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공학박사학위논문
Proper Orthogonal Decomposition-Based
Parametric Reduced Order Models
for Structural Analysis and
Design Optimization
구조 해석과 최적 설계를 위한
적합 직교 분해 기반의 파라메트릭 축소 모델
2015 년 2 월
서울대학교 대학원
기계항공공학부
이 재 훈
Proper Orthogonal Decomposition-Based
Parametric Reduced Order Models
for Structural Analysis and
Design Optimization
구조 해석과 최적 설계를 위한
적합 직교 분해 기반의 파라메트릭 축소 모델
지도교수 조 맹 효
이 논문을 공학박사 학위논문으로 제출함
2014 년 11 월
서울대학교 대학원
기계항공공학부
이 재 훈
이재훈의 공학박사 학위논문을 인준함
2014 년 12 월
위 원 장 :
부위원장 :
위 원 :
위 원 :
위 원 :
Proper Orthogonal Decomposition-Based
Parametric Reduced Order Models
for Structural Analysis and
Design Optimization
by
Jaehun Lee
A Dissertation
Submitted to the Department of Mechanical
and Aerospace Engineering
in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
at the
SEOUL NATIONAL UNIVERSITY
FEBRUARY 2015
Proper Orthogonal Decomposition-Based
Parametric Reduced Order Models
for Structural Analysis and
Design Optimization
Jaehun Lee
Department of Mechanical and Aerospace Engineering
Seoul National University
APPROVED:
Yoon Young Kim, Chair, Ph. D.
Maenghyo Cho, Supervisor, Ph. D.
Byeng Dong Youn, Ph. D.
Do-Nyun Kim, Ph. D.
Ki-Ook Kim, Ph. D.
To my wife with love.
Abstract
In this dissertation, parametric reduced order models for comprising the char-
acteristics of dynamics and the change of parameters were developed within
the finite element framework. The existing reduction techniques are appli-
cable to either dynamic characteristics or parameter variations only. Thus,
when the parameter changes in a dynamic system, the reduced system should
be reconstructed, which results in an inefficient computation. To this end, the
parametric reduced order models based on the proper orthogonal decompo-
sition were suggested.
First of all, based on the characteristics of the proper orthogonal de-
composition, enhanced reduced basis method was developed to treat multi-
ple loading conditions. Whereas existing methods have to reconstruct the
reduced basis as the external load changes, the developed method com-
bined with the global proper orthogonal decomposition needs not to con-
struct the basis again. The developed method was combined with the op-
timization strategy using the equivalent static load, and efficiency of the
optimization increased. In addition, to consider the change of parameter
in real-time, interpolation-based reduction technique consist of projection-
transformation-interpolation-recovery procedures was suggested. By combin-
ing with the moving least square approximation, the proposed method re-
covers the interpolated reduced model to the full system with high accuracy
compared to conventional Lagrange interpolation method.
On the other hand, to employ the parametric reduced order model to
the design optimization of large-scale dynamic system, not only the repeated
i
computations of optimization process including sensitivity calculation, but
the off-line computation that constructs approximated global response sur-
face also should be reduced. Therefore, by combining the parametric reduced
order model with substructuring schemes, both pre-computations and re-
peated computations in the optimization process were reduced. Thereby, the
efficiency of the design optimization of large-scale dynamic system was ex-
tremized. Accuracy and efficiency were verified by optimizing the system with
hundreds of thousands degrees of freedom and hundred-level design variables.
In addition, probabilistic analysis of dynamic system with uncertain param-
eters were performed.
The parametric reduced order model and the design optimization strategy
developed and verified in this dissertation can be further employed to other
various large-scale system for dynamic analyses and structural optimizations.
Keywords: Parametric reduced order model, Proper orthogonal decomposi-
tion, Structural optimization for dynamics, Moving least square, Parametric
substructuring scheme
Student Number: 2008-20778
ii
Contents
Abstract i
Chapter 1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Finite Element-Based Large-Scale System . . . . . . . 2
1.1.2 Limitations of Storage and Computing System . . . . 2
1.1.3 Repeated Computation . . . . . . . . . . . . . . . . . 3
1.1.4 Motivation for Reduced Order Model-Based Analysis
and Design . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Model Reduction Technique for Dynamics . . . . . . . 6
1.2.2 Parametric Reduced Order Model . . . . . . . . . . . 9
1.3 Objectives and Contributions . . . . . . . . . . . . . . . . . . 11
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2 Proper Orthogonal Decomposition-Based Model
Order Reduction Techniques 13
2.1 Review of Finite Element Formulation for Dynamics . . . . . 13
2.2 Proper Orthogonal Decomposition . . . . . . . . . . . . . . . 16
iii
2.2.1 Construction of Energy Functional . . . . . . . . . . . 16
2.2.2 Method of Snapshots . . . . . . . . . . . . . . . . . . . 17
2.2.3 Model Reduction Using Proper Orthogonal Decompo-
sition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . 22
2.3 Reduced Basis Method . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Reduced Basis Approximation . . . . . . . . . . . . . 31
2.3.2 Numerical Examples . . . . . . . . . . . . . . . . . . . 34
Chapter 3 Global Proper Orthogonal Decomposition and Re-
duced Equivalent Static Load 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Reduced Basis Method for Multiple Loading Condition . . . . 42
3.2.1 Global Proper Orthogonal Decomposition . . . . . . . 42
3.2.2 Mode of External Loads . . . . . . . . . . . . . . . . . 43
3.3 Structural Optimization Strategy Using Reduced Equivalent
Static Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Optimization Strategy Using Equivalent Static Load . 47
3.3.3 Mode of Equivalent Static Load . . . . . . . . . . . . . 49
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 4 Parametric Reduced Order Model: Interpolation
and Moving Least Square Method 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Parametric Reduced Order Model for Dynamics . . . . . . . . 64
iv
4.2.1 Dynamic System with Parameters . . . . . . . . . . . 64
4.2.2 ROM Construction at Operating Points . . . . . . . . 65
4.2.3 Transformation to Common Basis . . . . . . . . . . . 66
4.2.4 Matrix and Mode Interpolation . . . . . . . . . . . . . 69
4.3 Moving Least Square Method for Recovery . . . . . . . . . . 70
4.3.1 Moveing Least Square Method . . . . . . . . . . . . . 71
4.3.2 Computation at On-line Stage . . . . . . . . . . . . . 72
4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 5 Parametric Reduced Order Model with Substruc-
turing Scheme 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Review of Component Mode Synthesis . . . . . . . . . . . . . 94
5.2.1 Equation of Motion for a Substructure . . . . . . . . . 94
5.2.2 Fixed Interface Normal Modes . . . . . . . . . . . . . 95
5.2.3 Constraint Modes . . . . . . . . . . . . . . . . . . . . 96
5.2.4 Craig-Bampton Transformation Matrix . . . . . . . . 96
5.3 Interpolation of Transformation Matrix . . . . . . . . . . . . 98
5.3.1 Projection and Transformation of Fixed Interface Nor-
mal Modes . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Interpolation of Constraint Modes and ROM of Sub-
domain . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Parametric Component Mode Synthesis Method . . . . . . . 100
5.4.1 Synthesis of Component Modes . . . . . . . . . . . . . 100
5.4.2 Reduction of Interface Degrees of Freedom . . . . . . 101
v
5.4.3 Recovery Process to Full System . . . . . . . . . . . . 103
5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 104
Chapter 6 Stochastic Dynamic Analysis with Uncertain Pa-
rameters 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Dynamic Analysis of Uncertain Structures . . . . . . . . . . . 128
6.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 129
6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 130
Chapter 7 Conclusions 143
Bibliography 145
국문 요약 154
vi
List of Figures
Figure 2.1 Rib-skin-spar structure under dynamic load f(t). . . . 26
Figure 2.2 Dynamic loading profile. . . . . . . . . . . . . . . . . 26
Figure 2.3 Comparison of the deflection of the FOM and the
ROM: ‘40’ snapshots in [0∼0.02] sec. . . . . . . . . . 27
Figure 2.4 Comparison of the deflection of the FOM and the
ROM: ‘80’ snapshots in [0∼0.04] sec. . . . . . . . . . 28
Figure 2.5 Comparision of frequency responses of the FOM and
the ROM (50 snapshots) at position (1). . . . . . . . 29
Figure 2.6 Comparision of frequency responses of the FOM and
the ROM (50 snapshots) at position (2). . . . . . . . 30
Figure 2.7 Rib-skin-spar structure with 6 subdomains under tip
static load. . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 2.8 Comparison of optimal thicknesses of the FOM, parametrized
FOM and ROM. . . . . . . . . . . . . . . . . . . . . . 38
Figure 2.9 Comparison of objective function histories. . . . . . . 39
Figure 2.10 Comparison of computation time of the FOM, parametrized
FOM and ROM. . . . . . . . . . . . . . . . . . . . . . 39
vii
Figure 3.1 Cantilever beam with 4 subdomains under tip dy-
namic load. . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 3.2 Half sinusoidal loading profile. . . . . . . . . . . . . . 54
Figure 3.3 Comparison of optimal widths of the FOM and ROMs. 55
Figure 3.4 Rib-skin-spar structure with 6 subdomains under tip
dynamic load. . . . . . . . . . . . . . . . . . . . . . . 57
Figure 3.5 Sinusoidal loading profile. . . . . . . . . . . . . . . . 57
Figure 3.6 Comparison of optimal thicknesses of the FOM and
ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 3.7 Comparison of total computation time of the FOM
and ROMs. . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 3.8 Comparison of computation time of each steps. . . . 59
Figure 3.9 Wing box model with 20 subdomains under tip dy-
namic load. . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 3.10 Half sinusoidal loading profile. . . . . . . . . . . . . . 61
Figure 3.11 Comparison of optimal thicknesses of each optimiza-
tion methods. . . . . . . . . . . . . . . . . . . . . . . 61
Figure 3.12 Comparison of total computation time of the FOM
and ROMs. . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 4.1 Cantilever beam with 4 subdomains of plane stress
membrane element under tip impluse load. . . . . . . 77
Figure 4.2 Comparison of frequency responses of the FOM and
ROMs at position (1): 8 (mm) sampling range. . . . 78
viii
Figure 4.3 Comparison of frequency responses of the FOM and
ROMs at position (2): 8 (mm) sampling range. . . . 79
Figure 4.4 Comparison of frequency responses of the FOM and
the ROM at position (1): 16 (mm) sampling range. . 80
Figure 4.5 Comparison of frequency responses of the FOM and
the ROM at position (2): 16 (mm) sampling range. . 81
Figure 4.6 Comparison of frequency responses of the FOM and
the ROM at position (1): 24 (mm) sampling range. . 82
Figure 4.7 Comparison of frequency responses of the FOM and
the ROM at position (2): 24 (mm) sampling range. . 83
Figure 4.8 Average relative error of 1st∼8th eigenvalues accord-
ing to the sampling range. . . . . . . . . . . . . . . . 84
Figure 4.9 Average relative error of 1st∼8th eigenvalues for ran-
dom thickness input according to the sampling range. 84
Figure 4.10 Cantilever plate with 4 subdomains of under tip im-
pluse load. . . . . . . . . . . . . . . . . . . . . . . . . 85
Figure 4.11 Comparison of frequency responses of the FOM and
the ROM: linear sampling. . . . . . . . . . . . . . . . 86
Figure 4.12 Comparison of frequency responses of the FOM and
the ROM: quadratic sampling. . . . . . . . . . . . . . 87
Figure 4.13 Comparison of frequency responses of the FOM, the
ROM and Lagrange interpolation: cubic sampling. . . 88
Figure 4.14 Comparison of the relative errors of eigenvalues using
different polynomial order. . . . . . . . . . . . . . . . 89
ix
Figure 4.15 Wing box model with 8 subdomains under tip impluse
load. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Figure 4.16 Comparison of frequency responses of the FOM and
the ROM. . . . . . . . . . . . . . . . . . . . . . . . . 90
Figure 4.17 Average of relative eigenvalue errors and probability
density function of 1,000 random samples. . . . . . . 91
Figure 5.1 Rib-skin-spar structure with 8 subdomains under tip
dynamic load. . . . . . . . . . . . . . . . . . . . . . . 109
Figure 5.2 Comparison of mean of eigenvalue errors for 1,000 ran-
dom samples: The Craig-Bampton component mode
systhesis and the (a) linear, (b) quadratic and (c) cu-
bic interpolated ROM. . . . . . . . . . . . . . . . . . 110
Figure 5.3 Comparison of min-max of eigenvalue errors for 1,000
random samples: The Craig-Bampton component mode
systhesis and the (a) linear, (b) quadratic and (c) cu-
bic interpolated ROM. . . . . . . . . . . . . . . . . . 111
Figure 5.4 Comparison of mean of eigenvalue errors for 1,000 ran-
dom samples: The Craig-Bampton component mode
systhesis and the cubic interpolated ROM by chang-
ing sampling range: (a) 10∼15 (mm), (b) 10∼20 (mm)
and (c) 5∼20 (mm), . . . . . . . . . . . . . . . . . . . 112
Figure 5.5 Dynamic step loading profile. . . . . . . . . . . . . . 113
Figure 5.6 Comparison of optimal thicknesses of the FOM and
ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
x
Figure 5.7 Comparison of objective function histories. . . . . . . 114
Figure 5.8 Comparison of computation time of the FOM and
ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Figure 5.9 Wing box model with 85 subdomains under tip dy-
namic load. . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 5.10 Design variables of rib and spar. . . . . . . . . . . . . 116
Figure 5.11 Design variables of upper and lower skins. . . . . . . 117
Figure 5.12 Comparison of optimal thicknesses of the FOM and
ROMs of spar and upper skin . . . . . . . . . . . . . 118
Figure 5.13 Comparison of optimal thicknesses of the FOM and
ROMs of rib and lower skin . . . . . . . . . . . . . . 119
Figure 5.14 Comparison of objective function histories. . . . . . . 120
Figure 5.15 Comparison of computation time of the FOM and
ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Figure 5.16 Configureation of high-fidelity F1 front wing structure. 121
Figure 5.17 Half of F1 front wing with 96 subdomains under mul-
tiple dynamic loads . . . . . . . . . . . . . . . . . . . 122
Figure 5.18 Dynamic loads applied to each points . . . . . . . . . 123
Figure 5.19 Comparison of optimal thicknesses of the FOM and
ROMs of subdomain # 1∼41 . . . . . . . . . . . . . 124
Figure 5.20 Comparison of optimal thicknesses of the FOM and
ROMs of subdomain # 42∼96 . . . . . . . . . . . . 125
Figure 5.21 Comparison of objective function histories. . . . . . . 126
Figure 5.22 Comparison of computation time of the FOM and
ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
xi
Figure 6.1 Cantilever plate with 4 uncertain parameters. . . . . 133
Figure 6.2 PDF of elatic modulus of each substructures. . . . . 134
Figure 6.3 Average mean frequency responses of the FOM and
the ROM . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 6.4 Average maximum frequency responses of the FOM
and the ROM . . . . . . . . . . . . . . . . . . . . . . 136
Figure 6.5 Average minimum frequency responses of the FOM
and the ROM . . . . . . . . . . . . . . . . . . . . . . 137
Figure 6.6 Rib-skin-spar structure with 8 uncertain parameters . 138
Figure 6.7 Average mean frequency responses of the FOM and
ROMs . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 6.8 Average maximum frequency responses of the FOM
and ROMs . . . . . . . . . . . . . . . . . . . . . . . . 140
Figure 6.9 Average minimum frequency responses of the FOM
and ROMs . . . . . . . . . . . . . . . . . . . . . . . . 141
xii
List of Tables
Table 2.1 Geometric and material properties of rib-skin-spar struc-
ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Table 2.2 Relative error of the eigenvalues of frequency domain
ROM . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Table 2.3 Sampling strategy by combinations with repetitation . 37
Table 2.4 Comparison of weights of the FOM, parametrized FOM
and ROM . . . . . . . . . . . . . . . . . . . . . . . . . 37
Table 3.1 Problem condition of cantilever beam . . . . . . . . . 54
Table 3.2 Problem condition of rib-skin-spar structure . . . . . . 56
Table 3.3 Relative error (%) of objective function values . . . . . 56
Table 3.4 Problem condition of wing box model . . . . . . . . . 60
Table 4.1 Cases of sampling ranges . . . . . . . . . . . . . . . . . 77
Table 4.2 Upper and lower bound of each interpolation cases . . 85
Table 5.1 Upper and lower bound of each interpolation cases . . 109
Table 5.2 Problem condition of rib-skin-spare structure . . . . . 109
Table 5.3 Problem condition of wing box model . . . . . . . . . 115
xiii
Table 5.4 Problem condition of high-fidelity F1 front wing model 121
Table 6.1 Computation time of the FOM and ROMs of cantilever
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Table 6.2 Computation time of the FOM and ROMs of rib-skin-
spar structure . . . . . . . . . . . . . . . . . . . . . . . 138
xiv
Chapter 1
Introduction
1.1 Introduction
Due to the fast development of computers in the early 2000s, the finite el-
ement method which have been developing for half of a century became a
popular and attractive solution to lots of serious and complicated engineering
problems. In particular, in the field of mechanical and aerospace engineer-
ing, the demands for the structural analysis and design optimization increase
rapidly with the requirement of efficiency and accuracy. In this situation, the
finite element method is not just a simple option among various approaches,
but a certified universal solution to many scientific and engineering problems.
Also, various commercial finite element softwares provide the integrated pack-
ages that include multi-scale, multi-physics, geometric and material nonlin-
earities, nonlinear dynamics, multi-body dynamics and design optimizations,
instead of simple linear structural problems. Sometimes, the finite element
method is preferred to the method based on experimental approach. Because,
the former shows high performance with low costs, and does not have any
1
dependency on various environmental issues that the letter has.
1.1.1 Finite Element-Based Large-Scale System
The main feature of the FEM is a discretization of a continuous body. Hence,
the degrees of freedom and the accuracy of the analysis is determined by the
element size. Strictly speaking, when we analyze a continuous system using
the discretization, almost infinite number of degrees of freedom is needed to
replicate the original system. However, that approach is obviously impossible
in the simulation-based environment. Therefore, the system should be dis-
cretized within the range that computer memory permitted, or the designer
assigned. For example, some part of commercial aircraft is discretized from
millions to tens of millions degrees of freedom. Then the FE solution of the
aircraft part is almost exact as long as the external conditions are described
well. For the simulation of large-sized problem, two serious problems arise.
1.1.2 Limitations of Storage and Computing System
The first problem is the memory capacity of the computer. To make the com-
putation fast, all data should be stored in the main memory unit, in particu-
lar, dynamic random access memory (DRAM). If the size of data exceeds the
memory limit of the DRAM, most commercial packages automatically uses
secondary memory unit, usually, hard disk drive (HDD). That process results
in a large amount of extra time due to the slow speed of reading and writing
in HDD, even in solid state drive (SDD). There are two ways to solve this
problem: (1) Use additional RAM or super computing system that satisfies
the memory limits. (2) Sub-structuring approach which separates the mem-
2
ory into small ones. The solution is quite simple, but the above-mentioned
always should be recognized to the researcher during the analysis.
However, although we have succeeded in storing the system matrix on
the main memory unit, it takes a lot of computation time to calculate the
inverse or eigenvalue of the large-scale matrix. Even considering the sparsity
and the positive definiteness, it is very heavy to compute the inverse matrix
or eigenvalues and eigenvectors of the system with millions and ten millions
of degrees of freedom.
1.1.3 Repeated Computation
The second problem is a repeated computation, which is more serious than
the first one. The memory capacity may be solved by employing additional
financial supportings, but the second one may be not. During the analysis,
researchers are definitely faced with a repeated computation. In this case,
even though the size of the system is relatively small (< hundreds of thou-
sands and millions), tremendous computation time is required. The issues of
repeated computation are divided into three categories as the change of the
basic characteristics of the system.
The first case of repeated computation is the one that has a modification
(or modifications) to the linear elasto static characteristic. For example, for
the dynamic analysis in time or frequency domain, discrete time or frequency
steps are required, which results in a repeated calculation. In addition, if
the system is geometrically or materially nonlinear, a step-by-step analysis is
needed to satisfy the balance equation during the deformation. The third one
is a fully coupled (two-way) multi-physics problem: fluid-structure, acoustic-
3
structure, multi-scale problems, etc. In these problems, the variables of each
field affects to another simultaneously.
The second case is the one that has a parameter change. Geometric and
material properties are the basic parameters. And the parameters are nor-
mally constant during the analysis. However, for the design optimization that
the size or shape parameters change, repeated analysis is required for the it-
eration steps and sensitivity calculations. Moreover, when the uncertainty of
structures is considered, a lot of additional computation is needed, especially
for the Monte Carlo simulation.
The last case is a combination of the first and second cases. For instance,
design optimization for dynamics, stochasic analysis of a dynamic system, and
the parametric study of a nonlinear structure are representative examples.
The computation time of this case is a multiplication of computation times
of each cases. In the present dissertation, the third case is mainly considered,
especially the design optimization and uncertainty analysis for structural
dynamic systems.
1.1.4 Motivation for Reduced Order Model-Based Analysis
and Design
To overcome the above-mentioned problems regarding the computational
cost, reduced order models (ROM) and system reduction techniques are
needed. In other words, at the on-line stage where the analysis, iteration and
design optimization are mainly executed, the system size should be small
enough to handle the system without much difficulties. At the same time,
the accuracy of the ROM should not be less than the amount required. Gen-
4
erally, there are two ways to construct a small-sized model that also has high
accuracy.
First of all, one can initially construct a small-sized model before or in the
middle of discretizing the original system. Based on the geometry and physics
of the original system, mathematical and engineering assumptions are used
to derive the governing equation. For example, higher-order theories and
elements which have a high performance are the representatives including
the dimension reduction of beams and plates. Strictly speaking, the first
case is close to the construction of a simplified model, rather than the order
reduction of the full model. However, the size of final model is definitely small
compared to the original model. In addition, the assumptions and the derived
model are usually intutive and the computation is fast. For the drawbacks,
it is hard to generalize the method to all of the complex structural problems
since the assumptions are usually derived from specific characteristics of the
complexity.
The second approach is reducing the large-scale structure which consists
of the finite element. This method is generally applied to analyze and to de-
sign most of the structures. First, an initial continuous body is discretized
into a very fine mesh configuration. The discretized model has a lot of de-
grees of freedom, and physical responses are expressed almost close to ac-
tual behaviors. After generating the large-scale system, a model order reduc-
tion is followed by the obilique projection-based mathematical tools. This
method requires more resources than the first one as the second one have to
construct the full-sized finite element model. However, although the second
method lacks efficiency compared to the first one, it can be used to all of the
5
structural systems without any additional modification. Therefore, most of
the reduction techniques including the one developed in this dissertation are
based on the second approach.
By synthesizing the issues mentioned before, the reduced order model
(ROM) is a very attractive solution to various large-scale engineering prob-
lems. Thus many kinds of thechniques and methods have been developed,
especially for the issues in the first and second cases of repeated computa-
tions. For the third issue which is a combination of the first and second one,
various research are now being executed. Detail reviews of the reduced order
models are discussed in the following section.
1.2 Historical Review
1.2.1 Model Reduction Technique for Dynamics
In the structural dynamics point of view, system reduction methods are re-
garded as generalized eigenvalue problem since the global dynamic behavior
of structural system is governed by lower eigenmodes. From the classical
Rayleigh-Ritz method, numerous novel theories and methods have been de-
veloped, and still being continued.
Reduction methods are categorized into three groups: iterative eigen-
solvers [1, 2], degrees of freedom-based methods [3] and substructuring meth-
ods [4, 5]. Comprehensive reviews and assessments related to the model re-
duction method can be found in the references: [6, 7, 8, 9, 10, 11, 12]. First of
all, for the iterative eigensolvers based on [2], the implicitly restarted Arnoldi
6
method have been developed [13, 14] which is implemented in ARPACK [15],
and MATLAB function eigs [16]. The advantages of the iterative eigensolver
are efficiency and accuracy. However, if the size of matrix becomes large, the
storage problem arises and the efficiency decreases.
The second group is the degrees of freedom-based reduction techniques
which transform the degrees of freedom of the full order model to the primary
degrees of freedom selected initially. Hence, the reduced displacement vector
is in the same space of the full order model. Although the degrees of freedom-
based methods are inefficient compared to the other ones, the characteristic
of the transformation is easily combined with the experimental approach.
For example, if the experimental data are obtained at the specific sensor
position, the response of full system is recovered by the degrees of freedom-
based methods. After the work in Ref. [3], various improved methods (IRS,
IIRS, etc.) are developed [17, 18, 19]. Moreover, by combined with the node
selection method for the primary degrees of freedom [20], the performances
have also been improved.
Different from the previous two approaches, the substructuring methods
focus on the decomposition of domain and the synthesis of each substructures.
The main advantage of substructuring-based method is that the full system
does not need to be constructed during the calculation procedure. Also, since
the analyses of each subdomains are independent of each other, paralleliza-
tion is easily employed. The synthesis procedure is occured in three different
coordinates system: physical domain, modal domain and frequency domain
[12]. In this dissertation, the synthesis in the modal domain is mainly consid-
ered. Since the Craig-Bampton transformation had developed, other various
7
modes including free interface mode [21, 22] and attatchment mode [23] were
also developed. In Ref. [24], Qiu and his co-worker suggested a mixed mode
method which comprises both fixed and free interface mode. Also, optimal
modal reduction technique was presented in Ref. [25], which is replacing each
substructures by another smaller substructures.
The synthesizing processes are usually executed based on the compati-
bility of interface degrees of freedom. Therefore the size of reduced model is
always larger than the number of interface degrees of freedom. In 2001, Rubin
[22] proposed to reduce the interface degrees of freedom. Also, Benninghof
and Lehoucq developed the automated multilevel substructuring method
(AMLS) [27]. During the construction of the Craig-Bampton transforma-
tion matrix, the fixed interface normal mode should be selected appropri-
ately. Thus, by using the moment-matching, the important interior modes
are chosen, which shows better performance compared to the conventional
Craig-Bamption method [28]. Jakobsson and his co-worker [29] developed an
adaptive component mode synthesis by employing a posterior error estimator
to control the error of reduced solutions. To combine the component mode
synthesis with experiments, Butland and Avitabile [30] presented a test ver-
ified model which also improves the conventional component mode synthesis
method.
On the other hand, proper orthogonal decomposition (POD) is a universal
method that used in most of engineering field to reduced the order of various
problems. In particular, the POD method became popular after developing
snapshot methods for the eigenfunction [31, 32]. The characteristics of the
POD and model reduction are presented well in Ref. [33, 34]. Also, numerous
8
research have been performed regarding the POD [35]. Time-domain analysis
was extended to the frequency domain POD [36], and the balanced POD
developed by Willcox [37] for the aeroelastic application is noteworthy. Also,
randomly vibrating systems were analyzed by the proper orthogonal modes
(POMs) in Ref. [38].
1.2.2 Parametric Reduced Order Model
Previous section covers the reduction method for dynamics. In this section,
the reduced basis for the parametric variation are mainly treated. For modal
reduction methods, the basis of ROM for dynamics is chosen to be the eigen-
modes, because the eigenmodes represent well the characteristics of the dy-
namic behavior of structure. However, there is no basis which stands for the
change of parameters. Therefore, the POD is chosen to calculate the reduced
basis in which the full order model be transformed.
In fact, the reduced basis method is originally developed by Noor [39, 40]
for the nonlinear analysis of structures; the change of loading parameter was
mainly considered. Among numerous studies of the reduced basis method,
Balmes [41] suggested to use the displacement snapshots obtained by chang-
ing the thickness parameter of the plate structure. Basically, if the parameter
of a structure changes, the full system have to be reconstructed. Then the
computational time increases as the number of element increases. However,
if the full system is projected to the space generated by the snapshots taken
by changing the parameters, the reconstruction time is extremely reduced as
the reconstruting process is executed in the reduced space. After the work
of Balmes, many approaches have been developed including real-time solu-
9
tion strategies [42], a posterior error estimation, [42, 43] and an extention to
non-affine parameterization case [44].
On the other hand, the parametric ROM based on interpolation tech-
niques have been developed. In particular, by using the Grassmann manifold
in differential geometry, the ROM interpolation method was developed for
the CFD-based aeroelastic applications [45, 46]. As mentioned earlier, the
coupled systems require repeated coputations, which yields a lot of compu-
tational resources. Therefore, after constructing precomputed reduced model
database, the ROM at new operating points are obtained by interpolating
the database in near real-time. Various interpolations of matrix manifolds
and applications are presented in Ref. [47].
Other noteworthy techniques and methods are the matrix interpolation
based on the coordinates transformation [48], and the parametric ROM strat-
egy for damaged components based on the Talyor expansion [49, 50]. The
constructed ROM gave a good prediction to various structural problems,
however, unfortunatly, nither [48] or [49, 50] does not provide an accurate re-
covery process to the full system. In Ref. [48], the authors suggested two types
of snapshot techniques: using intact snapshots and weighted snapshots dur-
ing the computation of the POD. Although the second one is more accurate
than the first because of the weightings, the computational cost increases due
to the reconstruction of the snapshot matrix. The parametric ROMs using
the Talyor expansion are intuitive, but it cannot consider the large variations
of parameters, because there is no procedure of either the exponential map-
ping, or the coordinate transformation. Hence, we need a new method which
efficiently considers the parametric variation in a wide range.
10
1.3 Objectives and Contributions
The objective of this dissertation is to develop and apply a parametric re-
duced order model, in particular, for the design optimization of large-scale
structures for dynamic response. Different from constructing the ROM only,
the design optimization requires both the reduction of the construction of
database and the computation of dynamic response of the structure. To
achieve the above requirements, various approaches were developed and ver-
ified by applying developed methods to representative examples from small
structural component to complicated high-fidelity models. The major contri-
butions can be summerized as follows:
• Strategy to apply the reduced basis method under multiple loading
condition.
• Enhancing the efficiency of the equivalent static load-based optimiza-
tion technique by applying the reduced basis method combined with
global proper orthogonal decomposition.
• Projection-transformation-interpolation-recovery process for efficient para-
metric reduced order model combined with moving least square approx-
imation.
• Parametric reduced order model combined with substructuring scheme
which extremize the efficiency of structural design optimization
11
1.4 Thesis Outline
The dissertation is organized as follows. The POD-based reduced order mod-
els are derived and time and frequency responses are presented in chapter 2.
Also, the reduced basis method is reviewed with an optimization example un-
der a static load. In chapter 3, the reduced basis method under multiple load-
ing condition is developed by calculating the modes of multiple loads, which is
extended to optimization strategy using equivalent static load method. Chap-
ter 4 presents an interpolation algorithm for parametric reduced order model
including a recovery process to the full order model by using moving least
square method. Chapter 5 introduces parametric reduced order model com-
bined with optimization technique, which made a comprehensive reduction
to both off-line and on-line computations. Chapter 6 is devoted to stochastic
analyses of uncertain structure under dynamic loads using the parametric
ROMs developed in chapter 4 and 5. Finally, conclusions are provided in
chapter 7.
12
Chapter 2
Proper Orthogonal
Decomposition-Based Model
Order Reduction Techniques
2.1 Review of Finite Element Formulation for Dy-
namics
The reduction techniques studied in this dissertation are developed based
on the framework of the finite element method. Therefore, from the initial
continuous body, the equation of motion in the discretized system will be
derived. First of all, the initial configuration is represented by B and the
discretized one is B such that
B ≃ B =
Ne⋃e=1
Ωe, Ωe ⊂ B, (2.1)
where Ne is the number of total element and Ωe is the domain of an element.
Here, ∪ notation is used to represent the summation considering the inter-
13
element compatibility. The displacement vector is expressed as
u(x, t) =
Ne⋃e=1
ue(x, t), (2.2)
where u is in the Cartesian coordinates system. The equation of motion is
derived via the virtual work principle. For a single element in domain Ωe
with volume V and surface area S, the balance of internal and external work
becomes
∫Bδu · β dV +
∫Γσ
δu · τ dS +
Nσ∑i=1
δui · pi
=
∫B
(δu · ρ u+ δu · c u+ δε : σ
)dV, (2.3)
where β and τ represent the body forces and surface tractions, Γσ a boundary
where the surface traction is applied, pi is concentrated loads at a total of
n points, ρ is mass density, and c denotes a damping parameter similar to
viscosity. δu and δε are virtual displacements and their corresponding strains.
For the time derivatives, u = ∂u/∂t. The displacements are discretized as
follows:
ue = Nue, ue = Nue, ue = Nue, εe = Bue, (2.4)
where N is shape functions, ue represent nodal degrees of freedom of an
element, and
B = ∂N, (2.5)
where ∂ is a differencial operator for strain-displacement relations. We used a
bold font to represent a vector in N -dimensional Hilbert space. Substituting
14
Eq. (2.4) into Eq. (2.3) and integration in a element yields
δuTe
[ ∫Ωe
ρNTNdV ue +
∫Ωe
cNTNdV ue +
∫Ωe
BTσ dV ue
−∫Ωe
NTβ dV −∫Γσ
NTτ dS −Nσ∑i=1
pi
]= 0. (2.6)
where the concentrated load pi are assumed to be located at nodes. From
the first and second integral in Eq. (2.6), full system matrices of consistent
mass and damping are obtained as
M =
Ne⋃e=1
∫Ωe
ρNTNdV (2.7)
C =
Ne⋃e=1
∫Ωe
cNTNdV. (2.8)
Also, external forces are calculated as follows:
rext =
Ne⋃e=1
[ ∫Ωe
NTβ dV +
∫Γσ
NTτ dS +
Nσ∑i=1
pi
]= f . (2.9)
The internal force vector is defined as forces and moments applied to the
element by nodes. Also, the material is assumed to be linear elastic. Thus,
the internal force vector in the third integral in Eq. (2.6) becomes the mul-
tiplication of the element stiffness matrix and nodal displacement such that
rint =
Ne⋃e=1
∫Ωe
BTCBdV ue
= Ku. (2.10)
Finally, from Eqs. (2.1) and (2.7)∼(2.10) the finite element governing equa-
tion is derived for dynamics:
Mu(t) +Cu(t) +Ku(t) = f(t), (2.11)
15
where (M,C,K
)∈ RN×N × RN×N × RN×N . (2.12)
Thus, the total degrees of freedom of the system is N . Note that Eq. (2.11)
can be solved in both time and frequency domains.
2.2 Proper Orthogonal Decomposition
2.2.1 Construction of Energy Functional
The proper orthogonal decomposition (POD) is a statistical method which
finds the best and compact representation of given data. The proper orthogo-
nal mode (POM) obtained by the POD process becomes an orthogonal basis
which transforms the given data into the generalized coordinates. Finding
the proper orthogonal mode is the main procedure of the POD. As shown
in Ref. [34], the POD is also known as the Karhunen-Loeve decomposition
(KLD). First of all, the ensemble average which represents the profile energy
is defined as
J ≡⟨(ϕ, u)2
⟩= lim
T→∞
1
T
∫ T
0
[ ∫Bϕ(x)u(x, t)dx
]2dt, (2.13)
where u(x, t) is a given data that has zero means with respect to time. The
POD aims to seek a real function ϕ(x) that maximize the ensemble average:
max. J [ϕ] s.t, ∥ϕ∥2 = 1, (2.14)
where the constraint is imposed for the computation of an unique eigen-
function. By using a Lagrange multiplier, the functional J is changed to
16
unconstrainted one such that
J =⟨(ϕ, u)2
⟩− λ(∥ϕ∥2 − 1). (2.15)
Taking variation to Eq. (2.15) as δJ |ϕ = 0 yields∫BH(x,x′)ϕ(x′)dx′ = λϕ(x), (2.16)
where the auto-correlation function H is written as
H(x,x′) = limT→∞
1
T
∫ T
0u(x, t)u(x′, t)dt. (2.17)
2.2.2 Method of Snapshots
There are two main ways to solve Eq. (2.16). The first one is constructing
the sample covariance matrix. This method is proper to the problem that
has larger number of snapshots than that of the degrees of freedom of the
system (Ns>N) since the size of covariance matrix is the same to the number
of degrees of freedom. In this study, however, the number of the degrees
of freedom is much larger than that of snapshots. Therefore, the method
of snapshots developed in Ref. [31] are used. At time tk, the displacement
snapshot uk(x, t) = u(x, tk) is obtained by using either implicit or explicit
time integration method. Thus, the discrete form of the kernel function can
be written as
H(x,x′) ≃ 1
T
Ns∑k=1
[uk(x)uk(x
′)]∆tk, (2.18)
where Ns represents the number of snapshots. To apply the method of snap-
shots, the mode ϕ to be obtained is expressed as a linear combination of the
snapshots.
ϕi(x) =
Ns∑k=1
αkiuk(x)√∆tk, i = 1, 2, · · · , Nϕ ≤ Ns, (2.19)
17
where αki represent the coefficient of the snapshots and Nϕ is the number
of POMs. Eq. (2.19) shows the main characteristic of the POD; the system
response of the full order model (FOM) should be known priori in order to
obtain the modes which govern the system response. The displacement snap-
shots are taken from the response of FOM. Usually, the number of snapshots
is much smaller than the number of total time steps. (Ns ≪ NT )
From Eq. (2.16), the eigenvalue problem is derived by replacing the inte-
gration with respect to x in the domain B to vector inner product, and also,
by using the vector notation,
Hαi = λiαi, (2.20)
where
αi = [ α1i α2i · · · αNsi ]T (2.21)
λi = T λi. (2.22)
The kernel matrix H is represented as
H = WtFtWt (2.23)
Wt = diag(√∆t1,
√∆t2, · · · ,
√∆tNs) (2.24)
[Ft]i,j =
∫Bui(x)uj(x)dx
= uTi uj , (2.25)
where Wt is the weights of each snapshots. The size of the kernal matrix is
H ∈ RNs×Ns . Thus, the eigenvalue problem in Eq. (2.20) are solved without
any difficulties.
18
The above procedure of the eigen-problem is derived in different, but more
convenient way by constructing the snapshot matrix. From the snapshots
and weights obtained from the FOM, the following snapshot matrix can be
constructed.
X =[u1
√∆t1 u2
√∆t2 · · · uNs
√∆tNs
]. (2.26)
Then the multiplication of the snapshot matrix and its transpose yields the
kernal matrix (XTX = H). Therefore, Eq. (2.20) is changed as
XTXαi = λiαi. (2.27)
By solving Eq. (2.27), the eigenvectors are obtained. Successivly, the POM
is calculated from Eq. (2.19) such that
ϕi = Xαi, i = 1, 2, · · · , Nϕ. (2.28)
From the constraint in Eq. (2.14), the POM should be normalized. By as-
suming that the eigenvectors α are orthonomal (αTi αi = 1), the following
calculations are executed:
ϕTi ϕi = αT
i XTXαi
= αTi λiαi
= λi. (2.29)
Finally, the POM is normalized as follows:
ϕi =1√λi
Xαi, i = 1, 2, · · · , Nϕ. (2.30)
19
Corresponing matrix form of the POM is expressed as
Φ =[ϕ1 ϕ2 · · · ϕNϕ
]= XAΛ− 1
2 , (2.31)
where
A = [ α1 α2 · · · αNϕ ] (2.32)
Λ = diag(λ1, λ2, · · · , λNϕ). (2.33)
From the orthonormality of eigenvectors (ATA = I) the snapshot matrix is
derived as
X = ΦΛ12AT . (2.34)
When Nϕ is equal to Ns, the snapshot matrix expressed in Eq. (2.34) rep-
resents the singular value decomposition (SVD) of X. In other words, if the
orthonormality of the eigenvectors in Eq. (2.27) holds, the POD calculation
is the same to thin singular value decomposition. Generally, POD is an an-
other expression of thin SVD in Euclidean space. The calculation of SVD is
written as
X = U0Σ0VT0
= [ U Ur ]
[Σ 00 0
] [VT
VTr
]= UΣVT (Thin SVD), (2.35)
where U, V represent left and right singular vectors, respectively, and Σ
is the diagonal matrix of singular values. Comparing Eq. (2.34) and (2.35)
yields
Φ = U, Λ = Σ2, A = V. (2.36)
20
Thus, the eigenvalue problem in Eq. (2.27) is rewritten as
XTX = AΛAT . (2.37)
When the multiplication order of the snapshot matrices is reversed, another
eigenvalue problem is derived such that
XXT = ΦΛΦT , (2.38)
where the size of square matrix XXT is N . The meaning of Eq. (2.38) is the
covariance matrix mentioned at the front of this subchapter.
2.2.3 Model Reduction Using Proper Orthogonal Decompo-
sition
The physical meaning of the eigenvalues obtained by solving eigen-problem
in Eq. (2.27) is how much the mode participates in the system response. Let
the eigenvalues have the desending order of magnitude as
λ1 ≥ λ2 ≥ · · · ≥ λNϕ≥ 0. (2.39)
Thus total energy of the system can be expressed the summation of the
eigenvalues. The energy percentage captured by k-th mode can be expressed
as follows:
E(k) =λk∑Nϕ
i=1 λi
. (2.40)
By selecting R numbers of modes, the displacement of FOM is approximated
by
u(t) ≃ u(t) =
R∑k=1
ai(t)ϕi
= Tur(t), (2.41)
21
where the projection T is expressed as
T = [ ϕ1 ϕ2 · · · ϕR ], R ≤ Nϕ. (2.42)
The displacement variable in the generalized coordinates is given by
[ur]i = ai. (2.43)
External force is also projected by using the same projector of the displace-
ment as the Galerkin projection such that
f(t) ≃ f(t) = Tfr(t). (2.44)
Substituting Eqs. (2.41) and (2.44) into (2.11) yields
MTur(t) +CTur(t) +KTur(t) = Tfr(t). (2.45)
Considering the column orthogonality of the projector, TT is multiplied to
the left of Eq. (2.45) as
TTMTur(t) +TTCTur(t) +TTKTur(t) = TTTfr(t), (2.46)
such that
Mrur(t) +Crur(t) +Krur(t) = fr(t). (2.47)
Usually, R ≪ N , thus the ROM can be solved much faster than the FOM in
Eq. (2.11).
2.2.4 Numerical Examples
General Remark In the present dissertation, all computations were per-
formed by MATLAB R2013a [16] under Linux OS (Fedora 10). The CPU was
22
Intel i7 860 quad core with 2.80 GHz. The specification of the RAM is DDR3,
32 GB. There was not any memory swap to the hard disk drive during the
computation. All finite elements used in this dissertation were reproduced
by the author. In fact, both FOM and ROM were used the same elements.
Therefore, all the methods developed here are applicable to the other ad-
vanced elements in which the basic finite element routine is included.
First of all, the performance of the ROM was verified by comparing time
responses of the FOM and the ROM. The test model is rib-skin-spare struc-
ture as shown in Fig. 2.1. The dynamic load was applied to the tip of the
structure, and all degrees of freedom of the other end was fixed. In table
2.1, the material properties of the structure is presented. Flat shell element
with 6 degrees of freedom of each node was used [51], so the total degrees of
freedom of FOM is 1,158. The dynamic loading profile is presented in Fig.
2.2. The total analysis time was 0.4 sec and 800 time steps are generated.
In Fig. 2.3, two reduced models were constructed using 40 snapshots in
[0∼0.02] sec. The first one is reduced by using 4 POM and the second one
used 8 POM, respectively. Since the range of snapshot is not wide enough,
the time response of the ROM using 4 POM is not correct. If the range of
sampling time increases to [0∼0.04] sec as in Fig. 2.4, the ROM using 4 POM
shows good accordance to the FOM. Consequently, the performance of ROM
is determined by both the number of POM and the range of snapshots.
For the ROM constructed in frequency domain, an impluse load was ap-
plied to the tip of the structure. The range of frequency is [0:0.5:400] Hz,
totally 801 frequency points. The 50 snapshots were taken at [4:4:200] Hz.
As shown in table 2.2, the eigenvalues in the range of snapshots are almost
23
exact compared to the FOM. Also, the frequency responses of position (1)
and (2) shown in Fig. 2.5 and 2.6 are almost exact. The size of the ROM is
15 which indicates that the number of POM is greater than that of eigen-
values in the snapshot range. Note that if insufficient number of snapshots
are taken, the the reduced system cannot recover the accurate responses of
FOM. Therefore, if we want to use the POD for constructing the ROM, we
should know the range of eigenvalues roughly in priori.
24
Table 2.1 Geometric and material properties of rib-skin-spar structure
E (Pa) ν ρ (Kg/cm3) Thickness (m)
72e9 0.3 2700 2e-3
Table 2.2 Relative error of the eigenvalues of frequency domain ROM
Mode number Eigenvalue (Hz) Relative error (%)
1 10.37 1.59e-9
2 20.40 1.67e-9
3 37.24 2.03e-10
4 52.27 8.04e-10
5 98.46 4.07e-11
6 124.15 2.23e-9
7 159.70 2.50e-11
8 196.20 1.79e-10
9 197.30 4.53e-8
10 227.07 7.28e0
25
0 0.5 1 1.5 2 2.5 3 00.1
0.2
0
0.05
0.1
ClampedDynamic Load f(t)
Response (1) Response (2)
Figure 2.1 Rib-skin-spar structure under dynamic load f(t).
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
2000
4000
6000
8000
10000
time(sec)
F (
N)
0 0.005 0.01 0.015 0.020
5000
10000
Magnified
F(t)
Figure 2.2 Dynamic loading profile.
26
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.03
−0.02
−0.01
0
0.01
0.02
0.03
(a) position 1
time (sec)
Dis
pla
cem
ent (m
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.06
−0.04
−0.02
0
0.02
0.04
0.06
(b) position 2
time (sec)
Dis
pla
cem
ent (m
)
FOM ROM: ’4’ POM ROM: ’8’ POM
Figure 2.3 Comparison of the deflection of the FOM and the ROM: ‘40’
snapshots in [0∼0.02] sec.
27
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.03
−0.02
−0.01
0
0.01
0.02
0.03
(a)
time (sec)
Dis
pla
ce
me
nt
(m)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.06
−0.04
−0.02
0
0.02
0.04
0.06
(b)
time (sec)
Dis
pla
ce
me
nt
(m)
FOM ROM: ’4’ POM ROM: ’8’ POM
Figure 2.4 Comparison of the deflection of the FOM and the ROM: ‘80’
snapshots in [0∼0.04] sec.
28
0 50 100 150 20010
−10
10−5
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) x
0 50 100 150 20010
−12
10−10
10−8
10−6
10−4
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) y
0 50 100 150 20010
−10
10−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) z
0 50 100 150 20010
−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(d) φx
0 50 100 150 20010
−10
10−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(e) φy
0 50 100 150 20010
−10
10−8
10−6
10−4
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(f) φz
FOM ROM
Figure 2.5 Comparision of frequency responses of the FOM and the ROM
(50 snapshots) at position (1).
29
0 50 100 150 20010
−12
10−10
10−8
10−6
10−4
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) x
0 50 100 150 20010
−10
10−8
10−6
10−4
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) y
0 50 100 150 20010
−10
10−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) z
0 50 100 150 20010
−10
10−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(d) φx
0 50 100 150 20010
−8
10−6
10−4
10−2
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(e) φy
0 50 100 150 20010
−10
10−8
10−6
10−4
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(f) φz
FOM ROM
Figure 2.6 Comparision of frequency responses of the FOM and the ROM
(50 snapshots) at position (2).
30
2.3 Reduced Basis Method
2.3.1 Reduced Basis Approximation
Basically, the dynamic reduction method transforms the system into the ba-
sis generated by the dynamic characteristics: generalized eigenvectors of the
mass and stiffness matrices. On the other hand, the reduced basis method
transforms the full model into the basis characterized by the change of param-
eters. Actually, if the parameters that consist of the structural system change,
the finite element matrices have to be reconstructed every time. However, by
transforming the FOM into the basis constructed by the displacement snap-
shots obtained at specific operating points, the response can be obtained
without reconstructing the FOM. First of all, the set of operating points can
be defined as
SNpµ =
µ1,µ2, · · · ,µNs
, (2.48)
where Ns denotes the number of operating points and Np is the size of pa-
rameter such that
µi =[µ1i µ2
i · · · µNp
i
]T. (2.49)
For example, if we have three different sample of the elastic modulus, Ns = 3,
and if two different areas of the structure have each modulus, Np = 2.
The finite element equation of static problem is written as
K(µ)u(µ) = f(µ), (2.50)
where
K : RNp → RN×N , f : RNp → RN . (2.51)
31
The solution at the operating point µi is obtained as
u(µi) = K−1(µi)f , µi ∈ SNpµ . (2.52)
Totally, Ns number of displacement snapshots are obtained. Then, a low-
dimensional global approximation space can be spanned by the displacement
snapshots in Eq. (2.52), such that
WNp = spanζi ≡ u(µi), i = 1, 2, · · · , Ns. (2.53)
By applying the POD to the snapshots, the reduced basis that captures the
important characteristics of parametric variations can be obtained. To do so,
the kernal matrix in Eq. (2.20) are derived using the snapshots in Eq. (2.52)
as follows:
H = WµFµWµ (2.54)
Wµ = diag(√
δ1,√δ2, · · · ,
√δNs) (2.55)
[Fµ]i,j = uT (µi)u(µj), (2.56)
where δi denotes the weight determined from the relations between the op-
erating points. Also, the snapshot matrix can be constructed as in Eq. (2.26)
such that
X =[u(µ1)
√δ1 u(µ2)
√δ2 · · · u(µNs
)√δNs
]= ΦΛ− 1
2AT . (2.57)
Selecting R modes from the POM, the following projection matrix can be
obtained:
T = [ ϕ1 ϕ2 · · · ϕR ], R ≤ Nϕ. (2.58)
32
Note that the projection T does not depend on the parameter µ. The dis-
placement of the FOM can be approximated as
u(µ) ≃ u(µ) = Tur(µ). (2.59)
To obtain the reduced system, the stiffness matrix is affinely decomposed as
K(µ) =
Ns∑i=1
fi(µ)Ki, (2.60)
whereKi is independent of the parameter µ. The computation of Eq. (2.60) is
executed during the assembly process of the local and global stiffness matrices
as in Eq. (2.10). Substituting Eq. (2.59) into (2.50) yields
K(µ)u(µ) =
[ Ns∑i=1
fi(µ)Ki
]Tur(µ). (2.61)
Successively, multiplying TT to the left of Eq. (2.61) gives
TT
[ Ns∑i=1
fi(µ)Ki
]Tur(µ) = TT f . (2.62)
The LHS of Eq. (2.61) becomes
TT
[ Ns∑i=1
fi(µ)Ki
]T =
Ns∑i=1
fi(µ)TTKiT
=
Ns∑i=1
fi(µ)Kri
= Kr(µ), (2.63)
where
Kr : RNp → RR×R. (2.64)
Thus, for the new operating point µ∗ /∈ SNpµ , the following reduced system
can be solved with much smaller resources than that of FOM:
ur(µ∗) = K−1
r (µ∗)fr, (2.65)
33
where
fr = TT f . (2.66)
Finally, the displacement is recovered by the projection matrix T as in Eq.
(2.59)
2.3.2 Numerical Examples
The example used in this section is also the rib-skin-spar structure with a
more refined mesh configuration as shown in Fig. 2.7, whereas the static
loading was applied to the tip of the structure. Thus the material properties
except the thickness is also that presented in table 2.1. The total degrees of
freedom is 3,378 with 6 subdomains. The design variable is the thickness of
each subdomains. The objective function is weight and the maximum dis-
placement constraint was imposed. Details of the optimization problem is
presented as follows:
Minimize Weight (2.67)
s.t, 0.005 ≤ µ ≤ 0.012 (m) (2.68)
max(|u(µ)|) ≤ 0.01 (m). (2.69)
For the reduced basis method, totally 28 snapshots were taken by changing
the thicknesses of subdomains. The choice of thickness sample is presented
in table 2.3. In fact, the choice of snapshots can be determined by solving
optimization problem of sampling algorithm [52]. However the optimization
process requires repeated computations in almost full order level. Therefore,
in the present study, the sampling points are selected simply using the com-
binations and repetition of sample thicknesses. For example, if we have ‘3’
34
sampling thickness with ‘6’ design variables, the number of total samples are
6+1H3−1 = 7+2−1C3−1 =8!
6! · 2!= 28. (2.70)
Since the POD process are followed after obtaining the snapshots, even
roughly selected snapshots are acceptable if the number of snapshots are
so small. From the POD process in Eqs. (2.57) and (2.58), 22 POMs are
selected by calculating the rank of the singular value matrix. Thus the size
of ROM is 22, which is 0.6 % compared to the FOM.
In the present dissertation, all optimizations are executed by the con-
strained nonlinear optimization algorithm: sequential quadratic programming
of the ‘fmincon’ function in MATLAB. The tolerance of design variables are
set to be ‘1e-4’. The optimal thickness are presented in Fig. 2.8. The first
blue bar is the optimal thickness of the FOM. The second green bar is that
of parametrized model without reduction. As shown in Eq. (2.60), the affine
decomposition of the stiffness matrix without reduction process can reduce
the computational resources since the system construction at a new operat-
ing point is executed with a little additional computation. The third dark
red bar is the optima of the reduced model. The thicknesses of three different
methods are almost the same. Fig. 2.9 presents the histories of the objective
functions of each methods. The reduced model shows an identical conver-
gence compared to the FOM, which indicates the reduced basis method can
capture the global response surface of the FOM well. Table 2.4 represents
the optimum weights of each models. There is slight difference between the
FOM and ROM, but not significant. In Fig. 2.10, the parametrized FOM
and the ROM show extremely efficient computations compared to the FOM.
35
Since the construction of full system takes most of the computation, only the
affine decomposition makes the optimization efficient. However, if the design
variables, the size of FOM increase, or the loading condition is changed to
dynamics, the efficiency of the reduced basis method cannot be guaranteed.
In that case, we have to use other approach that can embrace the problem
mentioned.
36
Table 2.3 Sampling strategy by combinations with repetitation
Subdomains set 1 set 2 set 3 set 4 set 5 · · · set 28
Sub. 1 t1 t1 t1 t1 t1 · · · t3
Sub. 2 t1 t1 t1 t1 t1 · · · t3
Sub. 3 t1 t1 t1 t1 t1 · · · t3
Sub. 4 t1 t1 t1 t1 t1 · · · t3
Sub. 5 t1 t1 t1 t2 t2 · · · t3
Sub. 6 t1 t2 t3 t2 t3 · · · t3
Table 2.4 Comparison of weights of the FOM, parametrized FOM and ROM
- Initial FOM Para. w/o Redu. Para. w Redu.
Weight (Kg) 49.40 26.59 26.59 26.62
37
00.5
11.5
22.5
3 00.1
0.2
0
0.05
0.1
64
3
52
1
Static Load f1
3
Clamped
Figure 2.7 Rib-skin-spar structure with 6 subdomains under tip static load.
0 1 2 3 4 5 6 70
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Design variables
Th
ickn
ess (
m)
Full
Para. w/o Redu.
Para. w Redu.
Figure 2.8 Comparison of optimal thicknesses of the FOM, parametrized
FOM and ROM.
38
0 2 4 6 8 10 120.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Iterations
|Ob
j. f
un
ctio
n|
Full
Para. w/o Redu.
Para. w Redu.
Figure 2.9 Comparison of objective function histories.
1 2 30
10
20
30
40
50
60
70
61.57
4.82 1.52
tim
e (
sec)
1. Full
2. Para. w/o Redu.
3. Para. w Redu.
Figure 2.10 Comparison of computation time of the FOM, parametrized FOM
and ROM.
39
Chapter 3
Global Proper Orthogonal
Decomposition and Reduced
Equivalent Static Load
3.1 Introduction
The reduced basis method presented in Chap. 2 is applicable to the struc-
ture under a single loading condition. For a multiple loading condition, the
displacement obtained is not only the function of the parameter but the
function of the loading also. In this chapter, the reduced basis method under
the multiple loading condition is developed. In addition, an efficient struc-
tural optimization method for dynamics is proposed by combining the devel-
oped method with the optimization strategy using the equivalent static load
method.
41
3.2 Reduced Basis Method for Multiple Loading Con-
dition
3.2.1 Global Proper Orthogonal Decomposition
To obtain the reduced basis under multiple loading condition, the global-
POD method is employed to the matrices of the displacement snapshots.
The multiple loading is written as
F = [ f1 f2 · · · fNf ], (3.1)
where Nf represents the number of loads. Thus Eq. (2.52) is changed to
uj(µi) = K−1(µi)fj ,
i = 1, 2, · · · , Nsj = 1, 2, · · · , Nf
. (3.2)
Corresponding j-th snapshot matrix is constructed under the j-th load such
that
Xj =[uj(µ1)
√δ1 uj(µ2)
√δ2 · · · uj(µNs
)√δNs
]. (3.3)
The total snapshot matrix can be constructed, and successively the POD is
employed to the snapshot matrix X as follows:
X = [ X1 X2 · · · XNf ]
= ΦΛ− 12AT . (3.4)
The size of X is N -by-NS , where NS = Ns × Nf . From the Φ matrix, R
number of modes can be selected and become the reduced basis as follows:
T = [ ϕ1 ϕ2 · · · ϕR ], R ≤ Nϕ < NS . (3.5)
Similar to Eq. (2.62), Eq. (3.2) is reduced to
Ur(µi) = K−1r (µi)Fr, (3.6)
42
where
Ur(µi) = TU(µi) (3.7)
Kr(µi) = TTK(µi)T (3.8)
Fr = TTF, (3.9)
and
U(µi) =[u1(µi) u2(µi) · · · uNf
(µi)]. (3.10)
Note that U is used to express the displacement field, not the left singular
vectors in Chap. 2.
3.2.2 Mode of External Loads
Calculating the displacement snapshots under the multipling loads in Eq.
(3.2) requires computational resources which depends on the number of loads.
Therefore, we propose to reduce the number of the external loads. To do so,
the mode of external force is calculated by applying the POD to the multiple
loads. Note that if the direction of the external loads is fully random, the
mode of loads cannot be obtained, which only permits to solve Eq. (3.6)
without any reduction of external loads. However, if the loads have some
directions with random magnitudes, the POD can be used with much reduced
computations. The snapshot matrix and the POD of the external forces are
written as
XF =[f1√δ1 f2
√δ2 · · · fNf
√δNf
]= ΓFΛ
− 12
F ATF, (3.11)
43
where
ΓF =[γ1 γ2 · · · γNγ
], Nγ ≪ Nf . (3.12)
The number of multiple loads is relatively large compared to that of the mode
of multiple loads. Thus, the number of final reduced basis Rf is relatively
small compared to the Nf since the most energy is contained in the first few
modes. The main difference between the POD in Chap. 2 and present one
is the number of snapshots. For a time or frequency domain analysis, Ns is
much smaller than NT , or Nω. Thus, the final number of POM is similar,
or slightly smaller than that of snapshots. However, the snapshots of present
method are all multiple loads. So, we have to choose relatively small number
of modes from the POM. Thus, the modes of external force can be obtained
as follows:
Γ =[γ1 γ2 · · · γRf
], Rf ≤ Nγ . (3.13)
By using Eq. (3.13), Eq. (3.2) is converted to the following equation.
uj(µi) = K−1(µi)γj ,
i = 1, 2, · · · , Nsj = 1, 2, · · · , Rf
. (3.14)
Successively, the total snapshot matrix and the POD in Eq. (3.4) are changed
to
X = [ X1 X2 · · · XRf ]
= ΦΛ− 12AT . (3.15)
The remain parts are the same to the procedure from Eq. (3.5) to Eq.
(3.10). The computational gain of the present method is (i) initial sampling
time is reduced, (ii) the size of snapshot matrix X is also reduced, which
44
results in the fast computation of proper orthogonal mode Φ.
3.3 Structural Optimization Strategy Using Reduced
Equivalent Static Load
3.3.1 Problem Definition
In this chapter, we aim to optimize a structure under dynamic loading con-
dition. To do so, the parameter that is also the design variable should be
included in the equation of motion. From Eq. (2.11), a parameter dependent
equation of motion is expressed as
M(µ)u(t;µ) +C(µ)u(t;µ) +K(µ)u(t;µ) = f(t;µ), (3.16)
where t is the time variable in a bounded interval [t0, T ], and,
M : RNp → RN×N , C : RNp → RN×N ,
f : [t0, T ]× RNp → RN , u : [t0, T ]× RNp → RN .(3.17)
The optimization problem can be stated as ‘find the optimal design, µ∗’,
which satisfies
µ∗ = arg minµ∈F
W(µ). (3.18)
W(µ) is a cost function as a total weight of the structure. F is the set of the
feasible design variable defined as
F =µ ∈ RNp | µlb ≤ µ ≤ µub
. (3.19)
In this study, the constraints involve the design variable and the displacement
field only. The j-th inequility constraint is expressed as
cj(µ,u(t;µ)) ≤ 0, j = 1, 2, · · · , Nconst. (3.20)
45
For example, we require the maximum displacement be less than a allowed
limit such that
c1 = max(|u(t;µ)|)− u ≤ 0. (3.21)
Also, the maximum stress constraint is expressed as
c2 = max(|σ(t;µ)|)− σ ≤ 0. (3.22)
For the characteristic of dynamics, the first natural frequency condition is
assigned:
c3 = ω1 − ω1(µ) ≤ 0. (3.23)
Also, other natural frequencies and buckling constraints could be imposed.
The dynamic equation is solved by the time-discretized methods, explic-
itly or implicitly. In this study, Newmark-Beta scheme was chosen to solve
the equation under certain initial conditions. The initial condition and the ex-
ternal dynamic force are assumed to be independent to the design variables.
The external force is discretized as follows:
fi = f(ti), i = 1, 2, · · · , NT . (3.24)
Thus, the displacement field at every time steps can be obtained by solving
Eq. (3.16) and (3.17). After the displacement solution is obtained, and the
column matrices of the external force and the displacement field are written
as
F = [ f1 f2 · · · fNT ] (3.25)
U = [ u1 u2 · · · uNT ], (3.26)
where ui = u(ti).
46
Finally, the displacement solution obtained and the design variable are
substituted to the constraints shown in Eq. (3.20).
3.3.2 Optimization Strategy Using Equivalent Static Load
For the structural optimization under dynamic loadings, the FOM presented
in Eq. (3.16) have to be solved at every iterations and sensitivity compu-
tations. The problem is solving Eq. (3.16) requires a lot of computational
resources. To reduce the computational burden, the Equivalent Static Loads
(ESL) algorithm was developed by Choi and his co-workers (Ref. [53, 54, 55]).
The ESL is a static load set that makes the same displacement field as that
under a dynamic load. Thus, the optimization algorithm using ESL executes
multiple static optimizations instead of solving dynamic equation at every
function evaluations.
From the Eq. (3.16), the inertia and damping parts are moved to the
RHS. Then the sum of RHS is a equivalent load set such that
K(µ)u(t;µ) = f(t)−M(µ)u(t;µ)−C(µ)u(t;µ)
= feq(t;µ). (3.27)
Once the displacement field U is computed from the time integration, the
ESL is calculated by multiplying the stiffness matrix to the displacement field
obtained. Thus, Eq. (3.27) is changed as follows:
K(µ)U(µ) = Feq(µ). (3.28)
Note that the ESL can be calculated after obtaining the displacement field
from the time integration.
47
The main characteristic of the optimization using ESL algorithm is that
the update of the design variable of stiffness matrix is discriminated to that
of the design variable of ESL. Although this process can be questionable,
the discrimination of the design variable is nothing but an engineering as-
sumption. Actually, Stolpe [56] indicated that the optima of the ESL-based
method and that of full transient analysis could not be the same in general.
However, the optimal solution obtained by the ESL-based method satisfies all
the constraints under a dynamic loading condition. Therefore, eventhough the
procedure of optimization using ESL was not fully proved by the mathemat-
ical tool, we can use the ESL-based optimization considering the efficiency
of the algorithm.
The design variables in Eq. (3.28) are divided to µ and µ. During the
static optimization, µ is fixed while only µ is varied. Thus, the static solution
using each indices depend on the design variables can be represented as
U(µm,k) = K−1(µm,k)Feq(µm), (3.29)
where the superscript k denotes the iteration of design variable for the static
optimization. The superscript m represents the update of ESL. So, during
the static optimization process, m is fixed. Initially, all variables are the same
as follows:
µ1,1 = µ1 = µini. (3.30)
After the first static optimization process, the µ1,k is converged to the optima
µ1,∗ which is the initial design variable of the second static optimization
process such that
µm+1,1 = µm+1 = µm,∗, m ≥ 1. (3.31)
48
In Ref. [53, 54, 55], the convergence criteria was set as the convergence of
ESL, which is the physically same criteria of the convergence of the design
variables under the assumption of a linear elastic material.
3.3.3 Mode of Equivalent Static Load
The inner loop of the ESL optimization algorithm is exactly same to the static
optimization under a multiple loading condition. At the front of this chapter,
we proposed to calculate the mode of external loads to employ the reduced
basis method under the multiple loading condition. Thus, the global-POD
method combined with the reduced basis method can be employed to the
static optimization of the ESL algorithm. This combination of two methods
can reduce computational costs of the ESL optimization algorithm. Before
that, we need to clarify the meaning of the mode of external loads.
The meaning of the mode of ESL is clear. As shown in Eq. (3.27), the
ESL is composed of the external load, inertia and damping parts. The mode
of external load is straightforward. Since the magnitude does not affect to
the mode of external load, the direction vector of external load is the mode of
the external load. The inertia and damping parts are also clear. The inertia
is a multiplication of the mass matrix and the acceleration. The mass matrix
is not a function of time, so the mode of inertia is the same as that of
acceleration. Of course the mode of damping part is the same as that of
velocity. Therefore, by superposing the three modes, the mode of ESL is easily
derived. The good point of this method is that the acceleration and velocity
are necessarily computed during the time integration for updating ESL. So
the mode of ESL is easily obtained without additional, heavy computations.
49
Another way to obtain the mode of ESL is applying POD to the ESL
directly. In other words, each column of the ESL is regarded as a snapshot. So,
from the Eq. (3.11), the POM of the load is calculated. During the transient
analysis, the ∆t is constant, which results in all the same weighting factors.
δ1 = δ2 = · · · = ∆t. (3.32)
Thus, the snapshots matrix is expressed as follows:
Feq = [ feq,1 feq,2 · · · feq,NT ] (3.33)
= ΓeqΛ− 1
2eq AT
eq, (3.34)
where
Γeq =[γ1 γ2 · · · γNγ
], Nγ ≪ NT . (3.35)
The rest of the procedure is exactly same to Eqs. (3.13)∼(3.15) and Eqs.
(3.5)∼(3.10). To sum up, the static optimization part of the ESL algorithm
is reduced by applying the POD to the ESL and the reduced basis method.
3.4 Numerical Results
Example 1. Cantilever beam with 4 subdomains
First of all, simple cantilever Timoshenko beam example shown in Fig. 3.1
is studied. The beam is divided into 4 subdomains with 4 design variables.
At the free end, half sinusoidal load is applied as presented in Fig. 3.2. The
time interval is [0:0.001:0.2] sec. The objective function is the the width of the
beam. The thickness is 25.4 mm. Table 3.1 addresses the conditions for design
50
optimization. As shown in Fig. 3.3, the optimal design variables are almost
the same eath others. Consqeuently, the object functions of three methods
are the same.
Example 2. Rib-skin-spar structure
The rib-skin-spar structure in Chap. 2 is also investigated under dynamic
loading conditions (Fig. 3.4). The upper and lower bounds of design variables
are the same in Eq. (2.68). The other conditions are presented in table 3.2.
The dynamic loading profile is shown in Fig. 3.5. Thus total 5,000 time steps
are considered in [0:0.001:5] sec.
In Fig. 3.6 the optimal thicknesses of each methods are presented. ‘Full
tansient’ represents the FOM which executes transient analysis at every sensi-
tivity calculations. ‘ESL’ means conventional equivalent staticl load method,
and ‘ESL-L’ is the one that reduced the transient analysis by modal reduction
technique. ‘ESL w Param.’ and ‘ESL-L w Param.’ are the present methods
which reduced the static optimization by using the global-POD and by calcu-
lating the modes of equivalent static loads. Thus basically, the transient parts
can be reduced by the modal reduction and the static optimization parts are
reduced by the present method. In ‘ESL-L w Param.’, both reductions are
employed.
The optimum of all methods are similar except the thickness of subdo-
main 6, which indicates the sensitivity of design variable 6 is not significant.
Table 3.3 indicates the error of the optimal weights depending on the toler-
ance of design variables. When the tolerance is larger than 1e-3, the optimal
weight of each methods have small error since the solution did not be fully
51
converged. If the tolerance is small enough (∆µ ≤ 1e-4), the solution is re-
garded as the optimal one. In Fig. 3.7, the total computation time of each
methods are presnted. Compared to the FOM, all method using the ESL show
efficiency. Fig. 3.8 shows the computation time of each steps. First of all, for
‘ESL w Param.’ and ‘ESL-L w Param.’, almost no time was comsumed for
the matrix generation. Also the static optimization time is fast compared to
the other two methods. The reduction represents the step of taking snapshots
and calculating the modes of snapshots and ESLs. For the transient analysis,
the time of ‘ESL-L’ and ‘ESL-L w Param’ is small compared to the other two
methods, whereas the eigen-analysis is requried to execute the modal reduc-
tion. Finally the total time shows the efficiency of each method. Although
the efficiency cannot directly apply to the other structures and environments,
the trends of the efficiency of each steps ought to be maintained.
Example 3. Wing box model
Another example to verify is a wing box model with 20 subdomains as shown
in Fig. 3.9. To view the inside of the wing box, the configuration was plotted
separately. In fact, all components are exactly connected including the upper
and lower skins. The dynamic load is applied to the tip of the wing as showin
in Fig. 3.10 and the other end is clamped. The short half sinusoidal loading in
[0:0.001:5] sec was considered. The thicknesses of upper and lower skins, ribs
and spars are design variables. The material was assumed to be aluminum;
elatic modulus E = 72e9 Pa, density ρ = 2, 823 Kg/m3, and the Poisson’s
ratio ν = 0.33. The problem conditions for optimization are shown in table
3.4. The initial thicknesses are 0.015 m.
52
Fig. 3.11 is the optimal thicknesses of each approaches. As expected, all
designs of each methods correlate well, which also means the objective func-
tions are also similar with each other. Fig. designates the total computation
time of the ROMs. Different from the previous rib-skin-spar structure, effi-
ciency of parameterized methods decreased compared to Fig. 3.6. The reason
occurs from the number of design variables. Basically , the parameterized
methods have to obtain the snapshots by changing the design variables. In
that process, if the number of design variable increases, the number of ini-
tial samples also increases. Therefore, the efficiency of parameterized model
would decrese. Nevertheless, the ROM using the ESL and parameterization
is still appropriate alternatives compared to the FOM.
53
Table 3.1 Problem condition of cantilever beam
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz) |σmax| (Pa)0.681 2.54e-3 12.7e-3 0.101 30 310.2e6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1
−0.5
0
0.5
1x 10
−3
Sub. 1µ
1
Sub. 2µ
2
Sub. 3µ
3
Sub. 4µ
4
F(t)
Figure 3.1 Cantilever beam with 4 subdomains under tip dynamic load.
0 0.05 0.1 0.15 0.20
50
100
150
200
250
time(sec)
F (
N)
F(t)
Figure 3.2 Half sinusoidal loading profile.
54
0.5 1 1.5 2 2.5 3 3.5 4 4.50
1
2
3
4
5
6x 10
−3
Design Variables
Wid
th (
m)
Full transient
ESL
ESL w Param.
Figure 3.3 Comparison of optimal widths of the FOM and ROMs.
55
Table 3.2 Problem condition of rib-skin-spar structure
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz) |σmax| (Pa)49.40 0.005 0.012 0.01 8 2e9
Table 3.3 Relative error (%) of objective function values
∆µ ESL ESL-L ESL w Par. ESL-L w Par.
1e-2 1.19 0.62 0.02 0.22
1e-3 0.17 0.10 0.03 0.17
1e-4 2.3e-6 5.2e-4 2.5e-6 5.1e-4
56
0 0.5 1 1.5 2 2.5 3 00.10.2
0
0.05
0.1
64
3
3
52
1
1
Clamped Dynamic Load f(t)
Figure 3.4 Rib-skin-spar structure with 6 subdomains under tip dynamic
load.
0 1 2 3 4 5−50
0
50
time(sec)
F (
N)
F(t)
Figure 3.5 Sinusoidal loading profile.
57
1 2 3 4 5 60
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Design Variables
Th
ickn
ess (
m)
Full transient
ESL
ESL−L
ESL w Param.
ESL−L w Param.
Figure 3.6 Comparison of optimal thicknesses of the FOM and ROMs.
1 2 3 4 50
500
1000
1500
1194.93
159.18112.34
70.68 37.56
Tim
e (
sec)
1. Full transient
2. ESL
3. ESL−L
4. ESL w Param
5. ESL−L w Param.
Figure 3.7 Comparison of total computation time of the FOM and ROMs.
58
1 2 3 40
0.2
0.4
0.6
0.8
1
Matrix Generation
tim
e(s
ec)
0.73
0.97
0.02 0.02
1 2 3 40
0.1
0.2
0.3
0.4
0.5
Eigenvalue
tim
e(s
ec)
0.00
0.47
0.00
0.17
1 2 3 40
10
20
30
Newmark−Beta
tim
e(s
ec)
27.34
0.18
25.07
0.08
1 2 3 40
2
4
6
8
Reductiontim
e(s
ec)
0.00 0.00
7.64
3.64
1 2 3 40
50
100
150
Static optimization
tim
e(s
ec)
131.11110.72
37.9633.66
1 2 3 40
50
100
150
200
Total
tim
e(s
ec)
159.18
112.34
70.68
37.56
1: ESL 2: ESL−L 3: ESL w Param 4: ESL−L w Param.
Figure 3.8 Comparison of computation time of each steps.
59
Table 3.4 Problem condition of wing box model
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz) |σmax| (Pa)8699.9 0.005 0.02 0.005 3.5 2e9
24
68
1012
14
0
5
10
15
−4
−3
−2
−1
0
1
2
3
4
1
16
11
62
17
12
73
18
13
84
14
19
9
20
5
15
10
Dynamicload f(t)
Clamped
Figure 3.9 Wing box model with 20 subdomains under tip dynamic load.
60
0 1 2 3 4 50
100
200
300
400
500
600
700
800
time(sec)
F (
N)
0 0.1 0.2 0.3 0.4 0.50
200
400
600
800
Magnified
F(t)
Figure 3.10 Half sinusoidal loading profile.
0 5 10 15 200
0.002
0.004
0.006
0.008
0.01
0.012
Design Variables
Thic
kness (
m)
ESL
ESL−L
ESL w Param.
ESL−L w Param.
Figure 3.11 Comparison of optimal thicknesses of each optimization methods.
61
1 2 3 40
2000
4000
6000
8000
10000
tim
e (
sec)
8121.24
6998.28
4809.94
3325.96
1. ESL
2. ESL−L
3. ESL w Param.
4. ESL−L w Param.
Figure 3.12 Comparison of total computation time of the FOM and ROMs.
62
Chapter 4
Parametric Reduced Order
Model: Interpolation and Moving
Least Square Method
4.1 Introduction
When structural design optimization is executed by using the model order
reduction techniques, it should be reminded whether the reduced basis varies
according to the change of design variable or not. If not, the ROM might lead
to the wrong direction since the sensitivity calculated by the ROM could be
incorrect. In that case, the solution is just a better one than the initial design.
Therefore, to obtain the optimal design by using the ROM, the projection to
the reduced space should be calculated whenever the design variable changes.
In this chapter, we developed a mode interpolation scheme by using the
moving least square method. Also, the matrix interpolation method in Ref.
[48] is reinterpreted by the transformation to the global basis.
63
4.2 Parametric Reduced Order Model for Dynamics
4.2.1 Dynamic System with Parameters
The equation of motion for dynamic system is introduced again as in Chap.
3 such that
M(µ)u(t;µ) +C(µ)u(t;µ) +K(µ)u(t;µ) = f(t;µ). (4.1)
In this chapter, we used an overbar (·) instead of (·)r to express the reduced
system conveniently. Also, we do not discriminate the displacement of full
model u and the approximation u from the recovery shown in Eq. (2.41) for
convenience.
Since the reduced basis is the function of parameters, the projection is
written as
u(t;µ) = T(µ)u(t;µ), (4.2)
where, T ∈ RN×R is a projection matrix to transform the full system to the
generalized coordinates system. Substituting Eq. (4.2) into Eq. (4.1) yields
M(µ)¨u(t;µ) + C(µ) ˙u(t;µ) + K(µ)u(t;µ) = f(t;µ), (4.3)
where
M(µ) = TT (µ)M(µ)T(µ), C(µ) = TT (µ)C(µ)T(µ),
K(µ) = TT (µ)K(µ)T(µ), f(t;µ) = TT (µ)f(t;µ),(4.4)
also, (M(µ), C(µ), K(µ)
)∈ RR×R × RR×R × RR×R. (4.5)
If the transformation matrix T(µ) is obtained by the POD, orthogonality
condition should be satisfied such that
TT (µ)T(µ) = IR. (4.6)
64
Alternatively, the eigenmode can be used from the generalized eigenvalue
problem as follows:
K(µ)ϕi(µ) = λiM(µ)ϕi(µ), i = 1, 2, · · · , R, (4.7)
where ϕi(µ) is the i-th column vector of T(µ) as shown in Eq. (2.42). There-
fore, T(µ) becomes a mass orthogonal rectangular matrix, and the ROM
represents a modal truncation of the full system.
4.2.2 ROM Construction at Operating Points
The process of ROM based on interpolation technique can be divided into two
main steps: off-line and on-line procedures. In the off-line stage the ROMs
are constructed at the sample parameters (operating points). After finish-
ing the construction, the ROM at the new operating point is obtained by
interpolating the ROMs constructed in the off-line stage. Therefore the on-
line computation requires little amount of computational resources: usually
matrix summation and vector multiplication. The set of sample points are
defined as
SNpµ =
µ1,µ2, · · · ,µNs
, (4.8)
which is the same space of parameters shown in Chap. 2 and 3. Corresponding
parameter is written as
µi =[µ1i µ2
i · · · µNp
i
]T. (4.9)
In this chapter, we use a subscript to represent the index of operationg points
as follows:
Mi = M(µi), Mi = TTi MiTi. (4.10)
65
Using Galerkin projector to Eq. (4.1) yields
Mi ¨ui(t) + Ci ˙ui(t) + Kiui(t) = fi(t), i = 1, 2, · · · , Ns, (4.11)
Thus, there are totally Ns numbers of ROMs which are constructed at the
each parameters in Eq. (4.8). Consequently, the matrices of reduced system
can be interpolated by using a weighted interpolation function.
M(µ) =
Ns∑i=1
Wi(µ)Mi
C(µ) =
Ns∑i=1
Wi(µ)Ci (4.12)
K(µ) =
Ns∑i=1
Wi(µ)Ki.
The interpolation function Wi(µ) should satisfy the following conditions:
Ns∑i=1
Wi(µ) = 1, (4.13)
Wi(µj) = δij , i, j = 1, 2, · · · , Ns, (4.14)
where δij denotes the Kroneker delta.
4.2.3 Transformation to Common Basis
The interpolation in Eq. (4.12) is not, however, executed directly. Because
the local reduced matrices in Eq. (4.12) are not laid on the same coordi-
nates system. In other words, since the transform matrices of each reduced
system Ti depend on the parameter µi, the physical interpretation of the
reduced displacement vectors ui in generalized coordinates is not the same
with each other. Actually, T(µ) belongs to the Grassmann manifold in dif-
ferential geometry. Thus, the interpolation does not be guarenteed in curved
66
surface. Therefore, another coordinate transformation is required to make
the reduced displacement vectors be in the common coordinates system.
Let the second transformation matrix of i-th parameter to be Ri. The
coordinate transformation is expressed as follows:
ui = Riui. (4.15)
where Ri ∈ RR×R. Substituting Eq. (4.15) into Eq. (4.2) yields
ui = Tiui
= TiR−1i ui. (4.16)
To determine Ri, the global-POD method represented in Chap. 3 is intro-
duced. First of all, the original system is projected to the common basis as
ui = Su∗i . (4.17)
The common basis S is determined by the global-POD method of each pro-
jection matrix obtained at the operating points such that
X = [ T1 T2 · · · TNs ]
= ΦΛ− 12AT . (4.18)
By choosing R vectors from Φ, the common basis S is obtained as follows:
S =[ϕ1 ϕ2 · · · ϕR
]. (4.19)
Note that the common basis is automatically determined by the POD pro-
cedure. Usually, the methods based on the projection of manifolds to the
tangent plane require the initially given tangent vector in order to generate
67
the plane. Also, since the admissible range of interpolation is affected by the
initial tangent vector, the tangent vector should be in the center of the range
to make the performance of interpolation better. However, it is hard to find
out the best tangent plane initially. Thus the global-POD is applied to ob-
tain the basis which is the optimal because of the characteristic of the POD.
Moreover, there is no need to determine the tangent vector initially.
On the other hand, the global basis S is independent of the parameter µ.
Thus the projected displacement u∗i can be interpolated, whereas the original
displacement field cannot be interpolated directly. Therefore, if both ui and
u∗i have the same physical interpretation, the interpolation of ui is possible.
To do so,
S = TiR−1i . (4.20)
By multiplying ST to Eq. (4.20),
STS = STTiR−1i . (4.21)
Since the common basis is obtained by the POD procedure, the orthogonality
(STS = IR) holds, such that
IR = STTiR−1i , (4.22)
which results in
Ri = STTi. (4.23)
From Eqs. (4.16) and (4.23),
ui = TiR−1i ui
= Ti(STTi)
−1ui
= Qiui. (4.24)
68
Remark. The converse of Eqs. (4.20)∼(4.23) are only true when the following
row orthogonality is satisfied:
SST = IN (4.25)
TTT = IN . (4.26)
Generally, rows of rectangular eigenmode (or proper orthogonal mode) are
not orthogonal. Therefore, the projection and transformation procedure is
vaild and different from the projection using the common basis.
4.2.4 Matrix and Mode Interpolation
To realize the interpolation of the reduced matrices, projection and transfor-
mation processes are executed. Also, the FOM can be directly projected to
the basis obtained by multiplying the projection and transformation matri-
ces. Substituting Eq. (4.24) into Eq. (4.1) yields
Mi¨ui(t) + Ci
˙ui(t) + Kiui(t) = fi(t), i = 1, 2, · · · , Ns, (4.27)
where
Mi = QTi MiQi, Ci = QT
i CiQi,
Ki = QTi KiQi, fi(t) = QT
i fi(t).(4.28)
The reduced matrices except fi belong to the manifold of symmetric positive
definite matrix. Without reprojecting the reduced matrices to the common
69
basis S as in Ref. [48], the interpolation can be executed as follows:
M(µ) =
Ns∑i=1
Wi(µ)Mi
C(µ) =
Ns∑i=1
Wi(µ)Ci (4.29)
K(µ) =
Ns∑i=1
Wi(µ)Ki.
To recover the ROM to the full system, the projection Qi should also be
interpolated such that
Q(µ) =
Ns∑i=1
Wi(µ)Qi. (4.30)
Unlike the reduced system matrices, however, the projection Q(µ) belongs
to the Grassmann manifold. Thus, if the same interpolation function is used
as in Eq. (4.29), the admissible range of the interpolation would be smaller
than that of the system matrices. In fact, the interpolation of the system ma-
trices could be regarded as the interpolation of eigenvalues in curved surface.
But the projection matrix represents eigenvectors which are more strict to
interpolate than the eigenvalues.
4.3 Moving Least Square Method for Recovery
For the accurate interpolation of the projection matrix, the weights involved
global-POD method was suggested in Ref. [48].
X = [ T1W1(µ) T2W2(µ) · · · TNsWNs(µ) ]
= ΦΛ− 12AT . (4.31)
70
The drawback of the weights involved global-POD method is a repeated com-
putation of POD to obtain the common basis. In fact, the ROM at a new
operating point should be constructed fast in the on-line stage; this is the
key point of the parametric ROM. If tedious and complicated computations
should be executed in the on-line stage, there is no need to construct ROM
using the advanced technique. Therefore, in the present study, an interpola-
tion scheme using moving least square (MLS) approximation is devised for
the efficient computation maintaining the accuracy.
4.3.1 Moveing Least Square Method
The main difference of the moving least square with the (weighted) least
square method lies in the fact that the coefficients of the approximation
function depend on the input variable. The function of least square method
can be written as
qj(µ) = cTj b(µ), j = 1, 2, · · · , R, (4.32)
where qj ∈ RN×1 is a column vector of the projection Q(µ), b(µ) ∈ Rk×1
is the polynomial basis vector, cj ∈ Rk×N is unknown coefficients to be
determined and k denotes the number of polynomial basis. Details can be
expressed as
Q(µ) = [ q1(µ) q2(µ) · · · qR(µ) ] (4.33)
b(µ) = [ b1(µ) b2(µ) · · · bk(µ) ]T . (4.34)
The projection matrices and their columns which are computed at the sample
points µi are represented as follows:
Qi = [ q1i q2i · · · qRi ], i = 1, 2, · · · , Ns, (4.35)
71
The corresponding vector of unknown coefficients is obtained by minimizing
the error functional which is presented in Ref. [57] such that
cj =
[ Ns∑i=1
b(µi)bT (µi)
]−1[ Ns∑i=1
b(µi)qji
]. (4.36)
The moving least square method, however, the unknown coefficients are
the function of the input variable µ. Thus, Eq. (4.32) is changed to the
following:
qj(µ) = cTj (µ)b(µ), j = 1, 2, · · · , R, (4.37)
where
cj(µ) =
[ Ns∑i=1
w(µ,µi)b(µi)bT (µi)
]−1[ Ns∑i=1
w(µ,µi)b(µi)qji
]. (4.38)
In Eq. (4.38), w(µ,µi) is a weighting function. Among the various candidates,
the following inverse form is employed:
w(µ,µi) =1
(µ− µi)2 + ε2
, (4.39)
where the small scalar ε is included to avoid a singularity at µ = µi.
4.3.2 Computation at On-line Stage
From Eqs. (4.29), (4.33) and (4.37), the ROM of a new operating point µ∗ /∈
SNpµ is constructed as follows:
M(µ∗)¨u(t;µ∗) + C(µ∗) ˙u(t;µ∗) + K(µ∗)u(t;µ∗) = f(t;µ∗), (4.40)
where
f(t;µ∗) = QT (µ∗)f(t). (4.41)
Also, the displacement recovered to the full system is expressed as,
u(t;µ∗) = Q(µ∗)u(t;µ∗). (4.42)
72
4.4 Numerical Results
Example 1. Cantilever beam: plane stress element
To verify the developed method which consist of projection-transformation-
recovery processes, simple plane stress problem of cantilever beam structure
is considered. The geometry and material properties are the same to the
cantilever beam shown in Chap. 3, Fig. 3.1. The parameter is the thickness
of each subdomains. Total 160 bilinear plane stress elements were used as
shown in Fig. 4.1. The left end is fixed and at top of the right end, impluse
load is applied. To examine the frequency responses of the structure, two
different positions were selected, one for the center and the other for the
tip. The frequency range is [1:1:5,000] Hz, and the 21 number of frequency
snapshots was taken in [0:250:5,000] Hz.
First of all, we checked the performance of developed method by changing
the sampling range from 8 to 24 mm. In the sampling process, total 16 cases
are solved to construct the ROM: two upper and lower sample points of 4
subdomains (= 24). Then the 16 ROMs constructed are interpolated at a new
thickness (operating point) by using Lagrange interpolation function. Since
there are two samples for each thicknesses, a linear interpolation is executed.
After calculating frequency response of the interpolated ROM, the recovery
process is performed by using moving least square method. In this cantilever
beam example, the thickness of new operating point is set to be 20 mm. And
we changed the sampling range, and details are presented in table 4.1.
In Fig. 4.2, the frequency responses of the FOM, the ROM and the one
without coordinate transformation are compared. The frequency response of
73
the ROM shows a good agreement with that of the FOM. If the coordinates
transformation in Eq. (4.15) is not executed the interpolation is not vaild.
The same is true for the position 2 shown in Fig. 4.3. On the other hand, if
the sampling range increases, the slight difference occurs at 1,500∼2,000 Hz
and 4,000∼5,000 Hz as shown in Figs. 4.4 and 4.5. This gap becomes larger
as the sampling range increase to 24 mm (Figs. 4.6 and 4.7).
To examine the performance of the interpolated ROM, the errors occured
by the sampling range are studied. In Fig. 4.8, the average relative error of
eigenvalues from the first to eighth are presented according to the change
of sampling range. The thickness of new operating point is 20 mm and the
error increases nonlinearly. In addition, random thicknesses were set as new
operating points. Total 1,000 samples were investigated and min, max and
average errors are plotted in Fig. 4.9. Thus the interpolated ROM error can
be specified by setting the sampling range.
Example 2. Cantilever plate
The second example in Fig. 4.10 is cantilever plate with 4 subdomains. The
freqnecy response was examined in the range of [0:0.25:500] Hz, also the 201
snapshots of frequency response were taken in [0:2.5:500] Hz. Different from
the previous membrane example, the stiffness matrix is a function of linear
and cubic polynomial such that
K(µ) = µKshear + µ3Kbending. (4.43)
Then the addmissible range of sampling could be narrowed compared to the
membrane element which depends on linear function only. Thus we fixed
74
the sampling range and changed the polynomial order to interpolate. The
thickness of new operating point is fixed to be 7 mm. Fig. 4.11 shows the
results of linear interpolation. As shown in Eq. (4.43), the linear sampling
yields poor results in overall frequencies. For the quadratic sampling shown in
4.12, althogh the frequency response becomes much better compared to linear
case, still some misalignments are exists. The cubic sampling in Fig. 4.13
shows a good agreement to the FOM. For the Lagrange interpolation using
cubic polynomial, most frequency responses except the range around 150 Hz
shows poor results. As mentioned before, the interpolation of eigenvector is
more strict than that of eigenvalues. However, the developed method using
the moving least square accuratly predicts the frequency responses. Fig. 4.9
presents the comparison of the relative errors of eigenvalues. As expected,
the increase of polynomial order results in the decrease of the errors of ROM.
However, for the 8th mode, the error does not decrease but slightly increases.
Because, the 8th mode is rotational mode in xy plane. The prediction of the
rotational degree of freedom comes from the performance of the element used.
Therefore, in this plate bending problem, this problem is not significant. In
fact, the error is less than 0.5 %, which is quite small compared to the other
modes.
Example 3. Wing box model
The last example is wing box model with 8 subdomains showin in Fig. 4.15.
The geometry and material properties are the same to the one in Chap. 3.
The frequency range of the FOM is [0:0.05:100] Hz, and the 101 snapshots
were taken in [0:1:100] Hz. Fig. 4.16 shows the frequency responses of the
75
interpolated ROM with the 4.5 mm overall thicknesses. The linear sampling
was executed at the 4 mm for lower bound and 5 mm for upper bound, which
results in 256 ROM constructions totally. Whereas the Lagrange interpola-
tion was executed to interpolate the ROMs, the moving least square method
was employed for the recovery process. The two responses show good agree-
ments for all degrees of freedom. However, at 80∼100 Hz range, the some
misalignment observed since the sampling is linear. If we use cubic inter-
polation, more accurate results can be obtained, which also requires much
more computations in off-line stage. This issues will be presented in Chap. 5.
In Fig. 4.17, the average of relative eigenvalue errors of 1,000 random sam-
ples were computed within 5∼10 mm and corresponding probability density
function (PDF) was obtained. Thus we can predict the error of interpolated
ROM based on the PDF. If the sampling range increase, the relative error
also increase as shown in example 1.
76
Table 4.1 Cases of sampling ranges
Nominal: 20 (mm) Lower Upper Range
Case 1 16 24 8
Case 2 12 28 16
Case 3 8 32 24
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.05
0
0.05
0.1Sub. 4
µ4
Sub. 2µ
2
Sub. 3µ
3
Sub. 1µ
1Impulse
(1) Frequency Response (2)
Figure 4.1 Cantilever beam with 4 subdomains of plane stress membrane
element under tip impluse load.
77
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
FOM PROM: Range 8mm PROM w/o coord. trans.
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
Figure 4.2 Comparison of frequency responses of the FOM and ROMs at
position (1): 8 (mm) sampling range.
78
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
FOM PROM: Range 8mm PROM w/o coord. trans.
Figure 4.3 Comparison of frequency responses of the FOM and ROMs at
position (2): 8 (mm) sampling range.
79
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
FOM PROM: Range 16mm
Figure 4.4 Comparison of frequency responses of the FOM and the ROM at
position (1): 16 (mm) sampling range.
80
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
FOM PROM: Range 16mm
Figure 4.5 Comparison of frequency responses of the FOM and the ROM at
position (2): 16 (mm) sampling range.
81
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
FOM PROM: Range 24mm
Figure 4.6 Comparison of frequency responses of the FOM and the ROM at
position (1): 24 (mm) sampling range.
82
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
Frequency (Hz)
Magnitude (
dB
)
(a) x
0 1000 2000 3000 4000 5000−250
−200
−150
−100
−50
0
Frequency (Hz)
Magnitude (
dB
)
(b) y
FOM PROM: Range 24mm
Figure 4.7 Comparison of frequency responses of the FOM and the ROM at
position (2): 24 (mm) sampling range.
83
0 0.005 0.01 0.015 0.02 0.025 0.030
0.5
1
1.5
2
2.5
3
3.5
Sampling Range (m)
Err
or
(%)
Average Relative Error
Figure 4.8 Average relative error of 1st∼8th eigenvalues according to the
sampling range.
0 0.005 0.01 0.015 0.02 0.0250
0.5
1
1.5
2
2.5
3
3.5
4
Sampling Range (m)
Err
or
(%)
Min, Max error
Average error
Figure 4.9 Average relative error of 1st∼8th eigenvalues for random thickness
input according to the sampling range.
84
Table 4.2 Upper and lower bound of each interpolation cases
Nominal: 7e-3 (m) (1) (2) (3) (4) # of samples
Linear - 5.5e-3 8.5e-3 - 16
Quadratic - 5.5e-3 8.5e-3 10.0e-3 81
Cubic 4.0e-3 5.5e-3 8.5e-3 10.0e-3 256
00.1
0.20.3
0.40.5
0.60.7
0.8 0
0.2
0.4
−0.1
0
0.1
z
xy
Sub. 1µ
1
Clamped
Sub. 2µ
2 Sub. 3µ
3 Sub. 4µ
4
Impulse
FrequencyResponse
Figure 4.10 Cantilever plate with 4 subdomains of under tip impluse load.
85
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(a) z
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(b) φx
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(c) φy
FOM PROM: 1st order
Figure 4.11 Comparison of frequency responses of the FOM and the ROM:
linear sampling.
86
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) z
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) φx
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) φy
FOM PROM: 2nd order
Figure 4.12 Comparison of frequency responses of the FOM and the ROM:
quadratic sampling.
87
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) z
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) φx
0 50 100 150 200 250 300 350 400 450 500−150
−100
−50
0
50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) φy
FOM PROM: 3rd Poly. Lagrange Poly: 3rd
Figure 4.13 Comparison of frequency responses of the FOM, the ROM and
Lagrange interpolation: cubic sampling.
88
0 100 200 300 400 5000
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
Err
or
(%)
1st Poly.
2nd Poly.
3rd Poly.
Figure 4.14 Comparison of the relative errors of eigenvalues using different
polynomial order.
24
68
1012
14
0
5
10
15
−4
−3
−2
−1
0
1
2
3
4
Rib 4
Impulse
Skin 8
Skin 7
Skin 1
Skin 2
Spar 3
Rib 6
Spar 5
FrequencyResponse
Clamped
Figure 4.15 Wing box model with 8 subdomains under tip impluse load.
89
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)M
ag
nitu
de
(d
B)
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
0 20 40 60 80 100−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
FOM PROM
Figure 4.16 Comparison of frequency responses of the FOM and the ROM.
90
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
Random Samples
Rela
tive E
rror
(%)
0 0.5 1 1.5 20
0.02
0.04
0.06
0.08
0.1
Relative Error (%)
Eigenvalue Error
PDF of Relative Error
min.max.meanmedianstd.
0.091.950.670.620.29
Figure 4.17 Average of relative eigenvalue errors and probability density func-
tion of 1,000 random samples.
91
Chapter 5
Parametric Reduced Order
Model with Substructuring
Scheme
5.1 Introduction
As mentioned in the introduction of the dissertation, the design optimiza-
tion of large-scale structure is a challenging problem when the structure has
many number of design variables and is under dynamic loading conditions. In
this regard, the matrix interpolation combined with the moving least square
method is suitable for the relatively small number of design variables. As the
number of design variable increase, the number of sampling points increases
exponentially. Thus, in this chaper, the parametric ROM combined with sub-
structuring scheme is developed. Starting from the conventional component
mode synthesis, the parametric ROM of each substructure is constructed.
By using the proposed parametric ROM, the number of sampling points in-
creases algebraically. At the same time, the on-line and off-line procedure
93
presented in Chap. 4 is maintained, which also results in fast computations
in the on-line stage.
5.2 Review of Component Mode Synthesis
5.2.1 Equation of Motion for a Substructure
The equation of motion of a typical undamped component is written as
Ms(µ)us +Ks(µ)us = f s, s = 1, 2, · · · , Nd, (5.1)
where Nd denotes the number of subdomains, and the superscript s denotes
the index of a subdomain. In Eq. (5.1), we omitted the damping for the
convenience and it can be included in the computation procedure without
any difficulty. The parameter of whole system is defined as follows:
µ =[µ1 µ2 · · · µNp
]T. (5.2)
In general, the number of parameters Nd does not need to be equal to the
number of subdomains. However, to widen the admissible range of the in-
terpolation of the parameter, we assume that a subdomain has one design
variable. One subdomain can surely have more than two deign variables, but
the range of interpolation could be narrowed. Therefore, for a structural de-
sign optimization which requires a wide variation of the design varible, ‘1
design variable for 1 subdomain’ is proper. Thus Eq. (5.1) is changed as
Ms(µs)us +Ks(µs)us = f s, s = 1, 2, · · · , Nd. (5.3)
94
By partitioning the subdomain into interior and boundary, the equation of
motion is divided as,[Ms
ii(µs) Ms
ib(µs)
Msbi(µ
s) Msbb(µ
s)
] [usi
usb
]+
[Ks
ii(µs) Ks
ib(µs)
Ksbi(µ
s) Ksbb(µ
s)
] [usi
usb
]=
[0i
f sb
].
(5.4)
The subscripts i and b denote the interior and boundary degrees of freedom
of the subdomain. In this chapter, to avoid a confusion from the index of
vectors and parameters, the Greek alphabet is used for counting the number
of vectors and parameters instead of Latin index.
5.2.2 Fixed Interface Normal Modes
In the upper equation of Eq. (5.4), the following equation for the interior
part is derived by restraining all boundary degrees of freedom.
Msii(µ
s)usi +Ks
ii(µs)us
i = 0. (5.5)
Solving Eq. (5.5) gives a generalized eigenvalue problem such that
Ksii(µ
s)ϕsi,α(µ
s) = λsi,αM
sii(µ
s)ϕsi,α(µ
s), α = 1, 2, · · · , N si , (5.6)
where N si is the number of interior degrees of freedom of the subdomain s.
By selecting N sp (<N s
i ) modes as the ascending order of the magnitude of
the eigenvalues, the interior degrees of freedom are transformed to the gener-
alized coordinates. Corresponding displacement and transformation matrix
are expressed as
usi = Φs
ip(µs)us
p, (5.7)
where
Φsip(µ
s) =[ϕsi,1(µ
s) ϕsi,2(µ
s) · · · ϕsi,Ns
p(µs)
]. (5.8)
95
Φsip ∈ RNs
i ×Nsp and the subscript p is used to represent the transformation
from the interior degrees of freedom to the generalized coordinates.
5.2.3 Constraint Modes
In Eq. (5.4), the constraint modes can be calculated by assigning a unit
displacement to each degrees of freedom of the boundary of the subdomain.
Then the static deformation of a structure becomes the constraint modes. By
eliminating the inertia from Eq. (5.4), the static equations are derived such
that [Ks
ii(µs) Ks
ib(µs)
Ksbi(µ
s) Ksbb(µ
s)
] [Ψs
ib
Ibb
]=
[0i
f sb
]. (5.9)
The upper equation of Eq. (5.9) yields the constraint modes as follows:
Ψsib(µ
s) = −Ksii(µ
s)−1Ksib(µ
s), (5.10)
where Ψsib ∈ RNs
i ×Nsb and N s
b denotes the number of boundary degrees of
freedom of the subdomain s. The total degrees of freedom of a subdomain is
N s = N si +N s
b . From the constraint mode, the interior degrees of freedom is
represented in terms of the boundary degrees of freedom such that
usi = Ψs
ib(µs)us
b. (5.11)
5.2.4 Craig-Bampton Transformation Matrix
From Eqs. (5.7) and (5.11), the total displacement of a subdomain is trans-
formed as
us =
[usi
usb
]= Ts(µs)
[usp
usb
], (5.12)
where
Ts(µs) =
[Φs
ip(µs) Ψs
ib(µs)
0bp Ibb
]. (5.13)
96
By multiplying the transformation matrix to the system matrices in Eq. (5.3),
the reduced matrices are obtained as follows:
Ms(µs) = Ts(µs)TMs(µs)Ts(µs)
=
[Ipp Ms
pb(µs)
Msbp(µ
s) Msbb(µ
s)
], (5.14)
where
Mspb = Φs
piMsiiΨ
spb +Φs
pbMsib (5.15)
Msbp = (Ms
pb)T (5.16)
Msbb = Ψs
biMsiiΨ
sib +Ms
biΨsib +Ψs
biMsib +Ms
bb, (5.17)
in which the check (·) notation represents the transformation using the con-
straint mode.
The fixed interface modes are mass orthogonal, so the part of reduced
mass matrix related to the degrees of freedom of generalized coordinates are
identity. Successively, the reduced stiffness can be obtained as
Ks(µs) = Ts(µs)TKs(µs)Ts(µs)
=
[Λs
pp(µs) 0pb
0bp Ksbb(µ
s)
], (5.18)
where Λspp represents the corresponding eigenvalues of fixed interface normal
modes, and
Ksbb = Ψs
biKsib +Ks
bb. (5.19)
97
5.3 Interpolation of Transformation Matrix
5.3.1 Projection and Transformation of Fixed Interface Nor-
mal Modes
The fixed interface normal mode belongs to the Grassmann manifold since
it is the eigenvector of the interior of a subdomain. Thus, the interpolation
should be combined with coordinate transformation as shown in Chap. 4.
The common basis S is determined by the global-POD method such that
Xs =[Φs
ip,1 Φsip,2 · · · Φs
ip,Ns
]= Φ
sΛ− 1
2AT , (5.20)
where
Φsip,α = Φs
ip(µsα), α = 1, 2, · · · , Ns. (5.21)
By choosing N sp vectors from Φ
smatrix, the common basis of a subdomain
Ss is obtained as follows:
Ss =[ϕ1 ϕ2 · · · ϕNs
p
]. (5.22)
Corresponding coordinates transformation matrix for the interpolation is ex-
pressed as
Rsα = [Ss]TΦs
ip,α, α = 1, 2, · · · , Ns. (5.23)
Consequently, the displacement of interior degrees of freedom is transformed
to the generalized coordinates in which the interpolation is also possible.
usi,α = Φs
ip,αRsαu
sp,α
= Φsip,α([S
s]TΦsip,α)
−1usp,α
= Qsip,αu
sp,α. (5.24)
98
The notations used are the same to that presented in Chap. 4.
5.3.2 Interpolation of Constraint Modes and ROM of Subdo-
main
The constraint mode is the solution of the static problem as shown in Eq.
(5.9). In other words, the characteristics of the constraint mode is different
from the that of the fixed interface normal mode. The basis of displacement
variable projected by the constraint mode is the basis of interface degrees
of freedom, which indicates that all variables transformed by the constraint
modes are in the same coordinates systems. Therefore, another coordinates
transformation is not needed for the constraint modes. The constraint mode
at the parameter µα is expressed as
Ψsib,α = −[Ks
ii,α]−1Ks
ib,α. (5.25)
By combining Eqs. (5.24) and (5.25), the transformation matrix of a subdo-
main is constructed as
Tsα =
[Qs
ip,α Ψsib,α
0bp Ibb
]. (5.26)
Successively, multipling the transformation matrix to the system matrices of
a substructure yields the reduced mass and stiffness matrices at the operating
points: the reduced mass matrix is
Msα = [Ts
α]TMs
αTsα
=
[Ispp Ms
pb,α
Msbp,α Ms
bb,α
], (5.27)
99
and the reduced stiffness matrix is written as
Ksα = [Ts
α]TKs
αTsα
=
[Λ
spp,α 0pb
0bp Ksbb,α
]. (5.28)
Here, the tilde (·) notation is used to express that the matrix contains both
generalized coordinates system and Cartesian coordinates system.
After constructing the reduced model of a subdomain at the sampling
point of the parameter µsα, the interpolation is executed to approximate the
parameterized ROM. The interpolation of the reduced stiffness and mass
matrix are expressed as follows:
Ms(µs) =
Ns∑α=1
Wα(µs)Ms
α (5.29)
Ks(µs) =
Ns∑α=1
Wα(µs)Ks
α. (5.30)
Note that the all computation is performed in subdomain level.
5.4 Parametric Component Mode Synthesis Method
5.4.1 Synthesis of Component Modes
The ROMs of each substructures are synthesized based on the Craig-Bamption
component mode synthesis method. As shown in Fig. (*), the substructures
1 and 2 are sharing the boundary 3. Based on the interface compatibility, the
boundary vectors become
u1b = u2
b = u3b . (5.31)
100
The displacement vectors of two substructures and the component coupling
matrix are expressed as follows:u1p
u1b
u2p
u2b
=
I 0 0
0 0 I
0 I 0
0 0 I
u1
p
u2p
u3b
. (5.32)
The component stiffness and mass matrices are directly assembled using the
component coupling matrix. The assembled mass and stiffness are written as
M(µ) =
I1pp 0 M1
pb(µ1)
0 I2pp M2pb(µ
2)
M1bp(µ
1) M2bp(µ
2) M1bb(µ
1) + M2bb(µ
2)
(5.33)
K(µ) =
Λ
1pp(µ
1) 0 0
0 Λ2pp(µ
2) 0
0 0 K1bb(µ
1) + K2bb(µ
2)
, (5.34)
where
µ =[µ1 µ2
]T. (5.35)
Eqs. (5.33) and (5.34) represent the parameterized ROM that takes almost
few time to construct in the on-line stage: only matrix summation and direct
assembly processes are executed.
5.4.2 Reduction of Interface Degrees of Freedom
In Eqs. (5.33) and (5.34), the mass and stiffness matrices of interface degrees
of freedom are expressed as
M3bb(µ) = M1
bb(µ1) + M2
bb(µ2) (5.36)
K3bb(µ) = K1
bb(µ1) + K2
bb(µ2). (5.37)
101
The size of the M3bb and K3
bb is determined by the number of interface degrees
of freedom which depends on the finite element mesh. Therefore, for the large-
scale structures which have millions, or tens of millions degrees of freedom,
the degrees of freedom of interfaces cannot be ignored. To reduce the size of
interface degrees of freedom, Castanier, et al. (Ref. [26]) developed the modal
reduction technique based on the characteristic of constraint modes. In this
research, the interface degrees of freedom is also reduced by using the modal
reduction method. The eigen-problem of interface part is written as
K3bb(µ)ϕ
3α(µ) = λ3
αM3bb(µ)ϕ
3α(µ), α = 1, 2, · · · , N3
b . (5.38)
By selecting the N3q (<N3
b ) eigenvectors according to the low eigenvalues,
the transformation matrix of the interface degrees of freedom is obtained as
follows:
Φ3bq =
[ϕ31 ϕ3
2 · · · ϕ3N3
q
], (5.39)
where the subscript q represents the transformation from the interface degrees
of freedom to the generalized coordinates. Thus the transformation of the
displacement vector is written as
u1p
u2p
u3b
= Φ3(µ)
u1p
u2p
u3p
, (5.40)
where
Φ3(µ) =
I1pp 0 0
0 I2pp 0
0 0 Φ3bq(µ)
. (5.41)
102
Also, the mass and stiffness matrices in Eqs. (5.36) and (5.37) are reduced
as follows:
M3qq(µ) = [Φ3
bq(µ)]TM3
bb(µ)Φ3bq(µ) (5.42)
K3qq(µ) = [Φ3
bq(µ)]T K3
bb(µ)Φ3bq(µ). (5.43)
Finally, the mass and stiffness matrices which have the reduced the interface
degrees of freedom are express as
M(µ) = [Φ3(µ)]TM(µ)Φ3(µ) (5.44)
K(µ) = [Φ3(µ)]T K(µ)Φ3(µ). (5.45)
Note that the reduction of the interface degrees of freedom is executed with-
out any parameterizaion, or interpolation process. However, since the size of
the interface degrees of freedom is much small compared to that of the full
system, the eigenvectors in Eq. (5.38) is computed very fast.
5.4.3 Recovery Process to Full System
As presented in Chap. 4, the mass and stiffness are relatively easy to interpo-
late compared to the projection matrix. For the CB (Craig-Bamption) trans-
formation matrix, the fixed interface normal mode is interpolated by using
the moving least square method, whereas the constraint mode is interpolated
based on the Lagrange interpolation function. Thus, the CB transformation
matrix in Eq. (5.13) is interpolated separately such that
Qsik(µ
s) = fMLS(µs;µs
α,Qsik,α), α = 1, 2, · · · , Ns (5.46)
Ψsib(µ
s) =
Ns∑α=1
Wα(µs)Ψs
ib,α, (5.47)
103
where fMLS is a simplified expression for the moving least square process as
shown in Eqs. (4.32)∼(4.39). After finishing the recovery process in the sub-
domain level, the assembly process for the full projection matrix is required,
which is similar to the assembly of the mass and stiffness matrices. The rela-
tions between the displacement vectors in different coordinates systems are
expressed as u1i
u2i
u3b
= T(µ)Φ3(µ)
u1p
u2p
u3p
, (5.48)
where
T(µ) =
Q1ip(µ
1) 0 Ψ1ib(µ
1)
0 Q2ip(µ
2) Ψ2ib(µ
2)
0 0 Ibb
. (5.49)
Consequently, the multiplication of two transformation matrix is the full
projection matrix which is written as
P(µ) = T(µ)Φ3(µ)
=
Q1ip(µ
1) 0 Ψ1ib(µ
1)Φ3bq(µ)
0 Q2ip(µ
2) Ψ2ib(µ
2)Φ3bq(µ)
0 0 Φ3bq(µ)
. (5.50)
By multiplying P(µ) to the displacement obtained by solving the reduced
system, the displacement of full system is recovered.
5.5 Numerical Results
Example 1. Rib-skin-spar structure
First of all, the rib-skin-spare structure with 8 subdomains was investigated
(Fig. 5.1). The material properties and are the same presented in the previous
104
chapters, but the size and the number of subdomains are changed to deal with
more number of design variables. In the component mode synthesis (CMS)
process, the number of fixed interface normal mode is set to 10, and the
number of total interface mode is 12 for both Craig-Bampton CMS and the
developed method.
Figs. 5.2 and 5.2 represent the mean and min-max eigenvalue errors of
1,000 randomly generated thickness samples. As the order of interpolation
increases, the relative error decreas to the values of Criag-Bamption com-
ponent mode synthesis. The thicknesses of upper and lower bound of each
cases were presented in table 5.1. Note that the number of samples increase
algebraically different from in table 4.1. By using only cubic interpolation,
the eigenvalue error is converge to that of CB CMS. Fig. 5.4 was obtained by
changing the range of sample thicknesses maintaining the order of interpola-
tion as cubic. If the range is narrow, almost exact agreement can be seen in
(a). Even for (c) which has wide variation of the thicknesses, the error is still
below 0.1 %.
For the structural optimization, dynamic load is applied to the tip and
the loading profile is shown in Fig. 5.5. Time interval is [0:0.002:10] sec with
5,000 time steps. The objective function is the weight of the structure and
other conditions are presented in table 5.2. In Fig. 5.6, the optimal thick-
nesses of each method are presented: the FOM, modal reduction using 12
eigenmodes which solves the eigen-problem by using the Lanczos algorithm
(more specifically, implicitly restarted Arnoldi method) and the parametric
ROM (matrix interpolation with MLS recovery) with substructuring scheme.
The thicknesses of two ROMs show good agreements to that of the FOM.
105
The histories of the objective functions are shown in Fig. 5.7. In face, the
objective functions of each models were already converged within 10∼15 iter-
ations. After that, very slight decreases can be observed. By comparing each
computational time, the efficiency of present method can be seen, which is 7
% and 33 % compared to the FOM and the modal reduction.
Example 2. Wing box structure
The second example is wing box model (Fig. 5.9) with ‘85’ design vari-
ables and refined mesh configuration compared to the one shown in previous
chapers. Total degrees of freedom is 72,438 with 12,560 elements and de-
tails of condition for optimization were presented in talbe 5.3. Specific design
variables were shown in Figs. 5.10 and 5.11.
The optimal thicknesses of spar and upper skin were given in Fig. 5.12 and
that of rib and lower skin were shown 5.13. The optimal thicknesses of the
present ROM shows better agreement than that of the modal reduction with
the FOM. The histories of the objective functions (Fig. 5.14) are similar each
other, which indicates the robustness of the present parametric ROM. Nev-
ertheless, the present method is very efficient, even compared to the modal
reduction as shown in Fig. 5.15; 2.1 % and 20.2 % efficient compared to the
FOM and the modal reduction.
Example 3. High-fidelity F1 front wing
For the last example, high-fidelity front wing structure of fomular-1 machine
was considered shown in Fig. 5.16 from Ref. [58]. From that geometry model,
the author generated mesh configuration by using Hypermesh software [59].
106
Total degrees of freedom of the full model is 749,082. However, to optimize
the size of structural components, the symmetric boundary condition is ap-
plied, which results in the half model with 375,588 degrees of freedom in Fig.
5.17. The number of design variables is ‘96’. The material properties were
assumed to be carbon composites: elastic modulus E = 70e9 Pa, Poisson’s
ratio ν = 0.25, density ρ = 1600 Kg/m3 and the initial thicknesses are 8
mm for overall domains. The external loading profile is denoted in Fig. 5.18.
In Fig. 5.17, vertical down forces are applied to the red points, negative x-
directional forces are applied to cyan points and negative y-directional forces
are applied to yellow point. In fact, all nodes are under external forces which
are occured by aerodynamics, and to do so, the CFD-based fluid simula-
tion should be performed first. However, we simplified the external forces
to the nodal loads to avoid a complicated CFD computations since we can
still observe the efficiency of the present parametric ROM under the simpli-
fied loading conditions. The complicated loadings also can be applied to the
present method without any difficulties.
Fig. 5.19 and 5.20 show the optimal thicknesses of two different ROMs.
Both methods have good agreements with each other. There are slight dif-
ferences for the optimal thicknesses of subdomain 42∼96. However the ten-
dencies are almost the same. In Fig. 5.21, the histories of objective function
were compared, which is also similar to each other. Compared to the result
of the wing box problem in Fig. 5.14, the result of present F1 model con-
verged faster than that of the wing box model. Because, whereas the F1
model has distributed loading point, the wing box model has only 1 loading
point. Therefore, the design variables of the F1 are more sensitive than that
107
of the wing box model. The computation time of the parametric ROM is
22.7% compared to the modal reduction method. Considering the efficiency
increases as the degrees of freedom of FOM increase, estimated efficiency
compared to the FOM could be lower than 2.1 %.
108
Table 5.1 Upper and lower bound of each interpolation cases
(m) (1) (2) (3) (4) # of samples
Linear - 10e-3 15e-3 - 16
Quadratic - 10e-3 15e-3 20e-3 24
Cubic 5e-3 10e-3 15e-3 20e-3 32
Table 5.2 Problem condition of rib-skin-spare structure
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz) |σmax| (Pa)196.83 5e-3 20e-3 10e-3 12 3e9
00.5
11.5
22.5
3 00.5
00.10.2
85
4
7
2
1
Dynamic Load f(t)Clamped 3
6
Figure 5.1 Rib-skin-spar structure with 8 subdomains under tip dynamic
load.
109
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
102
(a)
Mode Number
Rela
tive E
rror
(%)
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
102
(b)
Mode Number
Rela
tive E
rror
(%)
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
102
(c)
Mode Number
Rela
tive E
rror
(%)
PROM w Substr. CB CMS
Figure 5.2 Comparison of mean of eigenvalue errors for 1,000 random sam-
ples: The Craig-Bampton component mode systhesis and the (a) linear, (b)
quadratic and (c) cubic interpolated ROM.
110
0 2 4 6 8 10 12 14 16 18 2010
−8
10−6
10−4
10−2
100
102
(a)
Mode Number
Re
lative
Err
or
(%)
0 2 4 6 8 10 12 14 16 18 2010
−8
10−6
10−4
10−2
100
102
(b)
Mode Number
Re
lative
Err
or
(%)
0 2 4 6 8 10 12 14 16 18 2010
−8
10−6
10−4
10−2
100
102
(c)
Mode Number
Re
lative
Err
or
(%)
PROM w Substr. CB CMS
Figure 5.3 Comparison of min-max of eigenvalue errors for 1,000 random
samples: The Craig-Bampton component mode systhesis and the (a) linear,
(b) quadratic and (c) cubic interpolated ROM.
111
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
(a)
Mode Number
Re
lative
Err
or
(%)
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
(b)
Mode Number
Re
lative
Err
or
(%)
0 2 4 6 8 10 12 14 16 18 2010
−6
10−4
10−2
100
(c)
Mode Number
Re
lative
Err
or
(%)
PROM w Substr. CB CMS
Figure 5.4 Comparison of mean of eigenvalue errors for 1,000 random samples:
The Craig-Bampton component mode systhesis and the cubic interpolated
ROM by changing sampling range: (a) 10∼15 (mm), (b) 10∼20 (mm) and
(c) 5∼20 (mm),
112
0 2 4 6 8 100
200
400
600
800
1000
1200
time(sec)
F (
N)
F(t)
Figure 5.5 Dynamic step loading profile.
0 1 2 3 4 5 6 7 8 90.005
0.01
0.015
0.02
Design Variables
Thic
kness (
m)
FOM
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.6 Comparison of optimal thicknesses of the FOM and ROMs.
113
0 5 10 15 20 25 300.75
0.8
0.85
0.9
0.95
1
Iterations
|Obj. function|
FOM
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.7 Comparison of objective function histories.
1 2 30
500
1000
1500
2000
1891.79
404.64
133.83
tim
e (
sec)
1. FOM
2. Modal Reduc. (Lanczos)
3. PROM w Substr.
Figure 5.8 Comparison of computation time of the FOM and ROMs.
114
Table 5.3 Problem condition of wing box model
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz)
8320.5 5e-3 20e-3 5e-3 3.5
24
68
1012
14
0
5
10
15
−0.5
0
0.5
Dynamicload f(t)
Clamped
Figure 5.9 Wing box model with 85 subdomains under tip dynamic load.
115
2
4
6
8
10
12
14
0
5
10
15
−0.5
0
0.5
1
2
26
6
3
7
27
30
11
8
28
31
4
12
34
16
9
13
32
29
35
5
17
38
14
33
36
10
18
21
39
22
15
37
19
41
23
45
20
40
44
24
43
25
42
Figure 5.10 Design variables of rib and spar.
116
2
4
6
8
10
12
14
0
5
10
15
−4
−2
0
2
4
46
47
49
48
54
55
51
50
56
57
62
63
53
59
52
58
64
65
61
60
70
66
67
71
72
73
69
68
74
75
79
78
76
77
81
80
83
82
85
84
Figure 5.11 Design variables of upper and lower skins.
117
0 5 10 15 20 250
0.005
0.01
0.015
0.02
(a) Spar
Design Variables
Thic
kness (
m)
0 5 10 15 200
0.005
0.01
0.015
0.02
(b) Upper Skin
Design Variables
Thic
kness (
m)
FOM
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.12 Comparison of optimal thicknesses of the FOM and ROMs of
spar and upper skin
118
0 5 10 15 200
0.005
0.01
0.015
0.02
(c) Rib
Design Variables
Thic
kness (
m)
0 5 10 15 200
0.005
0.01
0.015
0.02
(d) Lower Skin
Design Variables
Thic
kness (
m)
FOM
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.13 Comparison of optimal thicknesses of the FOM and ROMs of rib
and lower skin
119
0 5 10 15 20 25 300.4
0.5
0.6
0.7
0.8
0.9
1
Iterations
|Obj. function|
FOM
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.14 Comparison of objective function histories.
1 2 30
50
100
150
200
250
194.98
20.23
4.08
Tim
e (
h)
1. FOM
2. Modal Reduc. (Lanczos)
3. PROM w Substr.
Figure 5.15 Comparison of computation time of the FOM and ROMs.
120
Table 5.4 Problem condition of high-fidelity F1 front wing model
Weight (Kg) µlb (m) µub (m) |umax| (m) ω1 (Hz)
19.24 5e-3 10e-3 5e-3 24
Figure 5.16 Configureation of high-fidelity F1 front wing structure.
121
Figure 5.17 Half of F1 front wing with 96 subdomains under multiple dynamic
loads
122
0 2 4 6 8 100
10
20
30
40
50
60
70
80
time (sec)
F (
N)
z−dir. (red points)
x−dir. (cyan points)
y−dir. (yellow point)
Figure 5.18 Dynamic loads applied to each points
123
0 5 10 15 200
0.002
0.004
0.006
0.008
0.01
(a) Sub. # 1~20
Design Variables
Thic
kness (
m)
20 25 30 35 400
0.002
0.004
0.006
0.008
0.01
(b) Sub. # 21~41
Design Variables
Thic
kness (
m)
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.19 Comparison of optimal thicknesses of the FOM and ROMs of
subdomain # 1∼41
124
45 50 55 60 65 700
0.002
0.004
0.006
0.008
0.01
(c) Sub. # 42~69
Design Variables
Thic
kness (
m)
70 75 80 85 90 950
0.002
0.004
0.006
0.008
0.01
(d) Sub. # 70~96
Design Variables
Thic
kness (
m)
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.20 Comparison of optimal thicknesses of the FOM and ROMs of
subdomain # 42∼96
125
0 2 4 6 8 10 12 14 16 180.7
0.75
0.8
0.85
0.9
0.95
1
1.05
Iterations
|Obj. function|
Modal Reduc. (Lanczos)
PROM w Substr.
Figure 5.21 Comparison of objective function histories.
1 20
10
20
30
40
50
60
70
80
60.48
13.70
Tim
e (
h)
1. Modal Reduc. (Lanczos)
2. PROM w Substr.
Figure 5.22 Comparison of computation time of the FOM and ROMs.
126
Chapter 6
Stochastic Dynamic Analysis
with Uncertain Parameters
6.1 Introduction
All structural systems contain uncertain parameters which are occured by ini-
tial dificiencies, or during the manufacturing process. Due to these uncertain-
ties, the structures should be designed with proper safety factors. However,
for the efficient analysis and design of various structural system, probabilistic
approach is prefered by identifying the probability distribution of uncertain
parameters [60, 61]. In particular, if the structure has a defect, dynamic
response shows large deviations from the response of the intact structure
[62, 63]. Therefore we need to consider the dynamic uncertain parameters to
analyze and design the structural systems.
Among the various methods for the reliability analysis of engineering sys-
tems, Monte Carlo simulation technique is a simple and powerful tool. The
technique requires only a basic knowledge of probability and statistics. There-
127
fore, most of the probabilistic analysis execute the Monte Carlo simulation
to verify the new methods developed. In addition, if there is no exact solu-
tion of the problem dealing with the uncertainty, the Monte Carlo simulation
becomes a reference of the problem. In the present research, the parametric
ROM developed in the previous chapter 4 and 5, is employed to the dynamic
analysis of the uncertain structures. Without using the advanced techniques
for the uncertainty analysis, the parametric ROM combined with the Monte
Carlo simulation offers efficient and accurate solutions to various complicated
engineering systems.
6.2 Dynamic Analysis of Uncertain Structures
In the finite element framwork, the mass, damping and stiffness matrices
have the uncertain parameters. Thus, the equation of motion with uncertain
parameter have the same expression to Eqs. (3.16) and (4.1).
M(µ)u(t) +C(µ)u(t) +K(µ)u(t) = f(t). (6.1)
The uncertain properties of structural component have the upper and lower
bounds. Therefore, the beta distribution is proper to assum the probabitily
distribution of the engineering structure. Probability density function of stan-
dard beta distribution [60] is wrtten as
fX(x) =1
B(q, r)xq−1(1− x)r−1, 0 ≤ x ≤ 1,
= 0, elsewhere, (6.2)
where the beta function is
B(q, r) =
∫ 1
0xq−1(1− x)r−1dx. (6.3)
128
Corresponding cumulative distrubution function is expressed as follows:
FX(x) =1
B(q, r)
∫ x
0xq−1(1− x)r−1dx, 0 ≤ x ≤ 1. (6.4)
6.3 Monte Carlo Simulation
The computation procedure of the Monte Carlo simulation using the para-
metric reduced order model is stated as follows:
Step 1. Generating the set of random numbers from the computer.
Step 2. Substituting generated random numbers into the inverse cumulative
distrubution function of beta distribution.
Step 3. Obtaining the set of uncertain parameters from step 2.
Step 4. Substituting the set of uncertain parameters into the parametric
ROM.
Step 5. Performing dynamic analysis.
In this dissertation, there are two different parametric ROM as shown in
Chap. 4 and 5. Since the basic computational procedure is the same to that of
design optimization problem, the dynamic analyses of uncertain parameters
were performed without applying any modification to the parametric ROMs
in Chap. 4 and 5. The characteristics of both parametric ROMs are different.
Therefore, comparing the performance of both methods is also possible.
129
6.4 Numerical Results
Example 1. Cantilever plate
First of all, rectangular cantilever plate was investigated as shown in Fig. 6.1
with 4 uncertain parameters. The geometries and material properties are the
same to that of the cantilever plate in Chap. 4. The thickness of the plate is 7
mm and the uncertain parameter is the elastic modulus of each subdomains.
The norminal modulus is set to be 73.1e9 Pa and 10 % variation is assumed
in this example.
From the beta distribution, 1,000 random vectors of elastic modulus were
generated as shown in Fig. 6.2. The frequency response analysis was executed
in the range of [0:0.1:100] Hz. For the ROM, 21 snapshots of frequency re-
sponse were taken in [0:5:100] Hz. The coefficients of beta distribution are
simply assumed to be q = r = 3. In this example, the ROM in Chap 4. is
considered. Since the substructuring method is efficient with many numbers
of parameters.
Figs. 6.3, 6.4 and 6.5 represent average mean, maximum and minimum
responses of the FOM and the linearly interpolated ROM. For the mean
values, there is no difference between the FOM and the ROM. For the max
and min, overall distributions of the ROM have a good agreement to that of
the FOM except the peak points. In fact, the peak value itself does not have
significant meaning. And the ROM can express the migration of eigenvalues
according to the random samples. The computation times were presented in
table 6.1. The ROM took only 3.99 % compared to the FOM, which indicates
the robustness of the parametric ROM.
130
Example 2. Rib-skin-spar structure
The second example is the rib-skin-spar structure presented in Fig. 6.6. The
uncertain parameters are the elastic modulus of each subdomain, totally 8
parameters. The geometris and material properties are the same to that of the
structure in Chap. 5. The thickness, however, is constant value 15 mm. The
norminal elastic modulus is 72e9 Pa and the variation of uncertain parameter
is also assumed as 10 %. The frequency range is [0:0.2:100] Hz and also 20
snapshots were taken in [0:5:100] Hz. THe coefficients of beta distribution
are the same to that of example 1.
Both of the parametric ROMs in Chap. 4 and 5 were investigated. The
method in Chap. 4 is presented as ‘PROM: linear interp.’ and the one in
Chap. 5 is ‘PROM w Substr’. The ‘PROM w Substr’ method interpolated
by using cubic polynomial. As shown in table 6.1, the construction time of
both ROM is very different. For the first method, totally 28 = 256 numbers
of full model computations were performed to construct the ROM. However,
the second method have 32 numbers of subdomain computations. Therefore,
almost no time takes to construct the ROM. By contrast, the analysis time
of the first method is faster than that of the second method. Due to the
subdomain synthesis and the interface reduction, the analysis time increase
compared to the first method. The total time of the first and the second
method are 2.1 % and 0.9 % compared to the full model, which indicates
that both methods are very efficient.
In Fig. 6.7, average mean response looks similar with each other. However,
the degree of freedom of y-directional rotation of the method in Chap. 4 shows
131
slight differences. This becomes more serious for the maximun and minimum
responses as shown in Figs. 6.8 and 6.9. Therefore, we can conclude that the
performance of the parametric ROM with substructuring scheme is better
than that of the ROM in Chap. 4, especially when the structure has large
numbers of parameters.
132
Table 6.1 Computation time of the FOM and ROMs of cantilever plate
Time (sec) ROM FOM
Construction 1.40
Analysis 23.47
Total 24.87 622.90
00.1
0.20.3
0.40.5
0.60.7
0.8 0
0.2
0.4
−0.1
0
0.1
z
xy
Clamped
Impulse
FrequencyResponse
Sub. 1E
1 Sub. 2E
2 Sub. 3E
3 Sub. 4E
4
Figure 6.1 Cantilever plate with 4 uncertain parameters.
133
66 68 70 72 74 76 78 800
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
−3
E (GPa)
(a)
E1
66 68 70 72 74 76 78 800
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
−3
E (GPa)
(b)
E2
66 68 70 72 74 76 78 800
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
−3
E (GPa)
(c)
E3
66 68 70 72 74 76 78 800
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
−3
E (GPa)
(d)
E4
Figure 6.2 PDF of elatic modulus of each substructures.
134
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(a) z
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(b) φx
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(c) φy
FOM PROM: 1st order
Figure 6.3 Average mean frequency responses of the FOM and the ROM
135
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(a) z
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(b) φx
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(c) φy
FOM PROM: 1st order
Figure 6.4 Average maximum frequency responses of the FOM and the ROM
136
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(a) z
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(b) φx
0 20 40 60 80 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(c) φy
FOM PROM: 1st order
Figure 6.5 Average minimum frequency responses of the FOM and the ROM
137
Table 6.2 Computation time of the FOM and ROMs of rib-skin-spar structure
Time (sec) ROM w/o Substr. ROM w. Substr. FOM
Construction 614.6 2.3
Analysis 107.6 287.3
Total 722.2 289.3 33859.90
00.5
11.5
22.5
3 00.5
00.10.2
85
4
72
1
Clamped 3
6 Impulse
Frequency
Response
Figure 6.6 Rib-skin-spar structure with 8 uncertain parameters
138
0 10 20 30 40 50 60 70 80 90 100−200
−150
−100
−50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) z
0 10 20 30 40 50 60 70 80 90 100−200
−150
−100
−50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) φx
0 10 20 30 40 50 60 70 80 90 100−200
−150
−100
−50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) φy
FOM PROM w Substr. PROM: linear interp.
Figure 6.7 Average mean frequency responses of the FOM and ROMs
139
0 10 20 30 40 50 60 70 80 90 100−150
−100
−50
0
50
Frequency (Hz)
Magn
itude (
dB
)
(a) z
0 10 20 30 40 50 60 70 80 90 100−150
−100
−50
0
50
Frequency (Hz)
Magnitude (
dB
)
(b) φx
0 10 20 30 40 50 60 70 80 90 100−200
−100
0
100
Frequency (Hz)
Magnitude (
dB
)
(c) φy
FOM PROM w Substr. PROM: linear interp.
Figure 6.8 Average maximum frequency responses of the FOM and ROMs
140
0 10 20 30 40 50 60 70 80 90 100−250
−200
−150
−100
−50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(a) z
0 10 20 30 40 50 60 70 80 90 100−250
−200
−150
−100
−50
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(b) φx
0 10 20 30 40 50 60 70 80 90 100−300
−250
−200
−150
−100
Frequency (Hz)
Ma
gn
itu
de
(d
B)
(c) φy
FOM PROM w Substr. PROM: linear interp.
Figure 6.9 Average minimum frequency responses of the FOM and ROMs
141
Chapter 7
Conclusions
In this dissertation, parametric reduced order models for comprising the dy-
namic characteristics and the change of parameters were developed within the
finite element framwork. Based on the characteristics of the proper orthog-
onal decomposition, enhanced reduced basis method was developed to treat
multiple loading conditions. By calculating the mode of multiple loads, the
efficiency of constructing reduced basis was increased. In addition, efficient
design optimization strategy for dynamic response was suggested using re-
duced equivalent static load calculated by using the global proper orthogonal
decomposition.
To obain real-time, on-line parametric reduced order model, projection-
transformation-recovery procedure was developed by employing the global
proper orthogonal method for computing transformation matrix and by using
the moving least square approximation with recovery process. The accuracy
and robustness of the proposed method were decomstrated by the frequency
response analysis of various examples. Whereas the eigenvalues are interpo-
143
lated well, the eigenvectors consisting the basis of reduced space cannot be
accuratly interpolated by using conventional Lagrange interpolation function.
Therefore, moving least square method was employed to calculate accurate
projection matrix. Parametric studies provided the addmissible variation of
parameters to employ the proposed method within certain error bounds.
The computation on the off-line stage was also reduced by introducing
substructuring scheme to the parametric reduced order model. For the struc-
tural design optimization, computational time consumed in approximating
the global response according to the change of parameters is also significant.
The substructuring scheme facilitated to calculate the global response in a
subdomain level. Thus, both on-line and off-line calculations were reduced,
which results in the fast computation of large-scale structures contains many
design variables up to hundreds level. Considering the computations were ex-
ecuted in the desktop PC, extreme-scale problems could be solved by using
super computing system.
Based on the analysis and design optimization of high-fidelity model for
dynamic response performed in this dissertation, it is hoped that the present
optimization strategy and parametric reduced order model can be further
employed to other structural applications.
144
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국문 요약
본 논문에서는 동적 특성과 파라미터의 변화를 동시에 고려하는 유한요
소 기반의 파라메트릭 축소 기법을 개발하였다. 기존의 축소 기법은 동적
특성이나 파라미터 변화를 개별적으로 축소하기 때문에, 동적 시스템에서
파라미터가 변하면 축소 시스템을 재구성해야 하며, 이 경우 계산 효율성이
낮아지는 문제가 발생한다. 이를 해결하기 위해서 적합 직교 분해 기반의
파라메트릭 축소 기법을 제안하였다.
먼저, 적합 직교 분해의 특성에 기반하여, 하중의 축소를 통해 다중 하중
문제를 효율적으로 접근할 수 있는 축소 기저법을 제안하였다. 다중 하중
문제의 경우 기존의 방법으로는 하중이 변할 때 축소 기저를 재구축해야
하지만, 본 연구에서는 전역 적합 직교 분해 기법을 이용하여 축소 기저를
재구축 하지 않는 방법을 고안하였다. 이 방법은 등가정하중을 이용한 최적
설계 기법과 결합하여, 동적 시스템의 최적 설계 시, 계산 효율성이 증가
함을 확인하였다. 또한, 파라미터 변화를 실시간으로 고려하기 위해서, 투
영-좌표변환-보간-회복으로 구성되는 보간 기반의 축소 기법을 제안하였다.
이 방법은 이동 최소 자승법과 결합하여, 기존의 라그랑지 보간법에 비해
월등히 정확하게 축소 시스템을 전체 시스템으로 회복시킬 수 있는 것을
확인하였다.
한편, 파라메트릭 축소 모델을 대형 동적 시스템의 최적 설계 문제에 적
용하기위해서는,민감도계산을비롯한최적설계반복연산시간의축소가
필요할 뿐만 아니라, 반복 연산 이전에 근사화된 전역 반응면의 탐색 시간
또한 줄어들어야 한다. 따라서 기존의 파라메트릭 축소 모델과 부구조화
기법을 결합하여, 전역 반응면의 근사 시간과 최적 설계 반복 연산 시간을
동시에 줄임으로써 대형 동적 시스템의 최적 설계 효율을 극대화 하였다.
154
수십만 단위의 자유도와 백단위의 설계변수를 가지는 동적 구조 시스템의
예제를 통해서 제안한 방법의 정확성과 효율성을 검증하였다. 또한, 제안한
기법을 이용해서 파라미터의 불확실성에 의해 야기되는 동적 문제의 확률
분포 해석을 수행하였다.
본 논문에서 개발하고 검증한 파라메트릭 축소 모델은 다양한 대형 시스
템의 동적 구조 해석 및 설계에 널리 활용할 수 있을 것이라 생각한다.
주요어: 파라메트릭 축소 모델, 적합 직교 분해, 동적 구조 최적 설계, 이동
최소 자승법, 파라메트릭 부구조화 기법
학번: 2008-20778
155