HIGH
Incompressible CFD Solver
•ALE formulation
•Orthogonal subgrid subscale stabilization
•Choice of:
•Second-Order Accurate Fractional Step solver
•Monolithic solver
Structural Solver
•Non-linear large displacement/deformation capability
•Features advanced membrane elements including wrinkling
•Total Lagrangian model
Structural Deformation
Change in fluid Boundary conditions
Change in the pressure field
Coupled Fluid-Structure Interaction Problem
•Boundary conditions for the fluid are not known until the structure displacement is calculated
BUT
•Loads on the structure cannot be determined until the flow field has been solved for
Coupled “Fractional Step” Strategy
It follows the same rationale as the fractional step (pressure segregation) procedures used for the solution of the Navier-Stokes equations
•Structural Prediction
•Mesh movement step
•Fluid Solution
•Structural Correction
Prediction is done by SOLVING the structure subjected to a predicted pressure field (the simplest choice is the pressure at the end of the step before)
Error due to the coupling algorithm
Assuming that the pressure can be described in the form
and that the structural time integrator can be expressed in a form of the type
it is possible to express the solution of the coupled problem “in the future” as
11 1
1
nn n
n
n n n n+1 n+1
xy Ay +L f +L f y v
n+1 nex n
nn+1
y y= A +E yp p
ae ae aep t M x C x K x
where yn is an error term, for the coupling procedure to be stable this term must not grow without bounds
The amplification factor of the error term is
convergence is achieved when the amplification factor is less than one
Remark: The amplification factor does not depend on the particular time integration scheme selected
The basic scheme:
M
Mae
1 1 1 , , ,i i i i i iMx Cx Kx p x x x t
1 1 1 , , ,i i i i i i iM M x Cx Kx p x x x t Mx
can be replaced with
the procedure remains consistent, as there is no change when Δt→0
Inserting the assumed form of the pressure into the modified algorithm we have
now the scheme is stable when
1 1 11
i i i i i iae aeMx Cx Kx Mx C x K x
11
• By choosing an appropriate value for the procedure can be made stable irrespective of the mass ratio
• A suitable value can be estimated from the structure of the stiffness matrix of the fluid problem
flag flutter
-0.01
-0.005
0
0.005
0.01
0 5 10 15 20 25
time
tip d
isp
Example: Flag Flutter
Fcoupled simulation = 3.05Hz Fcoupled experiment = 3.10Hz
Fvon Karman = 3.7Hz
Example: 2D & 3D driven cavity with deformable base
PUMI 3D CFD SolverPUMI 3D CFD Solver
•Finite element unstructured compressible flow solver
•Edge-based data structure for minimum memory footprint and optimum performance
•Second order space accuracy
•Explicit multistage Runge-Kutta time integration scheme
•Convective stabilization through limited upwinding
•Implicit residual smoothing for convergence acceleration
•Parallel execution on shared memory architectures via OPEN-MP directives
Capabilities Overview:Capabilities Overview:
Algorithm OverviewAlgorithm Overview
3...10
kforxxt k
k
k
k GFΦ
NS equations in conservative form
kiki
i
i
i
i
i
ii
ii
ii
i
i
uqhu
pUu
pUu
pUu
U
e
U
U
U
2
2
1
33
22
11
3
2
1
0
GFΦ
ijkkvijiji
ivii ex
Tkqpeh
uTceuU
2,,,
2,
2
)()(
~)(
~)()(
~
xNxW
NxxNx
i
jj
jj
ΦΦΦ
Weak form of the NS equations
Finite element discretization
Wdxxt
xWk
k
k
k
0)(GFΦ
Weak semi-discrete form
nodek
k
k
kjji niford
xxNN ...10
~~~
GFΦ
The numerical fluxes are now approximated by
by introducing the mass matrix, the last expression can be solved for the time derivatives of the nodal variables
jkj
jkjk NxN FFF
~)(
~~
nodejk
jk
k
jjji niford
x
NNN ...10
~~~
GFΦ
jkjk
k
ji
ji
j
dx
NN
dNN
GFr
M
rMΦ 1
~~
~
To improve computational efficiency the residual is split two parts
k
jkj
ikkii
jk
ijkji x
NNwheredNNdNN
,,,
~~FFr i
(from this point on, no sum is assumed on i, and all sums on j are carried out for j≠i)
integrating by parts and rearranging
jk
ik
ijk
ikkii
jkkji
ikjki
ijkjki
wherednNN
dnNNdNNdNN
FFFF
FFFr i
~~~~
2
1
~~~,,
the expression must now be symmetrized to realize the benefits of the edge data structure
using the shape function property and after some manipulation
ikkii
jkkji
ikjki
ijkkji
ijkkjijki
dnNNdnNNdNN
dnNNdNNNN
FFF
FFr i
~
2
1~~
~~
2
1
,
,,
ij
ji NN 1
dnNNc
dnNNb
dNNNNd
cbd
kiiik
kjiijk
kjijkiijk
ik
ik
ijk
ijk
ijk
ijk
2
1
2
1
~~~
,,
FFFr i
Please remark that
icb
ddik
ijk
jik
ijk
nodeinternalanyfor0
thus, only one coefficient need be stored for each internal edge (pair of connected nodes, i.e. nodes belonging to the same element)
The scheme is conservative because for any given edge e connecting two internal nodes i-j, the total contribution the residual is zero
0FFFFrr jiee ij
kijk
ijk
ijk
jik
jik
ijk
ijk dddd
~~~~
When solving a viscous problem, nodal values the solution gradient are required to obtain the nodal diffusive fluxes. These can be recovered by means of a smoothing step. Using the regular FE interpolation for the gradients we set
jkji
jk
jkji
jkji
dNN
dNNdNN
ΦMΦ
ΦΦ
1
,
,
It is well known that the basic Galerkin discretization is inherently unstable (it is equivalent to a centered difference scheme). To overcome this limitation the interface fluxes are modified according to Roe’s upwind scheme
where the matrix represents the positive flux jacobian along the direction of the edge, evaluated at the Roe average state between states i and j
This scheme, while stable, provides only first order space accuracy. The amount of artificial dissipation must be reduced. Two additional states i+ and j- are introduced
ijuA
ijk
ijjk
ik
ijk ij uΦΦAFFF
u
~~
2
1~~
ijijijij
ij
ijij
ijk
ijjk
ik
ijk
ijk ij
ΦΦxxll
lu
uAFFFFu
~~
2
1~~~
from the backward and forward extrapolated differences
ijjjijii lΦlΦ
~~
the new interface states are calculated as
ijjjj
ijiii
kk
kk
114
1~~
114
1~~
ΦΦ
ΦΦ
where the parameter k controls the degree of approximation.
Near discontinuities the scheme must revert to first order. This is accomplished by limiting the degree of extrapolation
schemeorderhigh
schemeorderlows
kskss
i
ijiiii
ii
1
0
114
~~ΦΦ
There are many possible choices for the limiting parameter. As an example, we show here the van Albada limiter
12
,0max 22
iji
ijiis
Remark: It is in theory possible to achieve a higher accuracy by calculating the interface fluxes using the extrapolated values, i.e.
ijk
ijjk
ik
ijk ij uΦΦAΦFΦFF
u
~~
2
1~~
however there is usually little difference in practice, so this enhancement can de omitted without noticeable loss of accuracy
Time integration is performed using a n-stage Runge-Kutta scheme
this scheme is conditionally stable, the nodal allowable time step is calculated as
p
ah
au
hCFLt ii
i
2
2
,min
hi being the nodal size, the maximum fluid diffusivity and CFL the allowable Courant number
1
~)(
~
1
110
0
n
nt
jiijj
t
i
i
tt
ΨΦ
ΨrMΨΨ
ΦΨ
1
By means of implicit residual smoothing the allowable time step can be increased
which is solved using Jacobi iterations
ijj
ijii toconnectedforonly rrrr
j
j
jn
i
in 11
1
rr
r
Time derivatives (and solution gradients) can be solved for very efficiently using the following iterative process
11
0
~~~
~
mmm ΦMrΦΦM
rΦM
d
d
j
jiij dNNdM
Example: Transonic flow over a commercial airliner test model