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공학석사학위논문

Development of a Precision Vibration Analysis Framework for Structural Safety

Prediction of Gas Turbine Blades

가스터빈 블레이드의 구조 건전성 예측을 위한

고정밀 진동해석 프레임워크 개발

2018 년 2 월

서울대학교 대학원

기계항공공학부

김 용 세

i

Abstract

Development of a Precision Vibration Analysis Framework for Structural Safety

Prediction of Gas Turbine Blades

Yongse Kim

Department of Mechanical and Aerospace Engineering

The Graduate School

Seoul National University

Blades in the gas turbine engine are subjected to resonant excitation which causes

high cycle fatigue accumulation, and eventually may lead to failure of the blades. To

avoid this, the structural response associated with the resonant condition should be

predicted, and it is indispensable to accurately predict the dynamic characteristics of

the blade at its preliminary design process. In this thesis, an advanced vibration

analysis framework including the capability to predict the crucial physical

phenomena in gas turbine blades, i.e., geometric nonlinearity, high-speed rotational

and thermal effects, is developed. Three-dimensional co-rotational (CR) solid

element is employed for the geometric nonlinearity. On the other hand, a large

amount of discretized elements may be required for more accurate analysis of a

ii

complex blade configuration, and this causes significant increase in computational

cost. To overcome such problem, reduced order modeling based on the proper

orthogonal decomposition (POD-ROM) analysis is also developed.

The numerical examination is carried out aimed on the first-stage turbine blade of

75MW gas turbine engine under various operating conditions, i.e., high-speed

rotation and high temperature. The present analyses are validated by comparing with

the results obtained by the commercial software, ANSYS. As a result, it is found that

the present analyses show good correlation by comparison the natural frequencies

and mode shapes. And, by using the present POD-ROM, significant improvement in

computational cost is accomplished when compared with the full order model (FOM)

and ANSYS analysis. Also, the snapshot collection time for the initial POD-ROM

analysis is significantly improved by parallel computation base on the domain

decomposition.

Keywords: Gas turbine blade, High cycle fatigue, Vibration analysis,

Co-rotational element, Proper orthogonal decomposition,

Reduced order modeling, Domain decomposition

Student Number: 2016-20730

iii

Contents

Page

Abstract ..................................................................................................................... i

Contents ................................................................................................................. iii

List of Tables ............................................................................................................ v

List of Figures ......................................................................................................... vi

Chpater 1 Introduction ....................................................................................... 1

1.1 Background and Motivation ............................................................................ 1

1.2 Previous Researches ........................................................................................ 6

1.3 Objectives and Thesis Overview ................................................................... 11

Chpater 2 Theoretical Background .................................................................. 12

2.1 Ten-node Tetrahedral Solid Element.............................................................. 12

2.2 Geometrically Nonlinear Dynamics Based On the Co-rotational Formulation

............................................................................................................................. 17

2.2.1 Elemental Kinematics ............................................................................. 18

2.2.2 Inertial Load Vector and Tangent Matrices ............................................. 21

2.2.3 Governing Equation for Time-transient Analysis ................................... 23

2.3 Rotational and Thermal Effects ..................................................................... 25

2.3.1 Rotational Effect ..................................................................................... 25

2.3.2 Thermal Effect ........................................................................................ 28

2.4 Reduced Order Modeling Based On the Proper Orthogonal Decomposition 30

iv

2.4.1 Concept of the POD Method ................................................................... 30

2.4.2 POD-ROM for Structural Analysis ......................................................... 34

2.5 Parallel Computation Based on the Domain Decomposition ........................ 37

Chpater 3 Numerical Results and Discussion ................................................. 43

3.1 Non-rotating Condition .................................................................................. 48

3.2 Rotating Condition ........................................................................................ 51

3.3 Thermal Effect ............................................................................................... 56

Chpater 4 Conclusion and Future Works ........................................................ 59

4.1 Conclusion ..................................................................................................... 59

4.2 Future Works.................................................................................................. 61

References .............................................................................................................. 62

국문초록 ................................................................................................................. 66

v

List of Tables

Page

Table 3.1 Material properties used in the present validation [35] ........................... 45

Table 3.2 Finite element information ...................................................................... 46

Table 3.3 Comparison of the natural frequencies at non-rotating condition ........... 48

Table 3.4 Comparison of the natural frequencies at 3,600 rpm ............................... 51

Table 3.5 Comparison of the computation time in a single rotating condition ....... 53

Table 3.6 Comparison of the snapshot collection time............................................ 55

vi

List of Figures

Page

Fig. 1.1 Components of a gas turbine engine ............................................................ 3

Fig. 1.2 Primary elements of the forced vibration [2] ............................................... 3

Fig. 1.3 High cycle fatigue failure of the gas turbine blades [4] ............................... 4

Fig. 1.4 Example of Campbell diagram .................................................................... 4

Fig. 1.5 Forced vibration and structural response using FSI analysis ....................... 5

Fig. 1.6 Illustration of the fatigue crack initiation and propagation [9] .................... 8

Fig. 1.7 Modal analysis of the impeller blades [10] .................................................. 9

Fig. 1.8 Structural analysis of gas turbine blades based the on nonlinear finite

element method [15].................................................................................. 10

Fig. 2.1 Stresses on the three-dimensional solid element ........................................ 15

Fig. 2.2 Nodal information for the tetrahedral 10-node element ............................. 16

Fig. 2.3 Coordinates and elemental kinematics of the CR solid element [24] ........ 20

Fig. 2.4 Rotating spring-mass system ..................................................................... 26

Fig. 2.5 Effects of the stress stiffening and spin softening [26] .............................. 27

Fig. 2.6 Computing algorithm to couple heat conduction and vibration analyses .. 29

Fig. 2.7 Snapshot results and state locus ................................................................. 32

Fig. 2.8 Principal axes of the state locus ................................................................. 33

Fig. 2.9 Linear summation of POD modes .............................................................. 33

Fig. 2.10 Structural analysis based on the POD-ROM............................................ 36

vii

Fig. 2.11 Three phases of multilevel partitioning [34] ............................................ 39

Fig. 2.12 Coarsening algorithm of the METIS ........................................................ 40

Fig. 2.13 Refinement algorithm of the METIS ....................................................... 41

Fig. 2.14 Parallel computation process using the METIS ....................................... 42

Fig. 3.1 GT11N first stage turbine blade ................................................................. 44

Fig. 3.2 Discretization in the finite element representation .................................... 46

Fig. 3.3 Boundary conditions .................................................................................. 47

Fig. 3.4 Comparison of the mode shapes (1st – 4th) ................................................. 49

Fig. 3.5 Comparison of the mode shapes (5th – 8th) ................................................. 50

Fig. 3.6 Campbell diagram ...................................................................................... 53

Fig. 3.7 Turbine blade partitioned into 20 sub-domains by the METIS .................. 55

Fig. 3.8 Temperature conditions .............................................................................. 57

Fig. 3.9 Temperature gradient of the blade.............................................................. 57

Fig. 3.10 Comparison of the natural frequencies considering thermal effect .......... 58

1

Chpater 1

Introduction

1.1 Background and Motivation

A gas turbine is one component of the rotary power engine that drives combustion

gases at high temperature and pressure. This is mainly used as a power source for

power plants and aircrafts that require large thrust and efficiency. In the recent years,

demand for the gas turbines has been greatly increased as the global CO2 emission

regulations have been established and the proportion of natural gas sources has

increased. Thus, gas turbines with high efficiency and large capacity have been

developed continuously. And its global maintenance market is anticipated to grow at

compound annual growth rate (CAGR) of 4.14% from 2017 to 2021 [1].

Gas turbine engine components (Fig. 1) are exposed to severe environments due

to high efficiency and power requirements in modern advanced engine designs.

Under these conditions, the components should have sufficient structural reliability.

Especially, the blades, which are a core component in the gas turbine, are directly

contacted with heated and pressurized gases during operation and are subjected to

static loads due to high-speed rotation. At the same time, the blades are subjected to

periodic aerodynamic excitation in resonant conditions, as shown in Fig. 1.2. It is

due to the interaction with stationary disturbances such as vanes and struts. Such a

2

forced vibration in resonant conditions causes high vibratory stresses on the blades

[2]. This results in high cycle fatigue (HCF), which may eventually lead to failure of

the blades as shown in Fig. 1.3 [3-4]. These HCF problems averaged 2.5 cases per

one single gas turbine engine development, and US Air Force estimated an annual

cost of $ 2 billion in HCF calibration costs by 2020 [5].

There have been two approaches to prevent HCF problems [6-7]. The first

approach is to design to avoid dangerous resonances in the operating range. It is a

qualitative, standard design practice to avoid resonance using Campbell diagram

shown in Fig. 1.4. In general, the resonance should be avoided in the lower order

structural modes (first bending, torsion modes etc.). The second approach is to allow

resonance in the operating range and to quantitatively assess the associated response

level of the blades in resonant conditions. This approach is required to accurately

predict the structural response under the resonant condition by going through forced

vibration analysis using any available fluid-structure interaction (FSI) method. As

shown in Fig. 1.5., the vibratory stresses due to the aerodynamic excitations are

predicted and the structural safety with respect to HCF is evaluated based on the

predicted results.

In order to perform such approaches, it is essential to accurately predict the

dynamic characteristics of the blade at its preliminary design process. To do so,

application of the vibration analysis techniques with high precision and

computational efficiency is required.

3

Fig. 1.1 Components of a gas turbine engine

Fig. 1.2 Primary elements of the forced vibration [2]

4

Fig. 1.3 High cycle fatigue failure of the gas turbine blades [4]

Fig. 1.4 Example of Campbell diagram

5

Fig. 1.5 Forced vibration and structural response using FSI analysis

6

1.2 Previous Researches

Experimental methods are the most intuitive ways to predict the dynamic

characteristics of gas turbine blades, but it is not quite simple to precisely measure

those quantities under operating conditions with high speed and pressure. In addition,

redesign of the blades due to repetition of experiments may result in high cost and

development delays. Therefore, industries and research institutes have carried out

related research by applying the computational structural dynamics (CSD) analysis

based on the finite element method (FEM) during the design process.

Lee [8] predicted the natural frequencies of the turbine blades for vibrational

reliability assessment during the blade prototype development stage, and compared

with impact modal testing results. Then, Campbell diagram analysis was performed

to evaluate the resonance risk of each blade, and the safe operation limit speed of the

gas turbine engine was selected. Choi [9] found that fatigue failure occurred at the

root of the turbine blade because the transient event inside the combustion chamber

caused resonance in the nearby first stage turbine blade. Kim [10] predicted the high

frequency resonant conditions due to the diffuser vanes by utilizing Singh’s

advanced frequency evaluation (SAFE) diagram analysis, and predicted the

vibratory stress by performing an one-way FSI analysis under the associated resonant

conditions. Netzhammer [11] predicted the blade vibratory stress due to the

asymmetry of the turbine housing, and presented the geometric design parameters to

minimize the vibration amplitude. In the other studies, precise vibration analysis

7

played an important role in assessing structural reliability of the gas turbine blades

with regard to HCF [12-14].

Recently, to improve the performance of gas turbines, twisted-surfaced, high

aspect ratio, and light-weight blade designs have been developed. Also, in the case

of turbine blades, it features complex shape including the cooling holes and relevant

passages because the flow temperature entering the turbine inlet is designed to be

high. On this perspective, application of the geometrically nonlinear structural

analysis to gas turbine blades needs to be considered for precise structural response

prediction. Nonlinear structural analysis requires a large-size computation with a

considerable number of discretized elements and degrees of freedom. In addition,

the computational cost may be drastically increased according to the iterative

algorithm applied. Some studies was performed to predict nonlinear dynamics, but

application for the research was limited to relatively simple blade geometries [15-

17]. There has been still limited research on the dynamic characteristics of the gas

turbine blades with large degrees of freedom including nonlinearities.

8

Fig. 1.6 Illustration of the fatigue crack initiation and propagation [9]

9

(a) Resonant conditions

(b) SAFE diagram

Fig. 1.7 Modal analysis of the impeller blades [10]

10

(a) Blade mode (first bending) (b) Stress distribution due to a tip friction force

Fig. 1.8 Structural analysis of gas turbine blades based the on nonlinear finite

element method [15]

11

1.3 Objectives and Thesis Overview

In this thesis, an improved vibration analysis for dynamic characteristics of the

gas turbine blades will be developed. First, a full order model (FOM) analysis will

be developed. For the structural modeling of three-dimensional blades, a tetrahedral

10-node element will be developed. And a nonlinear analysis based on the co-

rotational (CR) formulation suitable for structures undergoing small strains and large

displacements will be developed and applied. In the present FOM analysis, both

stress stiffening and spin softening effects will be considered in order to include the

relevant effects due to a large rotational speed of the gas turbine blades. Also, thermal

effect including the prediction of stiffness change due to the temperature gradient in

the high temperature environment will be considered. Then, the proper orthogonal

decomposition-reduced order modeling (POD-ROM) method for fast computation

on the eigenvalues will be developed and applied. The advantage of POD-ROM is

that it is capable of conducting the modal analysis of the gas turbine blade dynamic

characteristics by using the reduced dynamical system. Thus, it will become possible

to reduce the computational cost. Also, when ROM is constructed, such advantage

is significant for a repeated computation, e.g., construction of Campbell diagram.

In order to examine the present vibration analysis, the first-stage turbine blades of

a 75MW GT11N gas turbine engine will be employed. And, variety of the operating

conditions will be considered. Then, the present analysis will be verified by

comparing with the results obtained by the commercial software

12

Chpater 2

Theoretical Background

2.1 Ten-node Tetrahedral Solid Element

The finite element method (FEM) is a numerical approximation method used in a

wide range of engineering problems. A structure is discretized into finite number of

elements to obtain the parameters required for design such as deformation, stress,

natural frequencies, etc. Among the elements used in finite element analysis, the

solid elements are the most suitable for precisely discretizing three-dimensional

structures compared to the beam and shell elements. In particular, tetrahedral solid

elements [18] are required to discretize curved-surface such as the gas turbine blade

structures. Thus, in the present structural analysis, a tetrahedral 10-node higher order

element is developed and applied.

The stresses applied to the three-dimensional solid element are shown in Fig. 2.1,

and the equilibrium equations are as follows.

3111 211 0bx y z

σσ σ ∂∂ ∂+ + + =

∂ ∂ ∂ (1)

3212 222 0bx y z

σσ σ ∂∂ ∂+ + + =

∂ ∂ ∂ (2)

13

13 23 333 0bx y z

σ σ σ∂ ∂ ∂+ + + =

∂ ∂ ∂ (3)

Then, the constitutive equation in accordance with Hooke’s law is expressed as

Eq. (4).

1111 1122 1113

2211 2222 2213

1311 1322 1313

(symmetric)klmn mnkl

D D DD D D

D

D D Dwith D D

σ

= ∈ = ∈

=

(4)

When the node information of the tetrahedral 10-node element is given in Fig. 2.2,

the shape function equation is as follows.

1 1 1 2 2 2 3 3 3

4 4 4 5 1 2 6 2 3

7 3 1 8 1 4 9 2 4 10 3 4

(2 1), (2 1), (2 1),

(2 1), 4 , 4 ,

4 , 4 , 4 , 4

N N N

N N N

N N N N

ζ ζ ζ ζ ζ ζ

ζ ζ ζ ζ ζ ζ

ζ ζ ζ ζ ζ ζ ζ ζ

= − = − = −

= − = =

= = = =

(5)

The relationship between the strain and the displacement of each node is expressed

as Eq. (6) by the relationship between the axial directions and the torsional strain.

14

1 1 1 2 2 2 10

0 0

0 0

0 0,

0

0

Tex y z x y z z

x i

y ix

y z i

z ei

y i ixy x

yz

iyzx

x

u u u u u u u u

u Nx xu Ny y

u Nz zBu Bu N Nu

y xy xNu uzzz y

u uzz x

γ

γ

γ

=

∂ ∂ ∂ ∂

∂ ∂ ∈ ∂ ∂ ∈ ∂ ∂

∈ ∂ ∂ ∈ = = = =∂ ∂ ∂∂ + ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ + ∂∂ ∂

∂ ∂+

∂ ∂

[ ]1 2 3 4 10

0

1,2 10,

i

i i

Ny

N Nz x

i B B B B B B

∂

∂ ∂ ∂ ∂

= =

(6)

Using B matrices of Eq. (6), stiffness matrix of the tetrahedral 10-node element

is expressed as Eq. (7).

1 1 1 1

1 2 3 40 0 0 0

1 (det(J))d6

e

e T e

V

e T

K B DBdV

K B DB ζ ζ ζ ζ

=

=

∫

∫ ∫ ∫ ∫ (7)

15

Fig. 2.1 Stresses on the three-dimensional solid element

16

Fig. 2.2 Nodal information for the tetrahedral 10-node element

17

2.2 Geometrically Nonlinear Dynamics Based On the Co-

rotational Formulation

Many structural analyses based on the nonlinear formulation have been developed.

Among those, co-rotational (CR) formulation suggested by Rankin [19] is suitable

for geometrically nonlinear analysis of structures undergoing small strain and large

displacement. The main advantage of such CR formulation is that the same method

can be applied depending on the number of nodes and degrees of freedom of existing

linear elements. In addition, the definition of existing linear elements can be

minimally modified and applied. In other words, it can be extended to the

geometrically nonlinear analysis using existing linear elements [20]. The CR

formulation was widely used in classical shell elements, two-dimensional planar and

beam elements, and it was extended to three-dimensional solid elements [21-23].

Recently, a three-dimensional nonlinear dynamic formulation based on the CR

approach for solid elements was developed by Cho [24]. For the present structural

analysis, a geometrically nonlinear analysis is developed by applying his formulation.

18

2.2.1 Elemental Kinematics

The CR method analyzes the overall behavior of the structure as rigid and pure

deformation. A CR coordinate system is defined to facilitate such behavior. Figure 1

shows the elemental kinematics and coordinate system of the nonlinear CR three

dimensional solid element. The derivation of the basic formula of structural analysis

starts from the concept of the virtual energy and is expressed as Eq. (8).

( ) ( ) ( )T T T TL G GL G LV q f f Bq q fδ δ δ= = = (8)

where Lq , Gq are the local and global displacement vectors, and Lf , Gf are the

local and global internal load vectors. The relation between local and global

displacement is as in Eq. (9). Equation (10) is derived from Eqs. (8) and (9).

L Gq B qδ δ= (9)

L T

Gf B f= (10)

By applying the variations to Eq. (10), the global tangent stiffness matrix is

obtained as shown in Eq. (11). The matrix 𝐵𝐵 is defined as Eq. (12), which is the

relationship between the projector matrix, 𝑃𝑃 and the rigid body rotation matrix, 𝐸𝐸.

Details on each matrix are described in Ref. 24.

19

Using Eqs. (11) and (12), the global tangent stiffness matrix will be represented as

Eq. (13). In addition, the rigid body rotation is expressed as Eq. (14).

( )L

TLT

G LG

f

B fK B K B

q∂

= +∂

(11)

TB P E= (12)

( )T T T TG LK B K B E GF P FG E= + − − (13)

Tr GGE qθ δ= (14)

where, 𝐹𝐹 is determined by the local formulation, which is a matrix to complement

the discontinuity of CR and deformed coordinates.

Using matrix, 𝐹𝐹 , nonlinear strain-displacement relationship matrix 𝐵𝐵0m

are

derived. Based on this relationship, the final form of the local element stiffness and

internal force vector is as follows.

( ) ( )0 0 0 0T TL m m m mV

K B CB B S B dV= +∫ (15)

( )0 ˆTL mV

f B s dV=∫ (16)

20

Fig. 2.3 Coordinates and elemental kinematics of the CR solid element [24]

21

2.2.2 Inertial Load Vector and Tangent Matrices

It is approached from the kinetic energy point of view to express dynamics. The

variation form for Eq. (17) is shown in Eq. (18). The global mass matrix 𝑀𝑀𝐺𝐺 of Eq.

(20) is defined using the relational expression Eq. (19). 𝑀𝑀𝐿𝐿 is local mass matrix.

The final form of the inertial load vector is derived from the kinetic energy by using

Lagrange’s equation of motion. Then, Eq. (21) is derived.

12

T

V

K u udVρ= ∫ (17)

12

T

V

K dVuuδ ρ ρ= − ∫ (18)

TGr s

u R N E qδ= (19)

G LTM E M E= (20)

( ) { }3

,1 ,

12

TGT TK G G G G G G G k

k r k

Mf M q M q q q G E

θ=

∂ = + − ∂

∑ (21)

In addition, the gyroscopic matrices and centrifugal stiffness is expressed by Eqs.

(22) and (23). 𝐶𝐶𝑎𝑎𝐾𝐾,𝐺𝐺

, 𝐾𝐾𝑎𝑎𝐾𝐾,𝐺𝐺

, 𝐾𝐾𝑏𝑏𝐾𝐾,𝐺𝐺

, and 𝐾𝐾𝑐𝑐𝐾𝐾,𝐺𝐺

are referred in Ref. 24.

22

,

, , ,

q

TK G a aGK G K G K G

G

fC M C C

∂ = = + − ∂

(22)

,, , , ,

K G a b cK G K G K G K G

G

K K K Kfq

∂= = + −

∂ (23)

23

2.2.3 Governing Equation for Time-transient Analysis

Hilbert Hughes Taylor (HHT)-α method based on the predictor-corrector type

algorithm is employed to solve the resulting nonlinear equation. In this section, the

superscript, n , denotes the time step at nt , and h denotes the size of the time step.

α , β , and γ denotes coefficients in HHT- α method. Displacements and velocities

are updated through Newmark time integration formulas, and relevant relationships

are as follows.

1 2 112

n n n n nG G G G Gq q hq h q qβ β

+ + = + + − +

(24)

( ){ }1 11n n n nG G G Gq q h q qγ γ+ += + − + (25)

Eqs. (24) and (25) are used by performing linearization of the velocity and

acceleration as follows.

2

2

12

nG G

hq q hq qhβ

∆ = ∆ − −

(26)

(2 )2

n nG G

hq q q qh

γ γ β γβ β β

−∆ = ∆ − + (27)

In order to obtain a tangent matrix, linearization of the inertial load vector is

required. The resulting tangent stiffness matrix is as follows.

Dyn, , ,2

1G G K G K GK M C Kh h

γβ β

= + +

(28)

24

The nonlinear governing equation of motion can be expressed as follows.

( ),( ) , , 0e G G Km G G G Gf f q f q q q− − = (29)

where ef , Gf denote the external and global internal load vectors.

Additionally, ,Km Gf is denote the inertial load vector, which is defined as follows.

, ,Km G K G G mf f M a= + (30)

where, ma denotes the translational acceleration of motion.

In HHT- α method, the time-transient equilibrium equation is rewritten as Eq. (31).

{ }1 1 1,(1 ) (1 ) 0n n n n ne G Km G G ef f f f fα α α+ + ++ − + − + − = (31)

When re-expressing Eq. (31) according to this procedure, it can include a

component of the terms from the previous step. Using the truncated Taylor expansion

for the internal load vector, 1nGf+ , the following derivation of the predictor can be

obtained.

,(1 )n nG Dyn G preK K Fα+ + = (32)

1,

2

,

(1 ) (1 )

(2 )2 2

n n n npre e G K G e

n n n n n nK G G G G G G

F f f f f

h hC q q M hq q

α α α

γ β γβ β

+= + − + − −

−+ − + +

(33)

25

2.3 Rotational and Thermal Effects

2.3.1 Rotational Effect

Stiffness of a structure will be increased as the stress acting in the plane generated

by the rotation increases. This will also increase the natural frequencies, which is

called the stress stiffening effect. On the other hand, the vibration of the rotating

body causes relative circumferential motions, which changes the direction of the

centrifugal force. This will act to reduce the stiffness, which is called the spin

softening effect [25]. In the rotating spring-mass system as shown in Fig. 2.4, the

equilibrium equation among the stiffness, centrifugal force, and mass of the spring

is Eq. (34). Considering the deformation effect, it is obtained as Eq. (35) and

rearranged as Eq. (36).

2G GuK M r = Ω (34)

( )2G Gu M rK u = Ω + (35)

( )2 2G G GuM M rK − Ω = Ω (36)

where, 2G GK M − Ω is the stiffness matrix based on the small strain assuming

deformation, and 2 GM Ω is the load vector based on the small deformation

26

assumption. This will reduce the effective stiffness matrix.

In the modal analysis, Eq. (37) is derived from Eq. (36) and stress stiffening effects,

stiffS , by centrifugal force.

( )2 2 0G Gf GstifK MS Mω + − Ω − = (37)

Figure 2.5 shows the change in natural frequencies by both stress stiffing and spin

softening effects.

Fig. 2.4 Rotating spring-mass system

27

Fig. 2.5 Effects of the stress stiffening and spin softening [26]

28

2.3.2 Thermal Effect

In order to consider the thermal effect within the structural analysis, steady-state

analysis for heat conduction based on finite element method [27] is developed. The

elemental stiffness matrix for heat conduction can be expressed as follows:

( )e TCV

K B diag k BdV= ∫ (38)

where, k is the conductivity which may be varied with respect to temperature.

And this will induce nonlinear characteristics within the governing equation for heat

conduction.

( ) CCK T T f∆ = (39)

where, Cf is the heat source vector.

Newton-Raphson iterative method is employed to solve the resulting nonlinear

equation. And, curve fitting technique using the second-order polynomials is used in

order to take the conductivity with respect to temperature into account. For coupling

the temperature effect within the structural vibration analysis, variation of the material

properties due to the temperature distribution is added. In this procedure, a curve fitting

using the second-order polynomials is also employed to consider the material properties

with respect to temperature. The relevant computing algorithm is shown in Fig. 2.6.

29

Fig. 2.6 Computing algorithm to couple heat conduction and vibration

analyses

30

2.4 Reduced Order Modeling Based On the Proper Orthogonal

Decomposition

In this section, the reduced order modeling based on the proper orthogonal

decomposition (POD-ROM) is described. The POD-ROM is a useful method for

efficiently approaching complex computations requiring iterative calculations.

The POD formulates an optimal basis required to represent a dynamical system.

It has been applied to many engineering and scientific systems including lower-

dimensional dynamics modeling [28, 29]. In the field of structural dynamics, several

studies was carried out to apply POD method to associate POD modes with the

normal modes of a dynamic system [30, 31].

2.4.1 Concept of the POD Method

POD is a method for extracting key characteristic information by time-response

(snapshot) results obtained from an experiment or numerical analysis based on a full

order model (FOM).

As shown in Fig. 2.7(a), assuming that there are 1( )u t , 2 ( )u t , and 3( )u t

snapshot results at three points, the overall variation of the snapshot results can be

shown in a space consisting of mutually orthogonal 1( )u t , 2 ( )u t , and 3( )u t axes

as shown in Fig. 2.7(b). In that way, in order to understand the temporal and spatial

variation of the snapshot at N points, the representation on the N-dimensional space

coordinates is called the state locus [32]. If there are snapshot results at more than

31

three points, there is a limit to the representation of the graphical state locus. POD

method constructs a new mutually orthogonal coordinate system to represent an

efficient state locus for the overall variations of snapshot results. Such new

coordinate is called the principal coordinate system. To obtain such principal

coordinate system, principal coordinate axes that contain the maximum information

of the state locus will be required. If the state locus shown in Fig. 2.7(b) exists in a

single plane (n plane), the state locus can be represented using two-dimensional

orthogonal coordinates of ( )tη and ( )tξ on the π-plane as shown in Fig. 2.8.

Since the state locus tends to increase to the right as shown in Fig. 2.8, the 1( )tα

axis, which contains the largest variation of the state locus, becomes the first

principal coordinate axis. Then, in the same way, the additional principal coordinate

axis 2 ( )tα orthogonal to the axis 1( )tα is employed.

On the N-dimensional spatial coordinates, the original state locus has N principal

coordinate axes, ( )tα , and the values obtained by projecting the state locus on these

principal coordinate axes represent the direction of the axis, ( )xΦ .

In other words, the information obtained by applying the POD method to the

snapshot results is the information, ( )tα and ( )xΦ about the time ( t ) and space

( x ) for each mode. And, the original results can be reconstructed with a linear sum

of these, as shown in Fig. 2.9.

32

(a) Time-response (snapshot) results at three points

(b) State locus of snapshot results

Fig. 2.7 Snapshot results and state locus

33

Fig. 2.8 Principal axes of the state locus

Fig. 2.9 Linear summation of POD modes

34

2.4.2 POD-ROM for Structural Analysis

Galerkin projection (GP) is widely used as a typical POD-ROM method in

structural analysis. GP reduces the matrices representing the physical quantities of

the total system by applying the major mode shapes.

For structural analysis, the displacement data can be obtained from a priori

numerical simulation with N degrees of freedom and S snapshots, and such data can

be assembled in N by S matrix.

( ) ( )

( ) ( )

11 1

1

S

SN N

q qW

q q

=

(40)

The method of snapshot is employed in order to define the eigenvalue problem.

Thus, the average operator is evaluated as a space average over the domain. Then, a

temporal-correlation function from the snapshots can be:

TC W W= (41)

Also, it is possible to formulate eigenvalue problem to compute the POD modes.

CQ Qλ= (42)

35

( 1 )

1

~

i ii

M

WQ

i Nλ

=

Φ = (43)

where, MN is the selected number of the POD modes.

Thus, the dynamic characteristics can be expressed as Eq. (44) using the POD

modes for information over a large-size of time.

1q( , ) ( ) ( )

N

i ii

x t t xα=

= Φ∑ (44)

Using Galerkin projection to the governing equation, the reduced order model for

structural free vibration analysis can be expressed as follows.

( ), 0,, ,T T

i G GMq K M q Kq q

whe KMre KMΦ Φ ΦΦ

Φ Φ

Φ + = + =

= Φ Φ =Φ Φ

(45)

( )2 0T

K M

X

X

X

ωΦ Φ Φ

Φ= Φ

− = (46)

where, ω is the natural frequency, X is the mode shape in full order model.

Figure 2.10 shows the POD mode extraction and reduced-order modeling in the

structural analysis.

36

Fig. 2.10 Structural analysis based on the POD-ROM

37

2.5 Parallel Computation Based on the Domain Decomposition

As mentioned in the previous section, the snapshot results, which are displacement

responses, are essential to perform the present POD-ROM analysis. For structures

with large number of degrees of freedom, considerable computational cost is

required to collect these snapshot results. In this section, the parallel computation

based on domain decomposition is described to quickly collect the snapshot results.

METIS algorithm [33], which was developed by Karypis, is useful for the partitioning

of a structure which has irregular domains and a number of nodes, and thus it is widely

used in CSD and CFD. The METIS algorithm consists of major three phases: coarsening,

initial partitioning and refinement. The relevant process is shown in Fig. 2.11.

Now the entire domain, 0G is transformed in to a sequence of smaller domains,

1G , 2G ,…, kG during the coarsening phase. Thus, kG is the coarsest domain.

Here, the domain is coarsen by going through the numbering and matching for the

boundary nodes, as shown in Fig. 2.12. During the initial partitioning phase, a k-way

sub-domain, kP , of kG is computed. Then, during the refinement phase, the sub-

domain, kP , of kG is finally projected back to 0G by going through the 1kP − ,

2kP − ,…, 1P , 0P sub-domains. Here, the boundary nodes are randomly selected,

and it is adjusted after confirming the balance of the sub-domains, as shown in Fig.

2.13.

In order to utilize the parallel computation, a global stiffness matrix of entire

domain is decomposed into a set of sub-domains which are partitioned using the

38

METIS algorithm. And, the local stiffness matrix of sub-domains is estimated by

each CPU. Then, the global stiffness matrix of entire domain is defined by

assembling the localized stiffness matrices. To assemble the localized matrices,

message passing interface (MPI) is implemented and relevant collective

communication algorithm (MPI_barrier, MPI_reduce) is applied. Also, Intel MKL

parallel direct sparse solver for clusters (parallel PARDISO) is employed to solve

equations for a sparsity of the system matrix. Figure 2.14 shows the relevant parallel

computation process.

39

Fig. 2.11 Three phases of multilevel partitioning [34]

40

Fig. 2.12 Coarsening algorithm of the METIS

41

Fig. 2.13 Refinement algorithm of the METIS

42

Fig. 2.14 Parallel computation process using the METIS

43

Chpater 3

Numerical Results and Discussion

In this chapter, validation of the present analysis is performed. First, the present

FOM and POD-ROM analysis is conducted and validated in non-rotating conditions.

Then, in the second section, the present analysis is performed in rotating conditions.

And the present POD-ROM analysis is performed for variety of the different rotating

conditions. In this procedure, the computational cost is compared to that consumed

by ANSYS. Finally, the present POD-ROM analysis with thermal effects is

conducted.

For the comparison of the present results, the predictions obtained by the

commercial software, ANSYS Structural 18.0, under the same conditions as the

present one is used. The relevant example is the first stage blade of a 75 MW Alstom

GT11N gas turbine as shown in Fig. 3.1. There are five cooling passages and relevant

holes. Material properties of the blade are summarized in Table 3.1.

The three dimensional turbine blade configuration is discretized by 10-node

tetrahedral elements, and the relevant grid is generated using commercial software,

MSC.PATRAN. The discretized finite element model and relevant information are

shown in Fig. 3.2 and Table 3.2, respectively. The boundary conditions are given in

Fig. 3.3. First, the translational and rotational directions are constrained to the

surface fastened by the centrifugal force. And, the normal direction to one side of the

44

dovetails is constrained to prevent rigid body mode. In order for the present POD-

ROM analysis, 50 snapshots and 20 POD modes are employed.

Fig. 3.1 GT11N first stage turbine blade

45

Table 3.1 Material properties used in the present validation [35]

Alloy In-738LC

Density [𝐤𝐤𝐤𝐤/𝒎𝒎𝟑𝟑] 8,420

Thermal expansion coefficient [/℃] 1,594

Temperature [℃] Young’s Modulus [𝐆𝐆𝐆𝐆𝐆𝐆] Poisson’s ratio

23.89 200.6 0.28

93.33 195.1 0.27

204.44 190.3 0.27

315.56 184.8 0.28

426.67 179.3 0.28

537.78 175.1 0.3

648.89 167.5 0.3

760.00 160.0 0.3

871.11 151.0 0.29

982.22 140.0 0.3

Temperature [℃] Thermal conductivity [W m-1 K-1]

204.44 11.8

315.56 13.7

426.67 15.6

537.78 17.7

648.89 19.7

760.00 21.5

871.11 23.3

982.22 25.4

1,093.33 27.2

46

Fig. 3.2 Discretization in the finite element representation

Table 3.2 Finite element information

Value

Element Tetra 10-node

Number of nodes 250,082

Degrees of freedom 750,246

47

Fig. 3.3 Boundary conditions

48

3.1 Non-rotating Condition

The present FOM and POD-ROM analysis are conducted for the non-rotating and

then those results are compared with those by ANSYS. Comparison of the natural

frequencies between ANSYS and the present results is shown in Table 3.3. When

compared to the reference results obtained by ANSYS, both the present FOM and

POD-ROM analysis show good agreement within 0.8% of the average discrepancy.

Figures 3.4 and 3.5 show the mode shapes in each mode obtained by present and

ANSYS analysis, and the present analysis results also show good correlation with

ANSYS. The present POD-ROM analysis has a good advantage in reducing the

computational cost compared to the present FOM analysis, sustaining the accuracy.

Table 3.3 Comparison of the natural frequencies at non-rotating condition

Natural frequencies (Hz)

Mode No. Ref. (ANSYS) FOM POD-ROM

1st 754.35 750.1 749.5

2nd 982.92 969.2 967.5

3rd 2308.2 2301.3 2301

4th 3520.5 3495.2 3491.2

5th 4239.4 4219.7 4213.5

6th 5860.3 5833.2 5832.6

7th 6155.5 6102.3 6093.2

8th 7256.1 7221.4 7225.1

Average discrepancy 0.66 % 0.72 %

49

Fig. 3.4 Comparison of the mode shapes (1st – 4th)

50

Fig. 3.5 Comparison of the mode shapes (5th – 8th)

51

3.2 Rotating Condition

The present FOM and POD-ROM analysis are conducted at the operating

conditions of the GT11N gas turbine, 3,600 rpm. Then, the present results are

compared with ANSYS prediction. The comparison of the natural frequencies of the

ANSYS and the present results is summarized in Table 3.4. When compared to the

results obtained by ANSYS, both the present FOM and POD-ROM analysis show

good agreement with the average discrepancy, less than 0.7%.

Table 3.4 Comparison of the natural frequencies at 3,600 rpm

Natural frequencies (Hz)

Mode No. Ref. (ANSYS) FOM POD-ROM

1st 768.54 764.6 763.3

2nd 991.67 978.5 980.44

3rd 2311.9 2305.3 2305.49

4th 3535.8 3511.3 3512.33

5th 4247 4229.8 4239.84

6th 5869.6 5842.2 5869.25

7th 6154.8 6104.8 6115.2

8th 7277 7242.8 7292.71

Average discrepancy 0.62 % 0.47%

52

In the present POD-ROM analysis, once the snapshot results at arbitrary rotation

conditions are collected, it is possible to conduct the modal analyses at various

rotation conditions. Such advantage makes it easier to construct Campbell diagram

for resonance evaluation. Figure 3.6 shows Campbell diagram based on the present

POD-ROM analysis results under various rotation conditions. As shown in Fig. 8,

the major resonant conditions (red circles) predicted for the number of excitation

sources at 3,600 rpm operating conditions. Also, when the rotational speed is

increased, the natural frequency of each mode is also increased by the effect of stress

stiffening and spin softening. However, there is a difference in natural frequencies

increase rate in each mode. This may be explained by the Southwell effect [36],

which is a phenomenon in which coupling between flapping and lead-lag modes

occur due to the difference in natural frequencies increase rate among the modes as

the blade rotates.

Moreover, the computation time required by the present analysis is compared to

that by ANSYS prediction. The relevant result is summarized in Table 3.5. As shown

in Table 3.5, the present analysis is significantly efficient as the computational time

is 39% less than that consumed by ANSYS. However, in order to extract the POD

modes in the initial present POD-ROM analysis, the snapshot results, which are the

displacement responses at arbitrary rotation conditions, should be collected. And the

present POD-ROM analysis accuracy is affected by the time step for the snapshot

and the levels of results collected.

53

Fig. 3.6 Campbell diagram

Table 3.5 Comparison of the computation time in a single rotating condition

Computation time

Present 53s

ANSYS 88s

54

Additionally, parallel computation based on the domain decomposition is

conducted to reduce the computational cost for collecting the snapshot results. As

shown in Fig. 3.7, an entire domain of blade is partitioned into 20 sub-domains by

the METIS and its global stiffness matrix is calculated by each CPU. Then, the

snapshot results are collected by going through Parallel PARDISO. The snapshot

collection time required by the parallel analysis is compared to that by serial analysis.

The relevant result is summarized in Table 3.6. The snapshot collection time

consumed by the present parallel analysis is reduced by about 80% compared to the

serial analysis.

55

Fig. 3.7 Turbine blade partitioned into 20 sub-domains by the METIS

Table 3.6 Comparison of the snapshot collection time

Computation time

Parallel (20 CPUs) 2,310s

Serial 11,295s

56

3.3 Thermal Effect

The temperature distribution is assumed to be 600℃ for the airfoil surface, 250℃

for the cooling passages. The relevant condition is shown in Fig. 3.8. The resulting

temperature distribution obtained by the present heat conduction analysis is shown

in Fig. 3.9. Using the temperature distribution, the modal analysis of POD-ROM is

conducted.

The rotating speed is chosen to be 3,600 rpm. Then, the present result is compared

with the ANSYS predictions. The comparison of the natural frequencies between the

ANSYS and the present results is shown in Fig. 3.10. When compared to the results

obtained by ANSYS, the present results including the thermal effect show good

correlation. Moreover, it is found that the thermal effect decreases the natural

frequencies of the blade, and reduction rate is larger than 4.1%. Such situation is

caused by the material softening effect due to the high temperature.

57

Fig. 3.8 Temperature conditions

Fig. 3.9 Temperature gradient of the blade

58

Fig. 3.10 Comparison of the natural frequencies considering thermal effect

59

Chpater 4

Conclusion and Future Works

4.1 Conclusion

In this thesis, an advanced vibration analysis framework for gas turbine blades

which feature enormous number of discretization including complex geometric

nonlinearity is developed. Geometrically nonlinear three dimensional co-rotational

solid element is employed for nonlinearity dynamics of the gas turbine blades. And,

the variations of dynamic characteristics due to the rotational and thermal effect is

considered. Then, reduced order modeling based on proper orthogonal

decomposition (POD-ROM) method is suggested for large-size structural analysis

which needs require high computational cost with considerable number of

discretized elements and degrees of freedom. In addition, parallelized domain

decomposition method is employed for rapid collection of the snapshot results in the

present POD-ROM analysis.

The numerical analysis is performed for realistic geometry of gas turbine blade

under operating conditions which contain the rotational effect, and thermal effect.

This result is validated by commercial software, ANSYS. As a result, the present

analyses are in good agreement with the average discrepancy, less than 0.8%. In

particular, the present POD-ROM analysis is achieved significant improvement in

60

the computation cost (less than 39%).

For the initial present POD-ROM analysis, however, the collection of snapshot

results is required, and its computation time may be increased in a large-size problem.

Thus, parallelized domain decomposition method based on the METIS algorithm is

employed, which significantly improves the snapshot collection time.

61

4.2 Future Works

In order to accurate structural prediction with respect to high cycle fatigue, forced

vibration analysis using fluid-structure interaction (FSI) method is required.

In the future, a forced vibration analysis with fluid dynamics interaction will be

developed, which is extended from the present vibration analysis. Given steady loads

and temperature distributions from the computational fluid dynamics (CFD) analysis,

more accurate dynamic characteristics will be predicted. Then, a structural safety

prediction for high cycle fatigue associated with vibratory stresses will be performed

by going through the forced vibration analysis using unsteady loads from the CFD

analysis.

62

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66

국문초록

가스터빈 블레이드의 구조 건전성 예측을 위한

고정밀 진동해석 프레임워크 개발

김용세

서울대학교 대학원

기계항공공학부

가스터빈 엔진의 블레이드는 공진조건에서 주기적인 가진력이

가해짐에 따라 고주기 피로가 발생하며, 이로 인해 블레이드의 고주기

피로 파손이 발생할 수 있다. 고주기 피로 문제를 방지하기 위해 관련된

공진 조건에서 구조 응답을 예측해야 하며, 이를 위해서는 설계

단계에서부터 블레이드의 동적 특성을 정밀하게 예측하는 것이

필수적이다.

본 논문에서는 기하학적 비선형성, 고속 회전 및 열 효과에 의한 동적

특성을 정밀하게 예측 가능한 진동해석 프레임워크를 개발하였다.

기하학적 비선형 해석을 위해 3 차원 co-rotational 요소를 적용하였다.

반면, 복잡한 블레이드 형상은 상당한 유한요소 개수와 자유도를 가진

대규모 구조해석으로서 높은 계산 비용이 발생한다. 이에 고유치 해의

신속한 계산을 위해 적합 직교 분해 기반의 차수 축소 모델링 (POD-

67

ROM) 기법을 개발 및 적용하였다.

개발한 진동해석은 75MW 급 가스터빈 엔진의 1 단 터빈 블레이드를

대상으로 모드 해석을 수행하고, 그 결과를 상용 소프트웨어 ANSYS 와

비교 및 검증하였다. 특히, POD-ROM 해석은 전체 자유도 (FOM)와

ANSYS 해석 대비 계산 비용이 상당히 향상되었다.

그렇지만, 초기에 POD-ROM 해석을 위해서는 절점에서의 변위

응답을 확보하는 snapshot 결과 수집을 선행해야 하며, 대규모 구조

해석에서는 더욱 snapshot 계산 비용이 증가한다. 이에 영역 분할

기법 중 METIS 알고리즘을 적용하여 병렬 계산을 수행하였고, 그

결과, snapshot 결과 수집에 소요되는 계산 비용이 현저히 감소하였다.

주제어: 가스터빈 블레이드, 고주기 피로, 진동 해석, Co-

rotational 요소, 적합 직교 분해, 차수 축소 모델링,

영역 분할

학번: 2016-20730

Chpater 1 Introduction 1.1 Background and Motivation 1.2 Previous Researches 1.3 Objectives and Thesis Overview

Chpater 2 Theoretical Background 2.1 Ten-node Tetrahedral Solid Element 2.2 Geometrically Nonlinear Dynamics Based On the Co-rotational Formulation 2.2.1 Elemental Kinematics 2.2.2 Inertial Load Vector and Tangent Matrices 2.2.3 Governing Equation for Time-transient Analysis

2.3 Rotational and Thermal Effects 2.3.1 Rotational Effect 2.3.2 Thermal Effect

2.4 Reduced Order Modeling Based On the Proper Orthogonal Decomposition 2.4.1 Concept of the POD Method 2.4.2 POD-ROM for Structural Analysis

2.5 Parallel Computation Based on the Domain Decomposition

Chpater 3 Numerical Results and Discussion 3.1 Non-rotating Condition 3.2 Rotating Condition 3.3 Thermal Effect

Chpater 4 Conclusion and Future Works 4.1 Conclusion 4.2 Future Works

References 국문초록

11Chpater 1 Introduction 1 1.1 Background and Motivation 1 1.2 Previous Researches 6 1.3 Objectives and Thesis Overview 11Chpater 2 Theoretical Background 12 2.1 Ten-node Tetrahedral Solid Element 12 2.2 Geometrically Nonlinear Dynamics Based On the Co-rotational Formulation 17 2.2.1 Elemental Kinematics 18 2.2.2 Inertial Load Vector and Tangent Matrices 21 2.2.3 Governing Equation for Time-transient Analysis 23 2.3 Rotational and Thermal Effects 25 2.3.1 Rotational Effect 25 2.3.2 Thermal Effect 28 2.4 Reduced Order Modeling Based On the Proper Orthogonal Decomposition 30 2.4.1 Concept of the POD Method 30 2.4.2 POD-ROM for Structural Analysis 34 2.5 Parallel Computation Based on the Domain Decomposition 37Chpater 3 Numerical Results and Discussion 43 3.1 Non-rotating Condition 48 3.2 Rotating Condition 51 3.3 Thermal Effect 56Chpater 4 Conclusion and Future Works 59 4.1 Conclusion 59 4.2 Future Works 61References 62국문초록 66