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공학석사 학위논문

FEA Based Weight Optimization of

Semi-Submersible Structure

Considering Buckling and Yield Strength

좌굴 및 항복강도를 고려한 반잠수식 구조물의

유한요소기반 중량 최적화

2017 년 2 월

서울대학교 대학원

조선해양공학과

김 재 동

FEA Based Weight Optimization of

Semi-Submersible Structure

Considering Buckling and Yield Strength

지도 교수 장 범 선

이 논문을 공학석사 학위논문으로 제출함

2017 년 2 월

서울대학교 대학원

조선해양공학과

김 재 동

김재동의 공학석사 학위논문을 인준함

2017 년 2 월

위 원 장 신 종 계 (인)

부위원장 장 범 선 (인)

위 원 노 명 일 (인)

i

Abstract

FEA Based Weight Optimization of

Semi-Submersible Structure

Considering Buckling and Yield

Strength

Kim JaeDong

Department of Naval Architecture and Ocean

Engineering

The Graduate School

Seoul National University

Semi-submersible structure is widely used to drill and produce

oil and gas in the ocean. This structure is very sensitive to weight

increase in terms of payload and stability. As the detailed design

progresses, design changes such as lower shape change due to the

ii

increase in weight of the upper equipment often lead to delivery delay.

Therefore, it is important to secure the weight margin by optimizing

the substructure at the initial design phase.

There have been many researches on the weight optimization of

ship structures in terms of strength. Since the strength assessment

procedures of ship are relatively simple, it is possible to repeat the

strength assessment process and optimize its weight in terms of

strength. However, semi-submersible structure is complicated

compared to ship, therefore it requires more complex procedures

than conventional methods of ship’s strength assessment. This

strength assessment process of semi-submersible is not currently

fully automated. Thus, the offshore structure including semi-

submersible structure has been optimized not considering the

strength but considering the motion and stability.

In order to perform the optimization considering the strength of

the structure, it is necessary to automate the strength assessment

process. For this reason, necessary processes related to stress such

as stress scanning, mapping, and combination is automated as a

preliminary study of optimization. The process for strength

assessment such as generating panel and assigning panel information

iii

is automated.

Based on the developed automatic strength check system, weight

optimization considering the strength of semi-submersible structure

is carried out. The weight is set as an objective function, and the

buckling and yield strength were set as constraints. The plate

thickness and beam sections are set as design variables. In order to

reduce the number of design variables and to exclude solutions of

unrealistic beam sections, design variables are discretized. In

addition, the steepest descent method is selected as the optimization

algorithm to minimize the analysis time. The number of FE analysis

is reduced by using the equation for analytically estimating the stress

change.

In case of optimizing the entire model at once, there are too many

design variables to deal with. In the optimization problem, when the

number of design variables increases, the optimum point may not be

reached exactly. To solve this problem, the design variables of each

optimization step are reduced by independently performing

optimization for each plane.

For the column model of the semi-submersible structure, the

optimization using the method presented in this paper is performed

iv

to confirm the convergence of the optimal solution.

Keywords : Semi-submersible, Steepest descent method, Weight

optimization, Buckling strength, Yield strength

Student Number : 2015-21162

v

Contents

1. Introduction ........................................................ 13

1.1. Research Background and Objective ........... 13

1.2. Previous Research ........................................ 15

2. Automatic Global and Local Strength Check System

(AGLOS) 17

2.1. General strength assessment procedure of

semi-submersible structures ...................... 17

2.2. Outline of the system .................................... 19

2.3. Global stress scanning .................................. 21

2.4. Global stress mapping to local model .......... 24

2.5. Stress combination ........................................ 27

2.6. Strength assessment ..................................... 29

3. Formulation of optimization problem .................. 31

3.1. Objective function and Constraints .............. 33

3.2. Design variables ............................................ 35

3.3. Steepest descent method ............................. 36

3.4. Stress estimation........................................... 38

3.4.1. Stress estimation method ............................. 38

3.4.2. Verification of stress estimation .................. 40

3.5. Discretization of design variables ................ 47

3.5.1. Discretization method 1 ................................ 48

3.5.2. Discretization method 2 ................................ 48

vi

3.5.3. Verification for discretization method ......... 50

4. Effect of thickness change on the other plane ... 56

5. Case Studies for semi-column model................ 61

5.1. Model Description ......................................... 61

5.1.1. Applied loads ................................................. 62

5.1.2. Boundary condition........................................ 63

5.1.3. Design variables ............................................ 64

5.1.4. Constraints ..................................................... 66

5.2. Performing optimization at once .................. 68

5.3. Performing optimization plane by plane ...... 71

5.4. Convergence of solutions ............................. 82

5.4.1. Convergence of plane by plane optimization 82

5.4.2. Convergence of plane separation cases ...... 84

5.5. Efficiency of stress estimation method ....... 88

6. Conclusion .......................................................... 91

7. Appendix ............................................................ 94

7.1. Stiffener Library ............................................ 94

8. Reference ......................................................... 102

vii

List of Figures

FIG. 1 SEMI-SUBMERSIBLE STRUCTURE ....................................... 14

FIG. 2 OPTIMIZATION OF SHIP STRUCTURE .................................... 16

FIG. 3 GENERAL STRENGTH ANALYSIS PROCEDURE OF SEMI-

SUBMERSIBLE STRUCTURE ...................................................... 19

FIG. 4 OUTLINE OF AGLOS .......................................................... 21

FIG. 5 GLOBAL HYDRODYNAMIC RESPONSES .................................. 23

FIG. 6 GLOBAL STATIC RESPONSES ............................................... 23

FIG. 7 VERIFICATION OF SCANNING MODULE ................................. 24

FIG. 8 STRESS MAPPING METHOD .................................................. 26

FIG. 9 VERIFICATION OF MAPPING MODULE ................................... 27

FIG. 10 STRESS COMBINATION FOR BUCKLING STRENGTH CHECK OF

PONTOON ................................................................................ 29

FIG. 11 AUTOMATIC PANEL GENERATION ..................................... 30

FIG. 12 ASSIGNMENT OF PANEL INFORMATION .............................. 31

FIG. 13 OVERALL PROCESS OF OPTIMIZATION ............................... 33

FIG. 14 PLATE THICKNESS VARIABLES ......................................... 35

FIG. 15 BEAM SECTION VARIABLES ............................................... 36

FIG. 16 STEEPEST DESCENT METHOD ........................................... 38

FIG. 17 PANEL SECTIONS WITHOUT BEAM OFFSET ........................ 40

FIG. 18 APPLIED LOAD FOR VERIFICATION MODEL ......................... 41

FIG. 19 APPLIED PRESSURE FOR VERIFICATION MODEL .................. 42

FIG. 20 A SIMPLE PLATE MODEL FOR VERIFICATION ...................... 42

viii

FIG. 21 PANEL AVERAGE AXIAL STRESS (PLATE THICKNESS

CHANGE) ................................................................................ 43

FIG. 22 PANEL AVERAGE TRANSVERSE STRESS (PLATE THICKNESS

CHANGE) ................................................................................ 43

FIG. 23 PANEL AVERAGE SHEAR STRESS (PLATE THICKNESS

CHANGE) ................................................................................ 44

FIG. 24 CHANGE OF LONG EDGE (SPAN) STIFFENER SECTIONS ...... 45

FIG. 25 PANEL AVERAGE AXIAL STRESS (BEAM SECTION CHANGE) 45

FIG. 26 CHANGE OF SHORT EDGE (SPAN) STIFFENER SECTIONS .... 46

FIG. 27 PANEL AVERAGE TRANSVERSE STRESS (BEAM SECTION

CHANGE) ................................................................................ 46

FIG. 28 DISCRETIZATION METHOD 1 ............................................. 49

FIG. 29 DISCRETIZATION METHOD 2 ............................................. 50

FIG. 30 APPLIED LOADS AND BOUNDARY CONDITIONS .................... 51

FIG. 31 DESIGN VARIABLES OF THE MODEL ................................... 52

FIG. 32 OPTIMIZATION RESULT FOR DISCRETIZATION METHOD 1 ... 53

FIG. 33 OPTIMIZATION RESULT FOR DISCRETIZATION METHOD 2 ... 54

FIG. 34 DESCRIPTION OF PLANES FOR EFFECT OF THICKNESS CHANGE

.............................................................................................. 57

FIG. 35 RATE OF STRESS CHANGE OF FOUR PLANES (30%

DECREASED) ........................................................................... 59

FIG. 36 RATE OF STRESS CHANGE OF FOUR PLANES (30%

INCREASED) ............................................................................ 60

FIG. 37 AVERAGE RATE OF STRESS CHANGE ................................. 61

FIG. 38 SEMI-COLUMN MODEL ..................................................... 62

ix

FIG. 39 APPLIED LOADS ................................................................ 63

FIG. 40 BOUNDARY CONDITIONS ................................................... 64

FIG. 41 DESIGN VARIABLES OF FIRST TWO PLANES ....................... 65

FIG. 42 DESIGN VARIABLES OF SECOND PLANE .............................. 65

FIG. 43 DESIGN VARIABLES OF THIRD PLANE ................................ 66

FIG. 44 OPTIMIZATION RESULT .................................................... 69

FIG. 45 BUCKLING STRENGTH USAGE FACTOR ............................. 69

FIG. 46 YIELD STRENGTH USAGE FACTOR OF PLATE ................... 70

FIG. 47 YIELD STRENGTH USAGE FACTOR OF STIFFENER ............. 70

FIG. 48 PLANE BY PLANE OPTIMIZATION ....................................... 72

FIG. 49 CHANGE OF PLATE THICKNESS (SET1) ............................ 73

FIG. 50 BUCKLING STRENGTH USAGE FACTORS (SET1) ................ 74

FIG. 51 YIELD STRENGTH USAGE FACTORS (SET1) ...................... 74

FIG. 52 OPTIMIZATION RESULT OF SET1 ...................................... 75

FIG. 53 CHANGE OF PLATE THICKNESS (SET2) ............................ 76

FIG. 54 BUCKLING AND YIELD STRENGTH USAGE FACTORS (SET2) 77

FIG. 55 OPTIMIZATION RESULT OF SET2 ...................................... 77

FIG. 56 CHANGE OF PLATE THICKNESS (SET3) ............................ 78

FIG. 57 BUCKLING AND YIELD STRENGTH USAGE FACTORS (SET3) 79

FIG. 58 OPTIMIZATION RESULT OF SET3 ...................................... 79

FIG. 59 RESULT OF PLANE BY PLANE OPTIMIZATION ..................... 81

FIG. 60 THREE STARTING POINTS ................................................. 83

FIG. 61 RESULT FROM THREE DIFFERENT STARTING POINTS ......... 84

FIG. 62 THREE PLANE SEPARATION CASES ................................... 85

FIG. 63 RESULT OF THREE DIFFERENT CASES ............................... 86

x

FIG. 64 CONVERGENCE OF CASE1 ................................................. 87

FIG. 65 CONVERGENCE OF CASE2 ................................................. 87

FIG. 66 CONVERGENCE OF CASE3 ................................................. 88

FIG. 67 APPLING TWO STRESS UPDATING METHOD ........................ 89

FIG. 68 DESIGN VARIABLES OF TEST MODEL ................................. 90

FIG. 69 OPTIMIZATION RESULT APPLYING TWO METHOD ............... 91

xi

List of Tables

TABLE 1 LOAD FACTORS FOR ULS (DNV-OS-C101) ............... 28

TABLE 2 DESCRIPTION OF VERIFICATION MODEL ........................... 41

TABLE 3 STIFFENER LIBRARY USED FOR VERIFICATION ................. 47

TABLE 4 CHANGE OF DESIGN VARIABLES OF STARTING POINT1

(METHOD1) ........................................................................... 53

TABLE 5 CHANGE OF DESIGN VARIABLES OF STARTING POINT2

(METHOD1) ........................................................................... 53

TABLE 6 CHANGE OF DESIGN VARIABLES OF STARTING POINT3 FOR

(METHOD1) ........................................................................... 54

TABLE 7 CHANGE OF DESIGN VARIABLES OF STARTING POINT1

(METHOD2) ........................................................................... 54

TABLE 8 CHANGE OF DESIGN VARIABLES OF STARTING POINT2

(METHOD2) ........................................................................... 55

TABLE 9 CHANGE OF DESIGN VARIABLES OF STARTING POINT3

(METHOD2) ........................................................................... 55

TABLE 10 CHANGE OF BEAM SECTIONS (SET1) ............................ 73

TABLE 11 CHANGE OF BEAM SECTIONS (SET2) ............................ 76

TABLE 12 CHANGE OF BEAM SECTIONS (SET3) ............................ 78

TABLE 13 FINAL WEIGHT AND CONSUMED TIME ............................ 91

TABLE 14 FLAT BAR SECTION LIBRARY ........................................ 94

TABLE 15 L SECTION LIBRARY ..................................................... 95

TABLE 16 T SECTION LIBRARY ..................................................... 98

xii

13

1. Introduction

1.1. Research Background and Objective

Semi-submersible structure is widely used to drill and produce

oil and gas in the ocean. This structure is very sensitive to weight

increase in terms of payload and stability. Unlike ship structure, basic

engineering design of semi-submersible structure is carried out by

overseas engineering companies. There are many parts to reduce

weight of structure. As the detailed design progresses, design

changes such as lower shape change due to the increase in weight of

the upper equipment often lead to delivery delay. Therefore, it is

important to secure the weight margin by optimizing the substructure

at the initial design phase.

The semi-submersible structure consists of an upper deck with

drilling equipment, a rig floor with a drilling tower, a crane holding

various pipes and risers, two pontoons and several columns

supporting the pontoons. The pontoons are immersed in water to

increase the stability of the whole structure, and the several columns

connecting the pontoons and upper deck reduce the area of water

plane and improve the performance by making the structure less

14

affected by waves and currents. However, as the structure is

complex, the required structural analysis method is also complicated.

Because global direct and local analysis are performed in different

models, it is necessary to automate the strength assessment process

for weight optimization.

Fig. 1 Semi-submersible structure

There are two objectives for this study. The first objective is to

develop an automated system for assessing the strength of semi-

submersible structures. The stress scanning, mapping, and

combination processes are required for the strength assessment, so

it is aimed to automate the entire strength assessment process. The

second objective is to develop a scantling optimization system for

stiffened plates of semi-submersible structure based on the

15

developed strength assessment automation system. The developed

system can decrease the unnecessary weight of the structure to

reduce the cost and prevent the critical situation such as the change

of the substructure design due to the modification of the topside

structure.

1.2. Previous Research

There have been many studies on weight optimization of the ships.

Nobukawa (1996) has optimized the material cost and the welding

cost by using the design variables of the ship’s plate thickness, the

stiffener flange, and the web dimension and longitudinal space

through the genetic algorithm. Yu et al. (2010) performed weight

optimization with ship’s plate thickness and seam layout as design

variables.

16

Fig. 2 Optimization of ship structure

Ma et al. (2013) performed optimization of the stiffened plate

structure. Optimization of the weight and the manufacturing cost of

stiffened plate as the objective function was performed and the

simulated annealing was selected as the optimization algorithm. The

reason why such studies are possible is that the process of assessing

the structural strength of ships is relatively simple. In addition, there

is a difficulty in practical application of the method of assessing the

buckling strength which is generally inapplicable in actual design

phase.

There are also studies that have been optimized for semi-

submersible structures. Park et al. (2015) performed weight

optimization of the TLP structure using simulated annealing.

However, this study is an optimization considering the stability and

motion of TLP structure, not the structural strength.

17

In this study, a study on the optimization of the strength of a

semi-submersible structure has been carried out. Automatic

strength check system has been developed prior to optimization. To

apply actual engineering practice, scanning, mapping and combination

modules has been developed. Furthermore automatic 3D panel

generation module has been developed so that buckling check can be

performed automatically. By applying this system, optimization

system for semi-submersible structure has been developed. A

stress estimation method has been devised to reduce the number of

FE analysis. Design variables has been discretized to reduce the

number of design variables and avoid unrealistic results of shape of

stiffener sections. In addition plane by plane optimization has been

performed to reduce the number of design variables in each

optimization phase.

2. Automatic Global and Local Strength Check

System (AGLOS)

2.1. General strength assessment procedure of semi-

submersible structures

18

The general analysis procedure for semi-submersible structures

is as follows. Global analysis and local analysis are performed in an

independent model. The global stress is obtained by global response

of structure from global analysis. The local stress is also obtained by

local bending due to local pressure from local analysis. In order to

consider the conservative situation, global stress is obtained by

performing scanning to find the maximum stress among global

responses. Global analysis results and the local analysis results

should be in one single result file to carry out load combination.

Therefore, the scanned global stress must be transferred to the local

model. This is called mapping process. After mapping process, global

stress is combined with local stress in a different mesh to perform

the final strength assessment. Since these tasks are not fully

automated at the present, research has been carried out to automate

these tasks.

19

Fig. 3 General strength analysis procedure of semi-submersible

structure

2.2. Outline of the system

The overall outline of the system is as follows. The results of the

finite element analysis of the global and local models are read from

the import module and converted into a database. This database

contains not only geometric information such as Node, Element, and

Material properties of the finite element model, but also information

such as stress results and pressure of elements. When the database

of the global model is input to the scanning module, maximum stress

20

scanning is performed for all the phase angles of all design waves.

The result of the scanning process is used to create a new global

model database and map it to the local model. The mapping process

is necessary to combine global and local stresses. The process of

combining two analysis results in another model into the result of the

same model is a mapping process. Through the stress combination

module, global and local stresses can be combined and the combined

stress results can be used to perform buckling and yield strength

assessment. In the strength assessment module, panels are

automatically generated using the finite element model information,

and stress and pressure information are assigned to perform buckling

and yield strength assessment.

21

Fig. 4 Outline of AGLOS

2.3. Global stress scanning

A design wave analysis approach is used for maximum stress

analysis of semi-submersible structures. Different design waves are

used to assess the strength of different parts of the structure and the

strength of the structure must be satisfied for all design waves. For

this, the stress scanning process is required. The stress scanning is

the process of finding the design waves and phase angles that gives

the greatest stress to each part of the structure among all design

waves. This is required in order to reduce the number of

22

combinations of global load cases and local load cases. That is, the

scanning plays a role of merging the global load cases into one load

case even if it would be a conservative approach.

Fig. 6(a) shows the static load case with the maximum hogging

moment. Fig. 6(b) shows the static load case with the maximum

sagging moment. Scanning can be used to determine which load case

has the maximum stress value for each position. In general, the

semi-submersible structure is vulnerable to two static and six

hydrodynamic responses as shown in Fig. 5 and Fig. 6. Thus, when

the scanning of the design waves causing this response is performed,

the maximum global stress of the structure can be confirmed.

23

Fig. 5 Global hydrodynamic responses

Fig. 6 Global static responses

DNV-Xtract, a FE post-processor of DNV, is used to verify the

developed module. The Fig. 7 is a comparison of the results of

24

scanning using AGLOS and DNV-Xtract. It can be seen from the

figure that the two results are exactly the same. The accuracy of the

developed stress scanning module was verified.

Fig. 7 Verification of scanning module

2.4. Global stress mapping to local model

In order to combine the global stress and the local stress, both

loads must be in the same model. To do this, it is necessary to

transfer the stress of the global model to the stress of the local model.

Stress mapping is a process that applies the scanned global stress to

the local model.

The finite element analysis is a method of dividing the model into

25

many elements and the stress can be obtained only at the Gaussian

point of each element. Generally, the position of Gaussian points of

global model and local model are not coincident, because the mesh

size of the global model is larger than that of the local model.

Therefore, it is inevitable for some error to be involved in the stress

mapping process. In this study, a Gaussian point of the global model

closest to the each Gaussian point of the local model is found, and the

stress of this global Gaussian point is mapped to the corresponding

local Gaussian point. In this case, since the stress of the Gaussian

point in a completely different plane can be mapped, the process of

discriminating the same plane is also performed by comparing the

normal vectors of the elements.

26

Fig. 8 Stress mapping method

Unlike scanning, mapping is a function that not supported by

DNV-Xtract. Therefore, verification with the commercial software of

the mapping module is impossible. However, the accuracy of the

mapping can be confirmed by comparing the stress contour of the

global and local mesh. The Fig. 9 shows a comparison of the results

obtained by mapping the stress distribution from the coarse mesh to

the fine mesh. As can be seen from the figure, it can be seen that

despite the difference in mesh size, the two models show almost

similar stress contours. The accuracy of the developed stress

27

mapping module is verified.

Fig. 9 Verification of mapping module

2.5. Stress combination

The strength assessment of semi-submersible structures is

performed by linear superposition of global stress due to dynamic

response of global model and local stress due to local extreme load

of local model. Semi-submersible structures generally use the Load

and Resistance Factor Design (LRFD) method. The LRFD is a design

method of the structure that multiplies the load by load factor to

account for the uncertainty of the load and multiplies the yield stress

of the material by the resistance factor to account for the uncertainty

28

of the material strength. Therefore, when combining the loads, the

load factors are multiplied to perform the load combination. The load

factors required in DNV-OS-C101 for ULS assessment is shown in

Table 1.

Table 1 Load factors for ULS (DNV-OS-C101)

Combination of

design loads

Load categories

G Q E D

a) 1.3 1.3 0.7 1.0

b) 1.0 1.0 1.3 1.0

a) Operating Condition

b) Temporary Condition

Load categories are:

G=permanent load

Q=variable functional load

E=environmental load

D=deformation load

Fig. 10 shows an example of a stress component combination in the

pontoon section. Depending on the parts of the buckling strength

assessment, the stress components to be combined can vary.

Continuous, longitudinal structural elements may be evaluated

utilizing linear superposition of the individual responses as illustrated

in Fig. 10. When transverse stress components are taken directly

from the local structural model, the transverse stresses from the

global model may normally be neglected.

29

Fig. 10 Stress combination for buckling strength check of pontoon

2.6. Strength assessment

To check the buckling strength, a panel which is the minimum unit of

assessment is required. The finite element model is composed of

elements and nodes, and the element type is divided into shell and

beam type. Using this information, a panel is formed by finding shell

elements closed with beam element or perpendicular shell elements.

The panel search is enabled by a recursive panel search algorithm

30

which continuously adds a neighboring element on the same plane if

there isn’t any beam element or any perpendicular shell element in-

between until there is any more elements to be added.

Fig. 11 shows the finite element model in element unit and the

generated panel model generated by the panel auto generation

module.

Fig. 11 Automatic panel generation

The scanned and combined stresses are applied to the

automatically generated panels to perform buckling and yield

strength assessments. The other panel information for buckling

31

strength check, for example, panel size, panel thickness, three mean

stress components, stiffener profile, pressure information are also

automatically found. The applied buckling strength assessment

method is DNV-RP-C201-Part1 and DNV.PULS which are widely

used in actual engineering practice. The yield strength is assessed

by the minimum plate thickness and section modulus of the stiffener

required by DNV-OS-C101.

Fig. 12 Assignment of panel information

3. Formulation of optimization problem

32

In Chapter 2, a study on the development of automation systems

for semi-submersible structures has been introduced. Using this

automated system, a study on weight optimization of semi-

submersible structure has been conducted. This optimization study

focuses on practicality. The greatest challenges of structural

optimization is the time it takes for structural analysis. In this study,

the gradient of the design variables is obtained through the simple

stress estimation equation, and the time required for the analysis is

minimized. In addition, by making beam sections used in the actual

design as library and treating them as discrete variables, design

variables are reduced and unrealistic solutions are prevented. The

steepest descent method is chosen as the optimization algorithm. The

entire process of this optimization system is shown in Fig. 13.

33

Fig. 13 Overall process of optimization

3.1. Objective function and Constraints

The objective function of this optimization is the weight of the

structure. The weight of the structure is obtained by adding the

weight of the plate and the weight of the stiffener.

34

Buckling and yield strength are considered as constraints. To

check the buckling strength, non-linear ultimate strength check code

DNV-PULS and DNV-RPC201-Part1, which are frequently used in

practice, are applied. The yield strength is checked using the

minimum plate thickness and the minimum stiffener section modulus

of DNV-OSC101.

The constraints are added to the objective function as a penalty

function, making it an unconstrained optimization problem.

Quadratic exterior penalty function is used as penalty function.

Quadratic penalty functions always yield slightly infeasible solutions.

Since the optimal solution is always an infeasible solution when the

constraint violation criterion is set to 1.0, the violation criterion is

increased to 0.95 so that the optimal solution becomes the feasible

solution.

The objective function of the modified non-constrained

optimization is as follows:

n

i

m

j

ijkk gGrfr1 1

)]([)(),( XXX

35

]0,95.0)([)]([

Factors sageStrength U)(

parameterpenalty :

structure ofweight total:)(

fucntion objectivened unconstrai:),(

XX

X

X

X

iij

i

k

k

gMAXgG

g

r

f

r

3.2. Design variables

Design variables are the thickness of the plate and the section of

the stiffener. The section of the stiffener is selected by number from

the pre-defined stiffener library. By making beam sections used in

the actual design as library and treating them as discrete variables,

design variables are reduced and unrealistic solutions are prevented.

Fig. 14 Plate thickness variables

36

Fig. 15 Beam section variables

3.3. Steepest descent method

The steepest descent method has an advantage of being able to

approach the optimal solution quickly, but the obtained optimal

solution does not guarantee the global optimal solution. Also, it is not

the best algorithm for dealing with discretized design variables.

Despite these drawbacks, there are two reasons for using the

steepest descent method.

The first reason is that the scantling of the basic design is

reasonable. It is acceptable to assume that the local optimal solution

obtained from this initial design point is a global optimum of a

reasonable level. Because design variables, objective functions, and

constraints are monotonic relations. There is, of course, the

37

possibility that the global optimal solution may be at a different point

but, this is not considered to enhance the practical aspect.

The second reason is time constraints. The application of

probabilistic optimization techniques such as genetic algorithms or

simulated annealing is limited by finite element analysis time.

Optimization using these methods consumes too much time that

practical value is lost.

For these reasons, the steepest descent method was used.

The steepest descent method can be summarized by the following

steps and Fig. 16 :

Step 1. Start with an arbitrary initial point 𝐗1 (𝑖 = 1)

Step 2. Find the search direction 𝐒𝑖

𝐒𝑖 = −∇𝜋𝑖 = −𝜋(𝐗𝑖)

Step 3. Determine the optimal step length 𝜆𝑖∗ in the direction 𝐒𝑖.

This step is called line search.

𝐗𝑖+1 = 𝐗𝑖 + 𝜆𝑖∗𝐒𝑖 = 𝐗𝑖 − 𝜆𝑖

∗∇π𝑖

Step 4. Test the new point 𝐗𝑖+1 for optimality. If 𝐗𝑖+1 is optimum,

stop the process. Otherwise, go to Step 5.

Step 5. Set the new iteration number 𝑖=𝑖+1 and go to Step 2.

38

Fig. 16 Steepest descent method

3.4. Stress estimation

3.4.1. Stress estimation method

In the gradient calculation of the design variables, the finite

element analysis is excluded and the stress change is analytically

estimated. In assessment of the buckling strength of semi-

submersible structures, the offset of stiffeners is not applied in the

local model as well as in the global model. In the buckling check

specified in the class rules, the membrane stress is obtained by the

finite element analysis. The bending stress is calculated by the beam

theory. It is modeled without offset in order to avoid overlap with this.

39

Therefore, the cross-sectional shape of all stiffened plates is as

shown in Fig. 17. In addition, the stress of the plate used in assessing

the strength of semi-submersible structures is the membrane stress.

Thus, the stress change of the plate may be estimated as a value in

inverse proportion to the change of the cross-sectional area. The

nominal stress is calculated by taking into account the change of the

sectional area of the plate and the sectional area of the stiffener. The

normal stress estimation equation is as follows.

revrev

orioriori

rev

orioriest

ABt

ABt

Area

Area

)(

)(

The shear stress is calculated by considering the change of the

sectional area of the plate only. The shear stress estimation equation

is as follows.

rev

orioriest

t

t

40

Fig. 17 Panel sections without beam offset

3.4.2. Verification of stress estimation

In order to verify the validity of the stress estimation equation, a

simple plate model is used for the estimation equation. This model

consists of 9 panels, and the detail of the model is shown in the Table

2.

41

Table 2 Description of verification model

Length of each panel 3600mm

Space between stiffeners 1200mm

Initial plate thickness 10mm

Longitudinal Stiffener Tbar254x102x22

Longitudinal Girder Tbar525x150x12x25

Transverse Girder Tbar305x127x42

Longitudinal direction load Fx : 300kN/m, Fy : 100kN/m (Fig. 18)

Transverse direction load Fx : 100kN/m, Fy : 300kN/m (Fig. 18)

Lateral Pressure 10kPa at y = 0, 20kPa at y = 10.8m (Fig. 19)

Fig. 18 Applied load for verification model

42

Fig. 19 Applied pressure for verification model

The results of the stress change obtained by the finite element

analysis and the stress estimation equation are compared with each

other by changing the plate thickness of the following simple

verification model.

Fig. 20 A simple plate model for verification

43

Fig. 21 Panel average axial stress (plate thickness change)

Fig. 22 Panel average transverse stress (plate thickness change)

44

Fig. 23 Panel average shear stress (plate thickness change)

Fig. 21 is a graph comparing the panel axial stress estimated by using

the stress estimation equation with the panel average axial stress of

the FE analysis, changing the plate thickness of the center panel from

5 mm to 15 mm as shown in Fig. 20. Fig. 22 compares the panel

average transverse stress and Fig. 23 compares the panel average

shear stress in the same manner. From these graphs, it can be

confirmed that the difference between the two graphs is small.

Secondly, the plate thickness of the model is fixed, and the

stiffener sections are changed and compared in the same way as

thickness change.

45

Fig. 24 Change of long edge (span) stiffener sections

Fig. 25 Panel average axial stress (beam section change)

46

Fig. 26 Change of short edge (span) stiffener sections

Fig. 27 Panel average transverse stress (beam section change)

47

Table 3 Stiffener Library used for verification

Library

Number Stiffener Sections

1 Tbar254x102x22

2 Tbar254x146x43

3 Tbar305x127x42

4 Tbar254x102x22

5 Tbar285x100x12

6 Tbar300x90x11x16

7 Tbar305x127x42

8 Tbar395x120x12x20

9 Tbar425x120x12x25

10 Tbar455x120x12x30

As a result, when the sectional area does not differ greatly from

the initial sectional area, it can be confirmed that the error is not large.

Therefore, it is considered that this stress estimation equation can

be used to obtain the search direction and step size of design

variables.

3.5. Discretization of design variables

Plate thickness variables are discretized in 0.5 mm increments,

and the stiffener sections are discretized into library numbers. Since

48

stiffener section design variables are selected in the library, they are

always discretized. However, the plate thickness variables can be

treated as a continuous variable in some cases. In this study, a

better method was found by attempting to deal with the plate

thickness variables in two different ways.

3.5.1. Discretization method 1

The first method is a method to determine the step size by

discretizing the plate thickness design variables of every search point

during the line search. Although this method may reduce the number

of calculations, there is a possibility of stopping without converging

near the optimum point. The plate thickness variable is discretized in

0.5 mm increments and the stiffener section is discretized into library

numbers.

3.5.2. Discretization method 2

In the second method, the step size is determined by continuously

treating the plate thickness design variables values of all search

points at the time of the line search, and the rounding is performed

49

by discretizing in 0.5 mm interval to determine next design point. The

stiffener section variables are discretized in the same way as the first

discretization method. This method can converge closer to the

optimal point, although the number of calculations can be relatively

increased.

Fig. 28 Discretization method 1

50

Fig. 29 Discretization method 2

3.5.3. Verification for discretization method

To verify the two discretization methods, the load is applied to the

in-plane load parallel to the plate and the pressure in the direction

perpendicular to the panel. A spring boundary condition is attached

to the four corners of the model, which is parallel and perpendicular

to the plate.

51

Fig. 30 Applied loads and boundary conditions

The design variables, plate thickness and stiffener section, are set

by design variables with the same values in the initial model. This

model is initially designed to have three plate thickness variables and

two stiffener section variables. Using this model, three different

design points are used as starting points and the convergence test is

performed.

52

Fig. 31 Design variables of the model

The first and second design point are different from each other in

the feasible area. The third design point is the starting point in the

infeasible area.

And the final solution of the three design points for the two

discretization methods was compared. The buckling strength was

assessed by DNV-RPC201-Part1 module. Fig. 32 shows the

optimization result for discretization method 1, and Fig. 33 shows the

results for discretization method 2.

53

Fig. 32 Optimization result for discretization method 1

Table 4 Change of design variables of starting point1 (Method1)

Design Variables Initial point Final point

DV1 15.0mm 12.0mm

DV2 10.0mm 11.5mm

DV3 20.0mm 12.0mm

DV4 625x150x12x25 356x127x33

DV5 315x100x12x15 254x146x43

Weight(kg) 8927.3 5417.9

Table 5 Change of design variables of starting point2 (Method1)

Design Variables Initial point Final point

DV1 25.0mm 12.5mm

DV2 20.0mm 12.5mm

DV3 19.0mm 12.0mm

DV4 625x150x12x25 358x120x12x21

DV5 315x100x12x15 290x127x42

Weight(kg) 10860.3 5834.2

54

Table 6 Change of design variables of starting point3 for (Method1)

Design Variables Initial point Final point

DV1 8.0mm 12.5mm

DV2 10.0mm 12.5mm

DV3 11.0mm 12.0mm

DV4 510x150x12x25 127x76x13

DV5 235x102x22 220x102x22

Weight(kg) 5097.7 4970.5

Fig. 33 Optimization result for discretization method 2

Table 7 Change of design variables of starting point1 (Method2)

Design Variables Initial point Final point

DV1 15.0mm 12.5mm

DV2 10.0mm 12.0mm

DV3 20.0mm 12.0mm

DV4 625x150x12x25 310x100x12x15

55

DV5 315x100x12x15 254x140x43

Weight(kg) 8927.3 5072.8

Table 8 Change of design variables of starting point2 (Method2)

Design Variables Initial point Final point

DV1 25.0mm 12.0mm

DV2 20.0mm 12.5mm

DV3 19.0mm 12.0mm

DV4 625x150x12x25 310x100x12x15

DV5 315x100x12x15 254x140x43

Weight(kg) 10860.3 5072.8

Table 9 Change of design variables of starting point3 (Method2)

Design Variables Initial point Final point

DV1 8.0mm 12.5

DV2 10.0mm 12.0

DV3 11.0mm 12.0

DV4 510x150x12x25 310x127x42

DV5 235x102x22 220x102x22

Weight(kg) 5097.7 5171.8

Discretization method 1 discretizes all search points and can be

noticed that the final objective values are different when starting at

different design points. This can be attributed to the fact that there

are many points that are treated as the local minimum of the objective

function due to the discretization variables.

56

On the other hand, the final objective values of the discretization

method 2 converged to almost the same value. Since the stiffener

section variables are still discretized, the final optimal solutions are

not fully converged but have a tolerance range that is sufficiently

acceptable. Therefore, the discretization method 2 was applied to all

the optimizations performed in the chapter.

4. Effect of thickness change on the other plane

If the number of design variables increases, the optimization may

not be able to get the optimal solution because the design space

grows exponentially. Depending on how the initial design variables

are divided, the number of design variables can be small or large. In

order to obtain a good optimal solution even in the case of a large

number of design variables, it is necessary to introduce a method of

reducing design variables. In order to reduce the design variables,

the optimization step should be divided among the variables that are

less influenced by the stress change due to change of the design

variables.

The sub-substructure of semi-submersible structure model is

57

composed of various planes. In this kind of model, the effect of

scantling changes of one plane on the stresses of other planes is not

significant. FE analysis was performed to plot the stress change of

four planes of the outer shell of the column model. As shown in the

Fig. 34, the plate thickness of the plane1 of the outer shell was

changed and the scantling of the remaining three planes was left

unchanged. Then the stress change of the four planes was plotted.

Fig. 34 Description of planes for effect of thickness change

When the thickness of Plane1 is reduced by 30% from the initial

thickness, only the stress of Plane1 increased approximately 28.5%,

and the stresses of the remaining planes change around 1% only. (Fig.

35). Conversely when the initial thickness is increased by 30%, the

stresses on the planes that did not change the thickness are changed

58

less than 1%. (Fig. 36).

The Fig. 37 shows the average stress change of the other planes

with increasing or decreasing the thickness of Plane1 by 10%, 30%,

50%. In conclusion, it is considered that the stress effect between

different planes is not large. Therefore, if there are many design

variables, it would be effective to reduce the design variables and

optimize them by dividing the optimization step by plane.

59

Fig. 35 Rate of stress change of four planes (30% decreased)

60

Fig. 36 Rate of stress change of four planes (30% increased)

61

Fig. 37 Average rate of stress change

5. Case Studies for semi-column model

5.1. Model Description

The optimization will only be performed on the outer shell that

subjected to the external pressure rather than the entire column

model.

62

Fig. 38 Semi-column model

5.1.1. Applied loads

The load applied to the target model is the combined load of the

three loads in the Fig. 39.

Load Case 1 : Internal Pressure 1P & 3P

Load Case 2 : Internal Pressure 2P

Load Case 3 : External Pressure

Load case1 is a load case which internal water pressure is applied to

1P and 3P tanks. Load case2 is a load case which internal water

pressure is applied to a 2P tank. Load case 3 is a load case which

63

external sea water pressure is applied outside the column. A linear

combination of these three load cases is used for this optimization.

Fig. 39 Applied loads

5.1.2. Boundary condition

The boundary conditions were set as shown in the Fig. 40. The

upper section of the column fixes the three directions x, y, z, and the

lower section fixes the three directions x, y, z as well as the upper

section and additionally fixes up to three moment directions.

64

Fig. 40 Boundary conditions

5.1.3. Design variables

The design variables for each plane are as follows. There are four

planes in total, but there is one set of symmetrical planes. Therefore,

the design variables in two planes are set by one set. The design

variables of the first two planes are shown in the Fig. 41. It consists

of four plate thickness variables and four beam section variables.

65

Fig. 41 Design variables of first two planes

The design variables of the second plane consist of four plate

thickness variables and three beam section variables.

Fig. 42 Design variables of second plane

The design variables of the last third plane consist of three plate

thickness variables and four beam section variables.

66

Fig. 43 Design variables of third plane

Thus, the design variables of the outer shell of the column have

22 plate thickness variables and 11 beam section variables, total 22

design variables.

5.1.4. Constraints

The constraints of this optimization are buckling strength and yield

strength. The Buckling Strength calculation is based on the rule of

DNV-RP-C201-Part1. In this Rule, the usage factor is calculated by

considering the strengths of the following 10 cases.

Buckling factor of unstiffened plates with transverse compression

Shear yield check of unstiffened plate

67

Check for shear force for stiffener

Plate induced buckling factor of stiffened plates

Stiffener induced buckling factor of stiffened plates

Torsional buckling of stiffener

Stiffener resistance for axial compression and lateral pressure

(4 factors)

Yield Strength is calculated by checking two usage factors of plate

minimum thickness and minimum section modulus of stiffener based

on rule of DNV-OS-C101.

Yield strength of plate

Yield strength of stiffener

The total number of panels in this model is 972, and the number of

constraints to be considered per one panel is twelve. Therefore, this

optimization problem is a complex problem in which a total of 11664

constraints must be calculated and considered.

68

5.2. Performing optimization at once

In the first case study, the optimization of the entire outer shell of

the column part is performed at once. The 22 design variables

described above are covered at one time.

The Fig. 44 shows the change in weight and objective function

value of the optimization process. The weight of the structure is

reduced by 22%.

The buckling strength and yield strength, which are constraints,

should be checked to ensure that the structure is optimized. The Fig.

45, Fig. 46 and Fig. 47 shows the results of assessing the buckling

strength and yield strength of the structure. If the optimization is

correct, the strength factor usage factor of all design variables should

approach close to 1.0. However, the buckling strength and yield

strength of some design variables show that there is still a chance of

weight reduction. This, it can be concluded that the final solution is

not optimal.

69

Fig. 44 Optimization result

Fig. 45 Buckling Strength Usage Factor

70

Fig. 46 Yield Strength Usage Factor of Plate

Fig. 47 Yield Strength Usage Factor of Stiffener

It is considered that the optimal solution is not obtained because

there are too many design variables to be dealt with at once and the

constraint conditions are complicated. Therefore, in the next case

study, optimization is performed utilizing a method that can reduce

design variables.

71

5.3. Performing optimization plane by plane

In the optimization performed above, the optimal design results

were not obtained due to excessive design variables. In this case

study, optimization is performed by dividing the previous model into

planes. The outer shell model consists of four planes. However, in

case of two symmetric planes, one set is set because they share

design variables. As shown in the Fig. 48, it is divided into three sets,

and stepwise optimization is performed for each set. There is a

possibility that the stress of Set1 may change while the scantling to

the last Set3 in the 1st cycle is changed. Thus, in the 2nd cycle, all

the sets are optimized once again to see if the optimal point has not

changed.

72

Fig. 48 Plane by plane optimization

The results of optimization for each set are as follows.

First, Set1 has four plate thickness variables and four beam

section variables. The change of the design variable value of Set1

after optimization is shown in the Fig. 49 and Table 10.

73

Fig. 49 Change of plate thickness (Set1)

Table 10 Change of beam sections (Set1)

No. Initial Beam Sections Final Beam Sections

1 250x90x10x15 200x95x13.5

2 300x90x11x16 250x90x12x16

3 700x200x10x20 250x90x12x16

4 250x13 250x13

The buckling strength and yield strength, which are constraint

conditions, are confirmed to confirm that the final solution is the

optimal solution (Fig. 50 and Fig. 51). The parts marked in red are

the parts where the usage factor is close to 1.0. It can be seen that

the weight is reduced enough to be close to the limit of the constraint

in all design variables. The Fig. 52 shows the change of weight and

objective function value of Set1.

74

Fig. 50 Buckling strength usage factors (Set1)

Fig. 51 Yield strength usage factors (Set1)

75

Fig. 52 Optimization result of Set1

The next set to check is Set2. Set2 has four plate thickness

variables and three beam section variables. The change of the design

variable value of Set2 after optimization is shown in the Fig. 53 and

Table 11. It can be seen that the thickness of the upper plate on which

a relatively small pressure acts is sufficiently reduced to 7mm.

76

Fig. 53 Change of plate thickness (Set2)

Table 11 Change of beam sections (Set2)

No. Initial Beam Sections Final Beam Sections

1 200x90x10x14 150x90x9

2 300x100x11x16 250x90x10x15

3 300x12 300x12

In order to confirm that Set2 converges to the optimal solution,

the buckling strength and yield strength, which are constraint

conditions, are checked. Set2 also shows that the weight is reduced

enough to approach the limit of the constraint in the final design point.

The Fig. 55 shows the change of weight and objective function value

of Set2.

77

Fig. 54 Buckling and yield strength usage factors (Set2)

Fig. 55 Optimization result of Set2

The final set to check is Set3. Set3 has three plate thickness

variables and four beam section variables. The change of the design

variable value of Set3 after optimization is shown in the Fig. 56 and

Table 12.

78

Fig. 56 Change of plate thickness (Set3)

Table 12 Change of beam sections (Set3)

No. Initial Beam Sections Final Beam Sections

1 550x150x12 300x100x14

2 300x125x12 225x95x14.5

3 285x100x12 150x90x9.5

4 200x10 200x10

In order to confirm that Set3 converges to the optimal solution,

the buckling strength and yield strength, which are constraint

conditions, are checked, either. Set3 also shows that the weight is

reduced enough to approach the limit of the constraint in the final

design point. The Fig. 58 shows the change of weight and objective

function value of Set3. In case of Set3, the point with the minimum

weight is different from the point with the minimum objective function.

79

This is because the point with the minimum weight approaches the

infeasible area and the constraint value increases. Weights are

increased in the process of getting back to the feasible area to find

the optimum point, but the objective function value is minimized.

Fig. 57 Buckling and yield strength usage factors (Set3)

Fig. 58 Optimization result of Set3

80

The results of the stepwise optimization are shown in Fig. 59. In

the first cycle, the optimization is performed for each set and the

weight decreased. In the second cycle, the optimal solution

converged without changing the design points in all sets. Unlike the

optimization results that deal with all 22 design variables at once, it

is confirmed that a better solution can be obtained by dividing design

variables by plane.

81

Fig. 59 Result of plane by plane optimization

82

5.4. Convergence of solutions

5.4.1. Convergence of plane by plane optimization

A robust optimization method yields the same optimal solution at

any starting point. In this respect, it is also important to check the

convergence of the optimal solution. Until now, optimization has been

performed for a single fixed design start point. In this case study,

two new starting points with increasing or decreasing weight are

added. Optimization was performed at three starting design points

including the new two design points and the optimization results were

compared. The Fig. 60 shows three different starting design points

to be used for optimization. Stating point 1 is the starting design point

used in the previous chapter, starting point 2 is the design point with

a weight increase of about 53 tons, and starting point 3 is a design

point with a reduction of 58 tons.

83

Fig. 60 Three starting points

Plane by plane optimization is performed for all three starting

points. The Fig. 61 shows the weight change for all 3 points. The final

solutions are not exactly the same, but it is confirmed that the weight

values converge to approximately similar values. As a result, almost

the same optimal solution is obtained for all 3 different starting points.

84

Fig. 61 Result from three different starting points

5.4.2. Convergence of plane separation cases

In the previous section, convergence of the final solution was

verified by simply changing the starting design point. In this case

study, not only the convergence according to the change of the

starting design point but also the convergence according to the

method of reducing the design variable by dividing the set is also

confirmed.

The Fig. 62 below shows three cases divided according to how

the set is divided. Case 1 is a case divided into three sets according

to the method that was tried in the previous section. Case 2 is a case

in which the planes in the XZ-plane are grouped in one set, and the

85

planes in the XY-plane are grouped in one set and divided into two

sets in total. The final case 3 is a case where all 4 planes are

optimized at once without dividing the set. As the set is subdivided,

the number of design variables to deal with each step decreases.

Fig. 62 Three plane separation cases

First, optimization was performed with different cases at the same

starting design point (starting point 2). The Fig. 63 shows the weight

change of the optimization process for three cases. This graph shows

that case 1 yields the best solution. The number of appropriate design

variables cannot be quantified, but the smaller the number of design

variables, the better the optimal solution can be found.

86

Fig. 63 Result of three different cases

Secondly, for each case, optimization is performed on the three

starting points mentioned in the previous section, and the

convergence of the results is confirmed. As a result of the

optimization, the solutions of Case 1 and Case 2 converged to similar

solutions in all three starting points, but in Case 3, the deviation of

the result according to the starting point was large. Therefore, if the

number of design variables increases, it is difficult to find the optimal

solution, and the final solution according to the change of the starting

point can be greatly changed. In Case 1 and Case 2, the deviation of

the final solution according to the starting point was not large, but

Case 1 converged into a better solution. Thus, it is concluded that it

is a good way to obtain a better optimal solution as well as

87

convergence of the optimal solution by dividing the set by reducing

the number of design variables for each optimization step.

Fig. 64 Convergence of Case1

Fig. 65 Convergence of Case2

88

Fig. 66 Convergence of Case3

5.5. Efficiency of stress estimation method

In all of the previous case studies, a stress estimation method was

applied to reduce the finite element analysis time. Additional case

study is conducted to show the efficiency of this stress estimation.

As shown in the Fig. 67, optimization is performed by dividing the

two methods of updating stress, and the final solution and the

consumption time of each optimization are compared.

89

Fig. 67 Appling two stress updating method

In this case study model, only the Set 1 is used because it is unable

to use the entire outer shell due to excessive time consumption. The

design variables are as follows:

4 plate thickness parameters

4 beam section parameters

90

Fig. 68 Design variables of test model

After the optimization, the result is shown as the Fig. 69. The

optimization result shows a difference of about 2% in weight. Based

on only the optimal solution results, it can be seen that optimization

using FE analysis yields a better solution. This is because the stress

estimation equation conservatively predicts the stress. However, as

shown in the Table 13, the time difference between two methods is

more than 25 times. If it takes more than eight hours to optimize only

one plane, it is practically impossible to perform optimization. From

this point of view, the stress estimation equation can greatly reduce

the time required for FE analysis, and the solution is also very

practical because it is not significantly different from the FEM case.

91

Fig. 69 Optimization result applying two method

Table 13 Final weight and consumed time

No. Final Weight Time Consumed

FEM 50.7 ton 487 min.

Stress Estimation 52.0 ton 19 min.

6. Conclusion

A study on FEA based weight optimization of semi-submersible

structures has been carried out. As a preliminary work of

optimization, full automation of the strength assessment process of

semi-submersible structures has been implemented. Each function

of the automation system was verified by DNV-Xtract. Using the

geometry information of the FE model, a panel, which is the basic unit

92

of buckling strength assessment, can be automatically generated. The

assessment code based on DNV-RPC201-Part1 has been developed

and the interface with DNV-PULS has been constructed so that the

strength assessment method used in the practice can be considered.

The assessment of buckling and yield strength was fully automated

by applying strength assessment codes to automatically generated

panels.

Optimization using the developed automated system is performed.

The optimization problem is formulated by focusing on the

optimization that can be used in practice. The design variables are

defined as the plate thickness and the beam section, and the method

of treating them as discrete variables is proposed.

Since the weight and strength of the structure are monotonic, the

steepest descent method is chosen as the optimization algorithm,

assuming that there is no local optimum. The biggest problem of

structural optimization is FEA time. In order to solve this problem, a

simple method of estimating the stress is used, and it is confirmed

that the error is not large when the scantling change is not large.

Optimization was performed on the outer shell of the column model.

Since the convergence of the solution can be deteriorated when the

93

number of design variables is large, the optimization method can be

applied to reduce the design variables. The convergence of the

solution is confirmed by performing optimization at various starting

design points.

In the last case study, the practicality and efficiency of the stress

prediction equation are confirmed. In the case of using the stress

estimation equation, the solution which is not much different from the

solution in the case of using the FEM could be obtained in a much

shorter time.

In this study, an optimization method that reflects all of the

engineering practices in semi-submersible structures is proposed

and applied to obtain a reasonable solution within a reasonable time.

94

7. Appendix

7.1. Stiffener Library

Table 14 Flat bar section library

Section Name Height(mm) Width at

Top(mm)

Width at

Bottom(mm)

C_FB80x10 80 8 8

C_FB90x10 90 8 8

C_FB95x10 95 8 8

C_FB100x9 100 9 9

C_FB100x10 100 10 10

C_FB110x10 110 10 10

C_FB120x10 120 10 10

C_FB120x12 120 12 12

C_FB125x12 125 12 12

C_FB150x12 150 12 12

C_FB160x12 160 12 12

C_FB180x11 180 11 11

C_FB200x10 200 10 10

C_FB200x12 200 12 12

C_FB220x12 220 12 12

C_FB200x16 200 16 16

C_FB250x13 250 13 13

C_FB250x14 250 14 14

C_FB300x12 300 12 12

C_FB280x16 280 16 16

95

C_FB300x16 300 16 16

C_FB320x16 320 16 16

C_FB400x16 400 16 16

C_FB_600x18 600 18 18

Table 15 L section library

Section Name Height

(mm)

Thickness

of web

(mm)

Width of

flange

(mm)

Thickness

of flange

(mm)

150x90x9 150 9 90 9

150x90x9.5 150 9.5 90 9.5

150x90x10 150 10 90 10

150x90x9 150 11 90 11

150x95x11.5 150 11.5 95 11.5

150x90x12 150 12 90 12

160x100x11 160 11 100 12

180x90x12 160 12 90 12

160x100x11.5 160 11.5 100 12

200x90x9x14 200 9 90 14

200x95x13.5 200 9.5 95 13.5

200x90x10x14 200 10 90 14

210x90x10x14 210 10 90 14

210x95x14.5 210 10.5 95 14.5

225x95x14.5 225 11 95 14.5

250x90x10x15 250 10 90 15

250x90x14.5 250 10.5 90 14.5

250x95x16 250 11.5 95 15

250x90x12x16 250 12 90 16

96

270x95x14 270 12 95 14

285x100x12 285 12 100 12

300x90x11x16 300 11 90 16

300x100x14 300 11.5 100 14

300x100x11x16 300 11 100 16

300x110x11x15 300 11 110 15

300x110x11.5x15 300 11.5 110 15

300x125x12 300 12 125 12

325x120x12 325 12 120 12

350x100x12x12 350 12 100 12

350x125x11x13 350 11 125 13

350x125x11.5x12 350 11.5 125 12

350x125x12x12 350 12 125 12

350x125x11x15 350 11 125 15

400x120x11x11 400 11 120 11

400x125x11.5x13 400 11.5 125 13

400x125x11.5x15 400 11.5 125 15

420x100x12x12 420 12 120 12

425x150x11x15 425 11 150 15

425x150x11x16 425 11 150 16

425x150x11.5x15 425 11.5 150 15

425x150x12x15 425 12 150 15

425x150x11.5x18 425 11.5 150 18

434x150x12x16 434 12 150 16

450x150x11x18 450 11 150 18

450x150x12x15 450 12 150 15

475x150x11.5x15 475 11.5 150 15

475x150x11x18 475 11 150 18

97

500x150x12x14 500 12 150 14

500x150x12.5x14 500 12.5 150 14

500x150x12x16 500 12 150 16

550x150x12 550 12 150 12

550x150x11.5x14 550 11.5 150 14

550x150x12x14 550 12 150 14

550x150x11.5x16 550 11.5 150 16

550x150x12x16 550 12 150 16

550x200x10x20 550 10 200 20

550x160x12x21 550 12 160 21

550x150x14x16 550 14 150 16

550x200x11.5x20 550 11.5 200 20

550x160x12x22 550 12.5 160 22

550x170x12.5x22 550 12.5 170 22

700x200x10x20 700 10 200 20

700x200x20 700 20 200 20

900x250x16x17 900 16 250 17

900x120x18x12 900 20 200 12

1080x200x16x20 1080 16 200 20

1130x200x16x20 1130 16 200 20

1186x200x16x20 1186 16 200 30

1286x200x16x20 1286 16 200 30

98

Table 16 T section library

Section Name Height

(mm)

Thickn

ess of

web

(mm)

Width

of top

flange

(mm)

Thickn

ess of

top

flange

(mm)

Width

of

bottom

flange

(mm)

Thickn

ess of

bottom

flange

(mm)

T315x11_125x

15 315 11 125 15 11 15

T315x11.5_125

x15 315 11.5 125 15 11.5 15

T340x11_125x

15 340 11 125 15 11 15

T315x12_125x

15 315 12 125 15 12 15

T340x11.5_125

x15 340 11.5 125 15 11.5 15

C_T300x12_15

0x15 300 12 150 15 12 12

T315x11.5_150

x15 315 11.5 150 15 11.5 15

T340x11_150x

15 340 11 150 15 11 15

T315x12_150x

15 315 12 150 15 12 15

T365x11.5_125

x15 365 11.5 125 15 11.5 15

T340x11.5_150

x15 340 11.5 150 15 11.5 15

C_T320x12_16

0x15 320 12 160 15 12 12

T365x11_150x

15 365 11 150 15 11 15

T340x12_150x

15 340 12 150 15 12 15

T365x11.5_150

x15 365 11.5 150 15 11.5 15

99

T390x11_150x

15 390 11 150 15 11 15

T365x12_150x

15 365 12 150 15 12 15

T390x11.5_150

x15 390 11.5 150 15 11.5 15

T415x11_150x

15 415 11 150 15 11 15

C_T350x12_18

0x15 350 12 180 15 12 12

T390x12_150x

15 390 12 150 15 12 15

T415x11.5_150

x15 415 11.5 150 15 11.5 15

T393x11_150x

18 393 11 150 18 11 18

T440x11_150x

15 440 11 150 15 11 15

T393x11.5_150

x18 393 11.5 150 18 11.5 18

T415x12_150x

15 415 12 150 15 12 15

T418x11_150x

18 418 11 150 18 11 18

T440x11.5_150

x15 440 11.5 150 15 11.5 15

T465x11_150x

15 465 11 150 15 11 15

T393x12_150x

18 393 12 150 18 12 18

T418x11.5_150

x18 418 11.5 150 18 11.5 18

T440x12_150x

15 440 12 150 15 12 15

T443x11_150x

18 443 11 150 18 11 18

T465x11.5_150

x15 465 11.5 150 15 11.5 15

100

C_T400x12_13

0x20 420 12 130 20 12 12

T418x12_150x

18 418 12 150 18 12 18

T443x11.5_150

x18 443 11.5 150 18 11.5 18

C_T450x12_15

0x16 450 12 150 16 12 12

T465x12_150x

15 465 12 150 15 12 15

T468x11_150x

18 468 11 150 18 11 18

T443x12_150x

18 443 12 150 18 12 18

T465x11.5_150

x18 468 11.5 150 18 11.5 18

T493x11_150x

18 493 11 150 18 11 18

T468x12_150x

18 468 12 150 18 12 18

T493x11.5_150

x18 493 11.5 150 18 11.5 18

T495x11_150x

20 495 11 150 20 11 20

C_T500x12_15

0x16 516 12 150 16 12 12

T493x12_150x

18 493 12 150 18 12 18

T495x11.5_150

x20 495 11.5 150 20 11.5 20

T495x12_150x

20 495 12 150 20 12 20

T520x11.5_150

x20 520 11.5 150 20 11.5 20

T520x12_150x

20 520 12 150 20 12 20

T545x11.5_150

x20 545 11.5 150 20 11.5 20

101

T522x11.5_150

x22 522 11.5 150 22 11.5 22

T520x12.5_150

x20 520 12.5 150 20 12.5 20

T545x12_150x

20 545 12 150 20 12 20

T522x12_150x

22 522 12 150 22 12 22

T547x11.5_150

x22 547 11.5 150 22 11.5 22

T545x12.5_150

x20 545 12.5 150 20 12.5 20

T522x12.5_150

x22 522 12.5 150 22 12.5 22

T547x12_150x

22 547 12 150 22 12 22

T547x12.5_150

x22 547 12.5 150 22 12.5 22

C_T500x12_20

0x20 520 12 200 20 12 12

C_T500x20_20

0x20 520 20 200 20 20 20

C_T700x20_20

0x20 720 20 200 20 20 20

C_T1000x12_3

00x20 1020 14 300 20 20 20

C_T1100x14_3

00x20 1100 16 300 20 14 14

C_T1150x16_3

00x20 1150 16 300 20 16 16

C_T1200x16_3

00x30 1216 16 300 30 16 16

C_T1300x16_3

00x30 1316 16 300 30 16 16

102

8. Reference

[1] NOBUKAWA, H., and G. ZHOU. "Discrete optimization of ship

structures with genetic algorithms." Journal of the Society of Naval

Architects of Japan 179 (1996): 293-301.

[2] Ma, Ming, Owen Hughes, and Jeom Kee Paik. "Ultimate

Strength based Stiffened Panel Design using Multi-Objective

Optimization Methods and Its Application to Ship Structures."

Proceedings of the PRADS2013. CECO (2013).

[3] Yu, Yan-Yun, et al. "A Practical Method for Ship Structural

Optimization." The Twentieth International Offshore and Polar

Engineering Conference. International Society of Offshore and Polar

Engineers, 2010.

[4] Veritas, Det Norske. "Column-Stabilised Units." No. DNV-

RPC103, (2012).

[5] Veritas, Det Norske. "Buckling strength of plated structures."

Recommended practice DNV-RPC201, (2010)

[6] Park, Yongman, Beom-Seon Jang, and Jeong Du Kim. "Hull-

form optimization of semi-submersible FPU considering seakeeping

capability and structural weight." Ocean Engineering 104 (2015):

714-724.

103

[7] Andric, Jerolim, et al. "FE based structural optimization

according to IACS CRS-BC." Proceedings of PRADS2016 4 (2016):

8th.

[8] Amir, Hossain M., and Takashi Hasegawa. "Nonlinear mixed-

discrete structural optimization." Journal of Structural

Engineering 115.3 (1989): 626-646.

[9] Rao, Singiresu S., and S. S. Rao. Engineering optimization:

theory and practice. John Wiley & Sons, 2009.

[10] Arora, Jasbir. Introduction to optimum design. Academic

Press, 2004.

[11] Det Norske Veritas, "SESAM-Sestra User Manual." Hovik,

Norway (2014).

[12] Det Norske Veritas, "SESAM-Xtract User Manual." Hovik,

Norway (2015).

104

초록

좌굴 및 항복강도를 고려한 반잠수식 구조물의

유한요소기반 중량 최적화

반잠수식 구조물은 해양에 있는 석유나 가스를 시추, 생산하기 위해

많이 사용되는 구조물이다. 이 구조물은 payload 및 안정성 측면에서

중량증가에 매우 민감한 구조물이다. 상세설계가 진행됨에 따라 상부

장비 중량 증가로 인해 하부 형상 변경과 같은 설계 변경으로 인해 종종

인도 지연까지 발생하기도 한다. 따라서 설계 초기부터 하부 구조물의

최적화를 통한 중량 여유분의 확보가 중요하다.

기존에는 선박의 강도관점 중량최적화에 관한 연구는 많이 수행

되어왔다. 선박의 강도평가 절차들은 비교적 단순하기 때문에 강도평가

과정을 반복하며 강도관점의 최적화를 수행하는데 큰 어려움이 없었다.

하지만 반잠수식 구조물은 선박에 비해 구조가 복잡하기 때문에 선박의

전통적인 강도평가 방법보다 복잡한 절차를 필요로 한다. 이 강도평가

과정은 현재 완전한 자동화가 되어있지 않다. 따라서 반잠수식 구조물을

포함한 해양구조물은 강도관점이 아닌 운동성능과 안정성측면을 고려한

최적화가 주로 수행되어왔다.

구조물의 강도를 고려한 최적화를 수행하기 위해 강도평가과정을

105

자동화가 선행되어야 한다. 이러한 이유로 최적화의 선행연구로서 응력

scanning, mapping, combination 등의 필요한 응력처리 과정들을

자동화하고, 좌굴평가의 기본단위인 panel을 자동으로 생성하여

강도평가에 필요한 전 과정을 자동화하였다.

개발된 강도평가 자동화 시스템을 기반으로 반잠수식 구조물의

강도를 고려한 중량최적화에 대한 연구를 수행하였다. 중량을

목적함수로 설정하고, 좌굴 및 항복강도를 제약조건으로 설정하였다.

설계변수는 판 두께와 보강재 단면으로 설정하였다. 보강재 설계변수

수를 줄이고 비현실적인 보강재 단면이 해로 선택되는 상황을 배제하기

위해 설계변수를 이산화하였다. 또한, 해석 시간 최소화를 위해 최적화

방법으로 최급강하법(Steepest descent method)을 선택하였다.

응력변화를 해석적으로 추정하는 식을 사용하여 유한요소 횟수를

절감하였다.

전체 모델을 최적화할 경우 설계변수가 과도하게 많아진다. 최적화

문제에서 설계변수가 많아질 경우 최적점에 정확히 도달하지 못하는

문제가 발생할 수 있다. 이 문제를 해결하기 위해 비교적 응력의 연성이

적은 평면별로 최적화를 독립적으로 수행하여 각 최적화 단계의

설계변수를 줄였다.

반잠수식 구조물의 Column모델에 대해서 본 논문에서 제시하는

방법을 적용한 최적화를 수행하여 최적해의 수렴성을 확인하였다.

106

주요어: Semi-submersible, Steepest descent method, Weight

optimization, Buckling strength, Yield strength

학 번: 2015-21162