CISM Lectures on Computational Aspects of Structural Acoustics and Vibration Udine, June 19-23, 2006

1. CISM Lectures on Computational Aspects of Structural Acoustics and Vibration Udine, June 19-23, 2006

2. Presenter: Carlos A. Felippa Department of Aerospace Engineering Sciences and Center for Aerospace StructuresUniversity of Colorado at Boulder Boulder, CO 80309, USA

3. Coupling Non-matching Meshes Carlos A. Felippa & K. C. Park Computational Aspects of Structural Acoustics and Vibration - Part 3Udine, June 19-23, 2006

4. Lecture Topics 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. 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. 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. Sources of Nonmatching Meshes • Non-matching cases, in ascending order of understanding difficulty • Nodes do not match (obvious) • Nodes match but freedoms do notrotational 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. Local-Global Analyis 1

9. Local-Global Analyis 2

10. Multiple Mesh Generators (courtesy SNL) with interface frame

11. Contact - 2D

12. Contact - 3D

13. Contact with Slip

14. Contact - 3D

15. Aerospace Example: F16

16. 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. In Flow Interaction Studies, Fluid Mesh is Finer

18. Here nodes & DOFs match but ...

19. Perverse Example: Quadratic-Cubic Coupling [With apologies to Bob Dylan] 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.

20. 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. 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. 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

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

25. FSI Nonmatching Mesh Used in Examples • Simplification because of slip-allowed condition: • Only normal displacements match • Only normal forces are in equilibrium

26. Mesh Coupling Primal Methods

27. 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. DI+MS Elimination: Structure as Master

29. DI+MS Elimination: Fluid as Master

30. 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. 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. 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. Mesh Coupling Dual Methods

34. Dual Methods - Outline Dual Methods: bring additional unknowns Global Lagrange Multipliers Variational form of mortar Localized Lagrange Multipliers

35. Interface Conditions (1)

36. Interface Condition (2)

37. Interface Condition (3)

38. Interface Condition (4)

39. FSI NonMatching Mesh Example (3) Compatibility Equilibrium Strong:uFx- uSx=0 Weak: ! (uFx- uSx) wu dy=0 Strong:fFx +fSx=0 Weak: ! (fFx+ fSx) wf dy=0

40. Interface Conditions (1)

41. Traction Interpolation Strongly equilibrated tractions should be equal and opposite at each interface point, see figure.

42. Traction Interpolation Difficulties An interface has two faces. Which one do you pick to interpolate l? “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. 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. LLM for FSI Interface

45. LLM for Cross Point

46. A Tutorial LLM Example: 2D, Plane Stress • Question: how do you connect the partitions so that the uniform stress state is exactly preserved?

47. Tutorial LLM Example (2) Interface treatment: connection frame and localized Lagrangemultipliers (Reproduced from last slide)

48. 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. Tutorial LLM Example (4) Frame configurations Multipliers (= interaction forces)for constant y-normal stress This is the Zero Moment Ruleor ZMR, so far stated as recipe.These frame configurationspreserve constant stress states “Bending Moment” diagram

50. Linear-Quadratic Coupling (1)