1. A comparison between one-sided and two-sided Arnoldi
based Model Order Reduction techniques [MORe] for fully
coupled structural-acoustic analysis.
Oral Presentation Session at the 153rd Meeting - ASA
R Srinivasan Puri, Denise Morrey
Oxford Brookes University,
Advanced Vehicle Engineering Group,
School of Technology, Wheatley Campus,
Oxford OX33 1HX, United Kingdom.
Jeffrey L. Cipolla
Principal Development Engineer,
ABAQUS Inc.
166, Valley Street, Providence, RI 02909-2499, U.S.A
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2. Contents
1. Problem Description.
2. Idea of Model Order Reduction (MORe).
3. Model Order Reduction: Moment Matching.
4. Moment Matching: One Sided Arnoldi (OSA).
5. Moment Matching: Two Sided Arnoldi (TSA).
6. Model Order Reduction: Computational Aspects.
7. Numerical Test Case & Results: ABAQUS Benchmark Problem.
8. Summary.
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3. Problem Description
Compute pressure level at drivers ear location (automobile or an aircraft interior)
under structural or acoustic excitation.
Classical fully coupled FSI Formulation [Zienkiewicz & Newton 19691, Craggs,19712] :
Structure Matrices Displacements
 Ms 0   u   Cs 0   u   Ks
  Kfs   u   Fs 

 Mfs Ma     +  0 Ca   p  +  0
 p       =  
       Ka   p   0 

Coupling term Fluid Matrix
Pressures
• The direct formulation is the most accurate method when spatially variable,
frequency-dependent trim material damping exists.
• Unsymmetric Mass, Stiffness Matrix increases computational expense.
• Modelling the final trim parts and joints leads to very high mesh density, and
results in huge computational time.
1
Zienkiewicz, O. C., and R. Newton, 'Coupled Vibrations of a Structure Submerged in a Compressible Fluid,' Proceedings of the International
Symposium on Finite Element Techniques, Stuttgart, 1969.
2
Craggs, A, 'The Transient Response of a coupled Plate-Acoustic System using Plate and Acoustic Finite Elements', Journal of Sound and
Vibration, 15, 509—528, 1971
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4. Idea of Model Order Reduction
• Undamped:
  Ms 0   Ks Kfs    u   Fs   u
 − ω 2
 Mfs Ma  +  0    =   y (ω ) = LT  
 
 
  Ka    p   0 
   p
 
=
 −ω +

2
Msa Ksa Fsa
,
 
,
• Projection to lower dimensional subspace: States
,
 u
.   = { x} = Vz + ε
 p
  T
y r (ω ) = Lr z (ω )
−ω
2
Mrsa + Krsa  = Frsa
 
Reduced States
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5. Model Order Reduction: Moment Matching
• How to pick projection matrix [V]?
➔Modal Approaches: Uncoupled and Coupled (Morand and Ohayon 19973,
Ohayon 20044) projection formulation : Need to solve eigenvalue problem.
- Moment Matching: Expand transfer function via Taylor series.
- Moment Matching: Match first moments for the transfer function of
the coupled system.
∞ ∞
∑ ∑
−1 −1
H (s) = (− 1) L ( K sa M sa ) K sa Fsa s =
i T i 2i
mi s 2 i
i= 0 i= 0
- Explicit moment matching is unstable. Therefore, implicitly match
moments via Arnoldi process.
 Su and Craig, 19915: choose projection matrix [V] to be the Krylov subspace
to provide moment matching property.
3
Morand, H. and Ohayon, R. 'Fluid Structure Interaction', ed. 1, John Wiley and Sons Ltd, 1995, ISBN-13: 978-0471944591.
4
Ohayon. R. 'Reduced models for fluid–structure interaction problems', International Journal for Numerical Methods in Engineering, 60,
139--152, 2004
5
T J Su, R R Craig Jr 'Krylov model reduction algorithm for undamped structural dynamics systems' Journal of Guidance and Control
Dynamics 14 1311-1, 1991
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6. Moment Matching: Krylov via One Sided Arnoldi
Algorithm:1: Complete set-up for SISO / SICO Arnoldi Process (R.W.Freund , 20006)
6
Freund, R.W 'Krylov subspace methods for reduced order modeling in circuit simulation' Journal of Applied Mathematics 123 (1-2); 395-
421, 2000.
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7. Moment Matching: Krylov via Two Sided Arnoldi
➔ Accuracy of the one-sided projection can be improved by using appropriate
two-sided techniques where twice the number of moments are matched.
➔ Output explicitly participates in the order reduction. Therefore SISO is
restricted strictly to SISO.
Algorithm:2: Two-Sided Arnoldi Process (Grimme, E.J. 1997 7 and Salimbahrami, B. 2005 8)
7
Grimme E.J. 'Krylov Projection Methods for Model Reduction', PhD Thesis, Dept. of Electrical Engineering, University of Illinois at Urbana
Champaign, 1997
8
Salimbahrami, B. ' Structure Preserving Order Reduction of Large Scale Second Order Models' PhD Thesis, Dept. of Electrical Engineering,
Technische Universitaet Muenchen, 2005
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8. MORe: Modal Projection and Moment Matching
Table 1 – Comparison between coupled response prediction techniques
9
Everstine, G. C.. 'A symmetric potential formulation for fluid structure interaction' Journal of Sound and Vibration , 79, 157—160, 1981
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9. Numerical Test Case: Benchmark Problem
➢ABAQUS Benchmark Problem : 'Acid-Test' 10 :1.10.2 Analysis of a point-loaded, fluid-
filled,spherical shell.
Model Description:
•The model is a semicircular shell and fluid mesh of radius 2.286 m.
• A point load on the symmetry axis of magnitude 1.0 N is applied to the shell.
• The shells are 0.0254 m in thickness and have a Young's modulus of 206.8 GPa, a Poisson's
ratio of 0.3, and a mass density of 7800.0 kg/m3.
•The acoustic fluid has a density of 1000 kg/m3 and a bulk modulus of 2.25 GPa.
●The response of the coupled system is calculated for frequencies ranging from 100 to 1000 Hz in
1 Hz increments.
➔Obtaining accurate solutions in this case requires that the resonances and modes be modeled
very accurately10.
Results for comparison:
Modal Expansion results from Stepanishen, P. and Cox 200010 : Compares results from Modal
Expansion and ABAQUS Direct and modal projection solutions.
10
Stepanishen, P. and Cox, L. 'Structural-Acoustic Analysis of an Internally Fluid-Loaded Spherical Shell: Comparison of Analytical
and Finite Element Modeling Results' NUWC Technical Memorandum, 2000 , Rhode Island: 00—118, USA
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10. Numerical Test Case: Benchmark Problem
ABAQUS Benchmark Problem : Simulated in ANSYS for initial comparison
Figure:4 : ANSYS Axisymmetric structural (left) and coupled (right) FE Mesh.
21907 Elements – Combination of ANSYS PLANE42 and ANSYS FLUID29 elements
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11. Benchmark Test Case: Initial Comparison
ABAQUS Benchmark Problem : ANSYS Comparison with closed form (Modal Expansion) Solution.
Figure:5 : Driving point didplacement [Log]
Modal Expansion Results (Velocities) also Available Online:
http://sufi.nchc.org.tw:2080/v6.5/books/bmk/default.htm
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12. Benchmark Test Case: Order Reduction via Arnoldi
➔ Comparison between ANSYS and MORe via one-sided Arnoldi (OSA)
Figure:6 : ANSYS and Arnoldi predicted Driving point displacement
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13. Benchmark Test Case: Order Reduction via Arnoldi
➔ Predicted fluid pressure at the centre of the acoustic domain via OSA.
Figure:7 : ANSYS and Arnoldi predicted fluid pressure
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14. Benchmark Test Case: Order Reduction via Arnoldi
➔ Local Error plot for fluid pressure at the centre of the acoustic domain via OSA approach.
Figure:8 : Local Error plot for fluid pressure: ANSYS and Arnoldi
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15. Comparison between one-sided and two-sided Arnoldi
ABAQUS Benchmark Problem : Local Error plot for driving point displacement.
Figure:9: Local Error plot for driving point displacement - Comparison between one-sided and
two-sided predicted results
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16. Comparison between one-sided and two-sided Arnoldi
ABAQUS Benchmark Problem : True and Relative Error (Convergence) at start frequency
Figure:10 : Comparison between one-sided and two-sided convergence pattern.
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17. Comparison between one-sided and two-sided Arnoldi
ABAQUS Benchmark Problem : True and Relative Error (Convergence) at end frequency
Figure:11 : Comparison between one-sided and two-sided convergence pattern for end frequency (1000Hz.).
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18. Results: Computational Times: Benchmark Test Case
Table 2 – Computational Times; Benchmark test case
➔ Breakdown of computational steps for the Arnoldi based moment matching approach:
• Extract Matrices
• Read Matrices and generate required (q) Arnoldi Vectors
• Perform reduced harmonic simulation and convergence
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19. Results: Computational Times: Benchmark Test Case
Table 3 – Split Computational Times – Benchmark test case
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20. Results: Initial comparison with Modal approaches
Table 4 – Computational times for Modal projection techniques.
➔ Analysis carried out using ABAQUS V6.7.
➔ 20441 acoustic elements and 333 shell elements used for the Benchmark problem.
➔ Frequency sweep: 100 to 1000 Hz.: 901 substeps.
Future Work:
➔ Accuracy comparison.
➔ Current implementation for undamped/damped Arnoldi projection framework in
Matlab/Mathematica.
11
ABAQUS V6.7 Theory Manual, ABAQUS Inc., USA
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21. Summary
➔ The Arnoldi process generates orthonormal projection matrix [V] spanning
the Krylov subspace to match the maximum number of low frequency
moments of the coupled structural-acoustic system matrices – Resulting
projected system is of much lower dimension.
➔ Only matrix-vector dot product is required (+ 1 LU Decomposition).
➔ Vectors are dependent on geometry (FE/FE Information), and can be
efficiently incorporated into optimization or sensitivity analysis.
➔ Complete output approximation is guaranteed for the one-sided Arnoldi
process i.e. The Arnoldi generated matrix [V] can match both displacements
on the structural domain, and sound pressure levels in the fluid domain.
➔ Better approximation properties were found with the application of two-
sided Arnoldi process. A MIMO version must be employed to match more
than one specific output.
➔ Extension to damped formulation can be made by the complex stiffness
approach or by the explicit participation of [C]. Complex stiffness: Arnoldi;
Future Work: Participation of [C]: First order transformation (or) Compute
vectors using second order Arnoldi (SOAR) / two-sided SOAR .
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22. End.
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