Using Hyper-Dual Numbers To Construct Parameterized Reduced...

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Using Hyper-Dual Numbers To Construct

Parameterized Reduced-Order ModelsMatthew R. Brake, Sandia National Laboratories

Jeffrey A. Fike, Stanford University

Sean D. Topping, University of ArizonaNovember 19, 2014

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed MartinCorporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2014-19346 C

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Motivation

Real manufactured systems differ from the idealized systems that are

analyzed using computational analysis tools.

Random variations in geometry and material properties

Uncertainty Quantification (UQ) techniques typically require very

many samples to assess the effects of these random variations.

Analysis of very many, slightly different systems

Analyzing geometric variations would require the generation of

very many, slightly different meshes

Creating one good quality mesh can dominate the analysis time

Creating very many meshes can be prohibitive

Parameterized Reduced-Order Models (PROMs) reduce the cost of

sampling compared to using high-fidelity, Full-Order Model.

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Parameterized Reduced-Order Models (PROMs)

Create parameterized models using a Taylor series expansion about

the nominal design:

f̄(x +∆x) = f(x) + (∆x)f ′(x) +(∆x)2

2f ′′(x) +

(∆x)3

6f ′′′(x) + . . . .

Quantity of interest f(x):

Mass and stiffness matrices from Finite-Element Analysis (FEA)

Outputs of FEA, such as displacements or natural frequencies

Perturbations∆x: variations in geometry or material properties

Terminology:

Parameterized Full-Order Model if applied to FEA quantities

Parameterized Reduced-Order Model (PROM) if applied to a

Reduced-Order Model (ROM)

Craig-Bampton (C-B) Component Mode Synthesis (CMS)

approach [Craig and Bampton 1968]

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Parameterization Strategies

For a system composed of multiple components, parameterization

can be introduced at several possible levels.

Component Matrices

Construct CMS Matrices

Assemble CMS System

Analyze CMS System

Assemble FE System

Analyze FE System

Parameterize Component Matrices

Parameterize FE System Matrices Parameterize CMS System Matrices

Parameterize CMS Component Matrices

Parameterize Eigenvalues Parameterize Eigenvalues

Finite Element Branch Component Mode Synthesis Branch

Parameterize Component Matrices

Evaluate effectiveness of creating PROMs at each level

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Derivative Calculation Methods

Finite-Difference Formulas:

Accuracy depends on particular formula used

Require several evaluations of perturbed systems

For geometric variations this requires several meshes

Captures actual behavior away from nominal design

May enable accurate representation of large perturbations

Hyper-Dual Numbers [Fike and Alonso 2011]:

Derivative calculations are exact to machine precision

Perturbations applied to non-real part of the number

Real part contains nominal design and is unperturbed

Requires only one mesh for the nominal design, and information

on how mesh would change with geometric variations

Derivatives produced using information at a single point, may

only be valid for small perturbations from nominal

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Hyper-Dual Numbers

Complex Numbers:

a+ bi

i2 = −1

Dual Numbers: [Study 1903]

a+ bε

ε2 = 0, ε 6= 0

Quaternions: [Hamilton 1843]

a+ bi + cj + dk

i2 = j2 = k2 = −1

ij = −ji = k

Hyper-Dual Numbers: [Fike and Alonso 2011]

a+ bε1 + cε2 + dε1ε2

ε21 = ε22 = (ε1ε2)2 = 0

ε1 6= ε2 6= ε1ε2 6= 0

ε1ε2 = ε2ε1

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Derivative Calculations using Complex Numbers

Taylor series for a real valued function subject to a perturbation in

the imaginary direction:

f(x + hi) = f(x) + hf ′(x)i − 1

2!h2f ′′(x)− h3f ′′′(x)

3!i + ...

f(x+hi) =

(f(x)− 1

2!h2f ′′(x) + ...

)︸︷︷︸

real

+ h

(f ′(x)− 1

3!h2f ′′′(x) + ...

)i︸︷︷︸

imaginary

First-Derivative Complex-Step Approximation: [Martins, Kroo, and Alonso 2000 and

Martins, Sturdza, and Alonso 2003]

f ′(x) =Im [f(x + hi)]

h+O(h2)

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Derivative Calculations using Hyper-Dual Numbers

Taylor series with a dual number perturbation:

f(x + hε) = f(x)︸︷︷︸real

+ hf ′(x)ε︸︷︷︸non-real

Taylor series with a hyper-dual perturbation:

f(x+h1ε1+h2ε2+0ε1ε2) = f(x)+h1f′(x)ε1+h2f

′(x)ε2+h1h2f′′(x)ε1ε2

Taylor series when including an ε3 perturbation:

f(x + h1ε1 + h2ε2 + h3ε3 + 0ε1ε2 + 0ε1ε3 + 0ε2ε3 + 0ε1ε2ε3)

= f(x) + h1f′(x)ε1 + h2f

′(x)ε2 + h3f′(x)ε3 + h1h2f

′′(x)ε1ε2

+ h1h3f′′(x)ε1ε3 + h2h3f

′′(x)ε2ε3 + h1h2h3f′′′(x)ε1ε2ε3

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Accuracy of Derivative Calculation Methods

10−20

10−10

100

1010

10−20

100

1020

Step Size

Err

or

10−30

10−20

10−10

100

(a)

(b)

Plots:

(a) First-Derivative

Calculations

(b) Second-Derivative

Calculations

Legend:

• Forward-Difference

Formula

∗ Complex-Step

Approximation

◦ Hyper-Dual Numbers

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Stepped Beam Example

L

W

lHH2

x

Consider variations in geometry and material properties of

middle section

Create parameterized models for various quantities:

Eigenvalues, mass and stiffness matrices, etc.

Vary order of Taylor series and derivative calculation method:

Expansion includes up to: second, third, or fourth derivatives

Derivatives computed using finite differences or hyper-dual

numbers

Fourth-Order Finite-Difference Parameterization (FDP)

Third-Order Hyper-Dual Parameterization (HDP)

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Varying Young’s Modulus of Middle Section

40 70

0

1

1.5

Na

tura

l F

req

ue

ncy,

kH

z

Elastic Modulus of Middle Section, GPA

100

0.5

Legend:

∗ Analytical Solution

• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues

· · · HDP, 3rd Order, FE

matrices

– · – HDP, 3rd Order, FE

eigenvalues

— HDP, 3rd Order,

CMS matrices

– – HDP, 3rd Order,

CMS eigenvalues

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Varying Material Density of Middle Section

1500 2750

0

1

1.5

Na

tura

l F

req

ue

ncy,

kH

z

3Density of Middle Section, kg/m

4000

0.5

Legend:


• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues


matrices


eigenvalues

— HDP, 3rd Order,

CMS matrices


CMS eigenvalues

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Varying Cross-Sectional Height of Middle Section

0.02 0.03

0

1

1.5

Na

tura

l F

req

ue

ncy,

kH

z

Cross-Sectional Height of Middle Section, m

0.04

0.5

Legend:


• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues


matrices


eigenvalues

— HDP, 3rd Order,

CMS matrices


CMS eigenvalues

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Varying Cross-Sectional Width of Middle Section

0.02 0.03

0

1

1.5

Na

tura

l F

req

ue

ncy,

kH

z

Cross-Sectional Width of Middle Section, m

0.04

0.5

Legend:


• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues


matrices


eigenvalues

— HDP, 3rd Order,

CMS matrices


CMS eigenvalues

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Varying Length of Middle Section

0.15 0.3

0

1

2

Na

tura

l F

req

ue

ncy,

kH

z

Length of Middle Section, m

0.45

Legend:


• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues


matrices


eigenvalues

— HDP, 3rd Order,

CMS matrices


CMS eigenvalues

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Varying Location of Middle Section’s Center

0.25 0.45

0

1

2

Na

tura

l F

req

ue

ncy,

kH

z

Location of Middle Section’s Center, m

0.65

Legend:


• FDP, 4th Order,

CMS matrices

◦ FDP, 4th Order,

eigenvalues


matrices


eigenvalues

— HDP, 3rd Order,

CMS matrices


CMS eigenvalues

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Summary, Part 1

Range of perturbations with < 1% error in 5th natural frequency:

Middle Section’s Middle Section’s Middle Section’s

Method Width, B2 Height, H2 Modulus, E2

FDP, 2nd Order CMS [-85%, 327%] [-32%, >1000%] [<-99%, >1000%]

FDP, 2nd Order Eigenvalues [-95%, 152%] [-38%, 70%] [-50%, 74%]

FDP, 4th Order CMS [-91%, 228%] [-69%, 844%] [-99%, >1000%]

FDP, 4th Order Eigenvalues [-93%, 231%] [-56%, 51%] [-66%, 92%]

HDP, 2nd Order FE [<-99%, >1000%] [-29%, >1000%] [<-99%, >1000%]

HDP, 2nd Order Full CMS [-5%, 327%] [-5%, 799%] [<-99%, >1000%]

HDP, 2nd Order Reduced CMS [-19%, 800%] [-19%, 800%] [<-99%, >1000%]

HDP, 2nd Order Eigenvalues [<-99%, 153%] [-40%, 57%] [-52%, 83%]

HDP, 3rd Order FE [<-99%, >1000%] [<-99%, >1000%] [<-99%, >1000%]

HDP, 3rd Order Full CMS [-13%, 14%] [-13%, 14%] [<-99%, >1000%]

HDP, 3rd Order Reduced CMS [-32%, 35%] [-32%, 35%] [<-99%, >1000%]

HDP, 3rd Order Eigenvalues [<-99%, 182%] [-42%, 52%] [-61%, 95%]

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Summary, Part 2

Range of perturbations with < 1% error in 5th natural frequency:

Middle Section’s Middle Section’s Middle Section’s

Method Density, ρ2 Location, ` Length,W

FDP, 2nd Order CMS [-72%, 67%] [-15%, 12%] [-34%, 21%]

FDP, 2nd Order Eigenvalues [-54%, 69%] [-15%, 17%] [-85%, 30%]

FDP, 4th Order CMS [-87%, 78%] [-21%, 17%] [-48%, 28%]

FDP, 4th Order Eigenvalues [-73%, 94%] [-18%, 26%] [-51%, 35%]

HDP, 2nd Order FE [<-99%, >1000%] [-5%, 3%] [-3%, 3%]

HDP, 2nd Order Full CMS [-6%, 67%] [-2%, 2%] [-33%, 2%]

HDP, 2nd Order Reduced CMS [-19%, 68%] [-9%, 8%] [-33%, 10%]

HDP, 2nd Order Eigenvalues [<-99%, 154%] [-12%, 14%] [-82%, 29%]

HDP, 3rd Order FE [<-99%, >1000%] [-4%, 4%] [-6%, 6%]

HDP, 3rd Order Full CMS [-13%, 14%] [-5%, 4%] [-5%, 5%]

HDP, 3rd Order Reduced CMS [-32%, 36%] [-13%, 12%] [-30%, 21%]

HDP, 3rd Order Eigenvalues [-64%, 78%] [-20%, 17%] [-32% 45%]

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Conclusions

Explored multiple ways of creating parameterized models:

Parameterization introduced at several points in the analysis

process, accuracy depends on type of variation:

Cross-section or material properties: parameterize FE matrices

Location or length of middle section: parameterize eigenvalues

Taylor series truncated at second-, third-, or fourth-derivative

terms

Increasing the order of the Taylor series results in more accurate

parameterized models

Derivatives computed using finite differences and hyper-dual

numbers

Finite-difference calculations tend to be more accurate for a

larger range of perturbations

For geometric perturbations, finite differences require several

meshes whereas hyper-dual numbers only require a single mesh

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Questions?

This work was funded in part by the U.S. Department of Energy’s Predictive

Science Academic Alliance Program (PSAAP) Center at Stanford University.

Hyper-dual number implementations can be downloaded from:

http://adl.stanford.edu/hyperdual

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http://adl.stanford.edu/hyperdual

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