i

TOPOLOGY OPTIMIZATION OF A CURVED THERMAL

PROTECTION SYSTEM

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science in Engineering

By

MUTHUMANIKANDAN PRITHIVIRAJ

B.E., Bharathidasan University, India 2000

2005

Wright State University

WRIGHT STATE UNIVERSITY

ii

SCHOOL OF GRADUATE STUDIES

Dec 14, 2005

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY

SUPERVISION BY Muthumanikandan Prithiviraj ENTITLED Topology Optimization of a

Curved Thermal Protection System BE ACCEPTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Engineering.

Ramana V. Grandhi, Ph.D.

Thesis Director

Richard J. Bethke, Ph.D.

Department Chair

Committee on

Final Examination

Ramana V. Grandhi, Ph.D.

Ravi C. Penmetsa, Ph. D.

Scott K. Thomas, Ph. D.

Joseph F. Thomas, Jr., Ph.D.

Dean, School of Graduate Studies

iii

Acknowledgement

I would like to thank my advisor Dr. Ramana Grandhi for his guidance and support and for

giving me the opportunity to work in Computational Design and Optimization Center

(CDOC). He was instrumental in directing me towards successful completion of this thesis.

My sincere thanks goes to Mr. Mark Haney for his guidance and constructive comments

throughout the project. I would like to thank Dr. Kim for giving me initial thrust and

guidance. His eagerness to guide me in the proper direction is highly appreciated. I would

like to express my gratitude to Dr. Ravi Penmetsa and Dr. Scott K. Thomas

for being a part of my thesis committee.

I was well supported by all the members at Computational Design and Optimization Center

(CDOC), Wright State University. I would like to thank my friends Rajesh, Nagaraj, Ed,

Jalaja and all for always supporting me. I would like to thank Chris for his help in making

this document grammatically correct and readable.

Finally, I would like to take this opportunity to thank my parents, my elder sister, and my

fiancée for providing me with motivation and support.

This research work was sponsored by Wright Patterson Air Force Base through the task,

“Design and Analysis of Advanced Materials in a Thermal/Acoustic Environment”.

iv

I would like to dedicate this thesis to

My parents,

Prithivirajan and Sasireka Prithivirajan

v

ABSTRACT

Prithivirajan, Muthumanikandan, M.S. Enginering., Department of Mechanical and Materials

Engineering, Wright State University, 2005. Topology Design of a Curved Thermal

Protection System.

The purpose of Thermal Protection System (TPS) is to protect the spacecraft from the

extreme environmental conditions during its re-entry into atmosphere. TPS undergoes harsh

thermal and acoustic loads at around 3500 0F and 180 dB, making its design critical. The

high aerodynamic heating of the TPS panel induces heavy thermal stresses and decreases the

natural frequencies. The design of TPS should be such that it can withstand heavy thermal

stresses due to aerodynamic heating and high acoustic loads. In the current thesis, topology

design of the TPS is performed to obtain the minimum weight configuration using

Evolutionary Structural Optimization (ESO) method, to maintain the fundamental natural

frequency, and to reduce maximum thermal stress for its stability towards acoustic and

thermal loads. Transient thermal analysis is performed to simulate the re-entry heating effect.

A coupled thermal-structural analysis is performed to obtain thermal stresses at elevated

temperatures. Topology Design algorithm using ESO is implemented successfully on a

commercial non-linear solver ABAQUS.

vi

TABLE OF CONTENTS

1. Introduction........................................................................................................................... 1

1.1 Background..................................................................................................................... 5

1.1.1 Density-Based Method............................................................................................. 5

1.1.2 Homogenization Method.......................................................................................... 6

1.1.3 Evolutionary Structural Optimization (ESO) Method ............................................. 8

1.1.4 Bi-directional Evolutionary Structural Optimization Method (BESO) ................. 10

1. 2 Research Approach ...................................................................................................... 11

2. Theory Behind Evolutionary Structural Optimization........................................................ 13

2.1 Derivation of Dynamic Control Parameter ................................................................... 13

2.2 Derivation of Modified Dynamic Control Parameter ................................................... 14

2.2.1 Equate Modal Displacement to Spatial Displacement........................................... 15

2.2.2 Calculation of Modified Dynamic Control Parameter........................................... 17

2.3 Implementation of Modified Dynamic Control Parameter in ABAQUS ..................... 18

2.4 Derivation of Static Control Parameter ........................................................................ 19

2.5 Frame Work for the Implementation of Combined Control Parameter in ABAQUS .. 21

2.6 Problems Occurred in the Evolution............................................................................. 24

2.6.1 Checkerboard Prevention Algorithm ..................................................................... 24

2.6.2 Occurrence of Local Mode in the Evolution Process ............................................ 27

2.6.3 Occurrence of Local Mode due to the Presence of Unremovable Region............. 29

3. Design of a Curved Thermal Protection System Panel....................................................... 31

3.1 Loads on TPS................................................................................................................ 32

3.2 Material Properties of Inconel 693 at Elevated Temperatures [11].............................. 33

vii

3.3 Sequentially Coupled Thermal-Structural Analysis ..................................................... 35

3.3.1 Theory Behind Uncoupled Heat Transfer Analysis in ABAQUS [10].................. 35

3.3.2 Superimposing Thermal Load from Heat Transfer Analysis on the Structural

Model .............................................................................................................................. 37

4. Case Studies ........................................................................................................................ 38

4.1 Case Study 1 ................................................................................................................. 38

4.2 Case Study 2 ................................................................................................................. 40

4.3 Case Study 3 ................................................................................................................. 43

4.4 Case Study 4 ................................................................................................................. 46

4.5 Case Study 5 ................................................................................................................. 50

4.6 Case Study 6 ................................................................................................................. 55

5. Results and Discussion ....................................................................................................... 59

6. Future work......................................................................................................................... 61

6.1 Performing Topology Optimization Considering Large Deformation ......................... 61

6.2 Topology Optimization by Considering Mode-Switching Phenomenon ..................... 61

APPENDIX............................................................................................................................. 64

Appendix A......................................................................................................................... 64

Appendix B......................................................................................................................... 78

References............................................................................................................................... 80

viii

LIST OF FIGURES

Figure 1.1 Equilibrium Surface Temperatures for a NASA Hypersonic Vehicle Concept

for Sustained Flight on Mach 8 at 88,000 ft

Figure 1.2 Positioning of Various TPS Panels on a Space Shuttle Orbiter

Figure 1.3 Geometric Representation of Design Variables Given in Equation 1.3

Figure 1.4 Flow of Conventional ESO Algorithm

Figure 1.5 ESO Design Approach

Figure 2.1 Derivation of Modified Dynamic Control Parameter

Figure 2.2 Implementation of Modified Dynamic Control Parameter in ABAQUS

Figure 2.3 Frame Work for the Implementation of Static Control Parameter in

ABAQUS

Figure 2.4 Frame Work for the Implementation of Combined Control Parameter in

ABAQUS

Figure 2.5 Solid Square Plate

Figure 2.5 Uniform grid of Square Q4 Elements

Figure 2.5 Initial Structure for the Checkerboard Check

Figure 2.6 Final Structure without Checkerboard Filter

Figure 2.7 Final Structure with Checkerboard Filter

Figure 2.8 Occurrence of Elemental Local Mode

Figure 2.9 Algorithm to Handle Elemental Local Mode in Evolution

Figure 2.10 Initial Model without Dense Mesh in Frame Region

Figure 2.11 Modified Initial Model to avoid Local Mode at Unremovable Region

Figure 3.1 Temperature vs Time History in Re-entry

Figure 3.2 Heat Transfer Model of Curved TPS with Transient Temperature Boundary

Condition

1

3

8

12

11

15

19

20

23

26

27

28

28

29

30

32

32

24

24

26

ix

Figure 3.3 Initial Model

Figure 4.1 Initial Structure for Frequency Maximization Problem

Figure 4.2 Final Structure at the end of 85 iterations

Figure 4.3 First Natural Frequency Maximization

Figure 4.4 Initial Model to obtain Fully Stressed Design

Figure 4.5 Final Model

Figure 4.6 Plot of max. von Mises stress vs Iteration

Figure 4.7 Final Structure Considering only Dynamic Control Parameter

Figure 4.8 Fundamental mode of the Final Structure when Natural Frequency is 465 Hz

Figure 4.9 Iteration vs First Natural Frequency

Figure 4.10 Final Structure of Thermally Loaded Structure Considering only Dynamic

Control Parameter

Figure 4.11 Natural Frequency vs Iteration of Thermally Loaded Structure

Figure 4.12 Iteration vs max. von Mises Stress

Figure 4.12 Iteration vs max. von Mises Stress

Figure 4.13 Fundamental mode of the Final Structure when Natural Frequency is

458 Hz

Figure 4.14 Structural Deformation of the Final Structure in the Presence of

Thermal Loads

Figure 4.15 Final Structure Considering Combined Control Parameter

Figure 4.16 Iteration vs max. von Mises Stress Considering Combined Control

Parameter

Figure 4.17 Natural frequency vs Iteration Considering Combined Control Parameter

Figure 4.18 Fundamental mode of the Final Structure when Natural Frequency is

739 Hz

Figure 4.19 Deformation of the Final Structure in the Presence of Thermal Loads

39

39

40

41

41

42

44

45

47

48

49

53

52

53

36

45

49

49

50

54

54

x

Figure 4.20 Fundamental mode of the Final Structure when Natural Frequency is

993 Hz

Figure 4.21 Deformation of the Final Structure in the Presence of Thermal Loads

Figure 4.22 Natural frequency vs Iteration by Removing Maximum Stressed Element

Figure 4.23 Max. von Mises stress vs Iteration by Removing maximum stressed element

57

58

56

56

xi

LIST OF TABLES

Table 3.1 Variation of modulus of elasticity with temperature 33

Table 3.2 Variation of thermal properties with temperature 34

Table 3.3 Variation of specific heat with temperature 34

Table 4.1 Change of Volume and Natural Frequency in the Evolution

Table 4.2 Change of Volume, Natural Frequency and Max. von Mises stress in the

Evolution

Table 4.3 Change of Volume, Natural Frequency and Max. von Mises stress in the

Evolution

Table 4.4 Change of Volume, Natural Frequency and Max. von Mises stress in the

Evolution

Table 5.1 Comparison of results at 900 Hz 59

46

57

52

48

1

1. Introduction

A space shuttle consists of a winged orbiter, two solid-rocket boosters, and an

external fuel tank. The orbiter of the space shuttle is used to carry payload in and out of space.

The orbiter experiences high aerodynamic heating due to air friction because of its high

speed re-entry. Hence, to decrease this heating effect, the speed of the orbiter is reduced

during re-entry by flying through sweeping S-curves. Even then, the spacecraft enters the

atmosphere with the magnitude of velocity approximately 17,000 mph. When the space

vehicle re-enters the atmosphere from Low Earth Orbit (LEO), it hits the extreme fringes of

the atmosphere broadside, using friction (or drag) to slow the vehicle. The descent of the

space vehicle brings it deeper into the thicker atmosphere, increasing the vehicle’s rate of

deceleration as well as the amount of heat that is generated. Figure 1.1 shows the

equilibrium surface temperatures for a NASA hypersonic vehicle concept for sustained flight

of Mach 8 at 88,000 ft [15]. The figure shows elevated temperatures at nose and wing

leading edges.

Figure 1.1 : Equilibrium surface temperatures for a NASA hypersonic

vehicle concept for sustained flight of Mach 8 at 88,000 ft.

2

A major challenge is the selection of materials and design of structures that can

withstand the aerothermal loads of high-speed flight. Aerothermal loads exerted on the

external surfaces of the flight vehicle consist of pressure, skin friction (shearing stress), and

aerodynamic heating (heat flux). Aerodynamic heating is extremely important because

induced, elevated temperatures can affect the structural behavior in several detrimental ways.

Thermal stresses are introduced because of restrained local or global thermal expansions or

contractions as explained by Thornton [15]. TPS protects the entire spacecraft from these

extreme thermal and acoustic loads [5]; its survival from these extreme conditions is critical

to the safety of the mission. Hence, an optimal TPS design for a spacecraft operating in

extreme environments of thermal and acoustic loading is of significant importance for

today’s space missions.

Thermal Protection System (TPS) of a space shuttle orbiter can be classified based on

the types of materials used for the design, they are: Flexible External Insulations (FEI),

Surface Protected Flexible Insulation (SPFI), Ceramic Matrix Composites (CMC), and

Metallic.

Flexible External Insulations (FEI)

Materials used for FEI are ceramic, silica, sewing threads, microfiber fleeces felts etc.

They are used on surfaces with limited aerodynamic or mechanical loads. The materials can

withstand a temperature range of 300 0C – 1200

0C.

Surface Protected Flexible Insulation (SPFI)

SPFI is composed of a FEI-type blanket covered by thin ceramic sheet plate. SPFI is

used on surfaces with higher aerodynamic or mechanical loads and it can withstand

temperature range of 300 0C – 1200

0C.

3

Ceramic Matrix Composites (CMC)

CMC is made of Carbon and Silicon Carbide and used in the regions of high thermal

loads until 1600 0C.

Metallic TPS

Ti-Al alloys are used in the design of Metallic TPS, the alloys can withstand a

temperature range of 300 0C to 1200

0C.

A curved Metallic Thermal Protection System has wide application for the

installation on a space shuttle orbiter to fit onto the outer surface of the vehicle structure.

Figure 1.2 shows the conceptual positioning of a metallic thermal protection system on the

body of a space shuttle orbiter, which needs a curvature in the panel.

Figure 1.2 : Positioning of Various TPS panels on a Space Shuttle Orbiter [12]

C/SiC - Ceramic Matrix Composite TPS

SPFI - Surface Protected Flexible Insulation

ULTIMATE - Metallic Thermal Protection System

A successful TPS design will not only maintain the underlying vehicle structure

within acceptable temperature limits, but must also be lightweight, durable, operable, cost-

effective, and re-usable [4]. The main disadvantage of metallic TPS is its weight, so it is

important to keep the weight of the metallic TPS as low as possible. Therefore, the goal of

4

this research is to find the optimum topology of a curved TPS panel, which has less weight

that can withstand acoustic and thermal loads. Future Space vehicles are planned with no

Thermal Protection System, i.e., the outer structure of the space vehicle will also act as a

thermal protective layer. Hence, current research for the design of TPS is concerned with

using Metallic Thermal Protection System because of its easier replacement and maintenance

costs. Its inherent durability, ductility, and design flexibility lowers its maintenance costs

significantly compared to other TPS systems. Inconel 693 [11], which is an alloy of Ni, Cr

and Al, is chosen for the design of Thermal Protection System. Inconel 693 demonstrates

appreciable material properties even at elevated temperatures, and it shows good corrosion

resistance and low thermal conductivity.

Topology optimization of solid structures involves the consideration of various

parameters, such as the number, location, and shape of holes and the connectivity of domain.

In short, it is the determination of the points in the design domain, whether it is a material

point or the void (no material). There are various methods for performing the topology

optimization, such as the Homogenization method [3], [14], Density-based method [20],

Evolutionary Structural Optimization method [1], and Bi-directional Evolutionary Structural

Optimization method (BESO) [4]. ESO method is chosen for the current research because of

the following advantages:

1) It is a hard-kill method and is easy to implement stress and displacement based

constraints.

2) It can be easily integrated with any commercial packages.

3) It can be used for nonlinear problems considering large deformation.

4) Topology design of a structure undergoing dynamic load conditions can be performed.

5

The optimization by ESO method is performed by deriving control parameters based

on the objective, such as increasing the load carrying capacity of the structure or driving the

natural frequency to the target level.

1.1 Background

Various topology optimization algorithms were developed in the literature to

determine optimal distribution of material in the design domain. Broadly topology

optimization will be classified into two types: continuous and discontinuous approaches. In

continuous approach, an element can appear or disappear at a particular point during the

evolution, but in discontinuous approach a deleted element will never reappear and it is

termed hard-kill method. Density-based methods and homogenization methods fall under

continuous approach, and the ESO method is classified as a discontinuous approach. In a

conventional design process, topology optimization is followed by shape and size

optimization. Different types of topology optimization techniques used are further described

here:

1) Density-Based method

2) Homogenization method

3) Evolutionary Structural Optimization Method (ESO)

1.1.1 Density-Based Method

The material properties such as young’s modulus and density of each finite element

are varied to obtain the desired objective in a density-based method. A heuristic relationship

is constructed between the design variable X and the material properties [20]. For example

6

Where

)(XE - Young’s modulus

0E - Initial Young’s modulus

)(Xρ - Density

0ρ - Initial density

X - Topology design variables which represent volume fraction

minT - Minimum value of the topology design variable

A - Real value supplied by user (typically: 2.0~3.0)

B - Real value supplied by user (typically: 0.0~1.0)

The optimization routine is executed to minimize the objective function; hence, the

design variable values X , at the end of the optimization routine, indicate presence or absence

of the particular finite element. The value of X could not be equal to 0.0 to avoid singularity

of the stiffness matrix; hence, elements which have X values nearly equal to 0 are removed.

1.1.2 Homogenization Method

In the homogenization approach, topology optimization is built around the

employment of composite materials as an interpolation of void and full material. Introducing

composites as part of a solution method in topology design, one has to deal with a number of

aspects of materials science and, specifically, methods for computing the effective material

parameters of composites. Homogenization method deals with the limits on the possible

effective material behavior and gives information on the optimal use of local material

(1.1)

(1.2)

0 0( ) (1 ) AE X E A E B X= + −

0

min

( )

1.0

X X

T X

ρ ρ=

≤ ≤

7

properties, such as orientation of an orthotropic material, layup of laminates, and

parameterization of stiffness tensor.

Introducing a composite material consisting of an infinite number of infinitely small

holes periodically distributed through the base material, the topology problem is transformed

to the form of a sizing problem where the sizing variable is the material density. The density

of material is, in itself, a function of a number of design variables which describe the

geometry of holes at the micro level, and these variables are optimized. Hence, one spatial

point or mesh element will have more than one design variable. To obtain “classical” designs,

explicit penalties on the density are typically needed to steer the design to a 0-1 format, i.e.,

void or material.

In the homogenization approach, the design of continuum structures relies on the

ability to model a material with microstructure. The composite porous medium consists of

many such cells, infinitely small and repeated periodically through the medium. In the

implementation of the homogenization approach to design a structure with composites, the

same flow of computations is used as isotropic materials. For example

Geometric variables angles , ,....... ( ),Lα β ∞∈ Ω angle ( ),Lθ ∞∈ Ω

Young’s modulus, )),(),.......,(),(()(~

xxxExE ijklijkl

θβα=

Density, ,),.......)(),(()( xxx βαρρ =

( ) ;0 ( ) 1,x d V x xρ ρΩ

Ω <= <= <= ∈ Ω∫

The density of material, ρ , is a function of a number of design variables that describe

the geometry of the holes at the micro level, and these variables are optimized. Topology

(1.4)

(1.3)

8

optimization using homogenization approach can be used to minimize the compliance. The

geometric representation of the design variables is shown in Figure 1.3.

1.1.3 Evolutionary Structural Optimization (ESO) Method

Evolutionary Structural Optimization (ESO) method is one of the discontinuous

approaches for topology optimization. ESO method can be employed to solve many kinds of

problems of size, shape, and topology. It is based on the simple concept of evolution, where

by slowly removing inefficient material from a structure, residual shape evolves towards an

optimum [1]. The main advantage of ESO is that the optimality constraints can be based on

stress, stiffness, frequency, or buckling. The inefficient elements are selected by deriving a

γγγγ

µ

x2

θθθθ

x1

y1 x2

Composite material

Scale1: Rank-1 material

Scale2: Rank-2 material

Figure 1.3 : Geometric Representation of Design Variables Given in Equation 1.3

9

control parameter or sensitivity number for each finite element in the model. The control

parameter value is based on the type of topology optimization problem to be solved. For

example, control parameters for frequency, buckling, and static problems are calculated

separately for each finite element. ESO method is based on the fully stressed design concept.

The flow of a general ESO algorithm looks similar to figure 1.4.

Figure 1.4 : Conventional ESO algorithm

Fine meshed initial structure

Analysis

Remove inefficient elements

Are constraints violated?

End

YES

NO

Determine the Control Parameters for each finite element

10

1.1.4 Bi-directional Evolutionary Structural Optimization Method (BESO)

Bi-directional Evolutionary Structural Optimization Method is similar to that of ESO

method except that it allows elements to be added, and also to be removed from the structure,

to evolve towards an optimum. During the addition process, virtual elements are considered

around the actual elements; the virtual elements, which give good characteristics for the

structure, will be converted to an actual element in the next iterations. For example, the

control parameter for addition process for an eigenvalue maximization can be derived using

the following equations. Rayleigh quotient is given as in Equation 1.5:

Change in Rayleigh quotient is given by:

,i lλ∆ is considered as the dynamic control parameter for the addition process in the

frequency maximization problem. The ith

eigenvalue would be increased by adding the lth

element that has the highest positive ,i lλ∆

Similarly, for the removal process

,i lλ∆ - Change in Rayleigh quotient for ith mode

and lth element

,,

i l iT E - Local & Global kinetic energy

, ,i l iS L - Local & Global strain energy

i

lii

lii

liTE

SLλλ −

+

+=∆

,

,

,

,

,

,

i i l

i l i

i i l

L S

E Tλ λ

−∆ = −

−

(1.5)

(1.7)

i

i

iE

L=λ

(1.6)

11

The ith

eigenvalue would be increased by removing the lth

element that has the

highest positive ,i lλ∆ , as in Equation 1.7.

1. 2 Research Approach

The main objective of this investigation is to implement topology optimization

algorithm using Evolutionary Structural Optimization in the commercial nonlinear solver in

order to minimize the weight of a curved Thermal Protection System panel subjected to

reentry heat and to maintain the fundamental natural frequency of the thermally loaded

structure above the acceptable range. Two types of control parameters are used for the

topology design: static control parameter keeps the maximum thermal stress below the yield

stress and dynamic control parameter maintains fundamental natural frequency of the

structure above a certain value. The former is derived by performing structural analysis in the

presence of transient thermal loads; the latter is derived by performing combined modal and

structural analysis. Conduction heat transfer analysis is performed to obtain the temperature

profile in each time step. The temperature profile is applied on the structural model to obtain

thermal stresses by performing sequentially coupled thermal-structural analysis [9]. The

implementation of ESO in ABAQUS is checked first by solving benchmark topology

optimization problems before applying it to the TPS model. The main goal for the

implementation of ESO in ABAQUS is to perform topology optimization based on the

nonlinear analysis to capture large deformation. The modified ESO algorithm is coded in

Python script [8], [17] and integrated with ABAQUS, since heavy thermal stresses during the

re-entry drives TPS panels to exhibit large deformation which can be captured by performing

nonlinear analysis. The project flow can be explained from the following flowchart:

12

Figure 1.5 : ESO Design Approach

Develop finite element model of the initial

structure with boundary conditions

Derive static control parameter

Perform combined modal and structural

analysis

Apply transient thermal loads to simulate

the re-entry condition

Perform structural analysis

Derive dynamic control parameter

Obtain combined control parameter by combining static

and dynamic responses with weighting factors

Remove inefficient elements from the design domain

based on combined control parameter

13

2. Theory Behind Evolutionary Structural Optimization

2.1 Derivation of Dynamic Control Parameter

Dynamic control parameter is used to improve the dynamic characteristics, i.e., the

fundamental natural frequency of the structure during evolution. Improvement of the

fundamental natural frequency in the design has many applications in the aircraft and space

structures. The dynamic control parameter used for the removal process in the conventional

ESO algorithm in Equation 1.7 can also be written in the form following Equation 2.2. As

explained earlier Equation 1.7:

,i i lE T , - are the global and local kinetic energy

,i i lL S , - are the global and local potential energy

The global and local kinetic and potential energy terms can be expressed as :

, ,

, ,2

Te e e

i l l i l

i l

KS

Φ Φ =

[ ] 2

,,

,

e

li

e

l

Te

li

li

MT

ΦΦ=

[ ]

[ ]

, ,

,

, ,

2 2

2 2

T Te e e

i i i l l i l

i l iT Te e e

i i i l l i l

K K

M Mλ λ

Φ Φ Φ Φ −⇒ ∆ = −

Φ Φ Φ Φ −

[ ] ,

2

T

i i

i

KL

Φ Φ=

Substituting Equation 2.1 in 1.5

,

,

,

i i l

i l i

i i l

L S

E Tλ λ

−∆ = −

−

[ ] ,

2

T

i i

i

ME

Φ Φ=

[ ] [ ] [ ] [ ] i

e

li

e

l

Te

lii

T

i

e

li

e

l

Te

lii

T

i

li

MM

KKλλ −

ΦΦ−ΦΦ

ΦΦ−ΦΦ=∆

,,

,,

,

(2.1)

By Simplifying

14

- is the dynamic control parameter to increase ith

natural frequency for the lth

element.

i

m - is the modal mass

- is the ith

natural mode of the lth

element

[ ]l

K ,[ ]l

M - are stiffness and mass matrices of the lth

element

Elements with low values of

of this conventional dynamic control parameter are.

1) Not easy to implement in commercial packages, since it requires element mass and

stiffness matrices.

2) No direct consideration of modal stiffness; hence, smooth change in the natural frequency

is not assured in the evolution.

3) Hard to apply for analysis involving nonlinearity.

2.2 Derivation of Modified Dynamic Control Parameter

The limitation of the conventional dynamic control parameter can be handled by the

modification developed by Kim et al.[4]. According to earlier briefing in Section 2.1,

dynamic control parameter is used to increase any interested natural frequency during

evolution. Kim et al., uses modal stiffness of each finite element as the dynamic control

parameter, and this can be easily derived using any commercial packages. The contribution

[13] are removed from the structure. Limitations

, ,

,

, ,

Te e e

i i i l l i l

i l iTe e e

i i l l i l

m K

m M

λλ λ

− Φ Φ ⇒ ∆ = − − Φ Φ

[ ] [ ] [ ] e

li

e

l

Te

lii

e

li

e

l

Te

liiii

e

li

e

l

Te

liii

li

Mm

MmKm

,,

,,,,

,

ΦΦ−

ΦΦ+−ΦΦ−=∆

λλλλ

[ ] i

T

ii Mm ΦΦ=

( ) , ,

1 Te e e e

i i l i l l i l

i

M Km

λ λ ∴∆ ≈ Φ − Φ (2.2) l

iα=

l

iα

l

iΦ

l

iα

Let

15

of each finite element towards modal stiffness of the whole structure is calculated and least

contributing elements are removed from the structure. The steps involved in deriving this

modified dynamic control parameter are shown in Figure 2.1. The modal stiffness of each

finite element is calculated by creating virtual von Mises Stress due to the displacement of

interested natural mode.

Figure 2.1 : Derivation of Modified Dynamic Control Parameter

2.2.1 Equate Modal Displacement to Spatial Displacement

The general equation of motion can be written as:

where

][M , ][K - are the global mass and stiffness matrices

x - is the spatial displacement

)(tF - is the force vector, expressed as

ik - is the modal stiffness

im - is the modal mass

Equation of Motion

Compute spatial displacements in modal coordinates

Equate the spatial displacements to mass-normalized mode shapes (To show mass-normalized mode shapes can be used as

displacement)

Calculate von Mises stress (dynamic control parameter) from stress-strain relationship

)(][][ tFxKxM =+&&

tjeFtF ω=)(

(2.3)

16

m - No. of nodes x No. of d.o.f

Equation 2.3 can be expressed in modal coordinates as:

For a static problem Equation 2.3, the spatial displacement x can be expressed as

By simplifying Equation 2.4 we get:

][]][[ 1Fkx

T

i ΦΦ= −

Here, ),...,(][ 1 Ni kkdiagk =

.......[][ 1 Ni ΦΦΦ=Φ ]

1Φ - Mode shape column vector

N - Number of natural modes considered

By Substituting force vector in Equation (2.5), x becomes:

Since mass-normalized mode shapes are orthogonal to the stiffness matrix,

substituting and in Equation (2.5) we get:

iKF Φ= ][

[ ] [ ] [ ] [ ] [ ]

1

1

1

1

1

][

][

][

][

nxi

T

N

i

T

i

i

T

nxnimxnmximxm

T

nxmnxnimxn

K

K

K

kKkx

ΦΦ

ΦΦ

ΦΦ

Φ=ΦΦΦ=−−

M

M

0][ =ΦΦ j

T

i K ii

T

i kK =ΦΦ ][

[ ] [ ]

1

1

0

0

nx

inxnimxn kkx

Φ=−

M

M

[ ]

1

1

0

0

/10000

0000

00/100

0000

0000/1

nx

i

nxnN

imxn k

k

k

k

x

Φ=

M

M

O

O

(2.5)

(2.6)

(2.7)

[ ] [ ][ ] [ ] [ ][ ] [ ] [ ] ( )T T TM x K x F tΦ Φ + Φ Φ = Φ Φ&& (2.4)

[ ] [ ]

[ ] [ ][ ]

T

T

Fx

K

Φ Φ=

Φ Φ

17

It can be proved that mass normalized modal displacements can also be considered as

spatial displacements for the force vector rF . From the modal displacements can be

corresponding strains, and stress vector are calculated by simple strain-displacement, and

stress-strain relationship.

2.2.2 Calculation of Modified Dynamic Control Parameter

von Mises stress for each element can be calculated by resultant of principal and

shear stresses. This von Mises stress is nothing more than the representation of modal

stiffness of each finite element towards interested natural frequency:

are normal stresses and are shear stresses, respectively,

of the lth

element in x, y, and z directions. This control parameter is used to remove the

elements based on their stress level (i.e., elements with minimum are removed from the

structure).

[ ]

10

1

0

nx

mxnx

Φ=

M

M

( )1

1

0

1

0

i N mxn

nx

= Φ Φ Φ

M

L L

M

1mxiΦ=

ix∴ = Φ

[ ]r i

Hence F K is true = Φ

(2.8) 2 2 2 2 2 2

, , , , , , , , ,

1( ) ( ) ( ) 6( )

2

vm

dl x l y l y l z l z l x l xy l yz l xz lσ σ σ σ σ σ σ τ τ τ= − + − + − + + +

, , ,x l y l z lσ σ σ, , , , ,, ,xy l yz l xz lτ τ τ

18

2.3 Implementation of Modified Dynamic Control Parameter in ABAQUS

The dynamic control parameter can be derived using any commercial package to

perform the topology optimization without accessing the stiffness and mass matrices directly

as it was done in conventional ESO methods. The framework discussed here is

implementation of topology optimization in a well-known non-linear solver, ABAQUS. The

finite element model of the design domain is created with the prescribed boundary

conditions; modal analysis is performed on the model to obtain ‘n’ number of natural

frequencies and mode shapes. The mass normalized modal displacements are read from each

nodal point and reapplied on the model to perform structural analysis. Mode shape vectors or

are applied on the structural model to obtain the von Misses stress for each element at the

‘Centroid.’ This corresponds to the dynamic control parameter mentioned in Equation 2.7.

Since at the time of structural analysis execution no matrix inversion was involved, less cost

(time) is involved in that process. During each iteration only one modal analysis is performed,

which involves a matrix inversion. When there is a large deformation in the structure due to

thermal loads, just the linear modal analysis will be substituted by a non-linear eigenvalue

solver like the Newton-Raphson method, and the same steps will be followed to derive the

dynamic control parameter.

19

2.4 Derivation of Static Control Parameter

Static control parameter is used for the stress-based design. One of the most

frequently used is von Misses stress criteria. The stress level at each element can be

measured by an average of all the stress components. Von Misses stress has been one of the

most frequently used criteria for isotropic materials.

)(6)()()(2

1,

2,

2,

22

,,

2

,,

2

,, lxzlyzlxylxlzlzlylylx

vm

sl τττσσσσσσσ +++−+−+−=

(2.9)

Perform structural analysis with modal displacement

boundary conditions

Create the Finite Element model for the design domain, apply

the boundary conditions, and mark the unremovable region

Perform modal analysis and obtain mass normalized

fundamental natural mode shapes

Apply the modal displacements as displacement boundary

conditions on each node of the model

Obtain the dynamic control parameter, ( vm

dlσ ) , for each finite

element as in equation 2.7 at ‘Centroid’

Extract mass normalized modal displacements from

modal analysis

Figure 2.2 : Implementation of Modified Dynamic Control Parameter in ABAQUS

20

are normal stresses and are shear stresses, respectively,

of the lth element in x, y, and z directions. This control parameter is used to remove the

elements based on their stress level (i.e., elements with minimum are removed from the

structure). Static control parameter is used to arrive at the fully stressed design [16] during

evolution. At the end of evolution using static control parameter, all the finite elements have

approximately uniform stress distribution.

Topology optimization using this static control parameter is successfully

demonstrated for combined thermal and structural loads by Li et. al.. Kim et al. demonstrated

ESO algorithm in thermal problems by removing gradually lowly-stressed material from the

structure, while evolving towards optimum.

Apply boundary conditions and loads on the structure

Compute von Mises stress of each finite element at the “Centroid”

Calculated von Mises stress is the static control parameter for obtaining fully stressed design using ESO

Remove minimum stressed elements from the structure

Figure 2.3 Frame Work for the implementation of Static Control parameter in ABAQUS

Model fine meshed initial structure

, , ,x l y l z lσ σ σ, , , , ,, ,xy l yz l xz lτ τ τ

vm

slσ

21

2.5 Frame Work for the Implementation of Combined Control Parameter in ABAQUS

A fine meshed structure is modeled in ABAQUS. Two types of models are generated,

one for thermal analysis with heat transfer elements (1-dof) and the other for structural

analysis with 3-D solid elements (3-dof). The transient thermal loads simulating reentry of

the spacecraft orbiter into Earth’s atmosphere are applied as time-dependant temperature

boundary conditions at each node on the upper surface of the TPS panel. Transient heat

transfer analysis is performed in ABAQUS only by considering conduction mode of heat

transfer. This type of methodology is proved to be more conservative in simulating the re-

entry condition, compared to the application of surface heat flux as shown by Blosser [7].

The nodal temperature data are read from the output file of heat transfer analysis, and

are applied on the structural model as a temperature field. Since stress depends on the

temperature field and temperature field is not dependent on the stress, this mode of analysis

is called sequentially coupled thermal-structural analysis. The von Mises stress derived for

each element at this point is the Static Control Parameter.

Modal analysis is performed on a thermally loaded structure. The fundamental natural

frequency is checked for the occurrence of local mode. If there is a sudden drop in

fundamental natural frequency, the local mode prevention algorithm, as explained in Figure

2.4 is applied. The elements exhibiting local mode are inspected and removed from the

structure. The mass normalized mode shapes extracted from the output of modal analysis are

re-applied on the structural model with the temperature field; static structural analysis is

performed to obtain the von Mises stress of each element, and it is the dynamic control

parameter.

22

The static and dynamic control parameters [19] are normalized and combined

together with their respective weighting factors, as shown in Equation 2.9.

C W R W Rs ds d= +

Here, Ws,W

d are weighting factors for static and dynamic control parameters

Rs , Rd are normalized static and dynamic control parameters

C is the combined control parameter for each element. Weights given to each control

parameter indicate the weights given to static and dynamic control parameters in the

evolution. So far, topology optimization using ESO method for thermal structures is

performed by considering lowly stressed elements as inefficient, i.e., by removing elements

with low static control parameter, which is not true for thermal structure for reducing thermal

stresses. When elements with high thermal stresses are removed from the structure during

evolution, it might help in reducing the maximum thermal stress, since this will allow free

thermal expansion. Hence, in Equation 2.9 by substituting Ws= -1 and W

d= 1, and removing

the elements with minimum C value, leads to removal of elements with maximum thermal

stress to move towards optimum. This might show a considerable reduction in the maximum

thermal stress during evolution. After deriving the combined control parameter for each

finite element, checkerboard filter is executed through the whole structure to avoid the

formation of checkerboard as explained in Section 2.6.

The flowchart for the implementation of the evolution using combined control

parameter is shown in Figure 2.4.

(2.10)

23

Start with the initial model

Perform heat transfer analysis using

transient thermal boundary condition

Obtain von Mises stress from the results

of the structural analysis, which is the

Static Control Parameter (vm

slσ )

Apply temperature field from heat

transfer analysis onto the structural

model and perform structural analysis

If

Vol < minVol

Perform modal analysis on the deformed

model to extract mass-normalized mode

shapes for interested natural mode

No

Stop

Check for the occurrence of elemental

local mode; if so use algorithm in Figure

2.9

Read the mass-normalized mode shape

vector of interested natural mode and

apply on each node, with temperature

field, followed by structural analysis

Vol - Volume of the structure in current iteration

Volmin -Minimum volume required for the structure

Yes

Figure 2.4 : Frame Work for the Implementation of Combined Control Parameter in ABAQUS

Remove elements with low value of the

combined control parameter ‘C’

Combine the static and dynamic control

parameter by the equation

C W R W Rs ds d= +

Run Checkerboard filter as explained in

Section 2.6 to avoid the formation of

checkerboard in the evolution

Obtain von Mises stress from the results

of the structural analysis, which is the

Dynamic Control Parameter ( vm

dlσ )

24

2.6 Problems Occurred in the Evolution

Most frequent problems that occurred during the evolution of optimum topology

using ESO are the formation of checkerboard pattern, occurrence of local mode in the

unremovable region, and occurrence of elemental local mode. Various algorithms for the

prevention of these troubles are discussed.

2.6.1 Checkerboard Prevention Algorithm

Checkerboard [2] is referred to as the phenomena of alternating presence of solid and

void elements ordered in a checkerboard-like fashion. The formation of a checkerboard

pattern is the common phenomenon in shape and topology optimization processes. Though

the origin of checkerboard pattern is not fully understood, the hypothesis is that the structure

with checkerboard pattern appears to be numerically stiff, which is not a practical possibility.

A simple example for checkerboard can be given from the following example; the stiffness of

a uniform grid of square Q4 elements in Figure 2.6 is numerically equal to the stiffness of

half the thickness of square plate in Figure 2.5.

Figure 2.5: Solid Square Plate Figure 2.6: Uniform grid of square Q4 elements

25

There were various algorithms formulated to arrest the formation of checkerboard.

One way to reduce it is by using higher order elements. Li et al.[2] developed a checkerboard

prevention filter by an average over the element itself and its surrounding direct neighbors.

The reference factor at each node is calculated by averaging the elements connecting

to the node:

∑=

=n

i

ikn 1

1αα

Where n - number of elements connected to the k-th node.

iα - is the reference factor for ith

node (can be von Misses stress)

The reference factor for each element will be calculated by averaging the reference

factor of each node corresponding to that particular element.

A sample run is performed to determine the validity of the checkerboard algorithm. A

2-D, 50 x 25 x 0.1m Aluminum Mitchell structure which is fixed at two corners is chosen

and a central pointed load is applied. The initial structure is meshed into 800 finite elements.

The structure evolves towards optimum by removing minimum stressed elements. 8 elements

are removed in each iteration. Element removal is stopped until the total number of finite

elements in the structure equals 200.

(2.11)

26

Figure 2.5 : Initial Structure for Checkerboard Check

Figure 2.6 : Final Structure Without Checkerboard Filter

27

2.6.2 Occurrence of Local Mode in the Evolution Process

During evolution, local mode occurs when finite elements are connected only to one

node and free at all other nodes. At that time the natural frequency in the modal analysis

drops to a very low value, due to the presence of local mode for that element. These

rotational elements have to be removed during evolution to maintain the natural frequency of

the structure. The algorithm to remove the rotational elements from the structure is as

follows: in each iteration, fundamental natural frequency is checked for its sudden drop

(<Nmin); if it does so, the absolute sum of modal displacement of each node is checked for

all finite elements, and if this value is greater than 1, those elements are considered as

rotational elements and are removed from the structure. The general algorithm for the

prevention of local mode can be explained in Figure 2.9.

Figure 2.7 : Final Structure with Checkerboard Filter

28

Figure 2.9: Algorithm to Handle Elemental Local Mode in the Evoluion

Perform modal analysis, extract fundamental natural frequency and mass-normalized modal

displacement

Is first natural Frequency <

Nmin

Yes

No

Find the modulus of modal displacement of all Nodes for each finite element

Delete the elements with modulus>1; these are the rotational elements in the structure

Elemental local mode

Figure 2.8 : Occurrence of Elemental Local Mode

29

2.6.3 Occurrence of Local Mode due to the Presence of Unremovable Region

If the mesh density of the frame region is the same as that of the support region or if

the frame region has coarse mesh, it leads to the formation of local mode in the unremovable

region, which may also lead to a sudden drop of fundamental natural frequency. To avoid

this phenomenon, topology optimization begins with dense meshed TPS, at least at the frame

region, since increasing mesh density of the whole design domain will consume more time

for each iteration. The increase of mesh density in the frame region will postpone the

occurrence of local mode in the evolution. Figure 2.10 shows the initial model and Figure

2.11 shows the modified frame region to avoid local mode in unremovable region for

proceeding with topology optimization.

Unremovable region

Frame region

Support region

Figure 2.10 : Initial Model without Dense Mesh in Frame Region

30

Frame region

Unremovable region

Fixed Corners

Support region

Figure 2.11 : Modified Initial Model to Avoid Local Mode at Unremovable Region

31

3. Design of a Curved Thermal Protection System Panel

A curved TPS panel shown in figure 2.12 is designed using ABAQUS. The top layer

is considered unremovable or non-designable region and the other layers, i.e., support, are

considered design regions. Inconel 693 is an alloy of Ni, Cr and Al is used for designing the

Thermal Protection System. Inconel 693 [11] with temperature dependant properties Young’s

Modulus as in Table 3.1, Coefficient of thermal expansion in Table 3.2, thermal conductivity

as in Table 3.3, density 7,770 Kg/m3 , and Poisson’s ratio 0.32 is used. The frame region is

densely meshed to avoid the local mode of the un-removable region. Twelve elements are

removed in all case studies in every iteration. The number of elements in the unremovable

region is 256; the number of elements in the frame region is 1024; and the number of

elements in the support region is 1536, which is the same for both structural and thermal

model.

Body of the Space Shuttle

Curved Thermal Protection System

Frame region

Un-removable region

Fixed Corners

Support region

Figure 3.3 : Initial Model

32

3.1 Loads on TPS

Thermal Protection System is subjected to transient thermal loads and acoustic loads

at the time of its re-entry into the earth’s atmosphere. The transient thermal loads can be

simulated by applying time-dependent temperature boundary conditions as shown in Figure

3.1, and it will be stable to acoustic vibration if the fundamental natural frequency of the

TPS panel is kept above the frequency of acoustic excitation (i.e., if the fundamental natural

frequency of TPS is designed >900 Hz it will be stable from most of the acoustic loads as

explained in [5]). The temperature-time plot shown in Figure 3.1 is nothing more than the

radiation equilibrium temperature, measured at the surface of the panel during the re-entry of

the space shuttle [7]. Transient conduction heat transfer analysis is performed, for thermal

load as shown in Figure 3.1.

Figure 3.1 : Temperature and Time History During Re-entry

°F

Time (sec.)

33

3.2 Material Properties of Inconel 693 at Elevated Temperatures [11]

Material properties such as modulus of elasticity, coefficient of thermal expansion,

thermal conductivity and specific heat changes with temperature. These changes should be

taken into account while performing thermal, structural and modal analyses. ABAQUS has

the capability of incorporating this temperature-dependent data into the model for

constructing the mass, stiffness, and thermal conductivity matrices. The changes in material

properties with temperatures are tabulated below.

Transient thermal load

Figure 3.2 : Heat Transfer model of TPS with Transient Temperature Boundary Condition

34

Temperature

(0C)

Modulus of

Elasticity

(GPa)

21 196

100 194

200 188

300 180

400 172

500 165

600 157

700 148

800 137

Table 3.1 : Variation of Modulus of Elasticity with Temperature

Temperature

(0C)

Thermal

Conductivity

(W/m 0C)

Coefficient of

Expansion

(µm/m/0C)

21 9.1 13.04

100 10.7 13.61

200 12.6 14.05

300 14.2 14.42

400 16.1 14.80

500 17.8 15.22

600 19.5 16.32

700 21.6 17.01

800 22.8 -

900 23.6 -

1000 25.2 -

1100 26.8 -

1150 27.5 -

Table 3.2 : Variation of Thermal Properties with Temperature

35

Temperature

(0C)

Specific Heat

(J/Kg 0C)

23 455

100 484

200 505

300 525

400 548

500 560

600 579

700 598

800 616

900 642

1000 662

1100 674

1150 678

Table 3.3 : Variation of Specific Heat With Temperature

3.3 Sequentially Coupled Thermal-Structural Analysis

A sequentially coupled thermal-structural analysis is used to capture thermal stresses

from the model. In a sequentially coupled thermal-stress analysis, the stress field in a

structure depends on the temperature field, but the temperature field can be found without

knowledge of stress response. It is usually performed by conducting an uncoupled heat

transfer analysis and a stress analysis.

3.3.1 Theory Behind Uncoupled Heat Transfer Analysis in ABAQUS [10]

Uncoupled heat transfer analysis is intended to model solid body heat conduction

with temperature-dependent conductivity, including internal energy, convection, and

radiation boundary conditions. It starts with basic energy balance, boundary conditions, finite

element discretization, and a time integration procedure. The basic energy balance equation

by Green and Naghdi is given by:

36

V S V

U d V q d S r d Vρ = +∫ ∫ ∫&

Where V is a volume of solid material, S is Surface area, ρ is the density of the

material, U& is the time rate of the internal energy, q is the heat flux per unit area of the body,

and r is the heat supplied externally into the body per unit volume.

C(T)=dU/dθ

Heat conduction is governed by the Fourier law; hence, heat flux f at position x can be

expressed as f kx

θ∂= −

∂, where k is conductivity matrix and is written as: k=k(θ ).

Energy balance Equation 3.1 can be combined with fourier law and obtained directly

by standard Galerkin approach as

. .V V S V

U dV k dV qdS rdVx x

δθ θρ δθ δθ δθ

∂ ∂+ = +

∂ ∂∫ ∫ ∫ ∫&

The body is approximated geometrically with finite elements; the temperature can be

interpolated in terms of shape function as

( )N NN xθ θ=

N NNδθ δθ=

The Galerkin approach assumes that variational field, δθ , is interpolated with same

functions; by substituting Equation 3.3 in Equation 3.2,

. . N

N N N N

V V S V

NN U dV k dV N qdS N rdV

x x

θδθ ρ δθ

∂ ∂+ = +

∂ ∂ ∫ ∫ ∫ ∫&

The backward difference algorithm for time integration is given by

( )(1 / )t t t t t

U U U t+ ∆ + ∆

= − ∆&

(3.1)

(3.3)

(3.2)

(3.4)

(3.5)

37

By substituting Equation 3.5 in Equation 3.4, the new temperature at each node can

be obtained in each time step by solving single equations. Consequently, since Equation 3.4

can be written in a linear form, the convergence is fast.

3.3.2 Superimposing Thermal Load from Heat Transfer Analysis on the Structural Model

These nodal temperatures are stored as a function of time in the heat transfer results

file. The temperatures are read into stress analysis as a predefined field; the temperatures

vary with position and time. These temperatures are interpolated to the calculation points

within the elements of the structural model. The structural model uses the temperature

dependent Young’s modulus and the coefficient of thermal expansion for the mass and

stiffness matrices. Mesh and node numbers in the heat transfer model should coincide exactly

with the structural model for superimposing the thermal output from the heat transfer

analysis onto the structural model.

38

4. Case Studies

Two representative case studies are performed to compare the implementation

of topology optimization in ABAQUS with benchmark solutions. The case studies following

these representative studies are the actual implementation of the Curved Thermal Protection

System.

4.1 Case Study 1

An aluminium plate of dimensions 0.15 x 0.1 m is discretized as shown in figure 4.1.

The plate is fixed at two corners on its diagonal. Young’s modulus is 70 GPa, Poisson’s ratio

is 0.3, thickness is 0.3 m, and density is 2700 Kg/m3. The plate is divided into 1350 plane

stress quadrilateral elements. Only eight elements are removed in each iteration at the end of

each finite element analysis. As mentioned earlier, only the fundamental natural frequency is

considered for improvement.

Initial fundamental natural frequency of the structure is 2441.5 Hz. After 85 iterations,

and a 50% removal (i.e., when the number of elements in the structure is 662), the natural

frequency is increased from 2441.5 Hz to 3388.2 Hz. The natural frequency and shape of the

plate after 85 iterations are approximately the same as obtained by Xie and Steven [13] using

conventional dynamic control parameter. The increase in the fundamental natural frequency

during evolution is shown in Figure 4.3.

39

Figure 4.1 : Initial Structure

Figure 4.2 : At the end of 85 iterations, the fundamental

natural frequency is 3388.2 Hz.

40

4.2 Case Study 2

An aluminium plate of dimensions 50 x 25 x 0.1 m is modeled as shown in Figure 4.4.

Two corners at the lower edge of the plate are fixed. Young’s modulus is 100 GPa, Poisson’s

ratio is 0.3. The plate is divided into 800 plane stress quadrilateral elements. 1000 Pa central

point load is applied on the bottom edge of the plate. Only eight elements are removed in

each iteration at the end of each finite element analysis. Only von Misses stress (i.e., static

control parameter) is considered as the criteria for element removal. The number of elements

in the final structure is 200.

Figure 4.3 : First Natural Frequency Maximization

41

Figure 4.4: Initial Model to obtain fully stressed design

Figure 4.5 : Final Model

42

Figure 4.6 : Change of Max. von Mises stress for the fully stressed design

43

4.3 Case Study 3

Topology Optimization of a Curved Panel Considering only Dynamic Control

Parameters

A curved thermal protection system shown in Figure 2.11 is optimized. Evolution is

performed to increase the dynamic characteristics of the structure while reducing the

structural weight. As explained earlier, finite elements that contribute less to the modal

stiffness compared to modal mass are removed from the structure. Modal displacements are

obtained by performing modal analysis without applying any thermal or mechanical loads.

Then, modal displacements are applied as a displacement boundary condition on the

structural model to obtain the contribution of each finite element towards structural stiffness,

corresponding to the interested natural mode. Hence, by removing the elements with low von

Misses stress value, evolution leads towards the topology with increased fundamental natural

frequency. This case study is performed to maintain the fundamental natural frequency in the

topology optimization by considering only modified dynamic control parameters as

explained in Figure 2.1. Figure 2.11 shows the initial model and Figure 4.7 shows the final

model, where the first natural frequency is considered for improvement. The plot of

fundamental natural frequency vs iteration history shows that initially there is an increase in

the natural frequency when the elements that contribute more to modal mass compared to

modal stiffness are removed; and it decreases when the elements that contribute less to modal

stiffness of the fundamental natural frequency compared to other elements are removed. The

fundamental natural frequency of the space structures should be maintained >900 Hz for its

stable performance in the presence of acoustic loads. The fundamental twisting mode of the

final structure is shown in Figure 4.8. Change of volume and natural frequency during

evolution shown in Table 4.1.

44

Figure 4.7.3 : View (a) of Figure 4.7.1

Figure 4.7 : Final Structure Considering only Dynamic Control Parameter

View (a) View (b) View (c)

Figure 4.7.1 : Isometric View Figure 4.7.2 : View (b) of Figure 4.7.1

Figure 4.7.4 : View (c) of Figure 4.7.1

45

Figure 4.9 : Iteration History Vs First Natural Frequency

Figure 4.8 : Fundamental mode of the Final Structure when Natural Frequency is 465 Hz

46

Volume (m3)

Natural

Frequency

(Hz)

0.06011 908.68

0.04625 1020

0.03839 1023

0.03344 980.4

0.02938 919.91

0.01988 726.89

0.01584 465

Table 4.1 : Change of Volume and Natural Frequency in the evolution

4.4 Case Study 4

Topology Optimization of a Thermally Loaded Curved Thermal Protection System

Considering only Dynamic Control Parameters

A curved thermal protection system, shown in Figure 2.11, is optimized in the

presence of thermal loads. The transient thermal load, as shown in Figure 3.1, is applied to

the upper surface of the TPS for 2700s. A sequentially coupled thermal structural analysis is

performed before beginning modal analysis. The algorithm, as explained in Figure 2.1, is

used to obtain the dynamic control parameter. The weighting factor of static control

parameter, Ws , is set to zero. Topology design is performed to increase the fundamental

natural frequency of the thermally loaded structure, without considering the thermal stress.

Figure 4.10 shows the final model with optimized controlled fundamental natural frequency.

Figure 4.11 shows the change in fundamental natural frequency with the iterations. Change

in volume during the evolution is tabulated with frequency and maximum von Mises stress in

Table 4.2. Figure 4.13 and 4.14 shows the fundamental mode shape and thermal deformation

of the final model.

47

Figure 4.10 : Final Structure of a Thermally Loaded Structure Considering only

Dynamic Control Parameter

View (a) View (b) View (c)

Figure 4.10.1 : View (a) of Figure 4.10.4

Figure 4.10.3 : View (b) of Figure 4.10.4 Figure 4.10.4 : Isometric View

Figure 4.10.2 : View (c) of Figure 4.10.4

48

Volume (m3)

Natural

Frequency

(Hz)

Max. von Mises

Stress

(GPa)

0.06011 908.68 6.58

0.04653 1003.7 5.44

0.03924 1034 5.05

0.03391 986.14 5.92

0.02869 873.68 5.19

0.02432 758.51 3.60

0.01949 591.37 3.25

0.01586 457.89 2.19

Table 4.2 : Change in Volume, Natural Frequency, and Max. von Mises stress in the

Evolution

Figure 4.11 : Natural Frequency Vs Iteration of a Thermally Loaded Structure

49

Figure 4.12 : Iteration vs Max. von Mises Stress

Figure 4.13 : Fundamental mode of the Final Structure when Natural Frequency is 457.89 Hz

50

Since static control parameter is not considered, there is a fluctuation in the maximum

von Mises stress during the evolution. However, there is a overall decrease in the maximum

thermal stress since removal of elements from the structure encourage free thermal expansion,

which in turn reduces thermal stress in the structure.

4.5 Case Study 5

Topology Optimization of a Thermally Loaded Curved Thermal Protection System

Considering both Static and Dynamic Control Parameters

Starting with the initial model as shown in Figure 2.11 and applying transient thermal

loads shown in Figure 3.1., ESO algorithm as shown in Figure 2.4 is applied to find the

optimum topology using combined control parameter as shown in Figure 4.14. Elements

Figure 4.14 : Structural Deformation of the Final Structure in the Presence of Thermal Loads

51

with low value of this combined control parameter are removed from the structure.

Inefficient elements are removed from the structure by combining normalized static and

dynamic control parameters with their respective weighting factors. Static control parameter

is used to remove minimum stressed elements, and dynamic control parameter is used to

remove elements contributing less to the stiffness of first natural frequency. Weighting

factors of both static and dynamic control parameters are assumed to be one in this study.

View (a) View (b) View (c)

Figure 4.15.1 : Isometric View Figure 4.15.2 : View (b) of Figure 4.15.1

52

Figure 4.15 : Final Structure Considering Combined Control Parameter

Volume (m3)

Natural

Frequency

(Hz)

Max. von Mises

Stress

(GPa)

0.06011 908.68 6.58

0.05293 925.59 7.86

0.04459 943.81 7.73

0.03886 864.12 7.52

0.03442 798.53 7.15

0.03077 788.43 6.48

0.02771 738.63 5.85

Table 4.3 : Change in Volume, Natural Frequency and Max. von Mises stress during

evolution

Figure 4.15.4 : View (c) of Figure 4.15.1 Figure 4.15.3 : View (a) of Figure 4.15.1

53

Figure 4.17 : Natural frequency vs Iteration Considering Combined Control Parameter

Figure 4.16 : Iteration vs Max. von Mises Stress Considering Combined Control Parameter

54

Figure 4.18 : Fundamental mode of the Final Structure when Natural Frequency is 738.63 Hz

Figure 4.19 : Deformation of the Final Structure in the presence of Thermal Loads

55

Comparing Figure 4.12 and Figure 4.16, the later shows there is a constant and steady

decrease in the maximum von Mises stress while including static control parameter. By

combining the static control parameter, controls increase in maximum thermal stress at the

early stage. Although there is a significant reduction in the maximum von Mises in the

evolution, more efficient structures with respect to maximum von Mises can be obtained by

removing elements with maximum thermal stress, since it will allow free thermal expansion.

4.6 Case Study 6

Topology optimization of a Thermally Loaded Curved Thermal Protection System

Considering Combined Control Parameter (Removing elements with maximum

thermal stress):

As explained earlier, removing elements with maximum von Mises stress can be

proved more efficient to reduce the thermal stresses in the structure. This can be formulated

in the combined control parameter by using the weighting factor for static, Ws = -1 and the

weighting factor for dynamic as Wd = 1, as shown in Equation 2.9. By removing elements

with low value, this combined control parameter will lead to the removal of elements with

maximum thermal stresses and elements that contribute less to the stiffness of fundamental

natural frequency.

56

Figure 4.20 : Fundamental mode of the Final Structure when Natural Frequency is 993.36 Hz

Figure 4.21 : Deformation of the Final Structure in the presence of Thermal Loads

57

Volume (m3)

Natural

Frequency

(Hz)

Max. von Mises

Stress

(GPa)

0.06011 908.68 6.58

0.05881 915.05 5.75

0.05754 931.77 5.57

0.05651 943.69 6.53

0.05542 946.29 2.35

0.05386 957 1.92

0.04965 993.36 2.08

Table 4.4 : Change in Volume, Natural Frequency and Max. von Mises stress during

evolution

Figure 4.22 : Natural frequency Vs Iteration by Removing Maximum Stressed Element

58

Figure 4.23 : Max. von Mises stress Vs Iteration by Removing Maximum Stressed

Element

Plots as shown in Figure 4.23 and Figure 4.22 give an idea that, there is a significant

reduction in the maximum von Mises stress, even in the early iterations as compared to the

previous case studies. Hence, it can be proved that the removal of elements with high von

Mises stress can evolve towards optimum with appreciable reduction in thermal stresses.

After performing some iterations, structure loses its symmetry because of the local mode in

unremovable region, since more elements are removed near it, hence further study has to be

conducted to maintain the symmetry of the structure, by modifying the combined control

parameter.

59

5. Results and Discussion

Our objective in performing the topology optimization is to reduce the weight or total

volume of the structure by keeping the fundamental natural frequency as high as possible and

maximum thermal stress low. The comparison of case studies is performed at around 900Hz

as shown in the table below

S.No

Types of Control

parameters

considered

Volume of the

structure (m3)

Max. thermal

stress (GPa)

Fundamental

Natural

frequency (Hz)

1

Dynamic control

parameter 0.0278 - 895.49

2

Dynamic control

parameter for a

thermally loaded

structure

0.0295 0.47 892.83

3

Combined control

parameter for a

thermally loaded

structure (removing

minimum stressed

elements)

0.0404 0.77 889.44

4

Combined control

parameter for a

thermally loaded

structure (removing

maximum stressed

elements)

0.0525 0.2 973

Table 5.1 : Comparison of results at 900 Hz

The comparison shows maximum increase in the natural frequency and maximum

reduction in thermal stresses while considering only dynamic control parameters for the

60

element removal. Though static control parameter has appreciable control over the maximum

thermal stresses as shown in Figure 4.16, it doesn’t show significant decrease in the

evolution; hence, static control parameter needs to be revised for thermal structures. This can

be achieved by considering maximum stressed elements for removal. The final case study

shows a design of structure with a rapid reduction in maximum thermal stress and steady

increase in the fundamental natural frequency by considering maximum stressed elements

and elements with low stiffness for the removal. For the topology optimization of structure

with mechanical load, removing minimum stressed elements might be helpful, but it is not

true for the structures dominated by thermal loads. Different methodology has to be followed

for obtaining optimum topology with reduced thermal stresses; this might be obtained by

removal of maximum stressed elements, which allows free thermal expansion and helps in

reducing the thermal stress during the evolution.

61

6. Future work

6.1 Performing Topology Optimization Considering Large Deformation

Dynamic characteristics of a structure i.e., natural frequencies and mode shapes

depends on the temperature and thermal deformation. Young’s modulus will change with

respect to temperature as shown in Table 3.1, which causes a change in the stiffness matrix,

and this correspond to the change in the natural frequency of the structure. When a structure

is subjected to very high temperature like TPS of the space shuttle orbiter, it might undergo

large deformation, at this time a significant change in the natural frequencies and mode

shapes can be observed in the presence of large deformation, and this change can be captured

by performing nonlinear analysis. Hence topology optimization should be performed by

substituting the linear Eigen value analysis by nonlinear analysis, in the deformed model

after computing the large deformation.

6.2 Topology Optimization by Considering Mode-Switching Phenomenon

Mode-switching is the phenomenon in which switching of natural mode occurs with

the structural modification. In the evolution, while removing inefficient elements from a 3-d

structure, mode-switching occurs. When there is a mode-switching between bending and

twisting, elements that will be useful for bending mode might be considered inefficient in the

twisting mode and will be considered for removal, when twisting mode occurs first. Hence a

drastic change in the natural frequency is observed during mode-switching. In order to obtain

an efficient topology design of the structure, it is necessary to consider this mode-switching

phenomenon in the evolution. This can be solved by incorporating Bi-directional

62

Evolutionary Structural Optimization (BESO) algorithm instead of ESO. Figure 6.1 and 6.2

shows the occurrence of mode-switching between bending and twisting modes. First mode is

twisting in 32nd

iteration and first mode is bending in 33rd

iteration, for frequency

maximization problem of the initial model as shown in Figure 2.11.

Figure 6.1: First Twisting mode in 32nd

Iteration

63

Figure 6.2: First Bending mode in 33rd

Iteration

64

APPENDIX

Appendix A

This appendix has python script to perform topology optimization in ABAQUS

#-----------Python Script to perform Topology optimization in ABAQUS------------------#

from abaqus import *

from abaqusConstants import *

import visualization

import mesh

import load

import odbAccess

from assembly import *

import regionToolset

import part

from Numeric import *

import material

import section

import assembly

import step

import interaction

import sketch

import xyPlot

import displayGroupOdbToolset as dgo

# Subfunctions

#Creation of input file to apply modal displacement boundary condition on each node

def appl_bc():

node_no1=0

test=open("BC.inp","w")

while node_no1<len(v_data):

test.write("*Nset, nset=_PickedSet"+str(node_no1+30)+",internal,instance=Part-1-

mesh-1-1\n")

test.write(" "+str(node_label[node_no1])+",\n")

test.write("*Boundary\n")

65

dataline1="_PickedSet"+str(node_no1+30)+","+str(1)+","+str(1)+","+str(v_data[node_no1][

0])+"\n"

test.write(dataline1)

dataline2="_PickedSet"+str(node_no1+30)+","+str(2)+","+str(2)+","+str(v_data[node_no1][

1])+"\n"

test.write(dataline2)

dataline3="_PickedSet"+str(node_no1+30)+","+str(3)+","+str(3)+","+str(v_data[node_no1][

2])+"\n"

test.write(dataline3)

node_no1+=1

test.close()

# Checkerboard elimination subroutine

def checkerboard(ele,max_von_stress):

from Numeric import *

cor_max_von_stress=zeros((len(ele)),Float)

element_data=zeros((len(ele),8),Int)

#Create the element matrices with nodal connectivities

for i in range(0,len(ele)):

node_connect=ele[i].connectivity

element_data[i][0]=node_connect[0]

element_data[i][1]=node_connect[1]

element_data[i][2]=node_connect[2]

element_data[i][3]=node_connect[3]

element_data[i][4]=node_connect[4]

element_data[i][5]=node_connect[5]

element_data[i][6]=node_connect[6]

element_data[i][7]=node_connect[7]

for t in range(0,len(element_data)):

point_value=[]

66

for tt in range(0,8):

rows=find(element_data,element_data[t][tt])

point_value=point_value+[float(summing(rows,max_von_stress))/len(rows)]

cor_max_von_stress[t]=sum(point_value)/8

return cor_max_von_stress

# Subroutine for element removal

def removal(elementsToRemove):

allRemovableElements = mdb.models['Structural-Model'].parts['Part-1-mesh-

1'].elements

set1=mdb.models['Structural-Model'].parts['Part-1-mesh-

1'].SetFromElementLabels(name='Removable1',elementLabels=tuple(elementsToRemove))

set2=mdb.models['Thermal-Model'].parts['Part-1-mesh-

1'].SetFromElementLabels(name='Removable2',elementLabels=tuple(elementsToRemove))

mdb.models['Structural-Model'].parts['Part-1-mesh-

1'].deleteElement(elements=set1,deleteUnreferencedNodes=ON)

mdb.models['Thermal-Model'].parts['Part-1-mesh-

1'].deleteElement(elements=set2,deleteUnreferencedNodes=ON)

mdb.models['Structural-Model'].rootAssembly.regenerate()

mdb.models['Thermal-Model'].rootAssembly.regenerate()

return 1

# Initialization of matrices

def Initialize_mat(no_mode,ele):

global_reac=zeros((no_mode,len(ele)),Float)

global_stress=zeros((no_mode,len(ele)),Float)

scale_von_stress=zeros((no_mode,len(ele)),Float)

max_von_stress=zeros(len(ele),Float)

temp_mode=zeros(no_mode,Float)

67

return global_reac,global_stress,scale_von_stress,max_von_stress,temp_mode

# Subroutine to submit structural analysis job

def exec_job_struc(jobname,modelname):

myJob1 = mdb.Job(name=jobname, model=modelname)

myJob1.writeInput()

test1=open(jobname+".inp")

test2=open("temp.inp","w")

line=test1.readline()

while(line.strip()!=''):

if (line.strip()=="*Static"):

test2.writelines(line)

line2=test1.readline()

test2.writelines(line2)

line2=test1.readline()

test2.writelines(line2)

test2.writelines("*INCLUDE,INPUT=BC.inp\n")

else:

test2.writelines(line)

line=test1.readline()

test2.close()

test1.close()

os.remove(jobname+".inp")

68

os.rename("temp.inp",jobname+".inp")

myJob2=mdb.JobFromInputFile(jobname,jobname+".inp")

myJob2.submit()

myJob2.waitForCompletion()

del mdb.jobs[jobname]

#Subroutine to submit modal analysis job

def exec_job_mod(jobname,modelname):

myJob1 = mdb.Job(name=jobname, model=modelname)

myJob1.submit()

myJob1.waitForCompletion()

del mdb.jobs[jobname]

#Subroutine to find the element labels for respective node labels for checkerboard subroutine

def find(element_data,x):

rows=[]

columns=[]

for row in range(0,len(element_data)):

for col in range(0,8):

if(element_data[row][col]==x):

rows=rows+[row]

columns=columns+[col]

return rows

#Subroutine to remove rotational elements, in order to avoid elemental local mode

def remove_rot_ele():

ele=mdb.models['Structural-Model'].parts['Part-1-mesh-1'].elements

69

displac=myOdb.steps['modal_analysis'].frames[1].fieldOutputs['U']

dispvalues=displac.values

remove_ele_rot=[]

v_data=[]

for v in dispvalues:

v_data+=tuple([v.data])

for i in range(0,len(ele)):

node_connect=ele[i].connectivity

for node_nos in node_connect:

dispsum=abs(v_data[node_nos][0])+abs(v_data[node_nos][1])+abs(v_data[node_n

os][2])

if dispsum>1:

remove_ele_rot=remove_ele_rot+[ele[i].label]

removal(remove_ele_rot)

# Subroutine to sum von Mises stress for specified number of elements

def summing(rows,max_von_stress):

sumsv=0

for i in range(0,len(rows)):

sumsv=sumsv+max_von_stress[rows[i]]

return sumsv

# End of Subfunction Paste

#---------------------------Execute the main optimization routine ------------------------------#

70

#Creation of element and node objects

ele=mdb.models['Structural-Model'].parts['Part-1-mesh-1'].elements

no_of_nodes=mdb.models['Structural-Model'].parts['Part-1-mesh-1'].nodes

#Instantiate surface objects for boundary condition and unremovable elements

nodes_bc=mdb.models['Structural-Model'].rootAssembly.surfaces['Surf-bc'].nodes

unrem_ele2=mdb.models['Structural-Model'].rootAssembly.surfaces['Surf-bc'].elements

unrem_ele=mdb.models['Structural-Model'].rootAssembly.surfaces['Surf-1'].elements

nodes_bc_label=[]

unrem_ele_label=[]

#Get the element labels for boundary condition object

for node_num in range(0,len(nodes_bc)):

nodes_bc_label+=[nodes_bc[node_num].label]

#Create a set object for displacement ‘Encastre’ boundary condition

sets=mdb.models['Structural-Model'].rootAssembly.SetFromNodeLabels(name='set-bc-

nodes',nodeLabels=(('Part-1-mesh-1-1', nodes_bc_label),('Part-1-mesh-1-1',

nodes_bc_label)))

#Extract element labels for unremovable elements

for num in range(0,len(unrem_ele)):

unrem_ele_label+=[unrem_ele[num].label]

for num in range(0,len(unrem_ele2)):

unrem_ele_label+=[unrem_ele2[num].label]

i=1

jobname='modal_analysis'

71

k=1

no_mode=1

step_no=1

max_stat=0

#Execute the element removal loop until Number of elements in structure >Nmin

while len(ele)>Nmin:

#Initialization of the matrices

global_reac,global_stress,scale_von_stress,max_von_stress,temp_mode=Initialize_mat(no_m

ode,ele)

element_data=zeros((len(ele),8),Int)

#Execute thermal analysis job in the thermal model

exec_job_mod('Thermal_analysis'+str(step_no),'Thermal-Model')

#Create the sequentially coupled thermal structural analysis step

mdb.models['Structural-Model'].StaticStep(initialInc=1.0, maxInc=1.0, minInc=1.0,

name='struc_analysis', previous='Initial')

#Set the fieldoutput request for von Mises stress, displacement and reaction forces

mdb.models['Structural-Model'].fieldOutputRequests['F-Output-

1'].setValues(variables=('S', 'U', 'RF'))

#Create thermal field for structural analysis

mdb.models['Structural-Model'].Temperature(absoluteExteriorTolerance=0.0,

beginIncrement=150, beginStep=1, createStepName='struc_analysis', distribution=

FROM_FILE, endIncrement=150, endStep=1, exteriorTolerance=0.05,

fileName='Thermal_analysis'+str(step_no)+'.odb', interpolate=OFF,

magnitudes=(0.0, ), name='Field-1')

#Create the frequency step

mdb.models['Structural-Model'].FrequencyStep(maxEigen=None,

name='modal_analysis',normalization=MASS, numEigen=1,previous='struc_analysis')

#Create the BC-1 boundary condition for modal analysis

72

mdb.models['Structural-Model'].EncastreBC(createStepName='Initial', name='BC-1',

region=sets)

#Submit job for modal_analysis after thermal structural analysis

exec_job_mod('modal_analysis'+str(step_no),'Structural-Model')

#Read the odb file of modal analysis

odbpath='modal_analysis'+str(step_no)+'.odb'

myOdb = session.openOdb(odbpath)

#Check for rotational elements by tracing the sudden frequency drop

while (myOdb.steps['modal_analysis'].frames[1].frequency < 10):

#Call rotational element removal subroutine

remove_rot_ele()

exec_job_mod('modal_analysis'+str(step_no),'Structural-Model')

odbpath='modal_analysis'+str(step_no)+'.odb'

myOdb = session.openOdb(odbpath)

#Delete the boundary conditions and step

del mdb.models['Structural-Model'].boundaryConditions['BC-1']

del mdb.models['Structural-Model'].steps['modal_analysis']

del mdb.models['Structural-Model'].steps['struc_analysis']

v_stress=myOdb.steps['struc_analysis'].frames[-1].fieldOutputs['S']

stat_control=v_stress.getSubset(position=CENTROID)

#Loop for each mode shapes

for modes in range(0,no_mode):

modaldisp=zeros((len(no_of_nodes),3),Float)

stress_label=zeros((len(ele),2),Float)

73

remove_ele=[]

#Get the modal displacement value from the ODB file

#Read the values of stress from the structural analysis step

disp=myOdb.steps['modal_analysis'].frames[-1].fieldOutputs['U']

#Get the displacement value at each node

dispvalue=disp.values

node_label=[];

v_data=[];

for v in dispvalue:

node_label+=tuple([v.nodeLabel])

v_data+=tuple([v.data])

elementsToRemove=[]

node_count=0

sum_norm=0

node_count_bc=0

#Create the structural analysis step

mdb.models['Structural-Model'].StaticStep(initialInc=1.0, maxInc=1.0, minInc=1.0,

name='struc_analysis', previous='Initial')

#Set the fieldoutput request for von Mises stress, displacement and reaction forces

mdb.models['Structural-Model'].fieldOutputRequests['F-Output-

1'].setValues(variables=('S','U','RF'))

mdb.models['Structural-Model'].Temperature(absoluteExteriorTolerance=0.0,

beginIncrement=150, beginStep=1, createStepName='struc_analysis', distribution=

FROM_FILE, endIncrement=150, endStep=1, exteriorTolerance=0.05,

fileName='Thermal_analysis'+str(step_no)+'.odb', interpolate=OFF,

magnitudes=(0.0, ), name='Field-1')

74

#Applying the displacement boundary conditions for all the other nodes

appl_bc()

#Submit the job for structural analysis

exec_job_struc('struc_anal'+str(step_no),'Structural-Model')

print "Just deleted struc_anal job"

#Delete the step struc_analysis

del mdb.models['Structural-Model'].steps['struc_analysis']

print "Just deleted struc_anal step"

#Read the odb file for structural analysis

myOdb_stress = session.openOdb('struc_anal'+str(step_no)+'.odb')

#Read stress and Reaction force after the completion of each job and store it in an

array for each element

stress=myOdb_stress.steps['struc_analysis'].frames[-1].fieldOutputs['S']

cen_stress = stress.getSubset(position=CENTROID)

reac_force=myOdb_stress.steps['struc_analysis'].frames[-1].fieldOutputs['RF']

reac_force_mod=[];

mod_stress_group=[];

for i in range(0,len(ele)):

rea_force_ele=[]

mod_stress=[]

#Get the node indices for each element, get reaction force for those nodal indices

and

#Give sequence of node indices that define the connectivity of the element

node_connect=ele[i].connectivity

75

element_data[i][0]=node_connect[0]

element_data[i][1]=node_connect[1]

element_data[i][2]=node_connect[2]

element_data[i][3]=node_connect[3]

element_data[i][4]=node_connect[4]

element_data[i][5]=node_connect[5]

element_data[i][6]=node_connect[6]

element_data[i][7]=node_connect[7]

global_stress[modes][i]=cen_stress.values[i].mises

for node_no in node_connect:

rea_force_ele+=tuple([reac_force.values[node_no].data])

#Find the modulus of reaction force list

sum=0

for d in rea_force_ele:

sum=sum+pow(d[0],2)+pow(d[1],2)+pow(d[2],2)

global_reac[modes][i]=pow(sum,0.5)

#Determine maximum static control parameter value

if max_stat<stat_control.values[i].mises:

max_stat=stat_control.values[i].mises

myOdb_stress.close()

#Scaling stress with respect to the reaction forces

for ele_no in range(0,len(ele)):

for modes in range(0,no_mode):

scale_von_stress[modes][ele_no]=(pow(global_reac[0][ele_no]/global_reac[m

odes][ele_no],1))*global_stress[modes][ele_no]

temp_mode[modes]=scale_von_stress[modes][ele_no]

76

max_von_stress[ele_no]=max(temp_mode)

#Form the combined control parameter with normalized static and dynamic control

parameters

for ele_no in range(0,len(ele)):

max_von_stress[ele_no]=max_von_stress[ele_no]/max(max_von_stress)+stat

_control.values[ele_no].mises/max_stat

#Incorporated to prevent the checkerboard phenomenon

max_von_stress=checkerboard(ele,max_von_stress)

# Convert the maximum von-mises stress array into a tuple

list_max_von_stress=[]

i=0

for ele_no in range(0,len(ele)):

stress_label[i][0]=round(max_von_stress[ele_no]*100)

stress_label[i][1]=i

i=i+1

list_stress_label=stress_label.tolist()

list_stress_label.sort()

remove_ele=[]

num_ele_rem=0

remove_ele=[]

j=0

list_global_stress=list_stress_label

#Loop to remove atleast 16 elements in each iteration

77

while len(remove_ele)<16:

if unrem_ele_label.count(ele[int(list_global_stress[j][1])].label)<1:

remove_ele+=[ele[int(list_global_stress[j][1])].label]

j+=1

j=j-1

k=j+1

#Loop to find out elements with von mises stress equal to the 16th

element

while list_global_stress[j][0]==list_global_stress[k][0]:

if unrem_ele_label.count(ele[int(list_global_stress[k][1])].label)<1:

remove_ele+=[ele[int(list_global_stress[k][1])].label]

k=k+1

#Call the subroutine for element removal

removal(remove_ele)

#Instantiate new objects for elements and nodes

ele=mdb.models['Structural-Model'].parts['Part-1-mesh-1'].elements

no_of_nodes=mdb.models['Structural-Model'].parts['Part-1-mesh-1'].nodes

myOdb.close()

step_no=step_no+1

#---------------------------------------------------end-----------------------------------------------#

78

Appendix B

This appendix has Python Script to read Odb file of each iteration and print the output of

natural frequencies and maximum von Mises stress into an output file

from abaqus import *

from abaqusConstants import *

import visualization

import mesh

import load

import odbAccess

from assembly import *

import regionToolset

import part

from Numeric import *

import material

import section

import assembly

import step

import interaction

import sketch

import xyPlot

import displayGroupOdbToolset as dgo

# Open an output file in write mode to print natural frequency

test=open("output_freq.dat","w")

# Execute the loop from starting iteration number to ending to read frequency from the odb

file and write into the output file

for i in range(start,end):

file_name='modal_analysis'+str(i)+'.odb'

myOdb=session.openOdb(file_name)

freq=myOdb.steps['modal_analysis'].frames[1].frequency

test.write(str(freq)+"\n")

test.close()

# Open an output file in write mode to print maximum von Mises stress

test=open("output_maxstress.dat","w")

79

# Execute the loop from starting iteration number to ending to read max. von Mises stress

from the odb file and write into the output file

for i in range(start,end):

file_name='struc_anal'+str(i)+'.odb'

myOdb_stress=session.openOdb(file_name)

stress=myOdb_stress.steps['struc_analysis'].frames[1].fieldOutputs['S']

cen_stress = stress.getSubset(position=CENTROID)

global_stress=zeros((len(cen_stress.values)),Float)

max_stress=cen_stress.values[0].mises

for i in range(0,len(cen_stress.values)):

global_stress[i]=cen_stress.values[i].mises

if global_stress[i]>max_stress:

max_stress=global_stress[i]

test.write(str(max_stress)+"\n")

test.close()

80

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