TOPOLOGY OPTIMIZATION OF A CURVED THERMAL PROTECTION SYSTEM

Transcription

1 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 Wright State University WRIGHT STATE UNIVERSITY i

2 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 Committee on Final Examination Richard J. Bethke, Ph.D. Department Chair 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 ii

3 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. iii

4 I would like to dedicate this thesis to My parents, Prithivirajan and Sasireka Prithivirajan iv

5 ABSTRACT Prithivirajan, Muthumanikandan, M.S. Enginering., Department of Mechanical and Materials Engineering, Wright State University, 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 F 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. v

6 TABLE OF CONTENTS 1. Introduction Background Density-Based Method Homogenization Method Evolutionary Structural Optimization (ESO) Method Bi-directional Evolutionary Structural Optimization Method (BESO) Research Approach Theory Behind Evolutionary Structural Optimization Derivation of Dynamic Control Parameter Derivation of Modified Dynamic Control Parameter Equate Modal Displacement to Spatial Displacement Calculation of Modified Dynamic Control Parameter Implementation of Modified Dynamic Control Parameter in ABAQUS Derivation of Static Control Parameter Frame Work for the Implementation of Combined Control Parameter in ABAQUS Problems Occurred in the Evolution Checkerboard Prevention Algorithm Occurrence of Local Mode in the Evolution Process Occurrence of Local Mode due to the Presence of Unremovable Region Design of a Curved Thermal Protection System Panel Loads on TPS Material Properties of Inconel 693 at Elevated Temperatures [11] vi

7 3.3 Sequentially Coupled Thermal-Structural Analysis Theory Behind Uncoupled Heat Transfer Analysis in ABAQUS [10] Superimposing Thermal Load from Heat Transfer Analysis on the Structural Model Case Studies Case Study Case Study Case Study Case Study Case Study Case Study Results and Discussion Future work Performing Topology Optimization Considering Large Deformation Topology Optimization by Considering Mode-Switching Phenomenon APPENDIX Appendix A Appendix B References vii

8 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 Figure 1.4 Figure 1.5 Flow of Conventional ESO Algorithm ESO Design Approach Figure 2.1 Derivation of Modified Dynamic Control Parameter 15 Figure 2.2 Implementation of Modified Dynamic Control Parameter in ABAQUS 19 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 Figure 2.5 Solid Square Plate 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 viii

9 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 ix

10 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 x

11 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 46 Table 4.2 Table 4.3 Table 4.4 Change of Volume, Natural Frequency and Max. von Mises stress in the Evolution Change of Volume, Natural Frequency and Max. von Mises stress in the Evolution Change of Volume, Natural Frequency and Max. von Mises stress in the Evolution Table 5.1 Comparison of results at 900 Hz 59 xi

12 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. 1

13 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 C C. 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 C C. 2

14 Ceramic Matrix Composites (CMC) CMC is made of Carbon and Silicon Carbide and used in the regions of high thermal loads until C. Metallic TPS Ti-Al alloys are used in the design of Metallic TPS, the alloys can withstand a temperature range of C to C. 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, costeffective, 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 3

15 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. 4

16 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) 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 5

17 E( X) = E A+ E (1 B) X 0 0 ρ( X ) = ρ X T min 0 X 1.0 A (1.1) (1.2) Where E (X ) - Young s modulus E 0 - Initial Young s modulus ρ (X ) - Density ρ 0 - Initial density X - Topology design variables which represent volume fraction T min - 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 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 6

18 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 α, β,... L ( Ω ), angle θ L ( Ω ), (1.3) ~ Young s modulus, E ( x) = E ijkl ( α( x), β ( x),..., θ ( x)), ijkl Density, ρ ( x) = ρ( α( x), β ( x),...), Ω ρ( x) dω <= V;0 <= ρ( x) <= 1, x Ω (1.4) 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 7

19 optimization using homogenization approach can be used to minimize the compliance. The geometric representation of the design variables is shown in Figure 1.3. Composite material Scale1: Rank-1 material µ x 2 θ x 2 y 1 γ x 1 Scale2: Rank-2 material Figure 1.3 : Geometric Representation of Design Variables Given in Equation 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 8

20 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. Fine meshed initial structure Analysis Determine the Control Parameters for each finite element Remove inefficient elements Are constraints violated? NO YES End Figure 1.4 : Conventional ESO algorithm 9

21 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: λ = Change in Rayleigh quotient is given by: i L E i i (1.5) Li + S i, l λ i, l = λi E + T i i, l (1.6) λ i, l i, l, i - Change in Rayleigh quotient for i th mode and l th element T E - Local & Global kinetic energy S L - Local & Global strain energy i, l, i λ i, l is considered as the dynamic control parameter for the addition process in the frequency maximization problem. The i th eigenvalue would be increased by adding the l th element that has the highest positive λ i, l Similarly, for the removal process Li S i, l λ i, l = λi E T i i, l (1.7) 10

22 The i th eigenvalue would be increased by removing the l th element that has the highest positive λ i, l, as in Equation 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: 11

23 Develop finite element model of the initial structure with boundary conditions Apply transient thermal loads to simulate the re-entry condition Perform structural analysis Perform combined modal and structural analysis Derive static control parameter 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 Figure 1.5 : ESO Design Approach 12

24 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: Li S i, l λ i, l = λi E T i i, l E L i i, T - are the global and local kinetic energy i, l, S - are the global and local potential energy i, l The global and local kinetic and potential energy terms can be expressed as : E = i T { Φi } [ M]{ Φi }, 2 L i = T { Φi } [ K]{ Φi }, 2 S T e e e { Φ, } i l K l { Φi, l } 2 i, l =, T i, l = e T e e { Φi, l } [ Ml ]{ Φi, l } 2 (2.1) Substituting Equation 2.1 in 1.5 i, l T e T e e { Φ i } [ K ]{ Φ i } { Φ i, l } K l { Φ i, l } λ = 2 2 λ By Simplifying i, l T T e e e { Φ i } [ M ]{ Φ i } { Φ i, l } M l { Φ i, l } 2 2 T e T e e { Φi } [ K]{ Φi } { Φi, l } [ Kl ]{ Φi, l } T e T e e { Φ } [ M ]{ Φ } { Φ } [ M ]{ Φ } λ = λ i i i, l l i, l i i 13

25 Let m i = T { Φ } [ M ]{ Φ } i i { T e e e }, K { Φ, } { T e e e, } M { Φ, } λimi Φ i l l i l λ i, l = λi mi Φ i l l i l λ i, l λimi = e T e e e T e e { Φi, l } [ Kl ]{ Φi, l } λimi + λi { Φi, l } [ M l ]{ Φi, l } e T e e m { Φ } [ M ]{ Φ } i i, l l i, l 1 T i i l i l l i l m i e e e e {, } ( λ M K ){, } λ Φ Φ l = α i (2.2) l α i m { } Φ l i i [ K ],[ ] l l - is the dynamic control parameter to increase i th natural frequency for the l th element. - is the modal mass - is the i th natural mode of the l th element M - are stiffness and mass matrices of the l th element l Elements with low values of α i [13] are removed from the structure. Limitations 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 14

26 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. 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 Figure 2.1 : Derivation of Modified Dynamic Control Parameter Equate Modal Displacement to Spatial Displacement The general equation of motion can be written as: [ M ]&& x + [ K] x = F( t) (2.3) where [M ], [K ]- are the global mass and stiffness matrices x - is the spatial displacement F (t) - is the force vector, expressed as F( t) = F k i - is the modal stiffness m - is the modal mass i jω t { } e 15

27 m - No. of nodes x No. of d.o.f Equation 2.3 can be expressed in modal coordinates as: [ Φ] T [ M ][ Φ ]&& x + [ Φ] T [ K][ Φ ] x = [ Φ] T [ Φ] F( t) (2.4) For a static problem Equation 2.3, the spatial displacement { x } can be expressed as T [ Φ] [ Φ]{ F} { x} = T [ Φ ] [ K][ Φ ] (2.5) By simplifying Equation 2.4 we get: { x} = [ Φ][ k ] 1 i T [ Φ] { F} (2.6) Here, k ] = diag( k,..., k ) [ i 1 N [ 1 N Φ ] = [{ Φ }...{ Φ i }...{ Φ }] { Φ 1 }- Mode shape column vector N - Number of natural modes considered { } { } By Substituting force vector F = [ K] Φ in Equation (2.5), { x } becomes: i { x} = 1 T [ Φ] [ k ] [ Φ] [ K] { Φ } = [ Φ] [ k ] mxn i nxn nxm mxm i mx1 mxn 1 i nxn Φ Φ Φ T 1 T i T N [ K] Φ i M [ K] Φi M [ K] Φi nx1 (2.7) Since mass-normalized mode shapes are orthogonal to the stiffness matrix, T T substituting Φ [ ] Φ = 0 and Φ [ K ]Φ = k in Equation (2.5) we get: i K j i i i { x} = [ Φ] [ k ] mxn 1 i nxn 0 M ki M 0 nx1 { x} = [ Φ] mxn 1/ k O / k 0 0 i O / k N nxn 0 M ki M 0 nx1 16

28 { x} = [ Φ] mxn 0 M 1 M 0 nx1 ( L L ) 0 M = { Φ1} { Φi} { Φ N} 1 mxn { } { x} = Φi { r} = [ ]{ Φ i} Hence F K is true M 0 nx1 = { Φ i } mx1 It can be proved that mass normalized modal displacements can also be considered as spatial displacements for the force vector { F r }. From the modal displacements can be corresponding strains, and stress vector are calculated by simple strain-displacement, and stress-strain relationship 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: vm σ (2.8) dl = ( σ x, l σ y, l ) + ( σ y, l σ z, l ) + ( σ z, l σ x, l ) + 6( τ xy, l + τ yz, l + τ xz, l ) 2 σ, σ, σ are normal stresses and τ xy, l, τ yz, l, τ xz, l are shear stresses, respectively, x, l y, l z, l of the l th 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). 17

29 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. 18

30 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 Extract mass normalized modal displacements from modal analysis Apply the modal displacements as displacement boundary conditions on each node of the model Perform structural analysis with modal displacement boundary conditions Obtain the dynamic control parameter, ( σ vm dl element as in equation 2.7 at Centroid ), for each finite Figure 2.2 : Implementation of Modified Dynamic Control Parameter in ABAQUS 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. vm σ sl = ( σ x l σ y, l ) + ( σ y, l σ z, l ) + ( σ z, l σ x, l ) + 6( τ xy, l + τ yz, l + τ 2, xz, l ) (2.9) 19

31 σ, σ, σ are normal stresses and τ xy, l, τ yz, l, τ xz, l are shear stresses, respectively, x, l y, l z, l of the l th 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. vm σ sl 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. Model fine meshed initial structure Apply boundary conditions and loads on the structure Compute von Mises stress of each finite element at the Centroid Remove minimum stressed elements from the structure Calculated von Mises stress is the static control parameter for obtaining fully stressed design using ESO Figure 2.3 Frame Work for the implementation of Static Control parameter in ABAQUS 20

32 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 reentry 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. 21

33 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 Rs + W R s d d (2.10) Here, W s, W d are weighting factors for static and dynamic control parameters R s, R d 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 W s = -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

34 Start with the initial model Perform heat transfer analysis using transient thermal boundary condition Apply temperature field from heat transfer analysis onto the structural model and perform structural analysis Obtain von Mises stress from the results of the structural analysis, which is the Dynamic Control Parameter ( σ ) Combine the static and dynamic control parameter by the equation C = W Rs + W R s d d vm dl Obtain von Mises stress from the results of the structural analysis, which is the Static Control Parameter ( σ ) vm sl Run Checkerboard filter as explained in Section 2.6 to avoid the formation of checkerboard in the evolution Perform modal analysis on the deformed model to extract mass-normalized mode shapes for interested natural mode If Vol < Vol min Yes 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 Vol min -Minimum volume required for the structure Stop No Remove elements with low value of the combined control parameter C Figure 2.4 : Frame Work for the Implementation of Combined Control Parameter in ABAQUS 23

35 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 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 24

36 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: α k = 1 n n i= 1 α i (2.11) Where n - number of elements connected to the k-th node. αi - is the reference factor for i th 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

37 Figure 2.5 : Initial Structure for Checkerboard Check Figure 2.6 : Final Structure Without Checkerboard Filter 26

38 Figure 2.7 : Final Structure with Checkerboard Filter 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

39 Elemental local mode Perform modal analysis, extract fundamental natural frequency and mass-normalized modal displacement Is first natural Frequency < N min No Yes Find the modulus of modal displacement of all Nodes for each finite element Figure 2.8 : Occurrence of Elemental Local Mode Delete the elements with modulus>1; these are the rotational elements in the structure Figure 2.9: Algorithm to Handle Elemental Local Mode in the Evoluion 28

40 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 29

41 Unremovable region Frame region Support region Fixed Corners Figure 2.11 : Modified Initial Model to Avoid Local Mode at Unremovable Region 30

42 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/m 3, 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 Un-removable region Frame region Support region Fixed Corners Figure 3.3 : Initial Model 31

43 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. F Time (sec.) Figure 3.1 : Temperature and Time History During Re-entry 32

44 Transient thermal load Figure 3.2 : Heat Transfer model of TPS with Transient Temperature Boundary Condition 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. 33

45 Temperature ( 0 C) Modulus of Elasticity (GPa) Table 3.1 : Variation of Modulus of Elasticity with Temperature Temperature ( 0 C) Thermal Conductivity (W/m 0 C) Coefficient of Expansion (µm/m/ 0 C) Table 3.2 : Variation of Thermal Properties with Temperature 34

46 Temperature ( 0 C) Specific Heat (J/Kg 0 C) 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 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: 35

47 ρ U& d V = q d S + r d V V S V (3.1) 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 θ expressed as f = k, where k is conductivity matrix and is written as: k=k(θ ). x Energy balance Equation 3.1 can be combined with fourier law and obtained directly by standard Galerkin approach as δθ θ ρu& δθdv +. k. dv = δθ qds + δθrdv x x V V S V (3.2) The body is approximated geometrically with finite elements; the temperature can be interpolated in terms of shape function as θ = N N N ( x) θ δθ = N N δθ N (3.3) 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 θ N N δθ { N ρu& δθdv +. k. dv = N qds + N rdv } x x V V S V (3.4) The backward difference algorithm for time integration is given by U& = ( U U )(1 / t ) t + t t + t t (3.5) 36

48 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 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. 37

49 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/m 3. 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 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 Hz to 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

50 Figure 4.1 : Initial Structure Figure 4.2 : At the end of 85 iterations, the fundamental natural frequency is Hz. 39

51 4.2 Case Study 2 Figure 4.3 : First Natural Frequency Maximization 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 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

52 Figure 4.4: Initial Model to obtain fully stressed design Figure 4.5 : Final Model 41

53 Figure 4.6 : Change of Max. von Mises stress for the fully stressed design 42

54 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

55 View (a) View (b) View (c) Figure : Isometric View Figure : View (b) of Figure Figure : View (a) of Figure Figure : View (c) of Figure Figure 4.7 : Final Structure Considering only Dynamic Control Parameter 44

56 Figure 4.8 : Fundamental mode of the Final Structure when Natural Frequency is 465 Hz Figure 4.9 : Iteration History Vs First Natural Frequency 45

57 Natural Volume (m 3 ) Frequency (Hz) 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, W s, 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. 46

58 View (a) View (b) View (c) Figure : Isometric View Figure : View (b) of Figure Figure : View (a) of Figure Figure : View (c) of Figure Figure 4.10 : Final Structure of a Thermally Loaded Structure Considering only Dynamic Control Parameter 47

59 Volume (m 3 ) Natural Frequency (Hz) Max. von Mises Stress (GPa) 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 48

60 Figure 4.12 : Iteration vs Max. von Mises Stress Figure 4.13 : Fundamental mode of the Final Structure when Natural Frequency is Hz 49

61 Figure 4.14 : Structural Deformation of the Final Structure in the Presence of Thermal Loads 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 Elements 50

62 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 : Isometric View Figure : View (b) of Figure

63 Figure : View (a) of Figure Figure : View (c) of Figure Figure 4.15 : Final Structure Considering Combined Control Parameter Volume (m 3 ) Natural Frequency (Hz) Max. von Mises Stress (GPa) Table 4.3 : Change in Volume, Natural Frequency and Max. von Mises stress during evolution 52

64 Figure 4.16 : Iteration vs Max. von Mises Stress Considering Combined Control Parameter Figure 4.17 : Natural frequency vs Iteration Considering Combined Control Parameter 53

65 Figure 4.18 : Fundamental mode of the Final Structure when Natural Frequency is Hz Figure 4.19 : Deformation of the Final Structure in the presence of Thermal Loads 54

66 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, W s = -1 and the weighting factor for dynamic as W d = 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. 55

67 Figure 4.20 : Fundamental mode of the Final Structure when Natural Frequency is Hz Figure 4.21 : Deformation of the Final Structure in the presence of Thermal Loads 56

68 Volume (m 3 ) Natural Frequency (Hz) Max. von Mises Stress (GPa) 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 57

69 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. 58

70 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 (m 3 ) Max. thermal stress (GPa) Fundamental Natural frequency (Hz) 1 Dynamic control parameter Dynamic control parameter for a thermally loaded structure Combined control parameter for a thermally loaded structure (removing minimum stressed elements) Combined control parameter for a thermally loaded structure (removing maximum stressed elements) 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 59

71 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. 60

72 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 61

73 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 32 nd iteration and first mode is bending in 33 rd iteration, for frequency maximization problem of the initial model as shown in Figure Figure 6.1: First Twisting mode in 32 nd Iteration 62

74 Figure 6.2: First Bending mode in 33 rd Iteration 63