(*) CIMNE: International Center for Numerical Methods in Engineering, Barcelona, Spain

Innovative Finite Element Methods For Innovative Finite Element Methods For Aeroelastic AnalysisAeroelastic Analysis

Presented by:

Gabriel Bugeda*Gabriel Bugeda*

MATEO ANTASME Meeting, 21/05/2007

Compiled by:

Roberto Flores*Roberto Flores*

Main Objectives of Task:Main Objectives of Task:• Analysis of thin walled structures with little or no bending stiffness subject to unsteady aerodynamic loads

•Development of efficient FE techniques for the non-linear (large strain & large displacement) analysis of membrane behavior, including wrinkling effects

•Improvements to FE flow solvers to allow for fast solution of complex flow patterns

•Robust coupling of structural solver and CFD codes for aeroelastic analysis

Structural FE Solver•Non-linear large displacement/deformation capability

•Features advanced membrane elements including wrinkling

•Implicit dynamic solver (allows for large time steps)

•Total Lagrangian formulation

Sail deployment

Inflated airbag showing wrinkles

CFD Solvers (I)

Implicit incompressible solver for low speed flows

•ALE formulation: Allows for mesh deformation

•Orthogonal subgrid subscale stabilization: Technique developed at CIMNE. Achieves stabilization with minimum numerical diffusion by using assumed forms for unresolved flow scales

•Choice of:

•Second-Order Accurate Fractional Step (pressure segregated) solver

•Monolithic solver

CFD Solvers (II)

Explicit compressible solver for high speed flows

•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

Edge oriented data structureEdge oriented data structure

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

jj

jj NxxNx ΦΦΦ

~)(

~)()(

~

Approximate solution using FE discretization

Weak semi-discrete form of the NS equations

nodek

k

k

kjji niford

xxNN ...10

~~~

GFΦ

The same finite element interpolation is used for fluxes:j

kjj

kjk NxN FFF~

)(~~

Solving for the nodal unknowns yields:

dNN jiM

rMΦ 1~)

~~~(

~~~ jk

ik

ijk

ik

ik

ijk

ijk

ijk

ijk cbd FFFFFFr i

dnNNcdnNNbdNNNNd kiiikkji

ijkkjijki

ijk 2

1

2

1,,

The coefficients dij and bij are non-zero only for pairs nodes connected by an edge ( i.e. nodes belonging to the same element).

The resulting algorithm is equivalent to a finite volume scheme in which the interface flux is the average of the nodal values of the edge. Furthermore, for any interior node:

0 ik

ijk

jik

ijk cbdd

thus, the scheme is conservative because the total contribution of internal edges to the residual is zero.

The basic scheme is equivalent to a centered finite difference stencil which is inherently unstable due to the odd-even decoupling phenomenon.

The interface fluxes are modified according to Roe’s upwind scheme in order to suppress instabilities:

ijjk

ik

ijk ij ΦΦAFFF

u

~~

2

1~~

ijiiii

ii kskss

11

4

~~ΦΦ

ijii lΦ

~

ijij ΦΦ~~

The factor k controls the extrapolation order for the interface fluxes, which can range from first to third order.

The coefficients si represent the flux limiters which revert the scheme to first order near discontinuities and sharp gradients. In areas where the flow field is smooth the high order scheme is used instead. The limiters are calculated from the ratio of the solution gradients at the ends of the edge.

•Solution of viscous problems at high Re numbers requires use of turbulence models and hybrid meshes to resolve the boundary layer

•Preparation of a suitable mesh is a lengthy task which cannot be easily automated

•To reduce computational costs and speed up the preprocessing stage a coupled Euler+Boundary Layer solved has been developed

•Uses boundary mesh of 3D volume to create a “virtual” hybrid boundary layer mesh (extruded prisms)

•In order to capture 3D effects no integral solution is sought, 3D boundary layer equations are solved directly

•Mapping of arbitrary 3D surface to a plane using unstructured surface mesh considered too involved Flux balances calculated in global coordinate system and projected to local curvilinear coordinates at each point.

•Cell-centered finite volume scheme

•Boundary layer solution coupled to external inviscid flow through transpiration boundary conditions

Coupled Euler+Boundary Layer SolverCoupled Euler+Boundary Layer Solver

nij

nj

ni lij

h

Ci Cj

ji

jiij

nn

nnn

ijijij lnC h

The flow of a conservative variable from cell i to cell j is then calculated as:

ijij CFnF

~d

ij

Finite Volume DiscretizationFinite Volume Discretization

Outer boundary of Euler 3D mesh

“Virtual” boundary layer cell

Solution scheme for boundary layer equationsSolution scheme for boundary layer equations

ttdt

dV i

ii *~

UCFU

ijij

Remove normal component

Correct momentum using continuity equation

dnUU

Uidn

i

0

Solve approximate momentum equation in global coordinate system

iiiii nUnUU ****

niiii UnUU **

n

This integral is calculated establishing the mass balance for the cell

Coupling of boundary layer solution with external flowCoupling of boundary layer solution with external flow

sUv Eulern

*

Remarks:

•As the boundary layer thickness is replaced with a transpiration velocity, the Euler mesh does not need to be replaced

•The scheme is not self-starting, for cells around a stagnation point a similarity solution for the flow near a stagnation area is used

•The FV scheme is cell centered whereas the FE algorithm is vertex centered, the variables can by transferred by means of:

j

cjjn

i V

V

Determine displacement thickness * and evaluate transpiration velocity

Coupled Fluid-Structural Solver

•CIMNE’s Kratos multiphysics development framework enables coupling of CFD solver with a FEA structural code to analyze dynamic fluid-structure coupling phenomena

•KRATOS has been completely developed in C++ using a modular object-oriented data structure to enable efficient coupling of single field solvers in a straightforward way

•Features a Python-Based programmable input

•Available coupling strategies:

•STRONG COUPLING: “SAFE” but often computationally expensive, requires iterative solving strategy

•LOOSE COUPLING: Often considered “UNSAFE”, computational efficiency is potentially very HIGH

Structural Deformation

Change in fluid Boundary conditions

Change in the pressure field

Coupled Fluid-Structure Interaction ProblemCoupled 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” StrategyCoupled “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 subject to a predicted pressure field (the simplest choice is the pressure at the end of the previous step)

Error due to the coupling algorithmError 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 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

M

Mae

Convergence is achieved when this factor is less than one

Remark: The amplification factor does not depend on the particular time integration scheme selected

The basic scheme: 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

Remark: Fluid and structural meshes need not be congruent, therefore loads on the structure are calculated remapping the flow solution. Loads are transferred by means of:

d)p(NNdNN vFj

Sij

Sj

Si nτt

where NS and NF represent the shape functions for the structural and fluid meshes respectively. This is a conservative mapping scheme in the sense that energy conservation is preserved.

Fcoupled simulation = 3.05Hz

Fcoupled experiment = 3.10Hz

Fvon Karman = 3.7Hz

Example: Flag Flutter

Example applications

Main topic of interest is structural membranes (e.g. inflatable structures & airbags)

Deployment of inflatable structure Airbag deployment

Contact algorithms have been implemented to analyze problems involving solids impacting the membranes

Solid contacting inflatable structure Solid impacting airbag

(blue ball is attached to membrane)

Thank you for your attentionThank you for your attention