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

NoteNoteNoteNote

Eugenio should be giving this talk, but

a schedule conflict came up ...

OutlineOutline OutlineOutline

Variational Framework for FIC A FIC Functional for Incompressible Continua

Initial Numerical Tests Conclusions

A Variational Framework for FIC

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

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

FIC in Residual Framework (1)

FIC in Residual Framework (1)

For those problems the residual framework of FIC is natural

FIC in Residual Framework

FIC in Residual Framework

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.

FIC Variational Framework (2)

FIC Variational Framework (2)

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

A FIC Functional for Incompressible Continua

Mr. L. E. BlobMr. L. E. Blob

Tonti diagram

Constitutively Split VersionConstitutively Split Version

deviatoric

volumetric

Split shown is only valid for isotropic material

FIC Modified Variable Table FIC Modified Variable Table

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

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

Modified Tonti Diagram Modified Tonti Diagram

Put a bar and herehere

Modified Functional (1) Modified Functional (1)

Modified Functional (2) Modified Functional (2)

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)

Ingredients Ingredients

After some song & dance with the split equilibrium equations, i can be expressed aspictured in the Tonti diagram of next slide

Tonti Diagram with Stabilization Variable

Tonti Diagram with Stabilization Variable

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

FEM Discretization FEM Discretization

Same C0 spaces used for displacements, pressuresand stabilization field (e.g. linear-linear-linear)

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)

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.

Initial numerical tests

1D Test Configuration 1D Test Configuration

Configuration (R) is Relevant to Confined Fluid

Configuration (R) is Relevant to Confined Fluid

The 1D Functional The 1D Functional

Starting with 1D Allows Symbolic Work

Starting with 1D Allows Symbolic Work

FEM computations were carried out symbolicallyusing Mathematica, starting with patch tests

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

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

Compressible material (), hydrostatic body load

Compressible material (), hydrostatic body load

Incompressible material (), hydrostatic body load

Incompressible material (), hydrostatic body load

Incompressible material (), centrifugal body load

Incompressible material (), centrifugal body load

Conclusions

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

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