Diapositiva 1

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Innovative Finite Element Methods For Aeroelastic
Analysis
Compiled by
Roberto Flores
Presented by
Gabriel Bugeda
() CIMNE International Center for Numerical
Methods in Engineering, Barcelona, Spain
MATEO ANTASME Meeting, 21/05/2007
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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

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• 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

Inflated airbag showing wrinkles
Sail deployment
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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

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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

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Edge oriented data structure
NS equations in conservative form
Approximate solution using FE discretization
Weak semi-discrete form of the NS equations
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The same finite element interpolation is used for
fluxes
Solving for the nodal unknowns yields
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
thus, the scheme is conservative because the
total contribution of internal edges to the
residual is zero.
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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 Roes upwind scheme in order to
suppress instabilities
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.
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Coupled EulerBoundary Layer Solver

• 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 EulerBoundary
  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

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Finite Volume Discretization
Virtual boundary layer cell
Outer boundary of Euler 3D mesh
The flow of a conservative variable from cell i
to cell j is then calculated as
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Solution scheme for boundary layer equations
Solve approximate momentum equation in global
coordinate system
Remove normal component
Correct momentum using continuity equation
This integral is calculated establishing the mass
balance for the cell
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Coupling of boundary layer solution with external
flow
Determine displacement thickness ? and evaluate
transpiration velocity

• 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

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Coupled Fluid-Structural Solver

• CIMNEs 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

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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

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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
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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 as
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 this factor is less
than one
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Remark The amplification factor does not depend
on the particular time integration scheme
selected The basic scheme
can be replaced with
the procedure remains consistent, as there is no
change when ?t?0
Inserting the assumed form of the pressure into
the modified algorithm we have
now the scheme is stable when
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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
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.
Example Flag Flutter
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Example applications Main topic of interest is
structural membranes (e.g. inflatable structures
airbags)
Deployment of inflatable structure
Airbag deployment
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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)
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Thank you for your attention