1. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures CISM Lectures on Computational Aspects of Structural Acoustics and Vibration Udine, June 19-23, 2006

2. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Presenter: Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace Structures University of Colorado at Boulder Boulder, CO 80309, USA

3. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Carlos A. Felippa & K. C. Park Coupling Non-matching Meshes Computational Aspects of Structural Acoustics and Vibration - Part 3 Udine, June 19-23, 2006

4. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Lecture Topics Partitioned Analysis of Coupled Systems: Overview 1. Partitioned Analysis of Coupled Systems: Overview 2. Synthesis of Partitioned Methods + 3. Mesh Coupling and Interface Treatment 4. Partitioned FSI by Localized Lagrange Multipliers

5. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Outline  Source of nonmatching meshes  Mesh coupling methods  Primal Methods  Dual Methods Warning: this lecture part is still disorganized. Had never collected the bits and pieces before.

6. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Sources of Nonmatching Meshes Non-matching spatial meshes may occur in coupled problems for various reasons: ¶ one of the physical subsystems may require a finer mesh for accurate results · teams using different programs construct or generate the meshes separately ¸ one or both subsystems are previously modeled for different reasons, for example incremental simulation of the structure construction process. First 2 cases are common in aerospace (next slides), last one in Civil

7. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Sources of Nonmatching Meshes Non-matching cases, in ascending order of understanding difficulty ¶ Nodes do not match (obvious) · Nodes match but freedoms do not rotational DOF on one mesh, not on other (e.g. beam-solid) displacements on one side, a potential on the other ¸ Nodes and freedoms match but element interpolation does not

8. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Local-Global Analyis 1

9. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Local-Global Analyis 2

10. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Multiple Mesh Generators (courtesy SNL) with interface frame

11. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Contact - 2D

12. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Contact - 3D

13. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Contact with Slip

14. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Contact - 3D

15. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Aerospace Example: F16

16. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures F16 Internal Structure Full F-16 mesh took several months to prepare It is rarely touched. Fluid mesh is regenerated frequently to look for different effects (e.g. buffeting) It is typically much finer than the structure mesh (structure DOF ~ 1M, fluid DOF: ~ 100M)

17. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures In Flow Interaction Studies, Fluid Mesh is Finer

18. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Here nodes & DOFs match but...

19. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Perverse Example: Quadratic-Cubic Coupling Everything matches Everything passes Just do what you think you should do. And someday maybe, Who knows, baby, I'll come and be cryin' to you. [With apologies to Bob Dylan]

20. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Classification of Mesh Coupling Methods Primal - no additional unknowns master-slave elimination penalty Dual - additional unknowns Lagrange multipliers adjoined and perhaps a “kinematic frame” Primal-Dual: begin as primal, end as dual

21. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Another Classification of Mesh Coupling Methods Variationally Based Preferred when formulation of each subsystem is variational, since symmetry of overall equations retained Non Variational

22. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Primal Methods for Displacement based FEM Rely on direct interpolation to establish multifreedom constraints (MFCs) MFCs are then applied by one of 3 techniques primal: master slave elimination primal: penalty dual: Lagrange multipliers These will be illustrated with a structural example

23. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures

24. FSI Application from Part 4 Dam under seismic action Non-matching fluid-structure-soil meshes

25. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI Nonmatching Mesh Used in Examples Simplification because of slip-allowed condition: Only normal displacements match Only normal forces are in equilibrium

26. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Mesh Coupling Primal Methods

27. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Primal Methods - Based On  Direct interpolation (collocation) followed by Master slave elimination or Penalty function adjunction  Shape function least square matching  Transition elements We will cover only direct interpolation with master-slave elimination (DI+MS)

28. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures DI+MS Elimination: Structure as Master

29. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures DI+MS Elimination: Fluid as Master

30. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures DI+MS Elimination: Difficulties  Generally fails interface patch test (explained later for LLM)  Monolithically couples fluid & structure, Changes data structures of system matrices because unknown vector is modified  If master is finer mesh, prone to spurious modes. Think of the structure boundary motion pictured here: structure moves but fluid does not notice

31. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures DI+Penalty Function: Difficulties  Generally fails interface patch test (explained later)  Strongly couples fluid & structure Less change in system matrices, but need to pick weights  May need scaling  Again prone to spurious modes if MFCs are insufficient to prevent them

32. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Primal Methods Assessment [from experience] Easy to understand and explain Can be rapidly implemented if access to & manipulation of component matrices is easy (for example, if Matlab is used as wrapper)  Too meddling with each subsystem and thus prone to failure

33. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Mesh Coupling Dual Methods

34. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Dual Methods - Outline Dual Methods: bring additional unknowns Global Lagrange Multipliers Variational form of mortar Localized Lagrange Multipliers

35. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Conditions (1)

36. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Condition (2)

37. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Condition (3)

38. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Condition (4)

39. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures FSI NonMatching Mesh Example (3) Compatibility Equilibrium Strong: u Fx - u Sx =0 Weak: ! (u Fx - u Sx ) w u dy=0 Strong: f Fx +f Sx =0 Weak: ! (f Fx + f Sx ) w f dy=0

40. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Interface Conditions (1)

41. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Traction Interpolation Strongly equilibrated tractions should be equal and opposite at each interface point, see figure.

42. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Traction Interpolation Difficulties An interface has two faces. Which one do you pick to interpolate  “Cross points” (points at which more than 2 interfaces meet, see figure) require branching decisions Difficulties multiply in 3D. Integration over curved surfaces becomes a nightmare. Coding of special cases is delicate.

43. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Simplifications: Mortar Method Pick one face as master and one as slave. If one face is more finely discretized than the other, make it the master one. “Lump” Lagrange multipliers at nodes of master face. Multipliers become delta functions (concentrated forces) and integrals are easy. Difficulties at cross points remain: connections may be ambiguous.

44. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures LLM for FSI Interface

45. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures LLM for Cross Point

46. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures A Tutorial LLM Example: 2D, Plane Stress * Question: how do you connect the partitions so that the uniform stress state is exactly preserved?

47. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Tutorial LLM Example (2) (Reproduced from last slide) Interface treatment: connection frame and localized Lagrange multipliers

48. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Tutorial LLM Example (3) Multiplier discretization decision: point forces (delta functions) at partition interface nodes Frame discretization decision: piecewise linear interpolation Remark: interpolating displacements is easier than interpolating multipliers (forces)

49. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Tutorial LLM Example (4) This is the Zero Moment Rule or ZMR, so far stated as recipe. These frame configurations preserve constant stress states Frame configurations Multipliers (= interaction forces) for constant y-normal stress “ Bending Moment” diagram

50. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Linear-Quadratic Coupling (1)

51. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Linear-Quadratic Coupling (2) Frame configurations

52. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Quadratic-Cubic Coupling (1)

53. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Quadratic-Cubic Coupling (1)

54. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Quadratic-Cubic Coupling (2) Note: It would be a serious error to place frame nodes at interior interface node locations although they match as regards location and DOFs Frame configurations

55. University of Colorado - Dept of Aerospace Engineering Sci.ences & Center for Aerospace Structures Stop  End of Part 3