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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




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 :




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




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


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


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


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


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


ROMs at position (2): 8 (mm) sampling range. . . . 79


the ROM at position (1): 16 (mm) sampling range. . 80







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


the ROM: linear sampling. . . . . . . . . . . . . . . . 86


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


the ROM. . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 4.17 Average of relative eigenvalue errors and probability

density function of 1,000 random samples. . . . . . . 91


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


ROMs. . . . . . . . . . . . . . . . . . . . . . . . . . . 113

x


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


ROMs of spar and upper skin . . . . . . . . . . . . . 118


ROMs of rib and lower skin . . . . . . . . . . . . . . 119



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


ROMs of subdomain # 1∼41 . . . . . . . . . . . . . 124


ROMs of subdomain # 42∼96 . . . . . . . . . . . . 125



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)ü(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 üi(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

Miüi(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∗)ü(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


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


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


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




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




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




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.



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

(%)


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

(%)


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


PROM w Substr.


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


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


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


PROM w Substr.


1 2 30

50

100

150

200

250

194.98

20.23

4.08

Tim

e (

h)

1. FOM


3. PROM w Substr.


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)


PROM w Substr.


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)


PROM w Substr.


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|


PROM w Substr.


1 20

10

20

30

40

50

60

70

80

60.48

13.70

Tim

e (

h)


2. PROM w Substr.


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


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


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


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


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

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