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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