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