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E. O ñ ate,* C. A. Felippa**, S. Idelsohn*

FIC Variational Stabilization of Incompressible Continua. E. O ñ ate,* C. A. Felippa**, S. Idelsohn*. * International Center for Numerical Methods in Engineering (CIMNE) Universidad Polit é cnica de Catalu ñ a, Barcelona, Spain.

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E. O ñ ate,* C. A. Felippa**, S. Idelsohn*

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  1. FIC Variational Stabilization of Incompressible Continua E. Oñate,* C. A. Felippa**, S. Idelsohn* * International Center for Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña, Barcelona, Spain ** Department of Aerospace Engineering Sciences and Center for Aerospace StructuresUniversity of Colorado, Boulder, CO , USA US National Congress in Computational MechanicsSan Francisco, CA July 23-26, 2007

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

  3. 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 hi into the governing continuum equations, before discretization How: hiexpand balance (residual) laws overa domain of finite size, retain first order hiterms 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 C0 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 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.

  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 (n=0), hydrostatic body load

  36. Incompressible material (n=1/2), hydrostatic body load

  37. Incompressible material (n=1/2), 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

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