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E. Oñate,* C. A. Felippa**, S. Idelsohn*
** Department of Aerospace Engineering Sciences and Center for Aerospace Structures
University of Colorado, Boulder, CO , USA
US National Congress in Computational MechanicsSan Francisco, CA July 23-26, 2007
* International Center for Numerical Methods in Engineering (CIMNE)
Universidad Politécnica de Cataluña, Barcelona, Spain
FIC Variational Stabilization ofFIC Variational Stabilization ofIncompressible ContinuaIncompressible Continua
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NoteNoteNoteNote
Eugenio should be giving this talk, but
a schedule conflict came up ...
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OutlineOutline OutlineOutline
Variational Framework for FIC A FIC Functional for Incompressible Continua
Initial Numerical Tests Conclusions
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A Variational Framework for FIC
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Basic Idea of FICBasic Idea of FIC
Inject steplengths hi into the governingcontinuum equations, before discretization
How: hi expand balance (residual) laws overa domain of finite size, retain first order hi terms
Developed by Eugenio Oñate & colleaguesat CIMNE since 1998
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Applications to Date
Applications to Date
To date most applications have been to problems in Computational Fluid Dynamics that model
advection, diffusion, reaction, turbulence, gravity dominated incompressible flows
with focus on stabilization of associated solution processes
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FIC in Residual Framework (1)
FIC in Residual Framework (1)
For those problems the residual framework of FIC is natural
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FIC in Residual Framework
FIC in Residual Framework
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FIC Variational Framework (1)
FIC Variational Framework (1)
For problems such as
acoustics, elastic solids, Lagrangian fluids, Lagrangian-Lagrangian FSI [e.g. PFEM]
a variational framework seems worth exploring as lack of convective terms means that standard variational principles & tools are available, and unified fluid-structure formulations may be possible.
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FIC Variational Framework (2)
FIC Variational Framework (2)
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How To Construct a Modified VP *
How To Construct a Modified VP *
Recipe: replace original variables by modified variables
(an example coming up)
* VP: Variational Principle, not Vice President
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A FIC Functional for Incompressible Continua
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Mr. L. E. BlobMr. L. E. Blob
Tonti diagram
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Constitutively Split VersionConstitutively Split Version
deviatoric
volumetric
Split shown is only valid for isotropic material
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FIC Modified Variable Table FIC Modified Variable Table
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No Free Lunch No Free Lunch
Modified variables bring extra baggage:steplengths and space derivatives
So: Inject FIC-modified variables only where they would do most good
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Applying the Rule Applying the Rule
For stabilizing the treatment of (near)incompressibility:
Pressure p and volumetric strain v are modifiedto build a FIC mixed functional
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Modified Tonti Diagram Modified Tonti Diagram
Put a bar and herehere
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Modified Functional (1) Modified Functional (1)
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Modified Functional (2) Modified Functional (2)
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Three is CompanyThree is Company
A 3-vector stabilization field i is introducedas third independent (primary) variable.
Physically, it turns out to be the negated pressure gradient: i p,i = 0.
NB. Introduction ofi has received several names in the literature, e.g. “orthogonal sub-scales’’ by Codina (2000)
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Ingredients Ingredients
After some song & dance with the split equilibrium equations, i can be expressed aspictured in the Tonti diagram of next slide
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Tonti Diagram with Stabilization Variable
Tonti Diagram with Stabilization Variable
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And It’s All Over Now, Baby Blue
And It’s All Over Now, Baby Blue
After more steps the final 3D FIC functional emerges
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FEM Discretization FEM Discretization
Same C0 spaces used for displacements, pressuresand stabilization field (e.g. linear-linear-linear)
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FEM Discretization FEM Discretization
Raw freedom count in 3D:
3 displacement components per node 1 pressure per node 3 pressure gradient components per node
Total: 7 DOF/node in 3D (5 in 2D, 3 in 1D)
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DOF Reduction DOF Reduction
By paying attention to the FIC steplength matrixrank, theory says that DOF count can be cut to
3 displacement components per node 1 pressure per node 1 pressure gradient per node
Total: 5 DOF/node in 3D (4 in 2D, 3 in 1D)Not yet tested, however, in 2D or 3D.
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Initial numerical tests
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1D Test Configuration 1D Test Configuration
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Configuration (R) is Relevant to Confined Fluid
Configuration (R) is Relevant to Confined Fluid
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The 1D Functional The 1D Functional
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Starting with 1D Allows Symbolic Work
Starting with 1D Allows Symbolic Work
FEM computations were carried out symbolicallyusing Mathematica, starting with patch tests
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Benefits of Symbolic Calculation
Benefits of Symbolic Calculation
Effect of parametric discretization choices can beimmediately observed in the solution and responsive actions taken
Solution components can be Taylor series expanded in the steplength to assess its effecton accuracy
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DOF Condensation Rule DOF Condensation Rule
If all pressure and pressure-gradient freedoms are statically condensed for <1/2, the coefficient matrix must reduce to that of the standard displacement model if the FIC steplength tends to zero
This led to some discretization rules on theformation of mass-like submatrices. As a side benefit the solution was nodally exact for certain loading conditions, such ashydrostatic body loads
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Compressible material (), hydrostatic body load
Compressible material (), hydrostatic body load
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Incompressible material (), hydrostatic body load
Incompressible material (), hydrostatic body load
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Incompressible material (), centrifugal body load
Incompressible material (), centrifugal body load
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Conclusions
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Conclusions (1) Conclusions (1)
Preliminary numerical experiments encouraging
Taking = 1/2 caused no problems.
Effect of FIC steplength and mass-likesubmatrix lumping clarified by symboliccomputations
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Conclusions (2) Conclusions (2)
However, 1D problems are benign
Demanding verification tests will come in 2D & 3DReduction of freedoms will be important there.
One target use: Lagrangian-Lagrangian FSI in PFEM codes, where it will have to compete with other stabilization methods