1. 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

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

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

4.  A Variational Framework for FIC

5. 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

6. 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

7. FIC in Residual Framework (1) For those problems the residual framework of FIC is natural

8. FIC in Residual Framework

9. 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.

10. FIC Variational Framework (2)

11. How To Construct a Modified VP * Recipe: replace original variables by modified variables ( an example coming up) * VP: Variational Principle, not Vice President

12.  A FIC Functional for Incompressible Continua

13. Mr. L. E. Blob Tonti diagram

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

15. FIC Modified Variable Table

16. No Free Lunch Modified variables bring extra baggage: steplengths and space derivatives So: Inject FIC-modified variables only where they would do most good

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

18. Modified Tonti Diagram Put a bar and here here

19. Modified Functional (1)

20. Modified Functional (2)

21. 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)

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

23. Tonti Diagram with Stabilization Variable

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

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

26. 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)

27. 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.

28.  Initial numerical tests

29. 1D Test Configuration

30. Configuration (R) is Relevant to Confined Fluid

31. The 1D Functional

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

33. 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

34. 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

35. Compressible material (  ), hydrostatic body load

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

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

38.  Conclusions

39. 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

40. 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