1. Computational Aspects of Structural Acoustics and Vibration
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 Structures University of Colorado at Boulder
Boulder, CO 80309, USA

3. Synthesis of Partitioned Analysis Procedures
C. A. Felippa
CISM Lecture Series on Computational Methods in
Structural Acoustics and Vibration - Part 2 Udine, June 19-23, 2006

4. General Comment on Lectures
Note that in an FSI simulation (say) I won’t talk on
how to do the structure
how to do the fluid
I assume you know how to do each piece by itself,
or to get existing software that do them.
My focus is how you may couple the pieces and
solve the coupled system.

5. 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

6. Learning by Repetition
Effective synthesis practice relies
on recognizing some common features
of coupled systems.
Important is to learn about:
software components
types of interaction
model systems

7. Coupled System Examples
Aeroelasticity
Underwater Shock
MEMS Resonator
Dam Under Seismic Action
Common Features favor Partitioned Analysis

8. Aeroelasticity

9. Aeroelasticity: Interaction Diagram

10. Lesson: Try Not to Reinvent the ]
Structure model
benefits from 50
years of FEM
Fluid model
benefits from 50
years of CFD

11. Underwater Shock (UWS)- Early 70s

12. Resonator (Cell Phone Chip Component)

13. Resonator: Interaction Diagram
System entirely modeled
in the frequency domain
Reduced Models useful

14. Dam Under Seismic Action
Covered in Part 4

15. Common Features of Examples (1)
Dynamic, two-way interaction
Similar physical scales these are not multiscale problems
Interfaces and partitions well defined, surface coupling only

16. Common Features of Examples (2)
Isolated components well understood
Software (commercial or public) for components often available
Treatment benefits from customization (different models & methods for different components)

17. Common Features of Examples (3)
In production projects, new modeling features are often the result of customer requests; e.g.
What happens if “unnamed things” inside the submarine go nonlinear under a strong shock?
Can you simulate a fast fighter maneuver?
Can turbulence be the cause of a recent crash?
Often the request can be “localized”

18. Common Features of Examples (4)
Often the request can be “localized”. For instance when Lockheed was asked by the Navy:
What happens if “unnamed things” inside a N-sub go nonlinear under a strong shock? The answer was to replace the linear structural analyzer (NASTRAN) by a nonlinear one (DYNA3D) The fluid and interaction software were not changed.

19. Starting Project from Scratch with Limited Resources?
Mine existing codes for software or data components
(e.g, get stiffness/mass mtx from NASTRAN or ANSYS, read in Matlab)
Synthesize an interfacing method and a time marching scheme (following slides)
Use a higher order language (for example Matlab) as “driver-wrapper” and postprocessor
Try realistic problems from start and compare with validation codes.
Dont waste time on fancy features (e.g. parallel processing) before the “core stuff” works

20. Partitioned Method Synthesis

21. Some Experience So Far (1975-date)
Structure-structure, undamped / lightly damped:
I-I, A-stable, 2nd-order accurate, single pass, schemes possible.
Control-structure interaction
I-I, C-stable, 1st-order accurate, single pass schemes possible.
A-stable for light or moderate damping. 2nd order requires iteration
Underwater shock, no cavitation
I-I, A-stable, 2nd-order accurate, single pass, schemes possible only by augmentation of either fluid or structure
Underwater shock with cavitation
I-E-I, C-stable, 2nd-order accurate, single pass, schemes possible Fluid volume done explicitly. No augmentation required.

22. Augmentation

23. Important Step Construct a model equation test system
The system contains as many differential equations as coupled components
Goal: identify primary physics with minimal number of parameters

24. Test System Example (Developed Later)

25. How Long Does It Take?
Going through a test system synthesis loop can be time consuming, even for an experienced engineer or scientist.
The amount of work strongly depends on many design variables are carried along. This is problem dependent (next slide)

26. Design Parameters in Test System
Structure-structure, undamped
Structure-structure, Rayleigh damped
Control-structure interaction
Underwater shock, no cavitation
Underwater shock with cavitation
Aeroelasticity with dynamic mesh
Flexible ship hydrodynamics
Electrothermomechanics
Counts are for minimum # of partitions

27. Recommendations to Save Work
Try to reduce number of physical parameters by looking at the essential physics & by forming dimensionless combinations
Try to reduce the number of integration & predictor parameters by using theory if available

28. Your Computer Can Help
Synthesis loop may take weeks or months if done by hand (or by “potshot” computations)
The use of Computer Algebra Systems (CAS) such as Mathematica o Maple can reduce the time to days or hours. Examples in notes
Why the gain? Faster algebra, reduction of human errors & integrated graphics facilities

29. Recent Research: Partition Localization
Basic goal: maximally isolate software modules doing component computations so they communicate only by interface forces aka Lagrange Multipliers
Four Possibilities, with last one covered in Part 4:
Distributed Global Lagrange Multipliers
Distributed Local Lagrange Multipliers
Collocated Global Lagrange Multipliers
Collocated Local Lagrange Multipliers

30. Localization Goals
Software reuse, including extrating equations from commercial codes
Nonmatching meshes
Multilevel parallelization

31. How Long Does It Take?
Going through a test system synthesis loop can be time consuming, even for an experienced engineer or scientist.
The amount of work strongly depends on many design variables are carried along. This is problem dependent. Some examples follow.

32. Test System Example (Developed Later)

33. Design Variables in Test System (1)
Structure-structure interaction, undamped: subsystem frequencies frequency coupling parameter stepsize
Total: 4 physical parameters integrator & predictor parameters
With Rayleigh damping: add 3

34. Recommendations to Save Work
Try to reduce number of physical parameters by looking at the essential physics & by forming dimensionless combinations
Try to reduce the number of integration & predictor parameters by using theory if available

35. Getting Computer Help
Synthesis loop may take weeks or months if done by hand (or by sample computations)
The use of Computer Algebra Systems (CAS) such as Mathematica o Maple can reduce the time to days or hours
Why the gain? Faster algebra, reduction of human errors & integrated graphics facilities

36. Auxiliary Mathematica Modules
Provided in notes to help with
Stability Analysis
Derivation of characteristic equation
Polynomial stability (Routh criterion, etc.)
Accuracy Analysis
Derivation of Modified Equation

37. A Warning
For second order coupled systems, stability depends on the computational path
The path dictates how auxiliary variables such as velocities and momenta are computed
Consequence: a tiny change in the guts of a program may suddenly affect stability
Good news: changes in computational path can be compensated by predictor changes (details in article)

38. Stability Analysis Example: Structure-Structure Interaction

39. SSI: Staggered Partition

40. SSI: Test System (1)

41. SSI: Test System (2)

42. SSI: Test System (2)

43. Theoretical Result Result obtained by Park (1980)
If both structures are treated by the
Trapezoidal Rule, and an optimal predictor used (adjusted for the computational path) then
The staggered solution method is unconditionally
stable and has second order accuracy

44. Verified by Mathematica 24 Years Later

45. Stable Predictors

46. Mathematica Code

47. Summary
For undamped structure-structure interaction, optimal staggered methods are known, which do not hinder stability or accuracy
If the coupled system is Rayleigh damped, the same methods are recommended
For generally damped coupled systems, or control-structure interaction, no general theory is available. Problems must be done case by case

48. Lecture Sources
Parts 1 and 2: Material of recent FSI course (Spr 2003) posted at
contains posted student projects and references to journal papers, including
those in CISM brochure:
(Felippa-Park-Farhat - CMAME 2001)
(Park-Ohayon-Felippa - CMAME 2002)
Will add these slides sets on return to Boulder
Part 3: a potpourri of bits and pieces, mostly unpublished
Part 4: two CMAME papers under preparation (Ross’ Thesis)

49. Stop
End of Part 2