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 Mechanics San Francisco, CA July 23-26, 2007 * International Center for Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña, Barcelona, Spain FIC Variational Stabilization of Incompressible Continua

NoteNote Eugenio should be giving this talk, but a schedule conflict came up...

Outline Outline  Variational Framework for FIC  A FIC Functional for Incompressible Continua  Initial Numerical Tests  Conclusions

 A Variational Framework for FIC

Basic Idea of FIC Inject steplengths h i into the governing continuum equations, before discretization How: h i expand balance (residual) laws over a domain of finite size, retain first order h i terms Developed by Eugenio Oñate & colleagues at CIMNE since 1998

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) For those problems the residual framework of FIC is natural

FIC in Residual Framework

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)

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. Blob Tonti diagram

Constitutively Split Version deviatoric volumetric Split shown is only valid for isotropic material

FIC Modified Variable Table

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 For stabilizing the treatment of (near) incompressibility: Pressure p and volumetric strain  v are modified to build a FIC mixed functional

Modified Tonti Diagram Put a bar and here here

Modified Functional (1)

Modified Functional (2)

Three is Company A 3-vector stabilization field  i is introduced as third independent (primary) variable. Physically, it turns out to be the negated pressure gradient:  i  p,i = 0. NB. Introduction of  i has received several names in the literature, e.g. “orthogonal sub-scales’’ by Codina (2000)

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

Tonti Diagram with Stabilization Variable

And It’s All Over Now, Baby Blue After more steps the final 3D FIC functional emerges

FEM Discretization Same C 0 spaces used for displacements, pressures and stabilization field (e.g. linear-linear-linear)

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 By paying attention to the FIC steplength matrix rank, 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

Configuration (R) is Relevant to Confined Fluid

The 1D Functional

Starting with 1D Allows Symbolic Work FEM computations were carried out symbolically using Mathematica, starting with patch tests

Benefits of Symbolic Calculation Effect of parametric discretization choices can be immediately observed in the solution and responsive actions taken Solution components can be Taylor series expanded in the steplength to assess its effect on accuracy

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 the formation of mass-like submatrices. As a side benefit the solution was nodally exact for certain loading conditions, such as hydrostatic body loads

Compressible material (  ), hydrostatic body load

Incompressible material (  ), hydrostatic body load

Incompressible material (  ), centrifugal body load

 Conclusions

Conclusions (1) Preliminary numerical experiments encouraging Taking = 1/2 caused no problems. Effect of FIC steplength and mass-like submatrix lumping clarified by symbolic computations

Conclusions (2) However, 1D problems are benign Demanding verification tests will come in 2D & 3D Reduction 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