stochastic subspace projection schemes for solving random

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HCF Conference, New Orleans, LO, March 8HCF Conference, New Orleans, LO, March 8--11, 200511, 2005

Stochastic Subspace Projection Schemes Stochastic Subspace Projection Schemes for Solving Random for Solving Random BliskBlisk Mistuning Mistuning

Problems in Jet EnginesProblems in Jet EnginesDr. Dan M. GhiocelDr. Dan M. Ghiocel

Email: dan.ghiocel@ghiocelEmail: [email protected]: 585Phone: 585--641641--03790379

Ghiocel Predictive Technologies Inc.Ghiocel Predictive Technologies Inc.http://www.ghiocelhttp://www.ghiocel--tech.comtech.com


To discuss the application of a powerful stochastic subspace projection scheme for solving mistuning problems in bladed-disks via reduced-order modeling (ROM).

The proposed stochastic subspace projection scheme called the Stochastic Perturbation Matrix (SPM) approach provides an efficient tool for accurately solving large (and small) random mistuning problems, for both LO and HO system modes.

Objective of this PresentationObjective of this Presentation


1. Computational Stochastic Mechanics Issues1. Computational Stochastic Mechanics Issues

2. 2. Stochastic Perturbation Matrix Subspace Projection Stochastic Perturbation Matrix Subspace Projection

3. 3. The 72 Blade Compressor The 72 Blade Compressor BliskBlisk ExampleExample

4. 4. Concluding RemarksConcluding Remarks

Content PresentationContent Presentation


Two Major Stochastic Modeling Aspects:Two Major Stochastic Modeling Aspects:

1. Develop Accurate Stochastic Approximation Models for High-Complexity Behavior Given Sample Dataset (Data-based Stochastic ROM) - Global and Local Accuracy in Statistical Data Space – ROM in Data Space- For both System Inputs and Outputs

2. Develop Fast Stochastic Simulation Models Given the Physics (PDE) and Stochastic Inputs (Physics-based Stochastic ROM) - Global and Local Accuracy in Physical Space – ROM in Physical Space- For System Outputs

3. Combine the Statistics-based and Physics-based Stochastic ROMs

1.1. Computational Stochastic Mechanics IssuesComputational Stochastic Mechanics Issues


StatisticsStatistics--based Stochastic ROM based Stochastic ROM

Implicit Formulation: Using joint PDF estimation of

y

jx −

y

jx −

==)(f),y(f)y(f

xxx

T],y[z x=

dy)y(yf]y[E xx ∫∞

∞−

=

--- Least-square fitting ( is explicit)

--- Stochastic Networks ( is implicit)Decomposes overall complex JPDF in localized simple JPDFs.

y

y

Limited

Refined

)()(ry xxx ε+=

Explicit Formulation: Using function approximation via nonlinear regression

Convergence: Minimizing Mean-Square Error (in Mean-Square sense)Causal relationship

)(r]y[E xx =

Convergence: Using Maximumum Likelihood Function (in Probability sense)Non-causal relationship

Second-Order (SO) Approximation of Stochastic Fields

High-Order (HO) Approximation of Stochastic Fields

Stochasticity defined by second-order random vector/field of residuals


HO Stochastic Field Expansion in Local JPDF Models HO Stochastic Field Expansion in Local JPDF Models

Data Space “State” Space

Non-Analytical Joint PDF

Space Transformation

NonNon--Analytical Analytical MPDFsMPDFswith Complex Stochasticwith Complex StochasticInterdependenciesInterdependencies

JPDF DecompositionJPDF Decomposition


Stochastic PhysicsStochastic Physics--based ROMbased ROM

The development of efficient Stochastic Physics-based ROMs includes:

1) Partitioning the original physical stochastic domain into subdomains(stochastic domain decomposition, substructuring)

2) Projecting the original stochastic solution onto reduced-size stochastic subspaces (stochastic projection, physics-based ROM)

Remark:Physics-based stochastic ROM are extremely robust in comparison with perturbation methods (Taylor expansion, Neumann expansion, etc.) that are limited to local variations within the convergence radius of functions

Example:CMU SNM Approach for mistuning uses a solution projection in an Eigen subspace.


Large Mistuning vs. Small MistuningLarge Mistuning vs. Small MistuningThe equations of motion in physical coordinates are

Small (Frequency) MistuningSmall (Frequency) Mistuning (standard ROM) assumes:- proportional variation with RV

Notes: - Matrix is diagonal (same value for the blade/sector DOFs). - Deviation is applied to E modulus (is the same for all DOFs) - “Local” blade modes maintain their shapes

Large (Mode Shape) MistuningLarge (Mode Shape) Mistuning (input calibration for ROM) assumes:- non-proportional variation with ( and ) SF

Remark:- Matrix is non-diagonal. It can be computed - Deviation is applied to (and ) (is different for each DOF)- “Local” blade modes may change their shapes.

εKIKK∆K === εε

εK∆K =

εε

εε

F]αZ∆M)M(G)(C∆KK[ a =++−+++ 2ωiω

K∆KKε 1−=

M

K M

Stochastic Stochastic ModelModel

2. Stochastic Perturbation Matrix Projection2. Stochastic Perturbation Matrix Projection


Stochastic Perturbation Matrix Subspace ProjectionStochastic Perturbation Matrix Subspace Projection

FxuxK =)()(

)(( (y( 1jkk

jjk xu)xε)x)xu −∑=

Stochastic FEA Problem:Stochastic FEA Problem:

FxuxKK =∆+ )()]([ kk

The GPA subspace projection of the stochastic solution is:

The expansion is fastThe expansion is fast--convergent if the probability densities of random convergent if the probability densities of random eigeneigenvalues of the matrix are highvalues of the matrix are highly overlapping; only few ly overlapping; only few terms are needed, or equivalently the GPM ROM size is very reducterms are needed, or equivalently the GPM ROM size is very reduced. ed.

For a random realization k of stochastic input vector x we can rewrite

)]([ k1 xKKK ∆+−

Note:

Few terms Few terms needed!needed!


Bimodal Long Tail

10x1x

10x1x3%3%

0.5%0.5%

PDF of Mistuned Rotor Blade Tip Amplitude Responses

3. The 72 Blade Compressor 3. The 72 Blade Compressor BliskBlisk ExampleExample


5 SPM Basis Vectors5 SPM Basis Vectors 14 SPM Basis Vectors14 SPM Basis Vectors

Stochastic Perturbation Matrix ROM Solution Stochastic Perturbation Matrix ROM Solution for A 72 Blade for A 72 Blade BliskBlisk SystemSystem

LO Bending Blade Mode Family MistuningLO Bending Blade Mode Family Mistuning


900 910 920 930 940 950 960 970 980 990 10000

1

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8

9

Blue: Nominal model Red: Perturbed model

900 910 920 930 940 950 960 970 980 990 10000

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Blue: Nominal model Red: Perturbed model (Direct method)Black: Perturbed model (CA method)

900 910 920 930 940 950 960 970 980 990 10000

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12


900 910 920 930 940 950 960 970 980 990 10000

1

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SPM ROM Solution for Modified SPM ROM Solution for Modified BliskBlisk SystemSystem

ExactExact

ConvergedConverged

LO Bending Blade Mode Mistuning for Modified Disk (Stiffer)LO Bending Blade Mode Mistuning for Modified Disk (Stiffer)5 SPM Vectors5 SPM Vectors

10 SPM Vectors10 SPM Vectors14 SPM Vectors14 SPM Vectors


Multiple Multiple Starting Starting Frequency PointsFrequency Points

Single Single Starting Frequency Point

SPM ROM Solution Robustness StudySPM ROM Solution Robustness Study

)t,*(K*)(K)t,*(K ∆ω∆+ω∆+ω=∆ω∆+ω)t,(K)(K)t,(K ∆ω∆+ω=∆ω


EO=1EO=1

Direct KSP ROM

Single Frequency Point SPM ROM SolutionSingle Frequency Point SPM ROM Solution

100 Simulations

4 basis vectors


EO=1

Single Frequency Point SPM ROM SolutionSingle Frequency Point SPM ROM Solution

10 Simulations


EO=2

Single Frequency Point SPM Solution Single Frequency Point SPM Solution

EO=3

Single Simulation

Single Simulation


Blade 22Blade 17 Blade 63

Application of SPM ROM Approach to Application of SPM ROM Approach to Maintenance of Geometrically Mistuned Maintenance of Geometrically Mistuned IBRsIBRs


SPM ROM for Studying Blade Geometry Variation EffectsSPM ROM for Studying Blade Geometry Variation Effects

Blade 22 Max

Max of All Blades

Blade 17 Max

DOEDOE


1.1. Stochastic projection schemes represent systematic mathematicalStochastic projection schemes represent systematic mathematicalprocedures for building powerful physicsprocedures for building powerful physics--based stochastic ROMs that based stochastic ROMs that are highly applicable to bladedare highly applicable to bladed--disk mistuning problems.disk mistuning problems.

2.2. For large mistuning problems, the proposed Stochastic PerturbatiFor large mistuning problems, the proposed Stochastic Perturbation Matrix on Matrix (SPM) approach is an extremely accurate and fast prediction tool(SPM) approach is an extremely accurate and fast prediction tool. SPM . SPM ROM has an extremely fast convergence, since the size of the reqROM has an extremely fast convergence, since the size of the required uired ROM is very reduced. For the illustrated (meanROM is very reduced. For the illustrated (mean--based PC) 72 blade based PC) 72 blade bliskblisksystem case study, the typical SPM ROM size for computing accurasystem case study, the typical SPM ROM size for computing accurate te results was from 5 to 20 equations. results was from 5 to 20 equations.

3. 3. The SPM ROM approach is perfectly fitted for solving large mistuThe SPM ROM approach is perfectly fitted for solving large mistuning ning problems, for both lowproblems, for both low--order and highorder and high--order system modes, including order system modes, including complex dynamic couplings in veering regions.complex dynamic couplings in veering regions. The author believes that the SPM ROM approach will play a gradually increasing role in future for solving difficult, large mistuning problems.

4. Concluding Remarks4. Concluding Remarks

stochastic subspace projection schemes for solving random

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